# embeddings_queue

created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae298ec-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "August 2009 SPE Production & Operations 397pseudoradial flow. Each of these flow regimes has a characteristic \nappearance on various diagnostic plots.\nFluid leakoff from a propagating fracture is normally modeled \nassuming one-dimensional linear flow perpendicular to the fracture face. Settari (2000) has pointed out that in some cases of moder-ate reservoir permeability the linear flow regime may not occur, even during fracture extension and early leakoff. During fracture extension and shut-in, the transient may already be in transition to elliptical or pseudoradial flow. In this case analyses based on an assumed pseudolinear flow regime will give incorrect results. In all cases an understanding of the flow regime and its relation to the fracture geometry is critical to arriving at a consistent interpretation of the fracture falloff test.\nDiagnostic Derivative Examples\nFor each analysis technique various curves are used to help define"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2996e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "closure, leakoff mechanisms, and after-closure flow regimes. On each plot the curves are labeled as the primary (y vs. x), the first derivative ( /H11128y//H11128x), and the semilog derivative ( /H11128y//H11128(lnx) or \nx/H11128y//H11128x). For convenience the primary curve is plotted on the left \ny-axis and all derivatives are plotted on the right y-axis for all Cartesian plots. For the log-log plot all curves are shown on the same y-axis.\nFor pre-closure analysis, and consistent identification of frac-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae299fa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "ture closure, three techniques are illustrated for each example: G-function, Square-root of shut-in time, and log-log plot of pres-sure change with shut-in time. All these analyses begin at shut-in. The instantaneous shut-in pressure (ISIP) is taken as the incipient fracture extension pressure for all cases. When there is significant wellbore afterflow (fluid expansion or continued low-rate injec-tion), or severe near-well pressure drop, the ISIP can be difficult to interpret accurately and may be too high to represent actual fracture extension pressure. In all the examples in the paper the pressures have been offset to an approximate ISIP of 10,000 psi to remove any relation to the original field test data. The follow-ing sections detail the data and analysis for the four major leakoff type examples.\nNormal Leakoff Behavior.  Normal leakoff is observed when the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29a7c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "Normal Leakoff Behavior.  Normal leakoff is observed when the \ncomposite reservoir system permeability is constant. The reservoir may exhibit only matrix permeability or have a secondary natural fracture or \ufb01  ssure overprint in which the \ufb02  ow capacity of the \nsecondary fracture system does not change with pore pressure or net stress. After shut-in the fracture is assumed to stop propagat-ing and the fracture surface area open to leakoff remains constant during closure.\nNormal Leakoff G-Function. As noted in previous papers, \nthe expected signature of the G-function semilog derivative is a straight-line through the origin (zero G-function and zero deriva-tive) (Barree 1998). In all cases the correct straight line tangent to the semilog derivative of the pressure vs. G-function curve must pass through the origin. Fracture closure is identified by the departure of the semi-log derivative of pressure with respect to \nG-function (G /H11128p\nw//H11128G) from the straight line through the origin."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29afe-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "w//H11128G) from the straight line through the origin. \nDuring normal leakoff, with constant fracture surface area and constant permeability, the first derivative ( /H11128p\nw//H11128G) should also be \nconstant (Castillo 1987). The primary pw vs. G curve should follow \na straight line (Nolte 1979). The example in Fig. 1  shows some \nslight deviation from the perfect constant leakoff but is a good example of the expected curve shapes with a clear indication of closure at G\nc = 2.31. The closure event is marked by the dashed \nvertical line [1].\nNormal Leakoff Sqrt(t) Analysis . The sqrt( t) plot has fre-\nquently been misinterpreted when picking fracture closure, even for the simplest cases. The primary p\nw vs. sqrt( t) curve should form"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29b8a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "w vs. sqrt( t) curve should form \na straight line during fracture closure, as with the G-function plot. Some users suggest that the closure is identified by the departure of the data from the straight line trend, similar to the way the G-function closure is picked. This is incorrect and leads to a later closure and lower apparent closure pressure. The correct indication of closure is the inflection point on the p\nw vs. sqrt( t) plot. \nThe best way to find the inflection point is to plot the first \nderivative of pw vs. sqrt( t) and find the point of maximum ampli-\ntude of the derivative. Many fracture-pressure analysis software packages plot the inverse of the actual first derivative and show the inflection point as the minimum of the derivative. The plot in Fig. 2,  shows that the slope of the pressure curve starts low, then"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29c16-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "increases and reaches a maximum rate of decline at the inflection point, then decreases again after closure. The first derivative curve in Fig. 2 is plotted with the proper sign. The dashed vertical line [1] is the G-function closure pick that is synchronized in time and pressure with the sqrt( t) plot. Clearly the consistent closure lies \nat the inflection point and not at the point of departure from the straight line tangent to the pressure curve.\nThe semilog derivative of the pressure curve is also shown on \nthe sqrt( t) plot. This curve is equivalent to the semilog derivative of \nthe G-function for most low-perm cases. The closure pick falls at the departure from the straight line through the origin on the semi-log derivative of the P vs. sqrt( t) curve. A single closure point must \nsatisfy the requirement on both the G-function and sqrt( t) plots.\nNormal Leakoff Log-Log Pressure Derivative. The log-log"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29ca2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "Normal Leakoff Log-Log Pressure Derivative. The log-log \nplot of pressure change from ISIP vs. shut-in time for the normal leakoff example is shown in Fig. 3.  The heavy curve is the pres-\nsure difference and the dashed curve is its semilog derivative with respect to shut-in time. The vertical dashed line is the unique closure pick from the G-function and sqrt( t) plot. It is common for \nthe pressure difference and derivative curves to be parallel immedi-ately before closure. The slope of these parallel lines is diagnostic of the flow regime established during leakoff before closure. In many cases a near-perfect \u00bd slope is observed, strongly suggesting linear flow from the fracture. In this example the slope is greater than \u00bd suggesting possible linear flow coupled with changing fracture/wellbore storage (See Appendix B). The separation of the \n0 5 10 15 20 25\nG (Time)750080008500900095001000010500\n0200400600800100012001400160018002000\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29d2e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 1, "chroma:document": "GdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\nFig. 1\u2014Normal leakoff G-function plot.1/24/200704:00 08:00 12:00\n1/24/200716:00\nTime750080008500900095001000010500\n01002003004005006007008009001000\n1Pressure\nDerivativesP vs. \u221at\n\u221atdP/d\u221at vs. \u221at\ndP/d\u221at vs. \u221atFracture Closure\nFig. 2\u2014Normal leakoff sqrt( t) plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29dba-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 2, "chroma:document": "398 August 2009 SPE Production & Operationstwo parallel lines always marks fracture closure and is the final \nconfirmation of a consistent closure identification.\nAfter closure the semilog derivative curve will show a slope \nof \u22121/2 in a fully developed reservoir pseudolinear flow regime and a slope of \u22121 in fully developed pseudoradial flow. In the example the derivative slope is \u22121 indicating that reservoir pseu-doradial flow was observed. The late-time data shows a drop in the derivative probably caused by wellbore effects such as gas entry and phase segregation. The use of the semilog derivative of the log-log plot for after-closure flow regime identification, as well as closure confirmation, is a powerful new addition to fracture pressure decline diagnostics.\nAfter-Closure Analysis for Normal Leakoff Example. The \nTalley-Nolte After-Closure Analysis (ACA) flow regime identi-fication plot for the normal leakoff example is shown as Fig. 4"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29e3c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 2, "chroma:document": "(Talley et al. 1999). The heavy solid line is the observed bottomhole pressure during the falloff minus the initial reservoir pressure. The slope of the semi-log derivative of the pressure difference function (dashed line) is 1.0 during the identified pseudoradial flow period. If a linear-flow period existed in this data set a derivative slope of \u00bd would exist. It is critical to remember that the slope of the pressure difference curve on this plot is determined solely by the guess of reservoir pressure used to construct the plot. The slope of the deriva-tive is not affected by the input reservoir pressure value.\nIf a pseudoradial flow regime is identified, then the Cartesian \nRadial Flow plot (Fig. 5)  can be used to determine reservoir \nfar-field transmissibility, kh/\n/H9262. The viscosity used is the far-field \nmobile fluid viscosity and h is the estimated net pay height. For the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29ec8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 2, "chroma:document": "analysis of the example data kh//H9262 = 299 md-ft/cp. For gas viscosity at reservoir temperature, kh = 7.9 md-ft. For the assumed net pay, \nthe effective reservoir permeability is 0.097 md. \nHorner Analysis for Normal Leakoff Example. If a pseudo-\nradial flow period is identified, then a conventional Horner plot can also be used to determine reservoir transmissibility. In Fig. 6 \nthe Horner slope through the radial flow data is 14411 psi. Using an average pump rate of 18.4 bpm, kh/\n/H9262 = 298 md-ft/cp. For the \nassumed gas viscosity kh = 7.9 md-ft. Using the same assumed \nnet gives k = 0.097 md. This result is consistent with the ACA \nresults. \nG-Function Permeability Estimate. An empirical correlation \nhas also been developed to estimate formation permeability from the G-function closure time when after-closure data is not avail-able. The correlation is described in detail in the Appendix. Fig. 7"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29f4a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 2, "chroma:document": "shows the G-function correlation permeability estimate for the observed closure time and other input parameters. The perme-ability estimate of 0.097 md is consistent with the Horner and ACA results. \nPressure-Dependent Leakoff.  Pressure dependent leakoff (PDL) \noccurs when the \ufb02  uid loss rate changes with pore pressure or net \neffective stress in the rock surrounding the fracture. PDL is not caused by the normal change in transient pressure gradient during leakoff. This is part of the normal leakoff mode and is handled by the one-dimensional linear \ufb02  ow solution of the diffusivity \nequation used to model fracture leakoff in a constant permeability system. The pressure dependence referred to here is a change in the transmissibility of the reservoir \ufb01  ssure or fracture system that \ndominates the \ufb02  uid loss rate. PDL is only apparent when there 2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 67890.001 0.01 0.1 1\nSquare Linear Flow (FL^2)234567892345678923456789\n10100100010000"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae29fd6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 2, "chroma:document": "10100100010000\n (m = 1) Delta-Pressure and Derivative\u0394P vs. FL2\nFL2d\u0394P/dFL2vs. FL2\u0394P= (pw\u2013pr)\nStart of Radial Flow\nFig. 4\u2014Normal leakoff ACA log plot.\n2 3 1\nHorner Time72507500775080008250850087509000925095009750\n (m = 14411) \n  (Reservoir = 7476)  1Pressure\nFig. 6\u2014Normal leakoff Horner Plot. \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8\nRadial Flow Time Function72007400760078008000820084008600880090009200\n (m = 4814.2) Results\nReservoir Pressure  = 7475.68 psi\nTransmissibility, kh/\u03bc = 298.94991 md*ft\nkh = 7.94014 md*ft\nPermeability, k = 0.0968 md\nStart of Pseudo Radial Time = 2.15 hours 1Pressure\nFig. 5\u2014Normal leakoff ACA radial flow plot.2 3 456789 2 3 456789 2 3 456789 2 3 456789\n0.1 1 10 100 1000\nTime (0 = 8.15)2345678923456789234\n101001000\n (m = \u20131)  (m = 0.632)  BH ISIP = 9998 psi 1Delta-Pressure and Derivative\u0394P vs. \u0394t\n\u0394td\u0394P/d\u0394tv s .  \u0394t\nFracture ClosureRadial Flow\nFig. 3\u2014Normal leakoff log-log plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a062-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 3, "chroma:document": "August 2009 SPE Production & Operations 399is substantial stress dependent permeability in a composite dual-\npermeability reservoir.\nG-Function for Pressure-Dependent Leakoff.  Fig. 8 shows \nthe G-function behavior expected for PDL. The primary pw vs. \nG curve is concave upward and curved while PDL persists. The semilog derivative exhibits the characteristic \u201chump\u201d above the straight line extrapolated to the derivative origin. The end of PDL and the critical fissure opening pressure corresponds to the end of the \u201chump\u201d and the beginning of the straight line representing matrix dominated leakoff. Fracture closure is still shown by the departure of the semilog derivative from the straight line through the origin. \nSqrt(t) Analysis for PDL. Interpretation of the sqrt(t) plot in \nPDL cases has often led to incorrect closure picks. Fig. 9 shows an \nexpanded view of the sqrt( t) plot for the example with the curves"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a0e4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 3, "chroma:document": "scaled for better visibility. Note that the semilog derivative is nearly identical in shape and information content to the G-function semilog derivative. It clearly shows the PDL \u201chump\u201d and closure, which has been synchronized to the G-function result. \nIncorrect closure picks on the sqrt( t) plot will not occur if the \nsemilog derivative is used. Problems arise when the first deriva-tive is used exclusively to pick closure. In PDL cases the obvious derivative maximum, or most prominent inflection point, is caused by the changing leakoff associated with PDL and does not indicate fracture closure. The false closure indication is shown on the plot. Many fracture diagnostic tests have been badly misdiagnosed because the early and incorrect closure was picked because of dependence on only the sqrt( t) plot. This example clearly illustrates why all available diagnostic plots must be used in concert to arrive \nat a single consistent closure event. \nLog-Log Pressure Derivative for PDL Example. Fig. 10 shows"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a17a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 3, "chroma:document": "Log-Log Pressure Derivative for PDL Example. Fig. 10 shows \nthe log-log plot for the PDL example. The normal matrix leakoff period, following the end of PDL, appears as a perfect \u00bd slope of the semilog derivative with a parallel pressure difference curve exactly 2-times the magnitude of the derivative. The parallel trend ends at the identified closure time and pressure difference. In this example a well-defined \u2212\u00bd slope, or reservoir pseudolinear flow period, is \nshown shortly after closure. The later data approach a slope of \u22121, \nwhich indicates pseudoradial flow has been established. \nAfter-Closure Analysis for PDL Example. The ACA log-log \nplot (Fig. 11)  shows both the reservoir linear and radial flow peri-\nods in their expected locations. \nFigs. 12 and 13  show the ACA Cartesian plots for the linear and \nradial flow analyses. Both give consistent estimates of reservoir pore pressure. The pseudoradial flow analysis gives a transmissibil-ity of 37.2 md-ft/cp and estimated permeability of 0.047 md."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a1fc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 3, "chroma:document": "Horner Analysis for PDL Example. For an average pump rate \nof 6.7 bpm the Horner plot gives kh/\n/H9262 = 35.72 md-ft/cp. The Horner \nestimated permeability is 0.046 md compared to 0.047 md from the ACA Radial Flow analysis. Pore pressure estimated from the Horner plot is also consistent with both the linear and radial analyses because a well-developed pseudoradial flow period does exist in this case. The vertical dotted line in Fig. 14  shows the start of pseudora-\ndial flow. If a pseudoradial flow period does not exist, extrapolation of an apparent straight-line on the Horner plot can give extremely \ninaccurate estimates of pressure and flow capacity. 0.0010.010.1110100\n0              2              4               6              8             10            12            14             16            18            20\nGcData Input\nrp 1\n\u03c6 0.09 V/V\nct 7.50E-05 psi\u20131\nE3 . 5 M p s i\n\u03bc 1c p\nGc 2.44\nPz 966.0 psiEstimated Permeability = 0.0974 mdPermeability, md\nFig. 7\u2014Normal leakoff permeability estimate."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a288-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 3, "chroma:document": "Fig. 7\u2014Normal leakoff permeability estimate.\n2 3 456789 2 3 456789 2 3 456789 2 3 4567890.1 1 10 100 1000\nTime (0 = 9.133333)23456789234567892\n101001000\n (m = 0.5) \n(m = \u20131) (m = \u20130.5)  BH ISIP = 10000 psi 1Pressure Difference and Derivative\u0394P vs. \u0394t\n\u0394td\u0394P/d\u0394tv s .  \u0394tFracture ClosureLinear Flow\nRadial Flow\nFig. 10\u2014PDL log-log plot. \n0 2 4 6 8 10 12 14 16 18 20\nG (Time)8250850087509000925095009750100001025010500\n01002003004005006007008009001000\n21\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\nFig. 8\u2014PDL G-function plot.\n1/24/200700:20 00:40 01:00 01:20 01:40\n1/24/200702:00\nTime825085008750900092509500975010000102 50\n0100200300400500\n1False ClosurePressure\nDerivativesP vs. \u221at\n\u221atdP/d\u221at vs. \u221at\ndP/d\u221at vs. \u221atFracture Closure\nFig. 9\u2014PDL Sqrt plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a30a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 4, "chroma:document": "400 August 2009 SPE Production & OperationsG-Function Permeability Estimate for PDL Example. The \nG-function permeability correlation for the PDL example is shown in Fig. 15. It also gives a consistent permeability of 0.045 md. The \nimpact of the accelerated leakoff during PDL gives an estimate of the composite reservoir effective permeability. Note that the injected fluid viscosity is used for the permeability estimate based on closure time. Fracture Tip Extension.  In very low permeability reservoirs \nthe decline in wellbore pressure observed after shut-in may be caused by the dissipation of the pressure transient established in the fracture during pumping. The near-well pressure decreases as the fracture closes, which results in a decrease of fracture width at the well. The closing of the fracture volumetrically displaces \ufb02 uid to the tip of the fracture, causing continued extension of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a396-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 4, "chroma:document": "the fracture length. Much of the pressure decline is therefore not related to leakoff but to the dissipation of the linear transient along the fracture length. \nG-Function Analysis for Tip Extension. During fracture tip \nextension the G-function derivatives fail to develop any straight-line trends. The primary P vs. G curve is concave upward, as is the first derivative. The semilog derivative starts with a large posi-tive slope, and the slope continues to decrease with shut-in time, giving a concave-down curvature (Barree and Mukherjee 1996; Barree 1998). Fig. 16  shows a typical case of fracture tip exten-\nsion with minimal leakoff. This is another case that is frequently misdiagnosed. \nSqrt(t) Analysis With Fracture Tip Extension. Many times the \nfirst break in the semilog derivative curve has been misinterpreted as a closure event. The mistake is often compounded by the use of the sqrt( t) plot.  Fig. 17  shows the sqrt( t) plot for the same data."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a63e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 4, "chroma:document": "The first derivative shows a large maximum very shortly after shut-in. This is often mistaken for closure. The semilog deriva-tive on the sqrt( t) plot helps to avoid this mistaken closure pick, \nand shows the same continuously increasing trend as seen on the G-function semilog derivative plot. In low permeability systems it is generally safe to assume that as long as the semilog derivative is still rising, the fracture has not yet closed. This is not true in 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nLinear Flow Time Function8000820084008600880090009200\n (m = 1438.5) \nResults\nReservoir Pressure = 8056.66 psi\nStart of Pseudo Linear Time = 15.9End of Pseudo Linear Time = 54.391 2PressureStart Linear Flow End Linear Flow\nFig. 12\u2014PDL ACA linear flow plot.2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 67890.001 0.01 0.1 1\nSquare Linear Flow (FL^2)234567892345678923456789\n10100100010000\n (m = 1) \n (m = 0.5) \n1 2 3Delta-Pressure and Derivative\u0394P vs. FL2\nFL2d\u0394P/dFL2vs. FL2\u0394P=(pw\u2013pr)\nStart Linear Flow\nEnd Linear Flow"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a6de-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 4, "chroma:document": "FL2d\u0394P/dFL2vs. FL2\u0394P=(pw\u2013pr)\nStart Linear Flow\nEnd Linear Flow\nStart Radial Flow\nFig. 11\u2014PDL ACA log plot.\n0.0 0.1 0.2 0.3 0.4 0.5 0.6\nRadial Flow Time Function8000820084008600880090009200\n (m = 11373) Results\nReservoir Pressure = 8068.81 psi\nTransmissibility, kh/\u03bc = 37.21984 m\nkh = 0.93764 md*ft\nPermeability, k = 0.0469 md\nStart of Pseudo Radial Time = 11.2 6 1Pressure\nFig. 13\u2014PDL ACA radial flow plot.1Horner Time8000810082008300840085008600870088008900\n (m = 43920) \n  (Reservoir = 8064)  1Pressure\nFig. 14\u2014PDL Horner plot.\n0.0010.010.1110100\n0              2              4              6               8             10            12            14             16            18            20\nGcData Input\nrp 1\n\u03c6 0.08 V/V\nct 6.00E-05 psi\u20131\nE5 M p s i\n\u03bc 1c p\nGc 2.9\nPz 841.0 psiEstimated Permeability = 0.0453 mdPermeability, md\nFig. 15\u2014PDL permeability estimate."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a760-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "August 2009 SPE Production & Operations 401very high permeability reservoirs and should always be checked \nusing the log-log pressure difference plot. \nLog-Log Pressure Derivative Analysis With Tip Extension. \nThe log-log plot of pressure change after shut-in is particularly useful for diagnosing fracture tip extension. Fig. 18  shows the \npressure difference and pressure derivative (semilog) for the tip extension example. \nIn Fig. 18, the pressure derivative departs from the early unit-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a7f6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "slope (storage) and establishes a \u00bc slope during fracture tip exten-sion. The pressure difference curve falls on a parallel \u00bc slope line separated by 4-times the magnitude of the derivative. The \u00bc slope signature is diagnostic of bilinear flow representing a continued dissipation of the linear pressure transient along the fracture length (extension and concomitant fluid flow) and some linear flow driv-ing minimal leakoff. For tip extension to occur the leakoff rate to the formation must be low. As long as the parallel \u00bc slope trend continues, the fracture has not closed and is still in the process of extending. Closure cannot be determined and no after-closure analysis can be conducted.\nHeight Recession or Transverse Storage.  There are two differ-\nent mechanisms that can generate a similar diagnostic derivative signature during fracture closure. Both are caused by an excess stored volume of \ufb02  uid in the fracture at shut-in relative to the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a882-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "expected surface area of the fracture for a planar, constant-height geometry model. \nTraditionally this signature has been called \u201cfracture height \nrecession\u201d. The usual model assumes that leakoff occurs only through a thin permeable bed and that the fracture extends in height to cover impermeable strata with no leakoff. At shut-in there is a large volume of fluid stored in the fracture and the leakoff rate relative to the stored volume is small, hence the rate of pressure decline is likewise small. As the fracture empties, the rate of leakoff \nrelative to the remaining stored fluid accelerates and the pressure declines more rapidly. If the fracture height changes during leakoff, the fracture compliance may also decrease, adding to the rate of pressure loss.\nHowever, the same signature is observed in many cases where"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a904-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "However, the same signature is observed in many cases where \nfracture height growth out of zone is not observed by tracers, inclinometer, or micro-seismic mapping. Some of these cases show treating behavior similar to PDL cases, with a tendency for rapid screenout and difficulty placing high proppant concentration slur-ries. These observations suggest that another mechanism may be responsible for the same diagnostic derivative signature. The alter-nate mechanism is called \u201ctransverse fracture storage\u201d. \nIn transverse fracture storage a secondary fracture set is opened"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2a990-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "when the fluid pressure exceeds the critical fissure-opening pres-sure, just as in PDL. As the secondary fractures dilate they create a storage volume for fluid which is taken from the primary hydraulic fracture. While the fracture storage volume increases, leakoff can also be accelerated so PDL and storage are aspects of the same coupled mechanism of fissure dilation. The relative magnitude of the enhanced leakoff and storage mechanisms determines whether the G-function derivatives show PDL or storage. Numerical model-ing studies indicate that the storage mechanism can easily domi-nate even large PDL.\nAt shut-in the secondary fractures will close before the primary"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2aa12-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "fracture because they are held open against a stress higher than the minimum in-situ horizontal stress. As they close fluid will be expelled from the transverse storage volume back into the main fracture decreasing the normal rate of pressure decline and, in effect, supporting the observed shut-in pressure by re-injection of stored fluid. Accelerated leakoff can still occur at the same time but if the storage and expulsion mechanism exceeds the enhanced leakoff rate then the only signature observed during falloff will be storage. In many cases a period of linear, constant area, constant matrix perme-ability dominated leakoff will occur after the end of storage. \nG-Function Analysis With Storage. The characteristic G-function \nderivative signature is a \u201cbelly\u201d below the straight line through the origin and tangent to the semilog derivative of p\nw vs. G at the \npoint of fracture closure. Fig. 19  shows an example of slight to"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2aa9e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "moderate storage. In Fig. 19, fracture closure, indicated by the same departure of the tangent line from the semilog derivative, occurs just after the end of the storage effect. \nSqrt(t) Analysis With Storage or Height Recession. The sqrt( t) \nplot (Fig. 20)  shows a clear indication of closure based on both the \nfirst-derivative inflection point and the semilog derivative curve. Picking closure in the case of storage is not generally a problem.\nThe storage model, whether caused by height recession or \ntransverse fractures, requires that a larger volume of fluid must be leaked-off to reach fracture closure than is expected for a single planar constant-height fracture. In either case the time to reach fracture closure is delayed by the excess fluid volume that must be lost. Any estimation of reservoir permeability will give an incorrect result if the uncorrected closure time (either G\nc or time-to closure 0 5 10 15 20 25 30\nG (Time)840086008800900092009400960098001000010200\n0255075100125150\n1\nP vs. G"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ab20-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 5, "chroma:document": "0255075100125150\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GPressure\nDerivatives\nFig. 16\u2014Tip extension G-function plot.1/25/200704:00 08:00 12:00 16:00\n1/25/200720:00\nTime840086008800900092009400960098001000010200\n0102030405060708090100\n1Incorrect ClosurePressure\nDerivativesP vs. \u221at xdP/dxvs. \u221atdP/dxvs. \u221at\nFig. 17\u2014Tip extension sqrt( t) plot.\n2 3 4 56789 2 3 4 56789 2 3 4 56789 21 10 100 1000\nTime (0 = 33.7)23456789234567892345\n101001000\n (m = 0.25)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative\u0394P vs. \u0394t\n\u0394td\u0394P/d\u0394tv s .  \u0394t\nFig. 18\u2014Tip extension log-log plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2aba2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "402 August 2009 SPE Production & Operationsin minutes, tc) is used. The observed closure time must be corrected \nby multiplying by the storage ratio, rp. The magnitude of rp can \nbe determined by taking the ratio of the area under the G-function semilog derivative up to the closure time, divided by the area of the right-triangle formed by the tangent line through the origin at closure. For normal leakoff and PDL the value of r\np is set to 1.0 \neven though the ratio of the areas will be greater than 1 for the PDL case. It is possible that the closure time for PDL leakoff is proportional to the composite system permeability including both the matrix and fractures. For severe cases of storage r\np can be as \nlow as 0.5 or less. \nLog-Log Pressure Derivative With Storage. Fig. 21  shows"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ac2e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "Log-Log Pressure Derivative With Storage. Fig. 21  shows \nthe log-log plot of pressure difference and semilog derivative for the storage case. Prior to closure, and while transverse storage is dominant, the semilog derivative approaches a unit slope, with the pressure difference curve nearly parallel. In some cases the two curves lie together on a single unit-slope line. In this case the curves are separated slightly and the slope is not exactly 1.0. After closure the reservoir transient signature is defined as in the previ-ously presented cases. All fracture storage effects are eliminated and the reservoir pseudolinear flow period is shown by a \u22121/2 slope with a pseudoradial flow period indicated by a \u22121 slope of the semi-log derivative. \nConclusions\nThe use of prefrac injection/falloff diagnostic tests has become"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2acb0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "commonplace. Many important decisions regarding fracture treat-ment designs and expectations of post-frac production are based on the results of these tests. In too many cases individual diagnostic plots and analysis techniques are misapplied, leading to incorrect \ninterpretations. The analyses presented here lead to the following conclusions: 1.  With consistent application of all available pressure decline \ndiagnostics, a single unambiguous determination of fracture closure time and pressure can be made. \n 2.  A single, unique closure event can be identified on all diag-\nnostic plots. \n 3.  The conventional analysis of the sqrt( t) plot, using the inflec-\ntion point identified by the first derivative, gives incorrect indications of closure for cases of PDL and tip extension and should not be relied upon.\n 4.  A modified sqrt( t) analysis, using the semilog derivative, is \nequivalent to the G-function analysis and helps avoid incorrect closure picks in cases of PDL and tip extension."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ad32-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "5.  Flow regimes can be identified using the semilog pressure \nderivative on the log-log plot of /H9004p\nwf \u2212 /H9004t during the shut-in \nperiod following the fracture injection test.\n 6.  As in conventional transient test analysis, a pseudolinear flow \nperiod is identified by parallel \u00bd slope lines, separated by 2x, on the log-log /H9004p\nwf \u2212 /H9004t plot up until fracture closure. \n 7.  Bilinear flow can be identified by parallel \u00bc slope lines \nseparated by 4x on the log-log /H9004pwf \u2212 /H9004t plot prior to fracture \nclosure.\n 8.  After closure the pseudolinear reservoir flow period is identi-\nfied by a \u22121/2 slope of the semilog derivative of the pressure difference on the log-log /H9004p\nwf \u2212 /H9004t plot, and a \u22123/2 slope of \nthe first derivative of the pressure difference with shut-in time on the same plot.\n 9.  Pseudoradial flow is identified by a \u22121 slope of the semilog \nderivative on the log-log plot.\n10.  When a stable pseudolinear flow period exists, the after-clo-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2adbe-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "sure Cartesian plot of the linear flow function can be used to estimate reservoir pressure.\n11.  When a pseudoradial flow period exists, both the conventional \nHorner analysis and after-closure radial-flow analysis can be used to determine reservoir transmissibility and pore pressure.\nNomenclature\n A\nf = fracture area, L2, ft2\n B = formation volume factor, L3/L3, RB/STB\n ct = total compressibility, Lt2/m, psi\u22121\n Cac = after-closure storage, L4t2/m, bbl/psi\n Cbl = bilinear \ufb02  ow constant, m/Lt5/4, psi\u00b7hr3/4\n Cfbc = before-closure fracture storage, L4t2/m, bbl/psi\n Cpl = pseudolinear \ufb02  ow constant, m/Lt3/2, psi\u00b7hr1/2 \n Cpr = pseudoradial \ufb02  ow constant, m/Lt, psi\u00b7hr \n FL = linear \ufb02  ow time function, dimensionless \n FR = radial \ufb02  ow time function, dimensionless 5 10 15 20\nG (Time)750080008500900095001000010500\n0200400600800100012001400160018002000\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\nFig. 19\u2014Storage G-function plot.1/24/200702:00 04:00 06:00 08:00\n1/24/200710:00"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ae4a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 6, "chroma:document": "1/24/200710:00\nTime750080008500900095001000010500\n0100200300400\n1Pressure\nDerivativesP vs. \u221at\nxdP/dxvs. \u221at\ndP/dxvs. \u221atFracture Closure\nFig. 20\u2014Storage sqrt( t) plot.\n2 3 456789 2 3 456789 2 3 456789 2 3 4560.1 1 10 100\nTime (0 = 9.416667)234567892345678923\n101001000\n (m = 0.5)  (m = \u20130.5)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative\u0394P vs. \u0394t\n\u0394td\u0394P/d\u0394tv s .  \u0394t\nFig. 21\u2014Storage log-log plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2af08-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "August 2009 SPE Production & Operations 403 g = loss-volume function, dimensionless \n G = G-function, dimensionless \n h = height, L, ft \n k = permeability, L2, md \n Lf = fracture half-length, L, ft \n mH = slope of data on Horner plot, m/Lt2, psia\n mL = slope of data on pseudolinear \ufb02  ow graph, m/Lt2, psia\n mR = slope of data on pseudoradial \ufb02  ow graph, m/Lt2, psia\n p = pressure, m/Lt2, psia\n pwf = fracture pressure measured at wellbore, m/Lt2, psia\n q = \ufb02 ow rate, L3/t, bbl/D\n Qt = total injection volume, L3, bbl\n rp = storage ratio, dimensionless\n Sf = fracture stiffness, m/L2t2, psi/ft\n t = time, hr\n ta = adjusted pseudotime, hr\n /H9252 = constant, dimensionless\n /H9004 = difference, dimensionless\n /H9262 = viscosity, m/Lt, cp\n /H9278 = porosity, dimensionless\n /H9273 = constant, dimensionless\nSubscripts\n a = adjusted\n c = closure\n D = dimensionless\n e = end of injection\n f = \ufb01 ltrate\n p = pumping\n w = wellbore\n z = process zone\n 0 = end of injection\nAcknowledgments"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2af9e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "z = process zone\n 0 = end of injection\nAcknowledgments\nThe authors would like to thank Kumar Ramurthy, Halliburton, \nMike Conway, Stim-Lab, and Stuart Cox, Marathon, for their discussion and contribution to the procedures described. Sincere thanks are also due to the many operators whose diligence in prefrac testing has allowed these diagnostic analysis procedures to be developed and tested.\nReferences\nBarree, R.D. 1998. Applications of Pre-Frac Injection/Falloff Tests in \nFissured Reservoirs\u2014Field Examples. Paper SPE 39932 presented at the SPE Rocky Mountain Regional/Low-Permeability Reservoirs Symposium, Denver, 5\u20138 April. DOI: 10.2118/39932-MS.\nBarree, R.D. and Mukherjee, H. 1996. Determination of Pressure Depen-\ndent Leakoff and its Effect on Fracture Geometry. Paper SPE 36424 presented at the SPE Annual Technical Conference and Exhibition, Denver, 6\u20139 October. DOI: 10.2118/36424-MS.\nCastillo, J.L. 1987. Modified Fracture Pressure Decline Analysis Includ-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b02a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "ing Pressure-Dependent Leakoff. Paper SPE 16417 presented at the Low-Permeability Reservoirs Symposium, Denver, 18\u201319 May. DOI: 10.2118/16417-MS.\nCinco-Ley, H. and Samaniego-V ., F. 1981. Transient Pressure Analysis for \nFractured Wells. J. Pet Tech  33 (9): 1749\u20131766. SPE-7490-PA. DOI: \n10.2118/7490-PA.\nCinco-Ley, H., Kuchuk, F., Ayoub, J., Samaniego-V ., F., and Ayestaran, \nL. 1986. Analysis of Pressure Tests Through the Use of Instantaneous Source Response Concepts. Paper SPE 15476 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 5\u20138 Octo-ber. DOI: 10.2118/15476-MS.\nCraig, D.P. and Blasingame, T.A. 2006. Application of a New Fracture-\nInjection/Falloff Model Accounting for Propagating, Dilated, and Closing Hydraulic Fractures. Paper SPE 100578 presented at the SPE Gas Technology Symposium, Calgary, 15\u201317 May. DOI: 10.2118/ 100578-MS.\nHagoort, J. 1981. Waterflood-induced hydraulic fracturing. PhD thesis,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b0b6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "Delft Technical University, Delft, The Netherlands.Koning, E.J.L. and Niko, H. 1985. Fractured Water-Injection Wells: A \nPressure Falloff Test for Determining Fracture Dimensions. Paper SPE 14458 presented at the SPE Annual Technical Conference and Exhibi-tion, Las Vegas, Nevada, USA, 22\u201325 September. DOI: 10.2118/14458-MS.\nNolte, K.G. 1979. Determination of Fracture Parameters from Fractur-\ning Pressure Decline. Paper SPE 8341 presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, USA, 23\u201326 September. DOI: 10.2118/8341-MS.\nSettari, A. 2000. Coupled Fracture and Reservoir Modeling. Presented at \nthe Workshop on Three Dimensional and Advanced Hydraulic Frac-ture Modeling, Fourth North American Rock Mechanics Symposium, Seattle, Washington, USA, 29 July.\nTalley, G.R., Swindell, T.M., Waters, G.A., and Nolte, K.G. 1999. Field"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b138-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "Application of After-Closure Analysis of Fracture Calibration Tests. Paper SPE 52220 presented at the SPE Mid-Continent Operations Symposium, Oklahoma City, Oklahoma, USA, 28\u201331 March. DOI: 10.2118/52220-MS.\nAppendix A\u2014Definition of Diagnostic \nFunctions\nThe G-Function.  The G-function is a representation of the elapsed \ntime after shut-in normalized to the duration of fracture extension. Corrections are made for the superposition of variable leakoff times while the fracture is growing. The form of the G-function used in this paper assumes high \ufb02  uid ef\ufb01  ciency in low-permeability \nformations. Under that assumption the surface area of the fracture is assumed to vary linearly with time during fracture propaga-tion. The dimensionless pumping time used in the G-function is de\ufb01 ned as:\n/H9004tt t t\nDp p=\u2212()/.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-l)\nThe elapsed total time from the start of fracture initiation (not"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b1ba-857d-11ee-bb1c-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "start of pumping) is t and the total pumping time (elapsed time from fracture initiation to shut-in) in consistent time units is t\nP. For \nthe assumption of low leakoff the dimensionless time ( /H9004tD) is used \nto compute an intermediate function:\ngt t tDD D/H9004/H9004 /H9004() =+() \u2212 \u23a1\u23a3\u23a4\u23a64\n3115 15 . ..   . . . . . . . . . . . . . . . . . .(A-2)\nThe G-function used in the diagnostic plots is derived from the \nintermediate function as follows:\nGt gt gDD/H9004/H9004() = () \u2212 \u23a1\u23a3\u23a4\u23a64\n0/H9266,  . . . . . . . . . . . . . . . . . . . . . . .(A-3)\nwhere g0 is the dimensionless loss-volume function at shut-in ( t = tp \nor /H9004tD = 0). All derivatives are calculated using a central difference \nfunction of pressure and G-function (normalized shut-in time).\nAfter-Closure Analysis and Flow Regime Identi\ufb01  cation.  After-\nclosure pressure decline analysis requires the identi\ufb01  cation of fully-\ndeveloped reservoir pseudolinear and pseudoradial transient \ufb02  ow"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b246-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 7, "chroma:document": "regimes. The \ufb02  ow regimes can be identi\ufb01  ed by characteristic slopes \non a log-log plot of observed falloff pressure minus reservoir pressure, (p\nw(t) \u2212 pi), vs. the square of the linear-\ufb02  ow time function ( FL2) and \nthe semilog derivative, (X*dY/dX), of the pressure difference curve Talley et al.). It is important to note that the guess of reservoir pressure, p\ni, used in construction of the \ufb02  ow regime plot severely impacts the \nslope and magnitude of the pressure difference curve. The pressure derivative, because of the difference function used to generate it, is not affected by the initial guess of reservoir pressure. \nThe linear-flow time function is defined by:\nFt tt\ntfor t tLcc\nc ,s i n() =\u2265\u2212 21\n/H9266.  .  . . . . . . . . . . . . . . . . .(A-4)\nThe linear-flow function also requires an accurate determina-\ntion of the time required after shut-in to reach fracture closure, tc."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b2d2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "404 August 2009 SPE Production & OperationsIn the pseudolinear flow period the slope of the derivative curve \non the log-log plot should be \u00bd. For the correct estimate of reser-voir pore pressure, the pressure difference curve should also have a slope of \u00bd and should be exactly twice the magnitude of the derivative. If a stable pseudolinear flow period is identified then a Cartesian plot of observed pressure during the falloff, p\nw(t), vs. \nFL should yield a straight line with intercept equal to the reservoir \npore pressure, pi, and with a slope of mL. \npt p m Ft twi L L c()\u2212= (),.   . . . . . . . . . . . . . . . . . . . . . . . . .(A-5)\nIf a pseudoradial flow period exists, the slope of the derivative \nand correct pressure difference curves on the log-log flow regime plot should both be 1.0 and the two curves should coincide. In the pseudoradial flow period, a Cartesian plot of pressure vs. F\nR should \nalso yield a straight line with intercept equal to pi and slope of mR."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b354-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "pt p m Ft twi R R c()\u2212= (),.   . . . . . . . . . . . . . . . . . . . . . . . . .(A-6)\nIn these equations, tc is the time to fracture closure with time \nzero set as the beginning of fracture extension, pi is the initial reser-\nvoir pressure, and mR is the Cartesian slope of the \u201ccorrect\u201d straight \nline. The radial-flow function ( FR) is given by (Talley et al.)\nFt tt\nttRcc\nc,l n , .() =+\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f=\u22451\n4116162/H9273/H9273/H9266.  . . . . . . . . . . .(A-7)\nIn the properly identified pseudoradial flow period the reservoir \nfar-field transmissibility can be determined from the slope, fracture closure time, and volume injected during the test.\nkh Q\nmtt\nRc /H9262=251 000, .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-8)\nIn Eq. A-8, the permeability, k, is in md, net pay thickness, h, \nis in feet, far-field mobile fluid viscosity,  /H9262, in cp, tc in minutes, \nand the volume injected during the test, Qt, is in bbls, and the \nslope, mR, is in psi."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b3e0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "slope, mR, is in psi. \nHorner Analysis.  The conventional Horner analysis uses a Cartesian \nplot of observed pressure vs. Horner time, ( tp + /H9004t)//H9004t, with all times \nin consistent units. The fracture propagation time is tp and the elapsed \nshut-in time is /H9004t. As shut-in time approaches in\ufb01  nity the Horner \ntime function approaches 1. A straight-line extrapolation of the Horner plot to the intercept at a Horner time of 1.0 gives an estimate of reservoir pressure. The slope of the correct straight-line extrapola-tion, m\nH, can be used to estimate reservoir transmissibility:\nkh q\nmH /H9262=() 162 6 1440..   . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-9)\nThe flow rate in Eq. A-9 is assumed to be in barrels per minute"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b462-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "and is the average rate for the time the fracture was extending. In both Eqs. A-8 and A-9, the viscosity is the far-field mobile fluid viscosity. The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity.\nThe major problem with the Horner analysis is that the results \nare only valid if the data used to extrapolate the apparent straight line are actually in fully developed pseudoradial flow. There is no way to determine the validity of the Horner analysis or to determine the flow regime within the Horner plot itself.\nG-Function Permeability Estimate.  An empirical function to \napproximate formation permeability has been derived from numer-ous numerical simulations of fracture closure. The correlation is based on the observed G-function time at fracture closure:\nkp\ncGE rfz\ntcp=\n\u239b\n\u239d\u239c\u239e\n\u23a0\u239f0 0086 0 01\n0 03819 6..\n../H9262"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b4ee-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "kp\ncGE rfz\ntcp=\n\u239b\n\u239d\u239c\u239e\n\u23a0\u239f0 0086 0 01\n0 03819 6..\n../H9262\n/H9278.   . . . . . . . . . . . . . . . . . . . . . .(A-10)The rate of fluid loss from the fracture before closure is dominated \nby the mobility of the injected fluid instead of the far-field viscosity. The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected fluid at leakoff temperature and the relative permeability to the leakoff fluid. The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid mobility. As a general rule, an assumed injected fluid viscosity of 1.0 cp is used to incorporate the effects of reservoir temperature and relative permeability in the invaded zone.\nNote that the correlation gives permeability and not kh. The \ntime to closure is related to fracture volume vs. created area so both the fracture height and length do not appear in the equation. The process zone stress, P\nz, is the net fracture extension pressure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b570-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "z, is the net fracture extension pressure \nabove closure pressure, pc, or pz = pISIP \u2212 pc. The net extension pres-\nsure and Young\u2019s Modulus provide a relationship between facture volume (width) and surface area during pumping.\nThe other parameters in Eq. A-10 are defined, except the stor-\nage ratio, r\np. This parameter represents the amount of excess fluid \nthat must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height pla-nar fracture. The storage ratio, r\np, is the ratio of the area under the \nG-function semilog derivative up until closure divided by the area of the triangle defined by the straight-line (normal leakoff) tangent to the semilog derivative at closure. For normal matrix leakoff and PDL, r\np is therefore 1.0. For the transverse storage and height recession \nsignature, rp is some value less than 1.0. The observed G-function \nclosure time ( Gc) is always delayed by the excess fluid volume in the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b5fc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "fracture for either height recession or transverse storage. Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility estimate. The closure time should be corrected by multiplying by r\np \nfor all permeability and transmissibility estimates. \nAppendix B\u2014Analytical Solutions and the \nLog-Log Diagnostic Graph\nLinear and Bilinear Flow Before Closure.  Hagoort (1981) developed \na before-closure analytical solution accounting for both storage and linear formation \ufb02  ow. Koning and Niko (1985) write the solution as\npwDtC ee r f c tDf b c Dt\nDD =\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f \u2212 () () /H9266/H9266/H9252/H9252/H9004 /H9004/H9004\n212\n2, \n . . . . . . . . . . . . . . . . . . . . . . (B-1)\nwhere\npwDtkhpt pt t\nqBDwe we()() ( )\n./H9004/H9004=\u2212+ ()\n141 2 /H9262,   . . . . . . . . . . . . . . (B-2)\n/H9004/H9004tkt\ncLD\ntf=0 0002637\n2.\n/H9278/H9262,  . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)\n/H9252/H9266=2"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b67e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 8, "chroma:document": "/H9252/H9266=2\nCfbcD,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)\nand\nCC\nch LC\nch LfbcDfbc\ntffbc\ntf==5 615\n20 8936\n22..\n/H9266/H9278 /H9278.  . . . . . . . . . . . . . . . . . . . (B-5)\nHere, Cfbc is the before-closure fracture storage constant, which is \ndefined as (Craig and Blasingame 2006)\nCA\nSfbcf\nf=2\n5 615.,   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-6)\nwhere Af is the fracture area (one wing), and  Sf is the fracture \nstiffness."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b70a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 9, "chroma:document": "August 2009 SPE Production & Operations 405Hagoort\u2019s solution predicts that the before-closure falloff is a \ncombination of fracture storage and linear flow. As fracture storage becomes small, C\nfbcD\u21920, linear flow will dominate the before-\nclosure pressure falloff. In most \"hard rock\" environments, CfbcD \nis small, and for a fracture-injection/falloff sequence with /H9004t >> te \nand long closure times, \npt pt t twe we() ( )\u2212+ \u221d /H9004 ,  . . . . . . . . . . . . . . . . . . . . . . . . . (B-7)\nwhere t = te + /H9004t, and pw(te) is the instantaneous shut-in pressure. \nDefine a fracture-pressure difference as\n/H9004/H9004pp t p ttw f we we=\u2212 + () ( ) ,  . . . . . . . . . . . . . . . . . . . . . . . (B-8)\nand a log-log graph of /H9004pwf vs. t will exhibit a \u00bd slope during \nlinear flow before closure. An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist, but field data suggest that during before-closure bilinear flow"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b78c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 9, "chroma:document": "/H9004/H9004ptwf\u221d()14,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-9)\nand a log-log graph of /H9004pwf vs. t will exhibit a \u00bc slope during \nbilinear flow before closure.\nPseudoradial, Pseudolinear, and Bilinear Flow After Closure.  \nCraig and Blasingame (2006) developed an analytical solution for a fracture-injection/falloff sequence with a propagating and closing hydraulic fracture, and they derived the \u201ccomplete\u201d after-closure impulse solutions accounting for fracture storage. The after-closure impulse solution for pseudoradial \ufb02  ow is written as \npt t pkhQpwsDCacpwe it ().( )() ( +\u2212 = +\u2212 /H9004141 2 24\n200/H9262pp\ntti\ne) . ( )+1\n/H9004\n . . . . . . . . . . . . . . . . . . . . . (B-10)\nFor after-closure pseudolinear flow,\npt t p\nhLQ\npwwe i\nft()\n.( ) .+\u2212\n=+/H9004\n141 2 24 0 0002637\n21 /H9266\nssDCacppi () ( )00\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f\n\u00d7+\u239b\n\u239d\u239c\u239e\n\u23a0\u239f/H9262\n/H9278ckt tte112\n/H9004,   . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-11)\nand for after-closure bilinear flow,\npt t pwe i()"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b818-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 9, "chroma:document": "and for after-closure bilinear flow,\npt t pwe i()\n.( ) (. ) (. )+\u2212\n=/H9004\n141 2 24 0 6125 0 000263714\nkkwQ\npp Cac fft\ni +\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f()034/H9262\n\u00d7\u239b\n\u239d\u239c\u239e\n\u23a0\u239f+\u239b\n\u239d\u239c\u239e\n\u23a0\u239f1114 34\n/H9262ck t tte /H9004.   . . . . . . . . . . . . . . . . . . . . . . . (B-12)For convenience, define a constant pseudoradial flow term as\nCkhQpwsDCacppprti = +\u2212()141 2 24\n200.( )() ( )/H9262.   . . . . . (B-13)\nFor pseudolinear flow, a constant term is defined as\nC\nhLQ\npwsDCacpl\nft=+141 2 24 0 0002637\n21\n0.( ) .\n() (/H9266\nppp ck i t 012\n\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f\u239b\n\u239d\u239c\u239e\n\u23a0\u239f )./H9262\n/H9278\n . . . . . . . . . . . . . . . . . . . . . (B-14)\nSimilarly, for bilinear flow a constant term is written as\nC\nkwQ\nbl\nfft=+141 2 24 0 6125 0 000263714.( ) (. ) (. )\n(ppp Cacck i t 034141\n\u2212\u239b\n\u239d\u239c\u239e\n\u23a0\u239f\u239b\n\u239d\u239c\u239e\n\u23a0\u239f ). /H9262/H9262\n . . . . . . . . . . . . . . . . . . . . . (B-15)\nWith the new definition, and defining a reservoir-pressure differ-\nence, /H9004pw = pw(t) \u2212 pi, the pseudoradial-flow impulse solution can \nbe written as"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b89a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 9, "chroma:document": "be written as\n/H9004pC twp r=\u22121.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-16)\nSimilarly, the linear-flow impulse solution can be written as\n/H9004pC twp l=\u221212/,   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-17)\nand the bilinear-flow impulse solution can be written as\n/H9004pC twb l=\u221234/.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-18)\nThe derivatives with respect to t are written for after-closure \npseudoradial flow as, \n\u2212\u2202\n\u2202=\u2212 /H9004p\ntCtw\npr()2,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-19)\nfor after-closure pseudolinear flow as,\n\u2212\u2202\n\u2202=\u2212 /H9004p\ntCtw pl\n()/\n232,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-20)\nand for after-closure bilinear flow as,\n\u2212\u2202\n\u2202=\u2212 /H9004p\ntCtwb l\n()/ 3\n474.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-21)\nThe semilog derivatives and the impulse derivatives (Cinco-Ley"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b91c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 9, "chroma:document": "The semilog derivatives and the impulse derivatives (Cinco-Ley \net al. 1986) are calculated by multiplying the after-closure deriva-tives by t and t\n2, respectively. Table B-1  shows the derivative terms \nfor each after-closure flow regime.\nLog-Log Diagnostic Graph.  Before-closure linear or bilinear \ufb02  ow \nare identi\ufb01  ed by the relationship between the fracture-pressure \nTABLE B-1\u2014AFTER-CLOSURE IMPULSE SOLUTIONS AND DERIVATIVES  \nFlow Regime  Impulse Solution  Derivative  Semilog Derivative  Impulse Derivative  \nBilinear     \nPseudolinear     \nPseudoradial"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2b9a8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 10, "chroma:document": "406 August 2009 SPE Production & Operationsdifference, /H9004pwf, and total time, t, but the after-closure \ufb02  ow regimes \nare identi\ufb01  ed by the relationship between the reservoir-pressure \ndifference, /H9004pw, and t, which requires knowing the initial reservoir \npressure, pi. However, by noting that\n\u2202\n\u2202=\u2212\u2202\n\u2202/H9004 /H9004 p\ntp\ntwf w,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-22)\na log-log diagnostic graph can be prepared that is independent of \ninitial reservoir pressure but can be used to identify before- and \nafter-closure flow regimes. Table B-2  shows the characteristic \nslopes of each flow regime for log-log graphs of /H9004pwf vs. t, \u2202\u2202/H9004ptwf/ \nvs. t, tp twf\u2202\u2202/H9004 / vs. t, and tptwf2\u2202\u2202/H9004 / vs. t. \nIn gas reservoirs, log-log diagnostic graphs can be prepared \nby plotting /H9004pawf vs. ta, \u2202\u2202/H9004ptawf a/ vs. ta, tp taa w f a\u2202\u2202/H9004 / vs. ta, and \ntp taa w f a2\u2202\u2202/H9004 / vs. ta where adjusted pseudopressure is defined as\npaz\npipdp\nzp\n=\u239b\n\u239d\u239c\u239e\n\u23a0\u239f\u2320\n\u2321\u23ae\n\u23ae/H9262"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ba2a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 10, "chroma:document": "paz\npipdp\nzp\n=\u239b\n\u239d\u239c\u239e\n\u23a0\u239f\u2320\n\u2321\u23ae\n\u23ae/H9262\n/H92620,  . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-23)\nand adjusted pseudotime is defined as\ntactidt\nctt\n=\u2320\n\u2321\u23ae\n\u23ae\u23ae()/H9262/H92620.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-24)\nBecause adjusted pseudopressure is a function of initial reser-\nvoir pressure, and if initial reservoir pressure is unknown, it\u2019s often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining TABLE B-2\u2014LOG-LOG GRAPH CHARACTERISTIC SLOPES \nBefore Closure  After Closure  \nLog-Log Graph  Bilinear  Linear  Bilinear  Pseudolinear  Pseudoradial  \n 1/4 1/2 \u2013 \u2013 \u2013 \n \u20133/4 \u20131/2 \u20137/4 \u20133/2 \u20132 \n 1/4 1/2 \u20133/4 \u20131/2 \u20131 \n 5/4 3/2 1/4 1/2 0 \nthe estimate using diagnostic and interpretive graphs plotted in \nterms of adjusted pseudovariables.\nRobert D. Barree  is president and principal investigator of Barree"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 10, "chroma:document": "& Associates, a consulting firm specializing in stimulation and well performance optimization. He has been involved in the development of hydraulic fracture technology since 1980 and is the primary author of the fully 3D hydraulic fracture simulator GOHFER. Barree is also the author of more than 50 technical publications and has served as SPE Distinguished Lecturer on the topic of new philosophies in hydraulic fracturing. He holds a BS degree in petroleum engineering from the Pennsylvania State University and PhD degree from Colorado School of Mines. Valencia Barree  is a partner at Barree & Associates LLC,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 10, "chroma:document": "primarily working in well-test analysis and software development for the Stim Lab Proppant Consortium (including Predict-K) and GOHFER. Previously, she has focused on software development for core test analysis and instrumentation under the company name Custom Scientific Tools. Developed software includes tools for the acquisition and analysis of relative permeability and capillary pressure data, as well as many other customized testing procedures. She holds a BS degree in mineral engineering mathematics from the Colorado School of Mines. David P . Craig  currently writes a technical blog at www."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-107877-PA Barree Method .pdf", "page": 10, "chroma:document": "refrac.net and is researching complex-fracture propagation models for Halliburton. In 2000, Craig was awarded the Henry Matson Technical Service Award from the Denver Section of SPE in \u201crecognition of significant contributions in engineering design and diagnostics of hydraulic fractures,\u201d and in 2006 he received the Award of Excellence for Outstanding Research from the Harold Vance Department of Petroleum Engineering at Texas A&M University. He holds a BS degree from Texas Tech University and MS and PhD degrees from Texas A&M University, all in petroleum engineering."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "SPE-187348-MS\nNew Variable Compliance Method for Estimating In-Situ Stress and Leak-Off\nfrom DFIT Data\nHanYi Wang and Mukul M. SharmaThe University of Texas at Austin\nCopyright 2017, Society of Petroleum Engineers\nThis paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bcc8-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nAbstract\nOver the past two decades, Diagnostic Fracture Injection Tests (DFIT) or Injection-Fall-off Fracture\nCalibration Tests have evolved into a commonly used and reliable technique to evaluate reservoir properties,\nfracturing parameters and obtain in-situ stresses. Since the introduction of DFIT analysis based on G-\nfunction and its derivative, this method has become standard practice for quantifying minimum in-situ stress\nand leak-off coefficient. However, the pressure decline model that underlies the G-function plot makes two\ndistinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or\ncompliance) is assumed to be constant during fracture closure. Fracture closure is a gradual process that"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bd4a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "starts when asperities at the fracture tip first come into contact with each other. As pressure declines due\nto leak-off, more and more of the fracture wall comes into contact. It is important to model this process\nquantitatively to obtain good estimates of in-situ stresses and leak-off. In this paper, we present a model that\naccounts for changes in fracture stiffness/compliance as the fracture closes, with leak-off that is dependent\non fracture pressure. The model is, therefore, capable of analyzing DFIT data from the end of pumping to\ndays or even weeks after shut-in.\nWe first review Nolte's original G-function model and examine the assumptions inherent in the model.\nWe then present a new global pressure transient model for pressure decline after shut-in which not only\npreserves the physics of unsteady-state reservoir flow behavior, elastic fracture mechanics and material\nbalance, but also incorporates the gradual changes of fracture stiffness/compliance due to the contact of"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bdd6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 0, "chroma:document": "rough fracture walls during closure. This global model allows us to analyze the whole spectrum of DFIT data\nby bridging before closure and after closure data seamlessly. Analysis of synthetic cases, along with field\ndata are presented to demonstrate how the coupled effects of fracture geometry, fracture surface asperities,\nformation properties, pore pressure and wellbore storage can impact fracturing pressure decline and the\nestimation of minimum in-situ stress.\nIt is shown that so-called \"normal leak-off\" behavior that is modeled using Carter leak-off is an\noversimplification that leads to significant errors in the interpretation of the data. All the before\nclosure analysis conducted under a \"normal leak-off\" assumption should be reexamined cautiously. Most\nimportantly, this article reveals that previous methods of estimating minimum in-situ stress often lead to\nsignificant over or underestimates, because of their failure to account for changes in fracture stiffness/"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2be58-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "2 SPE-187348-MS\ncompliance correctly as the fracture closes progressively from the edge to the center. Based on our modeling\nand simulation results, we propose a much more accurate and reliable method to estimate the minimum in-\nsitu stress, fracture pressure dependent leak-off rate and evaluate the compliance of the un-propped fracture.\nIntroduction\nDiagnostic fracture injection tests (DFIT), which have also been referred to as Injection-Falloff Tests,\nFracture Calibration Tests, Mini-Frac Tests in the literature, involve pumping a fluid (typically water), at\na constant rate for a short period of time, creating a relatively small hydraulic fracture before the well\nis shut in. The pressure transient data after shut-in is analyzed to obtain in-situ stresses and reservoir\nproperties. A typical pressure trend is qualitatively shown in Fig. 1. Conventionally, DFIT analysis has\nfocused on acquiring fracturing treatment design parameters, such as fluid efficiencies, leak-off coefficient"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bee4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "and fracture closure pressure (which is normally interpreted as minimum in-situ stress). However, in recent\nyears, DFIT analysis has been extended to obtain reservoir properties such as reservoir pore pressure, and\npermeability in unconventional reservoirs, where traditional pressure transient tests are impractical. The\nreservoir properties determined by DFIT are representative, because the created fracture can pass through\nnear-wellbore damage zone and provide a large volume of investigation for true formation properties. In\naddition, the net pressure trends in a DFIT can be also used to infer the induced fracture complexity in\ndifferent geological settings (Potocki 2012). Thoese valuable information obtained from DFITs provides\nkey input parameters for modeling hydraulic fracture propagation (Wang 2015; Wang 2016; Wang et al.\n2016), stimulation design (Ramurthy et al. 2011), development of reservoir models (Mirani et al. 2016;"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bf66-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "Loughry et al. 2015; Wang 2017) and post-fracture analysis (Fu et al. 2017). Without a realistic estimation\nof the fracturing parameters and reservoir properties, it would not be possible to optimize hydraulic fracture\ndesign and evaluate the economic viability of producing hydrocarbons from unconventional reservoirs.\nFigure 1\u2014Diagram showing sequence of events observed in a DFIT\nThe pressure decline data after shut-in can be divided into two distinct regions for analysis: before closure\npressure data and after closure pressure data (separated by the fracture closure pressure). Analysis models are\ndeveloped differently for these regimes. Before closure analysis (BCA) assumes normal leak-off behavior\nor uses the portion of normal leak-off data (a straight line on G function and square root of time plots) to\nestimate fracture geometry, leak-off coefficient or formation permeability (Mayerhofer et al. 1995; Nolte"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2bfe8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 1, "chroma:document": "1979 and 1986; Valko\u2019 and Economides 1999). A departure from the straight line indicates fracture closure\n(Castillo1987). After closure analysis (ACA) models have been developed by several authors (Benelkadi\nand Tiab 2004; Chipperfield 2006; Nolte et al. 1997; Soliman et al. 2005; Soliman et al. 2010) to estimate"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c074-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "SPE-187348-MS 3\npore pressure and infer formation permeability. Because both BCA and ACA only use some portion of the\nDFIT data, global consistency of interpretation can't be guaranteed, even if the individual analyses match\na certain portion of the data well. Liu et al. (2016) and Marongiu-Porcu et al. (2014) attempted to bridge\nBCA and ACA on a log-log plot and the closure pressure is picked at the end of 3/2 slope of the pressure\nderivative. However, their work is still based on G-function model's assumptions and the 3/2 slope just\narises from a spatial integration of Carter's leak-off (McClure 2017). Currently, no model is available to\nsimulate DFIT pressure decline under a single, coherent mathematic framework, and correctly capture the\ndynamic evolution of fracture stiffness/compliance during closure. In the following section, we are going to\nreview and examine the fundamental limitations of the G-function model, show that closure does not occur"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c0f6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "at a single point in time but is a continuous and gradual process, and derive a fully coupled global model\nthe fits the entire pressure curve during the entire fracture closure process.\nThe advent of fracturing pressure decline analysis was pioneered by the work of Nolte (1979 and 1986).\nWith the assumptions of power law fracture growth, negligible spurt loss, constant fracture surface area after\nshut-in and Carter's leak-off model (one-dimensional leak-off of fluid from a constant pressure boundary,\nthe solution to the diffusivity equation predicts that the leak-off rate will scale with the inverse of the square\nroot of time), a remarkably simple and useful equation for the pressure decline can be obtained:\n(1)\nHere, is the instantaneous shut-in pressure at the end of pumping, Pf is the fracture pressure at\ndimensionless time \u0394t D.tp is the total pump time.r p is the productive fracture ratio, which is the ratio of"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c178-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 2, "chroma:document": "fracture surface area that is subject to leak-off to the total fracture surface area. For low permeability,\nunconventional reservoirs, r p \u22481. To increase the readability of this article, r p will be assumed to be 1 for the\nrest of the discussion. CL is Carter's leak-off coefficient which is a constant. Sf is the fracture stiffness, which\ncan be calculated using Table 1 for different fracture geometries. It is assumed that Sf is a constant until the\nfracture closes instantaneously when the fluid pressure in the fracture reaches the closure pressure. Because\nfracture stiffness is the reciprocal of fracture compliance, they will be used interchangeably throughout this\narticle.\nTable 1\u2014Fracture stiffness expressions for different fracture geometry models\nE\u2032 is the plane strain Young's modulus and can be calculated using Young's Modulus, E, and Poisson's\nRatio, v:\n(2)\nThe dimensionless time \u0394t D is defined by:\n(3)\nWhere t is the generic time and t e is the time at the end of pumping. G-function is defined as"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c204-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "4 SPE-187348-MS\n(4)\nWhere the g-function of time is approximated by,\n(5)\nFrom Eq. (1), we can infer that for normal leak-off behavior (constant Carter's leak-off coefficient,\nconstant leak-off area and constant fracture stiffness during fracture closure), the pressure declines\nlinearly with G(\u0394t D). Castillo (1987) usedNolte's G-function for modeling the pressure declinebehavior\nand developed the straight-line plotof the G-function vs. pressure. The slope of this curve is used for the\ncomputation of the leak-off coefficient that is independent of pressure. Any departure from this straight-\nline is interpretedasclosure of the fracture.\nUnfortunately, plots of pressure versus G-function often yield curves with multiple points of inflectionthat\nhave been attributed to abnormal leak-off behavior (such as pressure dependent leak-off, fracture height\nrecession, closing of secondary transverse fractures, and fracture tip extension), which makes it difficult"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c286-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "to interpretthe changes in slope and identify fracture closure. So identification of fracture closure pressure\nand non-ideal behavior is usually done using plots of pressure and GdP/dG versus G-function (Barree and\nMukherjee, 1996; Barree, 1998, Barree et al., 2014), where the closure is picked at the tangential point\nbetween a straight line that passes the origin and the GdP/dG curve. This prevailing method of determining\nminimum in-situ stress (although has been widely accepted, but has never been theoretically proved) will be\nclassified and discussed as \"tangent line method\" in the following text. The fracturing pressure decline model\nthat underlying G-function plot suffers from two distinct and important issues: (1) leak-off is not pressure-\ndependent, i.e. a constant pressure boundary is assumed (2) fracture compliance/stiffness is assumed to be\nconstant during fracture closure. This is why G-function based models are only used for before closure"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c312-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "analysis and are not capable of analyzing DFIT data from the end of pumping to days or even weeks after\nshut-in, which requires bridging both before and after closure data seamlessly.\nModel Formulation\nThere are two essential components of the general DFIT model formulated in this paper. In this section, we\nwill present a generalization of the pressure transient model that couples the fluid pressure in the fracture\nwith the leak-off rate without making any assumptions about the pressure in the fracture that acts as a\nboundary condition for leak-off. In addition, special cases will be demonstrated to link previous models and\nour fully coupled approach.\nGlobal DFIT Model\nIn unconventional reservoirs, fluid leak-off during pumping is very small and no filter-cake is formed on the\nfracture surface because only water is injected. In addition, the duration of injection is very short (typically\n3-5 minutes) while the total shut-in time may be days or even weeks. In order to correctly capture fracturing"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c39e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 3, "chroma:document": "pressure response during a DFIT in unconventional reservoirs, the pressure dependent leak-off at the fracture\nsurface and the dynamic changes of fracture compliance have to be accounted for. Fig. 2 illustrates one-\ndimensional leak-off into a semi-infinite formation."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c420-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 4, "chroma:document": "SPE-187348-MS 5\nFigure 2\u2014Illustration of one-dimensional leak-off\nAssuming linear Darcy flow and a slightly compressible, single phase fluid in the reservoir, the\ndifferential form of the mass balance can be written as:\n(6)\nwhere P is the pressure, k is formation permeability, \u03bc f is fluid viscosity, \u03d5 is formation porosity and ct is\ntotal formation compressibility. Inside the fracture, the average fracture width \n  and fracturing net pressure\nPnet is related by the fracture stiffness:\n(7)\nFrom a material balance perspective (fluid compressibility is negligible compared to that of the fracture),\nthe rate of fluid leak-off(one wing of the fracture) into the formation, qf, after shut-in equals the rate of\nshrinkage of fracture volume (one wing of the fracture), V f, as pressure declines:\n(8)\nAnd\n(9)\nwhere Af is the fracture surface area of one face of one wing of the fracture. So Eq. (8) can be re-written as\n(10)\nAt the boundary of the fracture surface, both Darcy's flow and material balance have to be honored"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c4a2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 4, "chroma:document": "(11)\nRearranging Eq. (11), we get the boundary condition at the fracture surface\n(12)\nIn the above derivation, it is assumed that the whole fracture surface area is subject to leak-off, which is\nthe norm in low permeabity reservoirs. If only a portion of fracture surface is considered permeable, then"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c52e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "6 SPE-187348-MS\none just need to multiply a leak-off surface area ratio (leak-off surface area devided by total fracture surface\narea) on the left side of Eq. (12).\nWith initial condition (disregarding the pressure disturbance during the short injection period)\n(13)\n(14)\nwhere P 0 is the initial pore pressure.\nThe governing Eq. (6), plus the initial condition of Eq. (13)-(14) and boundary condition of Eq. (11)\nuniquely describes the pressure transient behavior during DFITs. In the above derivations, the influence of\nwellbore storage (WBS) is not included. For typical DFITs, untreated water is normally used as injection\nfluid. Since water is slightly compressible, the wellbore fluid will expand after shut-in with the declining\npressure. Ignoring the early time fluid transient behavior along the wellbore after shut-in and assuming\nfracturing pressure and wellbore pressure have reached quasi-equilibrium, leak-off volume into formation"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c5b0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "should equal the shrinkage of fracture volume plus the expansion of wellbore fluid. We can modify the leak-\noff rate from Eq. (10) to account for wellbore fluid expansion as follows:\n(15)\nWhere V W is half the wellbore volume (only one wing of the fracture needs to be modeled) and C W\nis the compressibility of water. Using Eq. (15) with Eq. (10), we can define the fracture-wellbore system\nstiffness as:\n(16)\nUnder extreme circumstances, when wellbore storage is negligible (V w Cw \u2248 0), fracture stiffness\ndominates the system stiffness, and when Vw CW are large and Af is small, WBS dominates the system\nstiffness. If WBS has to be included in the model, one just need simply replace Sf with SS in Eq. (12). It should\nbe emphasized there are two stages of fracture closure: mechanical closure, where the fracture surfaces come\ninto contact at asperities, and the other is hydraulic closure, where fracture pressure in the closed section"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c63c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 5, "chroma:document": "is disconnected from the open section or wellbore pressure. In the above derivations, we assume that the\nmechanically closed fracture still retains hydraulic conductivity because of its residual fracture width.\nSpecial Case 1: Constant Fracture Compliance and Carter's Leak-off.   If we assume that a constant\nshut-in pressure (ISIP) is applied at the fracture surface, the solution of Eq. (6) to obtain the leak-off or\nthe velocity Uf across fracture surface into an intact, homogenous formation can be found as ( Economides\nand Nolte 2000):\n(17)\nIf we take \u0394t out of Eq. (17), then the R.H.S of Eq. (17) essentially becomes Carter's leak-off coefficient\nCL, which is a constant parameter. The total leak-off rate qf from one wing of the fracture starting from\nshut-in is:\n(18)"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c6be-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "SPE-187348-MS 7\nCombining Eq. (18) and Eq. (10), we have:\n(19)\nEq. (19) is a simple first order differential equation, with the initial condition of Pf (\u0394t = 0) = ISIP, the\nsolution of Eq. (19) leads to the pressure declining proportionally to the square root of time:\n(20)\nEq. (20) is the fundamental basis for the square root of time plot. From Eq. (1) and (20), we can infer that\nfor normal-leak-off behavior with constant fracture stiffness and assuming Carter's leak-off, the pressure\nand its derivative will be straight lines on both G-function and the square root of time plots. The fracture\npressure continues declining even when it reaches the initial reservoir pore pressure. This non-physical\nprediction indicates that the assumption of Carter's leak-off and constant stiffness made in the derivation\nprevent Notle's G-function model and the square root of time model from capturing the actual pressure\nresponse during fracture closure and late time pressure transient behavior."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c74a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "Special Case 2: Constant Fracture Compliance and Fracture Pressure Dependent Leak-off.   When the\nleak-off rate is fracture pressure dependent, a closed analytic form of pressure decline is not available,\nbut the solution can be obtained using a superposition time. A detailed derivation of the pressure decline\nsolution is presented in Appendix A. And from Eq. (A.4), the fracture pressure at the nth time interval, Pf,n,\ncan be calculated explicitly:\n(21)\nwhere Pf,0 = P0, Pf,1 = ISIP, \u0394t 1 = 0. It is important to note that in this article, the term \"fracture pressure\ndependent leak-off\" (FPDL) specifically refers to a leak-off rate that depends on the declining fluid pressure\ninside the fracture, not to how the leak-off rate changes with net stress in the rock surrounding the fracture,\nwhich is often denoted as pressure dependent leak-off or PDL ( Barree et al. 2009). In some cases, excessive\npressure drop can be observed during early-time of shut-in in low permeability reservoirs, but it is the result"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c7cc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "of high apparent ISIP, wellbore storage and additional friction away from the wellbore, and not caused by\nPDL, which is unlikely to happen in nanodarcy permeability systems.\nVariable Fracture Compliance Due to Closure on Rough Surfaces and Asperities\nAs pressure declines inside the fracture after shut-in, the fracture will gradually close and the fracture\naperture will approach the scale of the surface roughness. If the fracture faces are perfectly parallel and\nsmooth, they will come into contact all at once when the fluid pressure inside the fracture declines to the\nfar field stress, and the fracture is then mechanically and hydraulically closed. However, there is abundant\nevidence to suggest that fractures retain their conductivity after the walls have come into contact (mechanical\nclosure). Fractures retain a finite aperture after mechanical closure due to a mismatch of asperities on the\nfracture walls. van Dam et al. (2000) presented scaled laboratory experiments on hydraulic fracture closure"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c858-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 6, "chroma:document": "behavior. They observed up to a 15% residual aperture (compared to the maximum aperture during fracture\npropagation) long after shut-in. Fredd et al. (2000) demonstrated fracture surface asperities can provide\nresidual fracture width and sufficient conductivity in the absence of proppants. Using sandstone cores from\nthe East Texas Cotton Valley formation, sheared fracture surface asperities that had an average height of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c8e4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "8 SPE-187348-MS\nabout 2.286 mm were observed. Warpinski et al. (1993) reported hydraulic fracture surface asperities of\nabout 1.016 mm and 4.064 mm for nearly homogeneous sandstones and sandstones with coal and clay-\nrich bedding planes, respectively. Sakaguchi et al. (2008) created tensile fractures in large rock blocks and\nmeasured the asperity height and distribution. Their work shows that the fracture surfaces can be assumed\nto be a fractal object and most of the asperities fall within 1 to 2 mm in height. Wells and Davatzes (2015)\nconducted topographic measurement on dilated fractures from core samples and found the asperity height\nranges from hundreds to thousands of micrometers. Bhide et al. (2014) created X-ray microtomographic\nimages from shear induced fractures and the roughness values obtained varied from 1.8 to 1.95 mm along\nthe length of the rock samples. Zou et al. (2015) conducted experiments on 20 fractured shale samples and"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c970-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "found the average asperity height to be 1.88 mm. An experimental study ( Zhang et al. 2014) on Barnett Shale\nsamples reveals that the surface topography of the displaced fracture can be altered because of rock failure,\nand the fracture surface exhibited parallel strips of crushed asperities. Field measurements ( Warpinski et al.\n2002) using a down-hole tiltmeter array indicated that the fracture closure process is a smooth, continuous\none which often leaves 20%-30% residual fracture width, regardless of whether the injection fluid is\nwater, linear-gel or cross-linked-gel. The microscopic measurement and modeling of surface roughness and\nmechanical properties of asperities can often be up-scaled to macroscopic contact laws that relate fracture\nwidth and the associated contact stress. Willis-Richards et al. (1996) proposed contact law to relate fracture\nwidth and the net closure stress for fractured rocks, based on the work of Barton et al. (1985):\n(22)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2c9fc-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "(22)\nwhere wf is the fracture aperture and, w0 is the contact width, which represents the fracture aperture when\nthe contact normal stress is equal to zero, \u03c3 c is the contact normal stress on the fracture, and \u03c3 ref is a contact\nreference stress, which denotes the effective normal stress at which the aperture is reduced by 90%. it should\nbe emphasized that the contact width w0 is determined by the tallest asperities, and the strength, spatial and\nheight distribution of asperities are reflected by the contact reference stress \u03c3 ref (e.g., if the tallest asperities\non two fracture samples are the same, then they should have the same W0, but the one with a higher median\nasperity height will have higher value of \u03c3 ref, provided other properties are the same).\nWith known fracture geometry, rcok properties and surface roughness, the question now is how to\nestimate fracture stiffness/compliance as a function of pressure, if it continuously changes during fracture"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ca88-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "closure. Wang and Sharma (2017) presented a method and general algorithms to model the dynamic behavior\nof hydraulic fracture closure on rough fracture surfaces and asperities, using linear elastic solutions that\ncoupled with contact law for three different fracture models (PKN, KGD and radial fracture geometry).\nGiven the fracture geometry, rock properties, contact parameters and minimum principal stress, their\napproach can predict the evolution of fracture aperture profile, total fracture volume and fracture stiffness\nas fracturing pressure declines. A brief discussion of modeling non-local dynamic fracture closure behavior\ncan be found in the Appendix B.\nAssuming a Young's modulus of 20 GPa, Poisson's ratio of 0.25, W0 of 2 mm, \u03c3 ref of 5 MPa for a\nPKN fracture geometry with 10 m fracture height, 35 MPa minimum in-situ stress, the evolution of the\nfracture width profile and contact stress distribution can be determined, as the fluid pressure inside the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cbaa-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 7, "chroma:document": "fracture gradually declines. The results are shown in Fig. 3 and Fig. 4. To demonstrate the impact of fracture\nroughness and surface asperities on fracture closure behavior, the case without surface asperities (fracture\nsurface is completely planar and smooth) is also included. The result shows that at relative high fracturing\nfluid pressure, the fracture asperities have negligible impact on fracture width distribution, and the contact\nstress is always concentrated at the tip of the fracture, where the contact stress is much higher than in the\nmiddle of the fracture. We can also see that the fracture surfaces do not contact each other like parallel\nplates. In fact, the fracture closes on rough surfaces starting from the tip, and closes progressively all the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cc40-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 8, "chroma:document": "SPE-187348-MS 9\nway from the edges to the center of fracture. As fluid pressure continues to decline, more and more of the\nfracture surfaces come into contact and these changes the subsequent fracture closure behavior. At lower\nfluid pressures, contact stresses start to counter-balance the in-situ stress and the fracture becomes stiffer\nand less compliant. If the fracture faces were perfectly parallel and smooth, the fracture width would have\ncollapsed to zero when the fluid pressure dropped to 35 MPa. The moment when all fracture surfaces come\ninto contact on asperities and the contact stress becomes non-zero on the entire fracture surface, the fracture\nis mechanically closed and this mechanical closure stress is higher than the minimum in-situ stress.\nFigure 3\u2014Fracture width evolution with and without asperities at different fluid pressure for a PKN geometry\nFigure 4\u2014Fracture width and the corresponding contact stress distribution at different fluid pressure for a PKN geometry"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ccc2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 8, "chroma:document": "The simulated fracture volume evolution as a function of fluid pressure is shown in Fig. 5. As can be\nseen, when the fluid pressure inside the fracture is relatively high, the fracture volume declines linearly\nwith pressure (indicates roughly constant fracture compliance/stiffness). However, as the pressure declines\nto a certain level (2 MPa higher than the input minimum in-situ stress of 35 MPa), the fracture volume\nand pressure departs from a linear relationship. The traditional method of estimating fracture stiffness using\nTable 1 can only be used when the fracture pressure is still relatively high and the asperities are only in\ncontact at the edges of the fracture. However, as pressure declines further and more and more fracture surface\narea comes into contact, the effective fracture stiffness becomes pressure dependent:\n(23)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cd44-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "10 SPE-187348-MS\nFigure 5\u2014Fracture volume and fracture stiffness evolution as fluid pressure declines for a PKN geometry\nThe relationship between Pf and Vf can be used to calculate the pressure dependent fracture stiffness\nand the result is also shown in Fig. 5. This pressure dependent fracture compliance/stiffness obtained\nfrom non-local fracture closure modeling (Wang and Sharma 2017) can be incorporated into Eq. (12) to\nmake Sf pressure dependent. The \"tangent line method\" for picking up minimum in-situ stress is based\non the underlying assumptions of G-function model that assumes perfectly smooth fracture surfaces, and\nthe stiffness remains constant until fluid pressure drops to minimum in-situ stress. This leads to a sudden\nincrease in fracture stiffness when all fracture surfaces get into contact simultaneously. Fracture stiffness\nonly remains constant at high fracture pressures. It starts to increase long before the fracturing pressure"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cdd0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "declines to the minimum in-situ stress (35 MPa in this case). McClure et al. (2016) proposed a \"compliance\nmethod\" for picking closure pressure on the G-function plot based on numerical simulations, where closure\npressure is picked at the point where the fracture stiffness starts to increase. Their method can overestimate\nminimum in-situ stress if the mechanical closure pressure is interpreted as the minimum in-situ stress.\nThere is compelling field evidence to indicate that this occurs in the field. Fig. 6 shows the normalized\ntiltmeter response plotted against wellbore pressure during the shut-in period of 2B well at three different\nstations from the GRI/DOE M-site. Each tiltmeter response is normalized by dividing by the maximum\ntilt (displacement, also an indication of fracture width) measured at that instrument during the test. The\ntiltmeter demonstrates that soon after shut-in, the measured displacement declines linearly with pressure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ce5c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "If the fracture surface area remains constant during shut-in, then the fracture volume is proportional to the\naverage displacement and should also decline linearly with the pressure within this period. After the wellbore\npressure declines to a certain level, the measured displacement vs pressure departs from a linear relationship.\nThis field measurement is consistent with the general trend shown in Fig. 5. As pressure continues declining,\nmore and more fracture surface area comes into contact with the rough fracture surfaces and asperities, and\nthe fracture stiffness increases gradually. Even though these data were measured from the end of injection to\nweeks of shut-in, we can still observe the existence of a residual fracture width that is supported by surface\nasperities and mismatches even after the fracturing fluid pressure drops below the minimum in-situ stress.\nSo considering the nature of mineralogical heterogeneity and the existence of thin laminated rock layers,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cee8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 9, "chroma:document": "the formation of asperities and tortuous fracture walls are inevitable during hydraulic fracture propagation."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2cf74-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "SPE-187348-MS 11\nFigure 6\u2014Normalized tiltmeter data from shut-in of 2B from the GRI/DOE M-site. Data Courtesy of Norm Warpinski\nSimulation Results and Analysis\nThe pressure decline behavior during DFITs in low permeability formations is governed by Eq. (6) (a PDE)\nwith Eq. (12) (an ODE) acting as the boundary condition. To solve this system of PDE with an ODE\nboundary condition, the method of lines (MOL) is used, by replacing the spatial derivative in the PDE with\nalgebraic approximations. The PDE can then be transformed into a system of ODEs, which can be solved\nsimultaneously and efficiently by well-established numerical methods (Schiesser and Griffiths 2009). Since\nthe MOL essentially replaces the problem PDEs with systems of approximating ODEs, the addition of other\nODEs is easily accomplished. In this section, our fully-coupled global DFIT model is used to investigate how\ndifferent fracture geometry, contact parameters and reservoir properties impact fracturing pressure response"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d000-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "and our interpretation. It should be mentioned that the discussion presented in this section is restricted to\nlow permeability formations where the leak-off rate and poroelastic effects are small.\nNormal leak-off behavior\nNormal leak-off behavior is a special case of our model that satisfies all the assumptions of Nolte's G-\nfunction model, and it can be clearly identified with straight lines of pressure and its derivatives on G-\nfunction and the square root of time plots. All previous BCA models assume that these straight lines indicate\nnormal leak-off behavior and hence, leak-off coefficient or formation permeability can be determined based\non this portion of the data. However, one key assumption in all BCA is Carter's leak-off. To examine the\nimpact of incorrectly assuming Carter's leak-off for the duration of the DFIT, we extend the concept of\n\"normal leak-off behavior\" to account for different fracture boundary conditions, while other assumptions"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d08c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 10, "chroma:document": "remain the same (e.g., constant fracture compliance/stiffness and constant surface area during closure). The\nsimulation results from three different leak-off models are compared using input parameters from Table\n2. Fig. 7 and Fig. 8 show the simulated pressure and its derivatives on G-function and square root of\ntime plots for different formation permeability. As expected, Carter's leak-off (using Eq. (1) or Eq. (20),\nthey are interchangeable) always produce straight lines on diagnostic plots. But when pressure dependent\nleak-off at the fracturing surface is considered (using Eq. (21) and our fully coupled simulation), we no\nlonger obtain straight lines. This is because as fracturing pressure declines, leak-off rate departs from the\nlinear relationship with \n , so the assumption of Carter's leak-off is violated. Carter's leak-off model\noverestimates the leak-off rate and pressure drop by assuming a constant ISIP at the fracturing surface and\nleads to larger errors as shut-in time gets longer."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d118-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 11, "chroma:document": "12 SPE-187348-MS\nFigure 7\u2014Comparison of \"normal leak-off\" behavior for different models, k=0.0001 md\nFigure 8\u2014Comparison of \"normal leak-off\" behavior for different models, k=0.001 md\nTable 2\u2014Input parameters for DFIT simulation with constant fracture compliance\nThe most important observation here is that the so-called \"normal leak-off behavior\" with straight lines\non G-function and the square root of time plots is not \"normal\" at all. For example, the closure time for\ntypical DFIT in a formation with 0.0001 md permeability is around 2~4 days. And in the above case shown\nin Fig. 7, the fracturing pressure and its derivative depart from the straight line after 4 hours of shut-in,\neven with constant fracture compliance/stiffness, surface area and reservoir properties. Many field data sets\ndo show \"normal leak-off behavior\" with straight lines that extend all the way to the point of downward\ndeflection. This occurs if some mechanisms that accelerate pressure drop (e.g., the gradual increase of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d1a4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 11, "chroma:document": "fracture stiffness) happen to offset the effects of declining pressure at the fracture surface, so the pressure\ndata fits the straight lines predicted by Carter's leak-off model coincidentally. So any DFIT analysis that is\nbased on the assumption of Carter's leak-off and straight lines on G-function and the square root of time\nplots should be re-examined."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d230-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "SPE-187348-MS 13\nVariable Fracture Compliance on Pressure Decline and In-Situ stress Estimation\nAs discussed earlier, G-function and previous BCA models assume Carter's leak-off behavior as if the\npressure in the fracture is a constant value (i.e., ISIP). In addition, these models also assume fracture\ncompliance/stiffness is constant and only changes abruptly at the moment of fracture closure, when\nfracturing pressure drops to the minimum in-situ stress. The failure to account for the coupled mechanisms\nof pressure dependent leak-off at the fracture surface and the gradual increase of fracture stiffness during\nclosure explains why these models are only useful on a small portion of the before closure data, and are\nnot able to simulate the fracturing response for the entire duration of the DFIT test. These simplifying\nassumptions also make their interpretation of DFIT data and estimation of minimum in-situ stress inaccurate."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d2bc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "In this section, our global DFIT model is used to investigate how each factor impacts the pressure decline\ntrend and our estimation of minimum in-situ stress and fracture compliance.\nPKN Fracture Geometry.   We first examine a PKN geometry using parameters listed in Table 2 and\nassuming w0 is 2 mm, \u03c3 ref is 5 MPa and reservoir permeability is 0.0005 md as the Base Case. Fig. 9 shows\nthe contact stress at different fracture width. As expected, when the fracture width is larger than the contact\nwidth w0, the contact stress is zero. However, when the fracture width is smaller than the contact width,\nthe contact stress and fracture width follows a hyperbolic relationship. Fig. 10 shows the corresponding\nfracture stiffness evolution for different W0 with the given fracture geometry and rock properties. The results\nindicate that as the fracture width decreases, the rough fracture walls will come into contact sooner if the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d33e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 12, "chroma:document": "contact width is larger, so the noticeable changes of fracture stiffness occur earlier when the contact width\nis larger. We can also observe that when the contact width is larger, the increase in fracture stiffness is more\ngradual and smooth. In the extreme case of W0 close to zero, as would be the case for relative flat fracture\nsurfaces, we can infer that fracture stiffness will increase abruptly as soon as the fluid pressure drops to the\nminimum in-situ stress (35 MPa in this case). This pressure dependent fracture stiffness obtained from non-\nlocal fracture modeling (Wang and Sharma 2017) can be directly put into the boundary ODE of Eq. (12) to\npredict fracture pressure decline for certain reservoir properties.\nFigure 9\u2014The relationship between contact stress and fracture width for different W 0"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d3ca-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "14 SPE-187348-MS\nFigure 10\u2014Fracture stiffness evolution for different W 0 with a PKN geometry\nFig. 11 shows the fracturing pressure and its derivatives for different W0 on G-function and square root\nof time plots. Relate with Fig. 9 and Fig. 10, we can see that the contact width impacts pressure decline\nresponse significantly, because it alters the evolution of fracture stiffness. Large contact width leads to a\nsmooth pressure decline trend while small contact width leads to steep changes in pressure decline rate and\npressure derivatives. We can also infer that if the contact width is close to zero and all fracture walls come\ninto contact simultaneously, then a sudden change in pressure decline rate and the pressure derivative spikes\non both G-function and square root of time plots are inevitable, which is unrealistic and never observed in\nfield cases. So the conventional assumption that a fracture closes on flat, smooth fracture surfaces where W0"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d456-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "= 0 does not reflect reality. In addition, different contact widths lead to different derivative slopes, so any\nBCA models that attempt to match this portion of data and estimate reservoir properties without considering\nthe variable fracture compliance is questionable. Fig. 11 also shows G-function and square root of time plots\ngive the same quantitative information only slightly different in scales. More importantly, the estimated\nminimum in-situ stress from pressure derivatives using the \"convention method\" and \"compliance method\"\ngive inconsistent results for different contact widths, with the same input parameters of fracture surface\narea, minimum in-situ stress and reservoir properties. A detailed comparison and a new approach to estimate\nminimum in-situ stress will be discussed later.\nFigure 11\u2014Pressure decline response for different W 0 with a PKN geometry\nNext, we examine how contact reference stress affects the pressure decline response. Fig. 12 and Fig. 13"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d4e2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 13, "chroma:document": "show the relationship between contact stress and fracture width for different contact reference stress and the\ncorresponding fracture stiffness evolution at different fracturing pressure. For the same contact width, the\nhigher the contact reference stress, the more rapid the increase of contact stress as the fracture width shrinks.\nPhysically, the contact reference stress represents how hard and strong the fracture surface asperities are.\nThe lower the contact reference stress, the more gradual the change in fracture stiffness as pressure declines."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d56e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 14, "chroma:document": "SPE-187348-MS 15\nEven though the contact reference stress does not have much impact on the pressure at which the fracture\nstiffness starts to changes noticeably, it does impact the fracture stiffness evolution, as shown in Fig. 13.\nFig. 14 shows the fracturing pressure and its derivatives for different contact reference stress on G-function\nand the square root of time plots. It can be observed that the contact reference stress has negligible influence\non early time pressure decline when fracturing pressure is still high. This is because only the fracture edges\nstart to come into contact during this period and the fracture stiffness remains roughly constant. However,\nafter this period as the fracture pressure declines further, more and more fracture surfaces come into contact\nand the contact reference stress begins to affect the pressure decline trend and the peak value of the pressure\nderivatives.\nFigure 12\u2014The relationship between contact stress and fracture width for different \u03c3 ref"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d636-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 14, "chroma:document": "Figure 13\u2014Fracture stiffness evolution for different \u03c3 ref with a PKN geometry\nFigure 14\u2014Pressure decline response for different \u03c3 ref with a PKN geometry\nBesides the contact parameters, fracture geometry also affects fracture stiffness and its evolution during\nclosure. Fig. 15 shows fracture stiffness evolution for different fracture height while all the other parameters"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d6c2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 15, "chroma:document": "16 SPE-187348-MS\nremain the same as the Base Case. Similar to what Table 1 implies, for a PKN fracture geometry, the smaller\nthe fracture height, the higher the initial fracture stiffness. We can also observe that smaller fracture height\nleads to noticeable changes in fracture stiffness at higher fracturing pressure. This is because smaller fracture\nheight results in smaller fracture width at the same net pressure, so the more fracture surfaces can come into\ncontact at a higher fracture pressure. When fracture height is 5m, the effective fracture stiffness is already\naffected by surface contact when fluid pressure is 39 MPa.\nFigure 15\u2014Fracture stiffness evolution for different fracture height with a PKN geometry\nFig. 16 shows the fracturing pressure and its derivatives for different fracture height on G-function and\nthe square root of time plots. The results indicate that fracture height impacts the pressure decline trend"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d74e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 15, "chroma:document": "significantly. Larger fracture height leads to later occurrence of the peak of the pressure derivative and this\nalso increases the peak value of the pressure derivatives.\nFigure 16\u2014Pressure decline response for different fracture height with a PKN geometry\nFig. 17 shows the fracturing pressure and its derivatives for different reservoir permeability on G-function\nand the square root of time plots. As expected, the pressure declines more rapidly when the reservoir\npermeability is large and the decline rate slows down as the difference between fracturing pressure and\ninitial reservoir pressure becomes smaller and smaller."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d7d0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 16, "chroma:document": "SPE-187348-MS 17\nFigure 17\u2014Pressure decline response for different reservoir permeability with a PKN geometry\nNext, we examine the impact of wellbore storage on the fully coupled pressure decline trend. The\nwater compressibility is assumed to be 4.35e-4 MPa\u22121. Fig. 18 shows the fracture-wellbore system stiffness\nevolution for different wellbore volume. As can be seen, when fracturing pressure is high, fracture stiffness\ndominates the system stiffness. However, as fracturing pressure continues declining, the fracture become\nless and less compressible and the role of wellbore storage becomes apparent. In general, the larger the\nwellbore volume, the more gradual and slower the increase in system stiffness will be.\nFigure 18\u2014Fracture-wellbore system stiffness evolution for different wellbore volume with a PKN geometry\nFig. 19 shows the corresponding fracturing pressure and its derivatives for different wellbore volumes"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d85c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 16, "chroma:document": "on G-function and square root of time plots. It can be observed that larger wellbore volume leads to more\ngradual pressure decline trends. A larger wellbore volume also delays the occurrence of fracture closure\nand lowers the peak of the pressure derivative curve. It can be seen that WBS has a small impact during\nearly time of shut-in when the system stiffness is still dominated by fracture stiffness, however, as more\nand more of the fracture surface comes into contact and the fracture becomes stiffer, WBS effects become\napparent, and the after-flow of fluid from wellbore to fracture long after shut-in decelerates the pressure\ndecline rate and extends the tail of the pressure derivative after its reaches the peak. So the WBS effect on\npressure decline response must be accounted for when interpreting and analyzing DFIT data."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d8e8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "18 SPE-187348-MS\nFigure 19\u2014Pressure decline response for different wellbore volume with a PKN geometry\nFig. 20 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-\nfunction and square root of time plots. It can be observed when the initial reservoir pressure is low; the\npressure declines more rapidly. This is also reflected by Eq. (18), a lower reservoir pressure leads to a higher\nleak-off rate. However, when the reservoir pressure is over pressurized with high initial pore pressure, the\npressure decline trend resembles a \"normal-leak-off behavior\". In this case (P 0=28MPa), one can still notice\nthat there exist a subtle \"bump\" in the pressure derivatives that indicates an increase in fracture stiffness as\nreflected in Fig. 10. For small values of \u03c3 ref, this increase can be too gradual to be noticeable on G-function\nand square root of time plots. This example deomonstrates that long closure time not only can be attributed"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d96a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "to low formation permeability, but also can be the result of small pressure difference between ISIP and\ninitial pore pressure.\nFigure 20\u2014Pressure decline response for different initial reservoir pressure with a PKN geometry\nIn all the above synthetic simulations, the input minimum in-situ stress remains 35 MPa while other\nparameters are modified from the Base Case. The pressure decline trend varies substantially for different\nfracture geometry, contact parameters, reservoir permeability, wellbore volume and reservoir pore pressure.\nAny interpretation of the pressure data to determine the minimum in-situ stress must properly account for\nall these parameters. Choosing minimum in-situ stress where the pressure or its derivative departs from a\nstraight line (Nolte 1979 and 1986) is not reliable, because it ignores the changes in fracture compliance/\nstiffness during closure and the pressure dependent leak-off at the fracture surface. In addition, pressure does\nnot decline linearly with G-function or"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2d9f6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 17, "chroma:document": "not decline linearly with G-function or \n  in most cases. The most widely used \"tangent line method\" to\npick up minimum in-situ stress using a tangential line with pressure derivatives ( Barree et al., 2009; Barree\net al., 2014) always underestimates minimum in-situ stress, especially in under-pressure reservoirs. Even\nthough this method has been the common practice in the industry, it is an emprical method and is subject to\nthe assumptions discussed earlier. The \"compliance method\" identifies closure pressure as the point at which\nfracture stiffness starts to change (McClure et al. 2016). As we have discussed earlier, the fracture closes"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2da78-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "SPE-187348-MS 19\nprogressively from the edges to the center, As a result the fracture stiffness begins to increase long before\nthe fracture pressure drops to the minimum in-situ stress. A comparison of different methods of estimating\nminimum in-situ stress (closure pressure is interpreted as minimum in-situ stress) for all previous synthetic\ncases is presented in Table 3.\nTable 3\u2014Comparison of different methods of estimating minimum in-situ stress for PKN fracture geometry\nExamining Eq. (6)~Eq. (16) closely, it is evident that the pressure decline is governed by a linear PDE\nsystem in the formation that is coupled with an ODE at the fracture surface. The minimum in-situ stress is\njust one of many implicit factors that shape the evolution of the fracture pressure. Previous DFIT models\nsimplified this PDE-ODE system assuming Carter's leak-off and constant fracture stiffness during fracture\nclosure, so that a closed form of an analytic solution could be obtained. Since our model shows minimum"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dafa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "in-situ stress implicitly coupled with pressure, it is not possible to analytically estimate the minimum in-situ\nstress from a single diagnostic plot. Based on the analysis of our synthetic cases, it is clear that the \"tangent\nline method\" underestimates minimum in-situ stress and the \"compliance method\" overestimates closure in\nall cases if the closure pressure is interpreted as the minimum in-situ stress.\nOur analysis shows that the answer lies between these two limiting cases, regardless of parameters\nused in simulation. To keep the analysis pratical and not resorting to a complex inverse problem, we\npropose a \"variable compliance method,\" where the minimum in-situ stress can be obtained by averaging\nthe dimensionless G-time or the square root of time of these two methods and extrapolating back to the\npressure curve that corresponds to the averaged G-time or the square root of time. An exception is when"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2db86-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "pressure derivative plot resembles the so-called \"normal leak-off behavior,\" the changes in fracture fracture/\ncompliance may not be detectable on diagnostic plots and the \"tangent line method\" gives an estimation of\nminimum in-situ stress within an acceptable error.\nThe estimated minimum in-situ stress using \"variable compliance method\" is also included in Table 3.\nWhen we compare the estimated minimum in-situ stress obtained from the different methods with the input\nminimum in-situ stress, Table 3 shows that the \"variable compliance method\" gives significantly lower\nerrors. The maximum error does not exceed a few 100 KPa, while the \"tangent line method\" yield errors\nthat are an order of magnitude larger.\nEven though we have focused on the PKN fracture geometry in this section, these conclusions are also\napplicable to a KGD fracture geometry. We can obtain the results for a KGD geometry if we replace fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dc12-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 18, "chroma:document": "height with fracture length in calculating the evolution of fracture stiffness, as discussed in the Appendix B."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dc94-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 19, "chroma:document": "20 SPE-187348-MS\nPKN Fracture Geometry with Multiple Layers.   In previous section, we have analyzed pressure decline\nresponse of DFIT in a single layer; however, it is possible that fracture can penetrate into adjacent layers\nwhere the minimum in-situ stresses are different. Fig. 21 illustrates the examples where fracture geometry\nspan across multiple layers. In a single layer case, fracture gradually closes from edges to center on rough\nfracture wall, while in multilayer cases; fracture can close first within the interval where the minimum in-\nsitu stress is the largest.\nFigure 21\u2014Illustration of multilayer cases with different minimum in-situ stresses\nAssuming the input parameters are the same as the base case in the previous section (provided in Table\n2), and fracture geometry and minimum in-situ stresses are illustrated in Fig. 21, the fracture stiffness\nevolution can be calculated using non-local fracture modeling (Wang and Sharma 2017), as shown in Fig."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dd20-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 19, "chroma:document": "22. \"Multilayer Case 1\" is the most common scenario where height recession can happen during fracture\nclosure process where fracture closes first within formation barriers (e.g., Layer 1 and Layer 3) where\nstresses are higher. Compared to the single layer case in Fig. 22, height recession (i.e., Multilayer Case 1)\nleads to earlier and steeper increase in fracture stiffness. The fracture stiffness evolution of the single layer\ncase and \"Multilayer Case 2\" are similar, because they have the same average stress.\nFigure 22\u2014Stiffness evolution for single and multilayer cases\nFig. 23 shows the fracturing pressure and its derivatives for single and multilayer cases on G-function\nand square root of time plots. It can be observed that the trend of pressure derivatives generally reflects\nthe evolution of fracture stiffness. The case of height recession (i.e., Multilayer Case 1) leads to earlier and\nsteeper increase in pressure derivatives on both G-function and square root of time plots."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dda2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "SPE-187348-MS 21\nFigure 23\u2014Pressure decline response for single and multilayer cases\nA comparison of different methods of estimating minimum in-situ stress for above single and multilayer\ncases is presented in Table 4. Again, we can oberve that the tangent line method under estimates while\ncompliance method overestimates minimum in-situ stress (i.e., 35 MPa in the target layer), and our \"variable\ncompliance method\" gives the least errors for all cases.\nTable 4\u2014Comparison of different methods of estimating minimum in-situ stress for single and multilayer cases\nBased on the disscussion by McClure et al. (2016), the \"compliance method\" gives an estimate of the\nmecanical closure pressure, and this closure pressure can be 1-2 MPa above the actual minimum in-situ\nstress, depending on fracture geometry and contact width. Our anlaysis in Table 3 shows that the differrence\nbetween the infered closure pressure and input minimum in-situ stress that is constrained within 1-2 MPa"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2de2e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "is only limited to a few cases where fracture height is large or contact width is small. For a typical DFIT,\nthe injeciton volume is normally only a few cubic meters and the resulting fracture height or radius is often\nlimited. In addition, depending on rock mineralogy and heterogeneity, the height of asperities can range\nfrom hundreds to thousands of micrometers (Bhide et al. 2014; Fredd et al. 2000; Sakaguchi et al. 2006;\nWells and Davatzes 2015; Warpinski et al. 1993; Zou et al. 2015), and, as shown by the field tiltimeter\nsurvey, the residual fracture width can be as large as 20-30% of the maximum fracture width during pumping\n(Warpinski et al. 2002). Remember is determined by the tallest asperities, so it is challenging to calibrate\nthe minimum in-situ stress using the \"compliance method\". In addtion, if the fracture has penetetrated\ninto barrier layers with higher minimum pricinpal stress (e.g., Multilayer Case1 in Table 4), the fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2deb0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 20, "chroma:document": "walls will come into contact at a higher pressure within the barrier layers and increase the overall fracture\nstiffness, so the \"compliance method\" can furthur overstimate the minimun in-situ stress. However, in all\nsimulated cases the presented \"variable compliance method\" provides a much more consistent estimation\nof minimum in-situ stress. Most importantly, the results are insensitive to input parameters such as fracture\nsurface roughness and fracture geometry, which significantly reduces the uncertainties associated with the\n\"compliance method\".\nRadial Fracture Geometry.   Next, we investigate the pressure decline response for radial fracture geometry,\nusing parameters from Table 2 and assuming W0 is 2 mm, \u03c3 ref is 5 MPa, fracture radius is 10 m and reservoir"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2df3c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 21, "chroma:document": "22 SPE-187348-MS\npermeability is 0.0005 md as the Base Case. Fracturing pressure decline response for different W0 and are\nshown in Fig. 24. The results and trends are similar to those obtained for the PKN geometry, a small contact\nwidth leads to steeper changes in pressure decline rate and the slope of pressure derivative, and increases\nthe peak value of pressure derivative.\nFigure 24\u2014Pressure decline response for different W 0 with a radial geometry\nFig. 25 shows the pressure decline response on G-function and the square root of time plots for different\ncontact reference stress. As in the PKN cases, a larger value of contact reference stress increases the pressure\ndecline rate after the fracturing pressure drops to a certain level and reduces the peak value of pressure\nderivatives.\nFigure 25\u2014Pressure decline response for different \u03c3 ref with a radial geometry\nFig. 26 shows the corresponding pressure decline response on G-function and the square root of time"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2dfc8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 21, "chroma:document": "plots. A larger fracture radius leads to lower fracture stiffness and this, in turn, results in a lower pressure\ndecline rate.\nFigure 26\u2014Pressure decline response for different fracture radiiwith a radial geometry."}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e054-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 22, "chroma:document": "SPE-187348-MS 23\nFig. 27 shows the fracturing pressure and its derivatives for different reservoir permeability on G-\nfunction and the square root of time plots. As expected, pressure declines more rapidly when the reservoir\npermeability is large. At early time, the pressure decline rate increases because an increase in the fracture\nstiffness dominates changes in leak-off rate. At late time when the fracture becomes less compliant, the\npressure decline rate decreases due to smaller and smaller differences between the fracture pressure and the\ninitial reservoir pressure. Eventually, the leak-off rate will drop to zero as the fracture pressure becomes\nequal to the reservoir pressure.\nFigure 27\u2014Pressure decline response for different reservoir permeability with a radial geometry\nFig. 28 shows the pressure decline response for different wellbore volumes on G-function and the square\nroot of time plots. Compared to Fig. 19, it can be observed that the WBS effect has stronger influence on"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e0e0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 22, "chroma:document": "pressure decline trend in our radial fracture geometry for the same wellbore volume. This is because the\nfracture volume and fracture surface are smaller than that in previous case of PKN fracture geometry, which\nintensifies the impact of WBS effect.\nFigure 28\u2014Pressure decline response for different wellbore volume with a radial geometry\nFig. 29 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-\nfunction and square root of time plots. Similar to Fig. 20, pressure declines rapidly in depleted reservoirs and\ndeclines gradually with much lower peak values of pressure derivatives in over-pressurized reservoirs. Also\nthe pressure decline trend resembles a \"normal-leak-off behavior\" when pore pressure is abnormally high."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e1a8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 23, "chroma:document": "24 SPE-187348-MS\nFigure 29\u2014Pressure decline response for different initial reservoir pressure with a radial geometry\nTable 5 shows a comparison of the different methods of estimating minimum in-situ stress (closure\npressure is interpreted as the minimum in-situ stress) for the above synthetic cases for a radial fracture\ngeometry. Again, the results indicate that the \"tangent line method\" always underestimates minimum in-\nsitu stress while the \"compliance method\" overestimates the minimum in-situ stress (if the closure stress\nis assumed to be equal to the minimum in-situ stress) if no correction is applied (except for the so-\ncalled \"normal leak-off behavior\" where the changes in fracture stiffness/compliance are undetectable). The\nestimated minimum in-situ stress using our proposed \"variable compliance method\" give the smallest errors\nin all simulated cases.\nTable 5\u2014Comparison of different methods of estimating minimum in-situ stress for radial fracture geometry."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e234-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 23, "chroma:document": "From our synthetic analysis presented in this section, we can conclude that the pressure decline response\nduring DFIT is affected by many parameters, such as fracture geometry, reservoir properties, fracture surface\nroughness, etc. So history match DFIT data can be non-unique if we don't have good knowledge (from other\nindependent analysis and measurements) of the constraints on these parameters. Nevertheless, our proposed\n\"variable compliance method\" gives the most reliable estimation of minimum in-situ stress regardless of\nthese parameters, so the uncertainties that associated with fracture geometry, surface roughness and reservoir\nproperties do not impact our estimation of minimum in-situ stress."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e2c0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "SPE-187348-MS 25\nField Data Interpretation\nField Case 1\nThe first field case to be analyzed comes from a horizontal well-A drilled through a shale formation. The\nmeasured depth is around 5500 m and a diagnostic fracture injection test is conducted at the toe of the\nhorizontalwellbore, with 2.35 m3 of water injection in 3 minutes, then the well was shut-in for 27 days. Fig.\n30 shows the pressure decline response on G-function and the square root of time plots. As in the previous\nsynthetic cases, even though G-function and the square root of time plots give exactly the same quantitative\ninformation, it is more intuitive to infer the shut-in time on the square root of time plot, while the shut-in\ntime is implicit in the value of G-function, as reflected by Eq. (3)~Eq. (5). This field case also exhibits linear\npressure derivatives and low peak value of pressure derivative with a long extended tail after it reaches the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e34c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "peak, we can infer that the impact of wellbore storage should be substantial and this reservoir is potentially\nover-pressurized.\nFigure 30\u2014Pressure decline response from Well-A on G-function and square root of time plots\nFrom the signature of pressure derivatives, the changes in fracture stiffness/compliance are undetectable,\nso the minimum in-situ stress is picked at 31.8 MPa as the \"tangent line method\" suggests. A closer look at\nthe square root of time plot, shows that the pressure drops significantly during the first one hour of shut-in.\nBy examining the pressure derivative curve, whose extrapolated value is not zero when \u0394t=0, we can identify\nthat the excessive pressure drop at the beginning is caused by tip extension or near-wellbore tortuosity. The\napparent ISIP is 45.5 MPa while the true ISIP is 34.5MPa. Remember net pressure is the pressuredifference\nbetween thefluid pressure inside the fracture and closure pressure. In reality, the pressure measured at the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e3ce-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "wellhead can be much higher than expected during fracture propagationduring theearly time of shut-in\nperiod, because significant pressure drop can happen in the near-wellbore region and along the horizontal\nwellbore, due to friction and tortuous/complex fracture path that initiated from perforation clusters. So\npressure data from this period cannot be used to infer fracturing parameters and reservoir properties. If a\nrate step down test can be performed before complete shut-in, the relationship between early-time excessive\npressure drop and flow rate can be quantified.\nLog-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis\nand assist the interpretation of closure stress (Barree et al. 2019; Hawkes et al. 2012; Soliman et al. 2015).\nEven though Well-A was shut-in for nearly four weeks, pseudo-radial flow is still absent, as shown in\nFig. 31. This is a clear evidence of extremely low formation permeability. Without pseudo-radial flow,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e45a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 24, "chroma:document": "formation pore pressure and permeability cannot be determined independently with enough confidence\nusing just after closure analysis alone. One has to analyze the whole spectrum of DFIT data to reach a\nconsistent interpretation. Using late time data from the linear flow regime, and plotting Pf vs \n  on\nCartesian coordinates, the intercept on the y-axis when \u0394t \u2192 \u221e is the estimated initial pore pressure, which\nis 29.1 MPa. This indicates it is indeed an over-pressurized reservoir. From geological and petrophysical"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e4dc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "26 SPE-187348-MS\nstudies, it is known that the thickness of the target formation is 24.4 m with an average porosity of 0.07\nand in-situ fluid viscosity of 0.257 cp, The Young's modulus is 38.9 GPa, the Poisson's ratio is 0.2 and\nformation total compressibility is 3e-3 MPa\u22121. Hydraulic fracture modeling indicates that the fracture is\nwell contained within the target formation with penny-shaped fracture geometry, and the fracture radius is\nroughly 12m. Based on this information, the pressure decline response can be matched globally using our\nfully coupled DFIT model (Eq. (6)-Eq. (16) with pressure dependent fracture stiffness obtained from our\nnon-local fracture closure model (Wang and Sharma 2017)), from the end of tip extension/near-wellbore\ntortuosity dominated period to the end of the test.\nFigure 31\u2014Log-log pressure derivative plot of Well-A\nFig. 32 shows the pressure decline response predicted by our fully coupled DFIT model and field data"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e568-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "on G-function and the square root of time plots. The results indicate that our simulated pressure matches\nextremely well with the field data for the entire duration of the test, excluding the first hour of shut-in. Our\nmatched reservoir permeability is 220 nd, which is within the range of other petrophysical measurements.\nThe matched contact width and contact reference stress are 0.7 mm and 3 MPa, respectively, and Fig. 33\nshows the corresponding fracture and fracture-wellbore system stiffness, based on the matched fracture\ngeometry, wellbore volume and contact parameters. Because the DFIT was conducted at the toe of a\nhorizontal well, it is no surprise that the wellbore storage effect is significant, considering such a large\ncontrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts\nto increase noticeably when pressure drops to 34 MPa, over 2 MPa above the minimum in-situ stress. It"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e5ea-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 25, "chroma:document": "is apparent that the so-called \"normal leak-off behavior\" is just a coincidence when the effect of pressure\ndependent leak-off at the fracture surface and variable fracture compliance counterbalance each other during\nthe fracture closure process.\nFigure 32\u2014Matched pressure decline response for Well-A on G-function and square root of time plots"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e676-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "SPE-187348-MS 27\nFigure 33\u2014Matched fracture and fracture-wellbore system stiffness for Well-A\nField Case 2\nThe second field case to be analyzed comes from a vertical well-B drilled through a shale formation. The\ntotal wellbore length is around 2000 m and a diagnostic fracture injection test is conducted with 4.7 m3 of\nwater injection in6 minutes, then the well was shut-in for 11 days. Fig. 34 shows the pressure decline response\non G-function and the square root of time plots. If we pick the fracture closure pressure using the \"tangent\nline method,\" both G-function and the square root of time plots give the same minimum in-situ stress of\n50.4 MPa. One can also notice that there is a slight \"bump\" in the pressure derivatives soon after shut-in. For\nshale formations, this early-time accelerated pressure drop is not likely to be caused by PDL, but attributed\nto tip extension and near wellbore tortuosity (continued after-flow of fluid from wellbore to fracture after"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e6f8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "shut-in). It should be observed that in the early-time pressure transient response, the pressure derivative\nstarts to deviate upwards at around 55 MPa. Using \"variable compliance method\" through averaging the G-\nfunction or the square root of time that corresponding to a pressure of 50.4 MPa and 55 MPa, respectively,\nwe can obtain the estimated minimum in-situ stress of 52.8 MPa by picking the pressure that corresponds\nto the averaged G-function or square root of time.\nFigure 34\u2014Pressure decline response from Well-B on G-function and square root of time plots\nFrom Fig. 35, we can observe that only linear flow emerged after closure. Using late time data from the\nlinear flow regime, the estimated initial pore pressure is 33.7MPa. Geological and petrophysical studies\nindicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid\nviscosity of 0.28 cp. The Young's modulus is 39.5 GPa, the Poisson's ratio is 0.25 and formation total"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e784-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 26, "chroma:document": "compressibility is 1.9e-3 MPa\u22121. Hydraulic fracture modeling shows that the fracture is well contained within\nthe target formation with roughly 34 m fracture half length. Based on this information and assuming PKN\ngeometry, the pressure decline response can be matched globally using our global DFIT model by adjusting\nreservoir permeability and contact parameters."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e810-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "28 SPE-187348-MS\nFigure 35\u2014Log-log pressure derivative plot of Well-B\nFig. 36 shows the predicted pressure decline response and field data on G-function and the square root of\ntime plots. The results indicate that our simulated pressure agrees well with field data and it even captures\nthe downward curve of pressure derivatives before closure. Traditionally, this phenomenon is interpreted\nas \"height recession\" or \"transverse storage,\" where the top and bottom barrier formations close first and\nsqueeze fluid into the target formation during the closure process. This can be regarded as a special case of\n\"variable compliance,\" as demonstrated in Fig. 23. Because fractures always close from their edges to the\ncenter on rough walls, this constantly alters fracture compliance/stiffness. This variable fracture compliance\ncoupled with fracture pressure dependent leak-off dictates our pressure decline trend for a given formation."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e892-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "In this field case, our matched reservoir permeability is 210 nd, and the matched contact width and contact\nreference stress are 1.2 mm and 1.1 MPa, respectively.\nFigure 36\u2014Matched pressure decline response for Well-B on G-function and square root of time plots\nFig. 37 shows the corresponding fracture and fracture-wellbore system stiffness as a function of fluid\npressure inside fracture. It can be observed that both fracture and system stiffness increase gradually as\npressure declines. Even though this test was conducted in a vertical well with relative moderate wellbore\nvolume, the wellbore storage effect on the system stiffness is still significant. Re-examining Eq. (16), we\ncan see that the relative influence of wellbore storage not only depends upon the ratio of wellbore to fracture\nvolume, but it also depends on fracture stiffness itself. For the same wellbore volume, the higher the fracture\nstiffness, the less compressible the fracture is, so the compressed fluid in the wellbore plays a bigger role to"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e91e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 27, "chroma:document": "compensate leak-off and pressure drop. In this field case, with fracture height contained in 5 m, the fracture\nstiffness should be at least 5.36 MPa/mm (From Table 1) even without considering the contact of fracture\nsurfaces. If the fracture geometry and reservoir properties can be constrained with enough confidence, the\nestimated contact parameters and fracture stiffness evolution can be used to reconstruct the dynamic fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2e9a0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "SPE-187348-MS 29\nclosure process and the evolution of fracture width distribution, thus, the un-propped fracture conductivity\nand its pressure/stress dependence can be inferred.\nFigure 37\u2014Matched fracture and fracture-wellbore system stiffness for Well-B\nConclusions and Discussion\nSince the introduction of Nolte's pioneering work, diagnostic fracture injection test (DFIT) has evolved into\na commonly used technique and are generally accepted as a reliable way to obtain fracturing parameters\n(e.g., minimum in-situ stress, leak-off behavior) and reservoir properties (e.g., initial pore pressure and\nrepresentative formation permeability). These key parametersare needed to run fracture models, post-\nfracture productionprediction and economical evaluations. In this article, we first reviewed Nolte's G-\nfunction model and the associated assumptions (adopted by many subsequent works), and derived the\nequations that underpin the interpretation of the square root of time plot in low permeability reservoirs. We"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ea2c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "demonstrated that the square root of time plot gives the same quantitative information obtained from the\nG-function plot.\nA new global DFIT model is proposed, which for the first time, has the capability to predict pressure\ndecline response across the entire duration of a DFIT test using a single, coherent mathematic framework in\nlow permeability formations. We show how each factor impacts the pressure decline response and the effects\nof previous overlooked coupled mechanisms are examined and discussed, along with synthetic and field\ncase studies. The content of this article promotes our conceptual understanding of pressure transient and\ncoupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions\nreached from the analysis presented in this paper include the following.\n1.Carter's leak-off, (leak-off rate is proportional to \n ), is applicable during stable fracture\npropagation, when net pressure is roughly constant and injection time is limited. However, during the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2eaae-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "fracture closure peroid, the fluid pressure in the fracture declines significantly, and the leak-off rate\nbecomes fracture pressure dependent and ceases to be proportional to \n .\n2. The so-called \"normal leak-off,\" with a straight line of pressure and its derivatives on G-function and\nthe square root of time plots, is not \"normal\" at all. This is sometimes observed in data sets when\nmechanisms that accelerate pressure decline (e.g., increase in fracture stiffness during closure) offset\nthose that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and\nwellbore storage effects). So any \"before closure\" analysis based on this conventional \"normal leak-\noff\" assumption to estimate fracture geometry and formation permeability should be re-examined.\n3. Hydraulic fractures do not close as two smooth parallel plates, but rather close progressively from\nedges to the center on asperities and roughness surfaces, as confirmed by our modeling results and"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2eb3a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 28, "chroma:document": "field tiltmeter measurements. The fracture stiffness increases (compliance decreases) gradually as"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ebf8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "30 SPE-187348-MS\nmore and more fracture surfaces come into contact, even before the fracture pressure drops to the\nminimum in-situ stress.\n4. It is shown that the pressure decline response during fracture closure is governed by a PDE-ODE\nsystem, where the pressure at the fracture surface should be solved simultaneously with the pressure\ninside the formation. Previous DFIT models assumed a constant fracture compliance/stiffness and\nCarter's leak-off oversimplified and decoupled this system, which prevents them from predicting\npressure decline across the entire duration of the DFIT test.\n5. The conventional \"tangent line method\" underestimates the minimum in-situ stress, especially\nin depleted reservoirs. The \"compliance method\" picks the mechanical closure pressure but\noverestimates the minimum in-situ stress (if the two are equated), especially when height recession\noccurs. Height recession (or transverse storage) is a just special type of variable compliance\nphenomenon."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ec84-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "phenomenon.\n6. The \"variable compliance method\" proposed in this study suggests that picking an average of the\ndimensionless G-time (or the square root of time) of the \"tangent line\" and \"compliance\" methods\nprovides a much more robust and reliable way of estimating the minimum in-situ stress when concave\npressure derivative exist. The uncertainties that associated with fracture geometry, surface roughness\nand reservoir properties do not impact our estimation of minimum in-situ stress using \"variable\ncompliance method\". Results using this method match well with field data.\n7. Fracture closure on asperities and rough fracture surfaces is a gradual process starting at the fracture\ntip. The model presented here can be used to define the relationship between fracture width and\ncontact stress using DFIT pressure data. If other matching parameters can be constrained with enough\ncertainty, this information can be used to infer unpropped fracture conductivity as a function of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ed06-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 29, "chroma:document": "effective stress. This adds tremendous value to DFIT analysis for completion design and estimates\nof well productivity and EUR.\nNomenclature\na = Half fracture height (PKN model) or half fracture length (KGD model), L, m\nAf =Fracture surface area (one face of one wing),L2, m2\nct =Formation total compressibility, Lt2/m, 1/Pa\ncW =Water compressibility, Lt 2/m, 1/Pa\nCL =Carter's leak-off coefficient, \n , \nE = Young's modulus, m/Lt 2, Pa\nE\u2032 = Plane strain Young's modulus, m/Lt 2, Pa\ng(\u0394t D) =Dimensionless g-function of time\nG(\u0394t D) =Dimensionless G-function of time\nhf =Fracture height, L, m\nISIP = Instant shut-in pressure, m/Lt2, Pa\nk = Formation permeability, L2, m2\nP = Pressure, m/Lt2, Pa\nPf =Fracturing pressure, m/Lt2, Pa\nPnet =Fracturing net pressure, m/Lt 2, Pa\nP0 =Initial reservoir pressure, m/Lt2, Pa\nqf =Leak-off rate (one wing), L3/t, m3/s\nrD =Normalized radius\nrp =Productive surface ratio\nrf =Local fracture radius, L, m\nRf =Fracture radius, L, m"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ed74-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "SPE-187348-MS 31\nSf =Fracture stiffness, which is the reciprocal of fracture compliance, m/L2t2, Pa/m\nSS =Fracture-wellbore system stiffness, m/L2t2, Pa/m\nt =Generic time,t, s\ntD =Dimensionless time\nte =Generic time at the end of pumping, t, s\ntp =Pumping time, t, s\n\u0394t = Total shut-in time, t, s\n\u0394tD =Dimensionless shut-in time, t, s\nxf =Fracture half-length, L, m\nUf =Flux velocity through fracture surface with constant fracturing pressure, L/t, m/s\nVf =Fracture volume (one wing), L3, m3\nVW =Half of wellbore volume, L3, m\nW0 =Contact width, L,m\nWf =Local fracture width, L, m\n =Averagefracture width, L, m\n\u03bcf =Fluid viscosity, m/Lt, Pa\u00b7s\nv = Poisson's ratio\n\u03c3c =Contact stress, m/Lt2, Pa\n\u03c3hmin =Minimum in-situ stress, m/Lt2, Pa\n\u03c3ref =Contact reference stress, m/Lt2, Pa\n\u03d5 = Formation porosity\nSubscript\nn = Variables at nth time interval\nj =Variables at jth time interval\ni =Variables at jth time interval\nReferences"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2edf6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "i =Variables at jth time interval\nReferences\nBarr, D.T., 1991. Leading-edge analysis for correct simulation of interface separation and hydraulic fracturing. PHD\ndissertation, MIT, Cambridge, MA.\nBarree, R. D. 1998. Applications of pre-frac injection/falloff tests in fissured reservoirs - field examples. Paper SPE 39932\npresented at the SPE Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 5-8 April.\nhttp://dx.doi.org/10.2118/39932-MS.\nBarree, R. D., Mukherjee.H. 1996. Determination of pressure dependent leakoff and its effect on fracture geometry.\nPaper SPE 36424 presented at the 71st Annual Technical Conference and Exhibition, Denver, 6-9 October. http://\ndx.doi.org/10.2118/36424-MS.\nBarree, R. D., Miskimins,J. L. Gilbert.J. V . 2014. Diagnostic fracture injection tests: common mistakes, misuses, and\nmisdiagnoses. SPE Production & Operations 30 (2):84\u201398, http://dx.doi.org/10.2118/169539-PA."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ee78-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 30, "chroma:document": "Barree, R.D., Barree, V .L. and Craig, D., 2009. Holistic fracture diagnostics: consistent interpretation of prefrac\ninjection tests using multiple analysis methods. SPE Production & Operations, 24(03), pp.396\u2013406. http://\ndx.doi.org/10.2118/107877-PA\nBarton, N., Bandis,S. Bakhtar.K. 1985. Strength, deformation and conductivity coupling of rock joints. International\nJournal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts  22 (3): 121\u2013140, http://\ndx.doi.org/10.1016/0148-9062(85)93227-9.\nBenelkadi, S., and Tiab, D. 2004. Reservoir Permeability Determination Using After-Closure Period Analysis of\nCalibration Tests. SPE Reservoir Evaluation & Engineering, V ol (7)03: 230\u2013237. http://dx.doi.org/10.2118/88640-PA\nBhide, R., Gohring, T., McLennan, J. and Moore, J., 2014. Sheared Fracture Conductivity. In Proceedings of 39th\nWorkshop on Geothermal Reservoir Engineering"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2eefa-857d-11ee-bb1c-0242ac1c000c
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metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "32 SPE-187348-MS\nCastillo, J.L. 1987. Modified fracture pressure decline analysis including pressure-dependent leakoff. Paper SPE\n164417 presented at the SPE/DOE Low Permeability Reservoir Symposium, Denver, CO, 18-19 May. http://\ndx.doi.org/10.2118/16417-MS.\nChipperfield, S. T. 2006. After-Closure Analysis to Identify Naturally Fractured Reservoirs. SPE Reservoir Evaluation &\nEngineering, V ol (9)01:50\u201360. http://dx.doi.org/10.2118/90002-PA\nEconomides, Michael J., Kenneth G. Nolte. 2000. Reservoir Stimulation, 3rd edition. Chichester, England; New York,\nJohn Wiley.\nEngland, A.H. and Green, A.E., 1963, April. Some two-dimensional punch and crack problems in classical elasticity. In\nProc. Camb. Phil. Soc (V ol. 59, No. 2, p. 489). http://www.doi.org/10.1017/S0305004100037099\nFredd, C. N., McConnell, S. B., Boney, C. L., & England, K. W. 2000. Experimental Study of Hydraulic Fracture\nConductivity Demonstrates the Benefits of Using Proppants. Paper SPE 60326 presented at the SPE Rocky"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ef7c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Mountain Regional/Low-Permeability Reservoirs Symposium and Exhibition, Denver, Colorado, 12-15 March. http://\ndx.doi.org/10.2118/60326-MS\nFu, Y ., Dehghanpour, H., Ezulike, D. O., and Jones, R. S. 2017. Estimating Effective Fracture Pore V olume From Flowback\nData and Evaluating Its Relationship to Design Parameters of Multistage-Fracture Completion. SPE Production &\nOperations (PrePrint). https://doi.org/10.2118/175892-PA\nGringarten, A. C., Ramey, H. J., and Raghavan, R. 1974. Unsteady-State Pressure Distributions Created by a Well with a\nSingle Infinite-Conductivity Vertical Fracture. SPE Journal. V ol(14)04:347\u2013360. http://dx.doi.org/10.2118/4051-PA\nHawkes, R., Bachman, R. C., Walters, D. A., Toussaint, F. L., and Settari, T. 2012. Reappraisal of the G Time Concept\nin Mini-Frac Analysis. Paper SPE 160169 presented at the SPE Annual Technical Conference and Exhibition, San\nAntonio, Texas, USA, 8-10 October. https://dx.doi.org/10.2118/160169-MS"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2eff4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Liu, G., Ehlig-Economides, C. and Sun, J. 2016. Comprehensive Global Fracture Calibration Model. Paper SPE\n181856 presented at the SPE Asia Pacific Hydraulic Fracturing Conference, Beijing, China, August 24-26. http://\ndx.doi.org/10.2118/181856-MS\nLoughry, D., Epps, D. and Forrest, J., 2015. Using Pad ISIP, DFIT, and ESP Data to Generate a Pore Pressure Model for\nthe Midland Basin. Paper presented at the Unconventional Resources Technology Conference, San Antonio, Texas,\nUSA, 20-22 July. https://doi.org/10.15530/URTEC-2015-2162973\nMarongiu-Porcu, Matteo, Albertus Retnanto, Michael J. Economides, Christine Ehlig-Economides. 2014. Comprehensive\nfracture calibration test design. Paper SPE 168634 presented at the SPE Hydraulic Fracturing Technology Conference,\nThe Woodlands, TX, February 4-6. http://dx.doi.org/10.2118/168634-MS\nMayerhofer, M. J., and Economides, M. J. 1997. Fracture-Injection-Test Interpretation: Leakoff Coefficient vs."}


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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Permeability. SPE Production & Facilities. V ol (12) 04:231\u2013236. http://dx.doi.org/10.2118/28562-PA\nMayerhofer, M. J., Ehlig-Economides, C. A., and Economides, M. J. 1995. Pressure Transient Analysis of Fracture\nCalibration Tests. Journal of Petroleum Technology. V ol(47)03:229\u2013234. http://dx.doi.org/10.2118/26527-PA\nMcClure, M.W. 2017. The Spurious Deflection on Log-Log Superposition Time Derivative Plots of Diagnostic Fracture.\nSPE Reservoir Evaluation & Engineering (In Press). https://doi.org/10.2118/186098-PA\nMcClure, M.W., Jung, H., Cramer, D.D. and Sharma, M.M., 2016. The Fracture Compliance Method for Picking Closure\nPressure from Diagnostic Fracture-Injection Tests. SPE Journal 21(04):1321\u20131339. http://dx.doi.org/10.2118/179725-\nPA\nMirani, A., Marongiu-Porcu, M., Wang, H., and Philippe, E. Production Pressure Drawdown Management for Fractured\nHorizontal Wells in Shale Gas Formations. 2016. Paper SPE 181365 presented at the SPE Annual Technical Conference"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f0ee-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "and Exhibition held in Dubai, UAE, 26-28 Sep. http://dx.doi.org/10.2118/181365-MS\nNolte, K. G. 1979. Determination of Fracture Parameters from Fracturing Pressure Decline. Paper 8341 presented\nat the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 23-26 September. http://\ndx.doi.org/10.2118/8341-MS\nNolte, K. G. 1986. A General Analysis of Fracturing Pressure Decline with Application to Three Models. SPE Formation\nEvaluation. V ol (1) 06:571\u2013583. http://dx.doi.org/10.2118/12941-PA\nNolte, K. G., Maniere, J. L., and Owens, K. A. 1997. After-Closure Analysis of Fracture Calibration Tests. Paper SPE\n38676 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October. http://\ndx.doi.org/10.2118/38676-MS\nPotocki, D. J. 2012. Understanding Induced Fracture Complexity in Different Geological Settings Using DFIT Net\nFracture Pressure. Paper SPE 162814 presented at the SPE Canadian Unconventional Resources Conference, Calgary,"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 31, "chroma:document": "Alberta, Canada, 30 October-1 November. http://dx.doi.org/10.2118/162814-MS\nRamurthy, M., Barree, R.D., Kundert, D.P., Petre, J.E. and Mullen, M.J., 2011. Surface-area vs. conductivity-type fracture\ntreatments in shale reservoirs. SPE Production & Pperations, 26(04), pp.357\u2013367. https://doi.org/10.2118/140169-PA"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "SPE-187348-MS 33\nSakaguchi, K., Tomono, J., Okumura, K., Ogawa, Y . and Matsuki, K., 2008. Asperity height and aperture of an artificial\ntensile fracture of metric size. Rock Mechanics and Rock Engineering, 41(2), pp.325\u2013341. http://dx.doi.org/10.1007/\ns00603-005-0102-3\nSchiesser, W.E. and Griffiths, G.W., 2009. A compendium of partial differential equation models: method of lines analysis\nwith Matlab. Cambridge University Press.\nSneddon, I. N. and Lowengrub, M., Crack Problems in the Classical Theory of Elasticity. John Wiley & Sons, Inc., New\nYork, 1969\nSoliman, M. Y ., Craig, D. P., Bartko, K. M., Rahim, Z., and Adams, D. M. 2005. Post-Closure Analysis to Determine\nFormation Permeability, Reservoir Pressure, Residual Fracture Properties. Paper SPE 93419 presented at the SPE\nMiddle East Oil and Gas Show and Conference, Kingdom of Bahrain,12-15 March. http://dx.doi.org/10.2118/93419-\nMS\nSoliman, M. Y ., Shahri, M., and Lamei, H. 2013. Revisiting the Before Closure Analysis Formulations in Diagnostic"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f5a8-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "Fracturing Injection Test. Paper SPE 163869 presented at the SPE Hydraulic Fracturing Technology Conference, The\nWoodlands, Texas, USA, 4-6 February. http://dx.doi.org/10.2118/163869-MS\nSoliman, M.Y ., Miranda, C., and Wang, H.M. 2010. Application of After-Closure Analysis to a Dual-Porosity\nFormation, to CBM, and to aFractured Horizontal Well. SPE Prod & Oper 25 (4): 472\u2013483. SPE-124135-PA. http://\ndx.doi.org/10.2118/124135-PA.\nValko', P. P., and Economides, M. J. 1999. Fluid-Leakoff Delineation in High-Permeability Fracturing. SPE Production\n& Facilities. V ol (14)02:110\u2013116. http://dx.doi.org/10.2118/56135-PA\nvan Dam, D.B., de Pater, C.J. and Romijn, R., 2000. Analysis of Hydraulic Fracture Closure in Laboratory Experiments.\nSPE Production & Facilities,15(03), pp.151\u2013158. http://dx.doi.org/10.2118/65066-PA\nWang, H. and Sharma, M.M. 2017. A Non-Local Model for Fracture Closure on Rough Fracture Faces and Asperities."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f634-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "Journal of Petroleum Science and Engineering, 154, 425\u2013437. https://dx.doi.org/10.1016/j.petrol.2017.04.024\nWang, H. 2015. Numerical Modeling of Non-Planar Hydraulic Fracture Propagation in Brittle and Ductile Rocks using\nXFEM with Cohesive Zone Method. Journal of Petroleum Science and Engineering, V ol (135):127\u2013140. http://\ndx.doi.org/10.1016/j.petrol.2015.08.010\nWang, H. 2016. Numerical Investigation of Fracture Spacing and Sequencing Effects on Multiple Hydraulic Fracture\nInterference and Coalescence in Brittle and Ductile Reservoir Rocks. Engineering Fracture Mechanics, V ol (157):\n107\u2013124. http://dx.doi.org/10.1016/j.engfracmech.2016.02.025\nWang, H. 2017. What Factors Control Shale Gas Production and Production Decline Trend in Fractured Systems: A\nComprehensive Analysis and Investigation. SPE Journal, V ol 22(02): 562\u2013581. http://dx.doi.org/10.2118/179967-PA\nWang, H., Marongiu-Porcu, M., and Economides, M. J. 2016. SPE-168600-PA. Poroelastic and Poroplastic Modeling"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f6b6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "of Hydraulic Fracturing in Brittle and Ductile Formations, Journal of SPE Production & Operations, 31(01): 47\u201359.\nhttp://dx.doi.org/10.2118/168600-PA\nWarpinski, N.R., Branagan, P.T., Engler, B.P., Wilmer,R., and Wolhart, S.L. 2002. Evaluation of a downhole tiltmeter\narray for monitoring hydraulic fractures. International Journal of Rock Mechanics and Mining Sciences, V ol (34)3-4:\n329.e1\u2013329.e13. http://dx.doi.org/10.1016/S1365-1609(97)00074-9\nWarpinski, N.R., Lorenz, J.C., Branagan, P.T., Myal, F.R. and Gall, B.L., 1993. Examination of a Cored Hydraulic Fracture\nin a Deep Gas Well (includes associated papers 26302 and 26946). SPE Production & Facilities, 8(03), pp.150\u2013158.\nhttp://dx.doi.org/10.2118/22876-PA\nWells, O.L. and Davatzes N.C., 2015. The History of Dilation Across Natural Fractures Due to Evolving Surface\nRoughness. In Proc. 40th Workshop on Geothermal Reservoir Engineering\nWillis-Richards, J., Watanabe,K. Takahashi.H. 1996. Progress toward a stochastic rock mechanics model of engineered"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 32, "chroma:document": "geothermal systems. Journal of Geophysical Research 101 (B8): 17481\u201317496, http://dx.doi.org/10.1029/96JB00882.\nWu, W. and Sharma, M.M., 2017. Acid Fracturing in Shales: Effect of Dilute Acid on Properties and Pore Structure of\nShale. SPE Production & Operations, 32(01):51\u201363. http://dx.doi.org/10.2118/173390-PA\nZhang, J., Kamenov, A., Hill, A.D. and Zhu, D., 2014. Laboratory Measurement of Hydraulic-Fracture Conductivities in\nthe Barnett Shale. SPE Production & Operations, 29(03), pp.216\u2013227. http://dx.doi.org/10.2118/163839-PA\nZou, Y ., Ma, X., Zhang, S., Zhou, T., Ehlig-Economides, C. and Li, H., 2015. The Origins of Low-Fracture Conductivity in\nSoft Shale Formations: An Experimental Study. Energy Technology, 3(12), pp.1233\u20131242. https://dx.doi.org/10.1002/\nente.201500188"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f7a6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "34 SPE-187348-MS\nAppendix A\nSolutions of Pressure Decline under Fracture Pressure Dependent Leak-off\nIn order to overcome the limitations of Nolte's G-function model and circumvent the use of a constant leak-\noff coefficient, Mayerhofer et al. (1995) presented a model that describes unsteady-state fluid flow from\nfractures of varying area into the formation, with the filter cake considered as a time and rate dependent\nskin effect, allowing for superposition of the injection history, filter cake deposition and associated rate\nconvolution. Valko\u2019 and Economides (1999) simplified Mayerhofer et al. (1995) model by assuming\nconstant leak-off rate during fracture propagation and constant fracture area during closure. A special before\nclosure analysis (BCA) plot with a straight line can be constructed to determine reservoir permeability and\nfracture face resistance, without acquiring the initial guess of permeability and fracture area. However, the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f832-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "process to generate the special BCA plot and predict fall-off pressure is complicated and cumbersome,\nprone to mistakes that can lead to dimension inconsistency ( Soliman et al. 2013). In addition, their approach\nused a constant-rate solution of an infinite-conductivity well introduced by Gringarten et al. (1974). The\nresults obtained by use of the constant-rate solution are questionable to account for the fracture pressure\ndependent leak-off behavior. In this appendix, we derive the solutions of pressure decline under fracture\npressure dependent leak-off using superposition based on a constant-pressure solution.\nThe basic principle of superposition is that the response of a system to a number of perturbations is equal\nto the sum of the responses to each of the perturbation as if they were present by themselves if the responses\nare linearly related to the perturbations. From Eq. (18), we can calculate the leak-off rate under a constant\nfracture pressure P f:\n(A.1)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f8aa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 33, "chroma:document": "fracture pressure P f:\n(A.1)\nDivide the shut-in time \u0394t into n time steps, and the leak-off rate at the nth time step can be determined\nbased on superposition:\n(A.2)\nThe pressure difference P f,j - P f,j-1 is the pressure in the fracture at time step j minus the pressure in the\nformation at the fracture-formation interface, which equals to the pressure in the fracture at the previous\ntime step. Combine Eq. (A.2) with Eq. (10) via material balance, we can get:\n(A.3)\nlet P f,0 = P 0, Pf,1 = ISIP, \u0394t 1 = 0 then for n \u2265 2, integrate Eq. (A.3) we can obtain:\n(A.4)\nEq. (A.4) can be solved explicitly with marching time interval, for example, when n = 2\n(A.5)"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f922-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 34, "chroma:document": "SPE-187348-MS 35\nOn closer observation of Eq. (A.5), we can realize that it is exactly Carter's leak-off solution to Eq. (19),\nbecause during the first time step, no superposition is needed. The fracture pressure dependent leak-off\ncoefficient can be recovered as:\n(A.6)\nIt should be mentioned that Eq. (A.4) is only valid when Sf is constant, where the linear response system\nis the prerequisite of using superposition principal."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2f99a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 35, "chroma:document": "36 SPE-187348-MS\nAppendix B:\nModel Non-Local Fracture Closure on Rough Surfaces and Asperities\nEngland and Green (1963) showed that the width of a 2D fracture with any arbitrary pressure/stress profile\nis given by:\n(B.1)\nwhere \u03b3 is a dummy variable, y is the distance to the middle of the fracture, a is the half fracture height\n(PKN model) or half fracture length (KGD model). Let PNet(y) be the net pressure/stress distribution along\nthe y-direction, which is the total pressure/stress acting on the fracture surfaces minus the far field stress\nin the fracture opening direction. To obtain F(\u03b3) and G(\u03b3), PNet(y) has to be divided into even and odd\nfunctions, such that PNet(y) = -f(y) - g(y), with f(y) being even and g(y) being odd. Then F(\u03b3) and G(\u03b3) are\ndetermined by:\n(B.2)\n(B.3)\nwhere \u03b2 is a dummy variable. If the P Net(y) is symmetric, then g(y) and G(\u03b3) = 0. For example, assuming a\nuniform pressure P Net inside the fracture, f(y) =-P Net, g(y) = 0, From Eq. (B.2):\n(B.4)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fa08-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 35, "chroma:document": "(B.4)\nSubstituting Eq. (B.4) into Equation (B.1), we have,\n(B.5)\nEquation (B.5) is the well-known equation for a 2D static pressurized crack. If a is the fracture half-\nheight, then Eq. (B.5) describes the fracture width distribution forthe PKN fracture geometry, and if a is\nthe fracture half-length, Eq. (B.5) describes the fracture width distribution for the KGD fracture geometry.\nWhenan analytic form of P Net(y) is not available a priori (e.g., coupled with other physics) or P Net(y) is\nnot acontinuous function, numerical integration to calculate the fracture width has to be implemented in a\ndiscretized manner along the fracture height or length. Fig. B1 shows a fracture that is discretized into a\nnumber of n segments. In each segment, there exists a uniform net pressure/stress that acts on the fracture\nsurfaces. Any net pressure/stress distribution can be approximated by representing it in this manner given\nenough number of n."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fa76-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 36, "chroma:document": "SPE-187348-MS 37\nFigure B1\u2014DiscretizePKN/KGD fracture into n segments with different net pressure/stress\nDefine integration operator I:\n(B.6)\nthe corresponding fracture width profile at the mth segment can be obtained by integrating Eq. (B.1) across\neach segment to account for non-local elastic behavior (Wang and Sharma 2017):\n(B.7)\nRepeat this calculation from the 1st segment to the nth segment, the entire fracture width profile under\narbitrary normal loading can be obtained. Fig. B2 shows an example where with a fracture half-height\ndivided into three segments with equal length, and a certain net pressure/stress is imposed on each segment\nindividually. As we can see, when the net pressure/stress is the same on all segments, an elliptical fracture\nwidth profile is obtained, which is also validated by Eq. (B.5). When the net pressure/stress varies along\ndifferent segments, the fracture width profile will depart from the smooth, elliptical curve. And P Netn has"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fae4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 36, "chroma:document": "more impact on the overall width distribution when it is imposed close to fracture center."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fb5c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 37, "chroma:document": "38 SPE-187348-MS\nFigure B2\u2014Fracture width distribution along fracture half-height across three segments: a = 30 m, E=20 GPa, v =0.25\nThe radial crack opening due to an excess pressure distribution in a linearly elastic material of infinite\nextent can be expressed as (Sneddon and Lowengrub 1969):\n(B.8)\nwhere Rf is the length of fracture radius, u and s are dummy variables and r D is the normalized radius, which\nis defined as\n(B.9)\nwhere rf is the local fracture radius. For a radius crack with a uniform pressure distribution P Net, Eq. (B.8)\ncan be integrated directly to get an analyticalsolutionfor the fracture width profile:\n(B.10)\nBarr (1991) proved that the double integral of Eq. (B.8) can be transformed into a single integral with\nthe aid of elliptic integrals:\n(B.11)\nF(\u03c6, m) is the elliptic integral of the first kind, which is defined as:\n(B.12)\nwhere \u03c6 are input parameters for elliptic integral, and \u03b8 is a dummy variable."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fbd4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 37, "chroma:document": "Similar to Fig. B1, a radial fracture can be discretized into n segments, as shown in Fig. B3. In each\nsegment, there exists a uniform net pressure/stress that acts on the opposite fracture surface. If n is large\nenough, the net pressure/stress distribution can be approximated to any distribution of P Net(r)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fc42-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 38, "chroma:document": "SPE-187348-MS 39\nFigure B3\u2014Discretized radial fracture (n segments) with different net pressure/stress acting along the radial direction\nDefine integration operators R1 and R2 as:\n(B.13)\n(B.14)\nand the corresponding fracture width profile at the mth segment under arbitrary P Net(r) can be obtained by\nintegrating Eq. (B.11) across each segment (Wang and Sharma 2017):\n(B.15)\nwhere\n(B.16)\n(B.17)\nRepeat this calculation from the 1st segment to the nth segment, the entire fracture width profile can be\nobtained. Fig. B4 shows an example where with a fracture radius divided into three segments with equal\nlength, and a certain net pressure/stress is imposed on each circular segment individually. When the net\npressure/stress is the same on all segments, it reproduces the same fracture width distribution as predicted\nby Eq. (B.10). And when the net pressure/stress varies along different segments, the fracture width within\na certain segment can be substantially impacted by the load imposed on other segments."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fd0a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 39, "chroma:document": "40 SPE-187348-MS\nFigure B4\u2014Fracture width distribution along fracture radius across three segments: R f = 30 m, E=20 GPa, v =0.25\nIn the above discussion, we have presented the method and associated algorithms to determine the\nfracture width profile for PKN, KGD and Radial fracture geometries, given an arbitrary distribution of net\npressure/stress. Now, if the fracture closes on rough surfaces and asperities, then the net pressure/stressis\nimpacted by the contact stress, which itself is a function of fracture width distribution. Therefore, the net\npressure/stressdistribution imposed on fracture faces and the entire fracture width distribution have to be\nsolved simultaneously. Because both the net pressure/stress distribution and fracture width distribution are\ndiscretized, they can be only solved iteratively. Fig. B5 shows the iterative scheme to solve the net pressure/\npressureand fracture width distribution.\nFigure B5\u2014Solution flow diagram for solving coupled linear fracture mechanics with contact problem"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf", "page": 39, "chroma:document": "In this Picard iteration scheme, \u03b1 r is a relaxation factor that is less than or equal to 1.0. It controls how\nmuch the net pressure/stress should be corrected within iterations. Smaller \u03b1 r can avoid over-correction\nand improve convergence stability, but more iterations may be required with small value of \u03b1 r. When the\nchanges in net pressure/stress between iterations is less than the tolerance value, the estimation of contact\nstress across the entire fracture faces is converged and the final net pressure/stress distributionwith the\ncorresponding fracture width profile can be obtained. So given initial fracture geometry, rock properties\nand contact law parameters, the fracture width profile can be uniquely determined under certain fracturing\npressure. By discretizing fracturing pressure within a certain range, the relationship between fracture volume\nvs. pressure and fracture stiffness vs. pressure can also be obtained."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fe0e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "SPE-185649-MS\nReinterpretation of Fracture Closure Dynamics During Diagnostic Fracture\nInjection Tests\nB. Zanganeh and C. R. Clarkson, University of Calgary; R. V. Hawkes, Trican Well Service Ltd.\nCopyright 2017, Society of Petroleum Engineers\nThis paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April 2017.\nThis paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents\nof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect\nany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2fe90-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may\nnot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.\nAbstract\nA fit-for-purpose, fully coupled stress-pore pressure simulation model (Abaqus\u00ae) is used to simulate\ndiagnostic fracture injection tests (DFITs) and generate before closure pressure responses. The simulated\nresponses are used to explain field observations, and to propose a new concept: progressive fracture closure.\nThe cohesive zone model (CZM) is used to model fracture propagation and closure associated with\nDFITs. The customized model in Abaqus\u00ae is capable of modeling all the physical processes involved\nin a typical DFIT including: porous media deformation? fluid flow inside the reservoir? hydraulic\nfracture initiation, propagation and closure? compliance change before and after closure; residual fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ff12-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "conductivity? and fluid flow inside the fracture and fluid interaction between the fracture and reservoir\n(leakoff).\nA key result obtained is that the previously-introduced fracture compliance method is demonstrated to be\nthe most reliable approach to identify fracture closure. Depending on the pressure distribution around the\nfracture, the closure pressure is usually found to be higher than the minimum principal stress. Based on the\ncontinuity equation, the compliance method is expanded to include the progressive fracture closure (PFC)\nconcept. PFC refers to the scenario where fracture closure occurs gradually along the length of fracture, from\nthe tip of the fracture to near the wellbore. Different estimates of closure pressure will be obtained early and\nlate in this process. Several field cases are presented which exhibit progressive fracture closure. A consistent\nclosure signature can be identified for these cases using the primary pressure derivative. This study further"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae2ff9e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 0, "chroma:document": "suggests that wellbore storage can mask the closure signature. Therefore, reducing the wellbore storage\neffect by using downhole shut-in makes it easier to identify fracture closure, in addition to accelerating the\ntest.\nA common DFIT fracture closure referred to as \"fracture height recession\" is reinterpreted to be caused\nby the PFC phenomenon. This finding has tremendous implications for interpretation of before closure flow\nregimes and associated reservoir behavior. This study also confirms that the compliance method, which\ncorresponds to fracture tip closure, provides more of a true measure of closure pressure than conventional\napproaches which correspond to fracture closure near the well."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30020-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "2 SPE-185649-MS\nIntroduction\nThe DFIT, also known as a \"minifrac,\" has become the standard pre-stimulation method for determining\nfracturing treatment properties, effective reservoir permeability and initial reservoir pressure in\nunconventional resources. A DFIT is an injection-falloff test conducted before the main hydraulic fracturing\ntreatment. The DFIT procedure involves breakdown of the rock and creation of a short fracture during the\ninjection of high pressure fluid (usually water), and recording of the pressure response during the shut-\nin period. The pressure response is analyzed to obtain fracturing treatment parameters such as breakdown\npressure, fracture closure pressure and leakoff characteristics. If the falloff period is sufficient, good\nestimates of formation permeability and reservoir pressure can be obtained.\nBefore-closure analysis (BCA) was pioneered by Nolte (1979), who introduced the G-function based"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae300a2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "on material balance and the Carter leakoff model (Howard and Fast 1957); Nolte noted that the shut-in\npressure has a linear relationship with G-function during closure. Barree and Mukherjee (1996) and Barree\n(1998) recommended using the G-function and its diagnostic derivatives (\n  and \n ) to identify fracture\nclosure. Using these diagnostics, closure can be picked as the deviation from a straight line on a plot of\npressure, and its derivatives, versus the G-function. Barree (1998) and Barree et al. (2009) also presented\nsignatures for non-ideal leakoff behaviors such as tip extension, pressure-dependent leakoff and height\nrecession/transverse storage.\nMohamed et al. (2011) and Marongiu-Porcu et al. (2011) presented a model to predict the falloff pressure\ntrend of an idealized DFIT (normal leakoff), and identified fracture closure using standard pressure transient\n(PTA) log-log diagnostic plots. They identified normal leakoff as a \n  slope on the Bourdet-derivative (with"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3011a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "slope on the Bourdet-derivative (with\nrespect to superposition time), and picked closure as the deviation from the \n  slope. Bachman et al. (2012)\npresented a systematic PTA approach, and a combination of diagnostic plots, to identify various flow\nregimes before and after fracture closure. Van Den Hoek (2016) and McClure et al. (2016) demonstrated\nlimitations of superposition time and Agarwal's time (1980) for analyzing DFITs where the pumping time\nis small. They stated that the Bourdet-derivative (with respect to superposition time and Agarwal's time)\nstarts to exceed one at shut-in times equal to roughly one-tenth of the pump time and that the \n  slope is\nnot related to fracture closure.\nVan Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments\non different rock types. They observed that fracture closure happens at the tip first, and then towards the\nwellbore. They reported a break on a plot of pressure versus G-function due to the decrease of fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30192-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "compliance at closure, and argued that, instead of a deviation from a straight line on the pressure derivative\n(with regard to G-function), the local minimum should be selected as closure pressure. Further, they stated\nthat plastic deformation can cause a fracture to remain open in the vicinity of the wellbore. McClure et\nal. (2014) and McClure et al. (2016) investigated the effect of fracture compliance on pressure behavior\nusing a numerical simulator. Their modeling of after-closure compliance behavior was based on the Barton-\nBandis (1985) model, which is an empirical relationship. They demonstrated that the previous \"fracture\nheight recession\" signature presented by Barree et al. (2009) is the natural behavior of closure caused by\na change in fracture compliance. Based the \"fracture compliance method,\" closure is picked as the start of\nupward deviation on the \n  curve. McClure et al. (2016) argued that previously-presented methods tend\nto underestimate the closure pressure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30214-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 1, "chroma:document": "to underestimate the closure pressure.\nVan Den Hoek (2016) presented a PTA approach for modeling pressure behavior in DFITs and\nwaterflood-induced fractures based on simplified numerical and analytical solutions. In his work, fracture\ngrowth rate was a predefined input into the simulator, the Carter model was used as the leakoff model for\nthe DFIT, and closure was modeled as a gradual decline of fracture compliance representing a combination\nof \"hinge\" (constant length) and \"zipper\" (length recession) closure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30296-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "SPE-185649-MS 3\nThe cohesive zone model (CZM) for modeling crack propagation was originally proposed by Dugdale\n(1960) and Barenblatt (1962). Recently, this method has been successfully used for modeling hydraulic\nfracture initiation and propagation during fracturing treatment (Yao 2012; Shen and Cullick 2012; Shin and\nSharma 2012; Chen 2012; Zielonka et al. 2014; Haddad and Sepehrnoori 2015). Some of the advantages\nof the CZM compared to conventional linear elastic fracture modeling are that it avoids a singularity at the\ncrack tip, the location of crack tip is not an input and is a direct output of the solution, and it is capable\nof modeling fracture tip material softening in quasi-brittle rocks such as shale ( Chen 2012; Haddad and\nSepehrnoori 2015). The CZM has been implemented in the Abaqus\u00ae finite element program and validated\nanalytically and experimentally by Zielonka et al. (2014) and Searless et al. (2016)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3030e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "In this study, the CZM implemented in the Abaqus\u00ae finite element program is used to model fracture\ninitiation, propagation and closure for DFITs. A pressure-dependent leakoff model is coupled with the\nprogram through user subroutines. The model is used to generate synthetic pressure responses, which in\nturn are compared to field observations and used to study fracture closure dynamics.\nTheory and Methods\nIn the following, key components of the simulation model used in this study are described, as is the model\nsetup used for generating synthetic DFITs.\nCohesive Zone Model (CZM)\nWith the CZM, the fracture is modeled as a gradual separation between two material (rock) surfaces. This\nseparation is modeled as a progressive degradation of cohesive strength along the cohesive layer, which is\na pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's\nGuide 2016; Chen 2012). With traction-separation behavior (Fig. 1(a)), prior to damage initiation, cohesive"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30390-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 2, "chroma:document": "elements exhibit a reversible linear elastic response (characterized by the cohesive layer stiffness, Kn) until\nthe traction reaches the maximum tensile strength (T n0) with a separation of \u03b40. After damage initiation,\nthe cohesive traction evolves from the maximum strength (T n0) down to zero where the element is fully\ndamaged with the separation of \u03b4f. The damage evolution is governed by the critical energy release rate ( Gc)\nthat is equal to the area under traction-separation curve. Fig. 1(b) is a schematic of fracture propagation\nin a cohesive layer, demonstrating the fully damaged cohesive zone (filled with fracturing fluid) and the\nfracture process zone (damage initiation and evolution).\nFigure 1\u2014(a) Typical traction-separation behavior; (b) Damage evolution in cohesive elements.\nDifferent guidelines have been presented for selecting the cohesive element mesh sizes and stiffness.\nTuron et al. (2007) suggested the following equation for the cohesive layer stiffness:"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30408-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "4 SPE-185649-MS\n(1)\nwhere Kn is the stiffness of the cohesive layer, E is the Young's modulus of the material and tadjacent  is\nthickness of the adjacent sub-laminate. They recommended \u03b1 values much larger than 1 (\u03b1>>1). Zielonka\net al. (2014) and Searless (2016) used \u03b1=100, and presented close matches with analytical solutions and\nexperimental results. Haddad and Sepehrnoori (2015) derived an optimum \u03b1 value of 60 (assuming tadjacent  =\n1) by conducting a sensitivity analysis. As will be shown in the next section, we have observed that cohesive\nlayer stiffness is a major parameter affecting fracture compliance behavior before and after closure.\nFluid Flow within the Fracture\nFig. 2 is a plan view schematic of a wing of a fracture showing the components of fluid flow in the fracture.\nThere are two components of fluid flow inside the fracture: tangential flow within the fracture gap (q frac)\nand normal flow (leakoff) from the fracture to the surrounding rock (q leak). The continuity equation for fluid"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30480-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "flow inside the fracture is:\n(2)\nwhere w is the fracture opening, and q inj is the injection rate into the fracture.\nFigure 2\u2014Plan view schematic of one wing of the fracture.\nTangential Flow\nThe tangential flow is formulated based on Poiseuille's law:\n(3)\nwhere w is the fracture opening, \u03bc is the fluid viscosity and Pf is the fluid pressure along the fracture length.\nLeakoff Model\nThe Carter leak-off model (Howard and Fast 1957) has been used extensively for modeling of hydraulic\nfracture propagation and in the analytical solutions for DFIT analysis. It is a one dimensional pressure-\nindependent leak-off model, applicable to high-viscosity fracturing fluids causing the formation of a low\npermeability filter cake on the fracture walls and is described by:\n(4)\nwhere qleak(x) is leakoff rate at point x, Ccarter is the Carter leakoff coefficient, t is the time elapsed since the\nstart of injection and \u03c4(x) is the time when the fracture is created at point x (exposure time). The Carter leakoff"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3050c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 3, "chroma:document": "coefficient depends on filter cake created on fracture walls, and it is not related to formation properties such\nas reservoir permeability and pressure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3057a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "SPE-185649-MS 5\nIn this study, a rigorous leakoff model based on the solution of the 1-D diffusion equation in a half-space\nwith an imposed pressure history at the boundary is used (Sarvaramin and Garagash 2015):\n(5)\nwhere qleak(x) is leakoff rate at point x, S is the storage coefficient and \u03b1l is the hydraulic diffusivity. Unlike the\nCarter model, this model (Sarvaramin and Garagash 2015) is pressure-dependent, and it is directly related\nto formation properties such as permeability, porosity and total compressibility. The final approximation\nof this model is similar to Howard and Fast (1957)'s solution for leakoff of low viscosity fracturing fluids,\nexcept that the pressure difference term is not constant, and it is a function of time and location.\nModel Setup and Description\nIn this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure\nin a DFIT. Fig. 3 provides a schematic of the simulation model. Minimum and maximum horizontal stresses"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae305fc-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "are acting in the Y and X directions, respectively. In the Base Case model, wellbore storage is neglected,\nand the fluid is injected directly at the injection point (perforation). In a more complex scenario, a block\nrepresenting the wellbore storage is considered. The fluid is injected into the wellbore block, which is\nconnected to the model using pipe elements. The pipe elements also model possible pressure drops during\nthe test.\nFigure 3\u2014Plan view schematic of the simulation model setup.\nIn order to consider the appropriate far-field boundary conditions and model propagation of the fracture in\nan infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where\nthe modeled region is small in size compared to the surrounding medium (formation), and they provide\nquiet boundaries in the simulation (Abaqus Analysis User's Guide 2016). Chen (2012) and Haddad and\nSepehrnoori (2015) have discussed the advantages of using infinite elements in improving the accuracy"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30674-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 4, "chroma:document": "and runtime of the simulation. For well-testing applications, we have observed improvements in the\nrepresentation of pressure transient behaviors, particularly in after-closure flow regimes and removal of\nunrealistic boundary effects, using infinite elements.\nThe cohesive elements are embedded in the formation rock, and act as the potential pathway for hydraulic\nfracture growth. During the injection of high pressure fluid, once the required criteria (as discussed in the\nprevious section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-\nin, the fracture aperture reduces to closure aperture (w closure), which is a predefined input into the simulator.\nAfter closure, the aperture can further decrease below the wclosure, but it will always be a positive value,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae306e2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "6 SPE-185649-MS\nmeaning the fracture retains a finite aperture and conductivity even after closure. In low permeability\nformations, even a few microns of residual aperture results in an infinite conductive fracture relative to\nmatrix permeability.\nTable 1 lists properties of the Base Case simulation model. In order to compare simulation results with\nfield observations, four other models with the same reservoir and rock mechanical properties, but different\nsettings, are considered (Table 2). In Model 2, the cohesive layer stiffness is reduced to observe its effect on\nfracture compliance and pressure behavior. In Model 3, the first cohesive element connected to the injection\npoint remains open during shut-in and closure. In Model 4, wellbore storage is included that is comparable\nwith real field data.\nTable 1\u2014Base Case model simulation properties\nInput parameter Value Input parameter Value\nPermeability, md 0.025 Initial porosity, % 10\nYoung's modulus, GPa 20.7 Fracture fluid viscosity, cp 1"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3075a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "Young's modulus, GPa 20.7 Fracture fluid viscosity, cp 1\nPoisson's ratio 0.25 Shmin, MPa 38.6\nTotal compressibility, KPa 4.68\u00d710-6 Shmax, MPa 44.1\nPumping time, sec 120 Initial pore pressure, MPa 24.8\nPumping rate (per unit thickness),\nm3.sec-11.5\u00d710-4 Maximum tensile strength, MPa 1.25\nCohesive layer stiffness, GPa.m-1 2070\nTable 2\u2014Differences in model setup and simulation settings\nBase Case Model 2 Model 3 Model 4\nCohesive layer stiffness,\nGPa.m-12070 20.7 2070 2070\nDisplacement boundary\ncondition applied to the first\ncohesive elementNo No Yes No\nWellbore storage and pipe\nelementsNo No No Yes\nResults and Discussion\nIn the following, synthetic before-closure signatures are generated for the Base Case and Models 2-4 and\ninterpreted for various physical phenomena. Field cases are then analyzed in the context of the simulation\nmodel results to understand the signatures that commonly occur in the field.\nBase Case"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae307d2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 5, "chroma:document": "Base Case\nFig. 4(a) illustrates the pressure behavior during injection for the Base Case simulation model. Because no\nwellbore is included in the Base Case model, pressure increases quickly during injection, until it reaches the\nbreakdown point pressure. At this point, damage initiates (fracture initiation), and then propagates along\nthe cohesive element (fracture propagation). The smooth pressure behavior during propagation indicates\nproper mesh size of the model and cohesive elements. Fig. 4(b) illustrates the fracture aperture profile during\npropagation and early shut-in time. Fracture tip extension occurs for approximately 1 meter after shut-in."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3084a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 6, "chroma:document": "SPE-185649-MS 7\nFigure 4\u2014Base Case model: (a) pressure profile during injection and fracture propagation; (b)\naperture profile during injection and illustration of tip extension during the early shut-in period.\nFig. 5(a) illustrates fracture opening at a typical element along the length of the fracture (e.g. x=1)\nduring pressure falloff. As the fluid inside the fracture leaks off to the surrounding formation and pressure\ninside the fracture decreases, the fracture opening gradually reduces to the closure aperture. The fracture\naperture further reduces after the closure, but will always be a positive value, meaning the fracture retains\na finite aperture and conductivity even after closure. As demonstrated in Fig. 5(a), the rate of aperture\nchange with time before closure, \n , is significantly different from after closure behavior,\n. The simulated response is similar to the experimental results of Van Dam et al. (2002).\nBased on the continuity equation of fluid flow in the fracture ( Eq. 2), the change of"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae308c2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 6, "chroma:document": "at the time of closure\ncauses an abrupt change in pressure. The same behavior is observed for change of the aperture with pressure\n(Fig. 5 (b)) in our simulated case. This is in agreement with the concept of compliance contrast at closure\npresented previously by Van Dam et al. (2002) and McClure et al. (2014).\nFigure 5\u2014Base Case model: fracture aperture behavior at x=1 during falloff with (a) time and (b) pressure.\nProgressive Fracture Closure.   Fig. 6 illustrates the accumulated leakoff volume from the fracture face\nalong the fracture length for a typical time during falloff (e.g. 15 minutes). The accumulated leakoff volume\nis higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this\nresults in a non-uniform pressure profile and effective stress (on fracture walls) along the fracture length\n(Fig. 6). The effective stress is maximum at the fracture tip (x=11) and gradually decreases along the fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3093a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 7, "chroma:document": "8 SPE-185649-MS\n(up to x=5) where it remains nearly constant up to the perforation (x=0). Therefore, fracture closure first\noccurs at the tip and then moves toward the wellbore, which is referred to herein as \"progressive fracture\nclosure\". Fig. 7 illustrates the fracture aperture over time during progressive closure. As expected, closure\nfirst occurs at the tip (x=11) moving toward x=5, where fracture suddenly closes up to perforation (x=0).\nFigure 6\u2014Accumulated leakoff volume and effective stress profile at fracture face along the fracture.\nFigure 7\u2014Progressive fracture closure from the tip of the fracture to the perforation.\nThe progressive closure causes an increase in the pressure derivative. Fig. 8 illustrates the pressure falloff\ntrend, and its semi-log derivative \n , on a G-function plot. Using the Carter leakoff model, we would\nexpect a linear trend on the \n  curve; however, due to the pressure-dependency included in the leakoff\nmodel used herein (Eq. 5), the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae309b2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 7, "chroma:document": "model used herein (Eq. 5), the \n  curve slightly deviates from the straight line with a concave downward\ntrend. The start of the upward deviation (derivative spike) on the \n  plot is correlated with tip closure, and\nthe end of derivate spike indicates a fully closed fracture (closure at the wellbore)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30a34-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 8, "chroma:document": "SPE-185649-MS 9\nFigure 8\u2014Base Case model: before-closure analysis using G-function and derivative plots.\nThe PTA plot in Fig. 9 includes the Primary Pressure Derivative (PPD), Bourdet-derivative with respect\nto Agarwal's time (\n ) and Bourdet-derivative with respect to shut-in time (\n ). The early time\nfluctuations on the derivative plots are caused by fracture tip extension. According to Mattar and Zaoral\n(1992), the PPD for any type of reservoir effect or flow regime will always either be a constant or decreasing,\nand any non-reservoir effect will appear as an increase in the PPD. The non-reservoir effects in the Base\nCase model are fracture tip extension and fracture closure. After the early time tip extension signature\n(fluctuations), the start of PPD violation (upward trend) is caused by fracture tip closure. The end of PPD\nviolation corresponds to a fully closed fracture. Closure pressure may be estimated at the start and end"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30ab6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 8, "chroma:document": "of the PPD violation as 38.7 MPa and 38.3 MPa, respectively. In real field tests, it is likely that stress\nconditions near wellbore area are significantly affected during drilling and completion. Therefore, the\ndifference between closure pressure at the perforation and fracture tip can be larger.\nFigure 9\u2014Base Case model: before-closure analysis using PTA plot.\nConsistent with previous claims (Van Den Hoek 2016; McClure et al. 2016) it is observed that \nstarts to deviate from \n  at \u0394t = 0.1 tp, and the closure time is not long enough for the \n  slope to establish."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30b2e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 9, "chroma:document": "10 SPE-185649-MS\nModel 2\nThe main difference between this model and the Base Case is that the \n  ratio used in the cohesive layer\nstiffness calculation (Eq. 1) is reduced from 100 to 1. As shown in Fig. 10, even after closure, the fracture\naperture continues to decrease from the closure aperture to a residual aperture. The difference between\n and \n  is also less than for the Base Case.\nFigure 10\u2014Model 2: fracture aperture behavior during falloff with (a) time and (b) pressure.\nSimilar to the Base Case, progressive closure from the tip of the fracture to near the wellbore causes a\nclear break in pressure, and a spike in the pressure derivative on the G-function and PTA plot; however, its\nmagnitude is smaller than for the Base Case model (see Fig. 11). Fracture tip closure starts at 39.4 MPa,\nwellbore closure occurs at 38.4 MPa, and the progressive closure period lasts for 5 minutes.\nFigure 11\u2014Model 2: before-closure analysis using (a) G-function (b) PTA plot."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30bb0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 9, "chroma:document": "Low values of cohesive layer stiffness are not realistic. But even for low cohesive layer stiffness, a small\nchange in aperture behavior along the length of fracture caused an abrupt change in pressure behavior. While\nthe progressive fracture closure signature (or previously known as height recession/transverse storage) is\nusually observed in field data, a very sharp increase in derivative for a fully closed fracture is not common.\nWith the next two models, a few scenarios are presented that can affect this sudden pressure change."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30c32-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 10, "chroma:document": "SPE-185649-MS 11\nModel 3\nIn this model, the first cohesive element connected to the injection point remains open during closure.\nThis was achieved by applying appropriate boundary conditions to the first element. Fig. 12 presents\nBCA analysis plots for Model 3. The overall pressure behavior (Fig. 12(a)) is smooth and similar to field\nobservations. Also, the progressive closure signature (upward deviation in derivative plots) is smooth.\nFigure 12\u2014Model 3: before-closure analysis using (a) G-function (b) PTA plot.\nModel 4\nIn this case, wellbore storage is included in the model by addition of a well block and pipe elements. Fig.\n13 provides pressure transient analysis for this model, without any smoothing applied to the derivative.\nAs demonstrated by the schematics included in the figure, at early time, the afterflow caused by wellbore\nstorage increases both the fracture aperture and length identified by a unit slope on Bourdet-derivative."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30cb4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 10, "chroma:document": "The transition from the unit slope occurs when the fracture walls start to close (the so-called hinge closure\nsuggested by Van Den Hoek 2016). Due to low leakoff, this can cause additional tip extension during the\nearly closure (abrupt changes and fluctuations) as fluid is pushed toward the tip of fracture. After this\ntransition, leakoff becomes the dominant factor affecting the pressure transient and similar flow regimes\nto previous models appear."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30d36-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "12 SPE-185649-MS\nFigure 13\u2014Model 4: before-closure analysis using PTA plots.\nIn the presence of wellbore storage, the first fracture element connected to the wellbore can stay open\nwithout applying any boundary conditions (Model 3). This can be explained based on the continuity equation\n(Eq. 1). Initially it was assumed that the qinj term is zero during shut-in. In the presence of afterflow caused by\nwellbore storage, the qinj term is no longer zero and can act to reduce the abrupt change in the \n  term during\nclosure. Therefore, progressive fracture closure stops at the wellbore without causing an abrupt change in\npressure and derivatives (as with the Base Case model and Model 2). Also, as mentioned earlier, Van Dam\net al. (2002) demonstrated that plastic deformation can cause the fracture to remain open in vicinity of\nthe wellbore. While plastic deformation is not modeled in this study, it can be another reason why a very"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30db8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "sharp spike in derivative plots is not observed in field data. Nonetheless, the start of progressive closure (tip\nclosure) can be used to estimate closure pressure and minimum horizontal stress.\nAs suggested by Cramer and Nguyen (2013), using downhole shut-in accelerates fracture closure and\ntime to reach radial flow after closure. These advantages can be explained with Model 4. In the presence\nof wellbore storage, fracture length and volume increase significantly during the early falloff period. With\nlarger fracture length, a much longer time is required for radial flow to be established in the formation.\nFurther, re-supply of fluid into the fracture due to afterflow, coupled with a larger volume, delays fracture\nclosure significantly. In addition to the benefits suggested by Cramer and Nguyen (2013), reducing afterflow\nby using downhole shut-in results in a clearer signature of fracture closure (abrupt change in pressure and\nderivatives)."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30e30-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 11, "chroma:document": "derivatives).\nIn many field cases, a unit slope on the Bourdet-derivative is observed. Hawkes et al. (2013) introduced\na composite permeability concept to explain after-closure behavior. Bachman et al. (2015) expanded this\nidea and stated this behavior can happen before or after closure. A simpler explanation for unit slope after\nclosure is the effect of wellbore storage. Depending on the re-supply rate of fluid from the wellbore, and its\nmagnitude relative to leakoff, a unit slope after closure can exist.\nField Example 1\nThe analysis for this dataset is provided in Fig. 14 and Fig. 15. The pressure values are recorded using\nsurface gauges. No smoothing is used for the purpose of comparison with Model 4. The overall trend and"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30ea8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 12, "chroma:document": "SPE-185649-MS 13\nsignatures are very similar to Model 4 except that the progressive closure period is longer. The early unit\nslope (Fig. 14) and fluctuations indicate fracture volume increase and tip extension due to afterflow. The\ntransition from a unit slope represents the start of hinge closure. Similar to Model 4, tip extension occurs\nduring this period as hinge closure pushes the fluid to the tip of the fracture. Progressive fracture closure\nstarts with PPD violation (or upward deviation on derivative plots) at 22.7 MPa and ends at 19.5 MPa. The\nprogressive closure period lasts around 5 hours and the difference between closure pressure estimates is\nsignificant.\nFigure 14\u2014Field Example 1: before-closure analysis using PTA plot.\nFigure 15\u2014Field Example 1: before-closure analysis using the G-function plot.\nField Example 2\nThis well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very\nsimilar to Model 4 (Fig. 16). The upward deviation at the end of the"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30f34-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 12, "chroma:document": "slope was of particular interest in the\nprevious analysis, and no explanation was provided at the time for this feature."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae30fac-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "14 SPE-185649-MS\nFigure 16\u2014Field Example 2: before-closure analysis using PTA plot (modified from Hawkes et al., 2013; Well 058L).\nThe early time unit slope indicates fracture expansion and tip extension due to wellbore storage. At the\nend of the unit slope, hinge closure starts and is accompanied by tip extension. After this period (considering\nthat falloff time in this case is large enough), the \n  slope appears on the Bourdet-derivative. Consistent\nwith our modeling results, the upward deviation at the end of the \n  slope, that is consistent with the start\nof the PPD violation, corresponds to the progressive fracture closure period. The start of the PPD violation\nindicates tip closure (52.8 MPa) and the end of PPD violation represents a fully closed fracture (50.9 MPa).\nThe progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.\nField Example 3\nIn this test, pumping time was relatively long (19 minutes). Combined with the low permeability of the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3102e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "formation, this probably resulted in a long fracture requiring more time to close. The pressure values\nare recorded using surface gauges and significant ambient temperature variations between day and night\noccurred during the falloff period. As shown in Fig. 17(a), there are severe fluctuations in the G-function\nderivative plot. These functions can result in misinterpretation of the falloff period (e.g. near wellbore\ncomplexities, tip extension, pressure dependent leakoff, etc.); however, when ambient temperature is\nplotted, it is observed that pressure fluctuations are a direct effect of ambient temperature variation. To\nremove the temperature effect from the surface pressure data, the following equation is used to account for\nfluid volume change with pressure and temperature:\n(6)\nwhere Preading is the recorded pressure at surface, T is the current ambient temperature, T0 is the ambient\ntemperature at time of shut-in, \u03b2 is the thermal expansion coefficient of wellbore fluid and Cf is the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae310a6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 13, "chroma:document": "compressibility of wellbore fluid. In this equation, temperature dependency of \u03b2 and Cf is neglected which\ncan further increase its accuracy."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31150-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 14, "chroma:document": "SPE-185649-MS 15\nFigure 17\u2014Field Example 3: (a) effect of ambient temperature variation on derivative plot (b) removing\nthe effect of ambient temperature using Eq. 6 and BCA based on height recession signature.\nAs demonstrated in Fig. 17(b), applying Eq. 6 to recorded pressure values reduces the ambient\ntemperature effect significantly without using a larger derivative window. Closure pressure is estimated to\nbe 21.5 MPa, based on conventional interpretation of fracture height recession/transverse storage.\nFig. 18 provides before-closure analysis based on PTA plot and using the progressive closure concept.\nAlthough closure time is much longer compared to pumping time, a \n  slope is not clear in this test. After the\ntip extension period, the start of the PPD violation at 25.2 MPa represents tip closure. Due to the very low\npermeability of the formation, and large fracture created during pumping, the progressive closure period"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae311d2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 14, "chroma:document": "is very long (around 6 days). The end of the PPD violation at 23.3 MPa indicates a fully closed fracture.\nThe closure pressure estimate based on the conventional interpretation of height recession (Fig. 17(b))\nunderestimates the pressure by 3.7 MPa.\nFigure 18\u2014Field Example 3: before-closure analysis using PTA plots.\nConclusions\nThe concept of compliance change at the time of closure is validated in this study. We demonstrated that\neven for a soft material with low cohesive layer stiffness (not realistic) this feature exists. Therefore, the\nfracture compliance method (McClure et al. 2016) is a reliable approach to identify fracture closure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3124a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "16 SPE-185649-MS\nFor planar fractures, closure is a transient process, starting from the tip of the fracture to the vicinity of\nthe wellbore. We refer to this process as \"progressive fracture closure\". The common signature referred to as\n\"fracture height recession/transverse storage\" is reinterpreted to be caused by this phenomenon. The start of\nprogressive closure (tip closure) is picked as the closure pressure, which is consistent with the compliance\nmethod. Progressive fracture closure explains the increase of Bourdet-derivative after the 3/2 slope. It can\nalso provide information about the leakoff characteristics, fracture dimensions and stress distribution around\nthe fracture. The application of progressive fracture closure to evaluate these properties will be discussed\nin a follow up paper.\nThe primary pressure derivative (PPD) is a powerful tool for identifying progressive fracture closure.\nThe start of a PPD violation corresponds to tip closure, and the end of the violation indicates full closure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae312cc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "Other PTA derivative plots are useful in the identification of flow regimes before the start of progressive\nclosure and explaining other PPD violations. For instance, PPD violations happening before closure during\nthe unit slope on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are\ncaused by fracture expansion and tip extension due to wellbore storage.\nIn previous studies, the magnitude and duration of the tip extension period has been underestimated.\nThe use of the cohesive zone model in this study enabled us to capture this process accurately and the\nsimulations are consistent with field observations. The model results illustrate that tip extension can occur\nduring early fracture expansion period due to wellbore storage and during the hinge closure (and can last\nfor several hours).\nThe afterflow caused by wellbore storage can mask the \n  change at closure, and its transition from"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31344-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "change at closure, and its transition from\nthe tip of the fracture to near the wellbore. Further, it may result in excessive tip extension and a larger\nfracture which delays after closure radial flow. Therefore, removing wellbore storage using downhole shut-\nin not only accelerates closure and radial flow, but also provides a clear signature of compliance change\nand progressive closure.\nA simple method is presented for removing the effect of ambient temperature change on pressure values\nwithout applying large derivative windows for smoothing. The temperature effect may be interpreted\nincorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results\nin loss of accuracy and removal of signatures such as tip extension and PPD violation.\nAcknowledgements\nThe authors would like to thank Dassault Systemes Simulia Corporation for providing Abaqus\u00ae program\nand Seven Generations Energy Corporation for providing field data. The authors would also like to"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae313c6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 15, "chroma:document": "acknowledge Dr. Jack R. Jones for his valuable advice and comments. The sponsors of Tight Oil Consortium\nare acknowledged for their support. Partial funding for this project was obtained through an NSERC\nCollaborative Research and Development grant (CRDPJ 452752-13) led by Clarkson.\nNomenclature\nCcarter Carte leakoff coefficient, m3/sec0.5\nCfWellbore fluid compressibility, Pa-1\nEYoung's modulus, Pa\nGG-function time, dimensionless\nGcCritical energy release rate, Pa.m\nKnStiffness of the cohesive layer, Pa/m\nmSlope of straight line, dimensionless\nPPressure, Pa\nPfFluid pressure in the cohesive element, Pa"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3143e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "SPE-185649-MS 17\nqfracGap flow rate within the cohesive element, m3/sec\nqinjInjection rate into the cohesive element, m3/sec\nqleakLeakoff rate from cohesive element, m3/sec\nSStorage coefficient, Pa-1\nShmaxMaximum horizontal stress, Pa\nShminMinimum horizontal stress, Pa\ntaAgarwal's time, dimensionless\ntadjacent Thickness of the adjacent sub-laminate, m\nTn0Maximum tensile strength, MPa\ntpPumping time, sec\nwFracture opening (aperture), m\nwclosure Closure aperture, m\nGreek Variables\n\u03b1Coefficient for calculation of cohesive layer stiffness, dimensionless\n\u03b1lHydraulic diffusivity, m2/sec\n\u03b2Thermal expansion coefficient, \u00b0C-1\n\u03b40Separation at damage initiation, m\n\u03b4fSeparation at failure, m\n\u0394tShut-in time, sec\n\u03c4(x) Exposure time, sec\n\u03bcFracturing fluid viscosity, cp\nReferences\nAbaqus Analysis User's Guide 2016. Dassault Systemes Simulia Corp., Providence, RI.\nAgarwal, R.G. 1980. A New Method to Account for Producing Time Effects when Drawdown Type Curves are Used to"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae314b6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "Analyze Pressure Buildup and Other Test Data. Presented at the SPE Annual Technical Conference and Exhibition,\nDallas, Texas, 21-24 September. SPE-9289-MS. http://dx.doi.org/10.2118/9289-MS\nBachman, R.C., Walters, D.A., Hawkes, R.A. et al. 2012. Reappraisal of the G Time Concept in Mini-Frac Analysis.\nPresented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8-10 October. SPE-160169-\nMS. http://dx.doi.org/10.2118/160169-MS\nBachman, R., Afsahi, B., and Walters, D. 2015. Mini-Frac Analysis in Oilsands and their Associated Cap Rocks Using\nPTA Based Techniques. Presented at the SPE Canada Heavy Oil Technical Conference, Calgary, Alberta, Canada, 9-11\nJune. SPE-174454-MS. http://dx.doi.org/10.2118/174454-MS\nBarenblatt, G. 1962. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Adv. Appl. Mech. 7: 55\u2013129.\nhttp://dx.doi.org/10.1016/S0065-2156(08)70121-2\nBarree, R. D. 1998. Applications of Pre-Frac Injection/Falloff Tests in Fissured Reservoirs-Field Examples. Presented"}


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topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3152e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "at the SPE Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, Colorado, 5-8 April.\nSPE-39932-MS. http://dx.doi.org/10.2118/39932-MS.\nBarree, R. D. and Mukherjee, H. 1996. Determination of Pressure Dependent Leakoff and Its Effect on Fracture Geometry.\nPresented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 6-9 October. SPE-36424-MS.\nhttp://dx.doi.org/10.2118/36424-MS.\nBarree, R.D., Baree, V .L., and Craig, D.P. 2009. Holistic Fracture Diagnostics: Consistent Interpretation of Prefrac\nInjection Tests Using Multiple Analysis Methods. SPE Production & Operations 24(3): 396\u2013406. SPE-107877-PA.\nhttp://dx.doi.org/10.2118/107877-PA\nBarton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation and Conductivity Coupling of Rock Joints.\nInternational Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 22(3): 121\u2013140. http://\ndx.doi.org/10.1016/0148-9062(85)93227-9."}


created_at: 2023-11-17 19:17:23
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 16, "chroma:document": "dx.doi.org/10.1016/0148-9062(85)93227-9.\nChen, Z.R. 2012. Finite Element Modeling of Viscosity-Dominated Hydraulic Fractures. J. Petrol. Sci. Eng. 88\u201389:\n136\u2013144. http://dx.doi.org/10.1016/j.petrol.2011.12.021"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31628-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "18 SPE-185649-MS\nCramer, D. D. and Nguyen, D. H. 2013. Diagnostic Fracture Injection Testing Tactics in Unconventional Reservoirs.\nPresented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 4\u20136 February.\nSPE-163863-MS. http://dx.doi.org/10.2118/163863-MS\nDugdale, D. 1960. Yielding of Steel Sheets Containing Slits. J. Mech. Phys. Solids 8(2): 100\u2013104. http://\ndx.doi.org/10.1016/0022-5096(60)90013-2\nHaddad, M., and Sepehrnoori, K. 2015. Simulation of Hydraulic Fracturing in Quasi-Brittle Shale Formations Using\nCharacterized Cohesive Layer: Stimulation Controlling Factors. Journal of Unconventional Oil and Gas Resources 9:\n65\u201383. http://dx.doi.org/10.1016/j.juogr.2014.10.001\nHoward, C.C., and Fast, C.R. 1957. Optimum Fluid Characteristics for Fracture Extension. API Drilling and Production\nPractice 24(1): 261\u2013270. API-57-261.\nMarongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J. 2011. Global Model for Fracture Falloff Analysis."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae316aa-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "Presented at the SPE North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas, 14-16\nJune. SPE-144028-MS. http://dx.doi.org/10.2118/144028-MS\nMattar, L. and Zaoral, K. 1992. The Primary Pressure Derivative (PPD) A new Diagnostic Tool in Well Test Interpretation,\nJ. Can. Pet. Technol. 31(4). http://dx.doi.org/10.2118/92-04-06\nMcClure, M.W., Blyton, C.A.J., Jung, H. et al. 2014. The Effect of Changing Fracture Compliance on Pressure Transient\nBehavior During Diagnostic Fracture Injection Tests. Presented at the SPE Annual Technical Conference and\nExhibition, Amsterdam, The Netherlands, 27\u201329 October. SPE-170956-MS. http://dx.doi.org/10.2118/170956-MS\nMcClure, M.W., Jung, H., Cramer, D.D. et al. 2016. The Fracture-Compliance Method for Picking Closure\nPressure From Diagnostic Fracture-Injection Tests. Soc. Pet. Eng. 21(4): 1321\u20131339. SPE-179725-PA. http://\ndx.doi.org/10.2118/179725-PA"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3172c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "dx.doi.org/10.2118/179725-PA\nMohamed, I.M., Nasralla, R.A., Sayad, M.A. et al. 2011. Evaluation of After-closure Analysis Techniques for Tight\nand Shale Gas Formations. Presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The\nWoodlands, Texas, 24-26 January. SPE-140136-MS. http://dx.doi.org/10.2118/140136-MS\nNolte, K.G. 1979. Determination of Fracture Parameters From Fracturing Pressure Decline. Presented at the SPE\nAnnual Technical Conference and Exhibition, Las Vegas, Nevada, 23-26 September. SPE-8341-MS. http://\ndx.doi.org/10.2118/8341-MS\nSarvaramini, E., Garagash, D.I. 2015. Breakdown of a Pressurized Fingerlike Crack in a Permeable Solid. J. Appl. Mech\n82(6). http://dx.doi.org/10.1115/1.4030172\nSearles, K.H., Zielonka, M.G., Ning, J. et al. 2016. Fully-Coupled 3D Hydraulic Fracture Models: Development,\nValidation, and Application to O&G Problems. Presented at the SPE Hydraulic Fracturing Technology Conference,"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae317ae-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "The Woodlands, Texas, 9-11 February. SPE-179121-MS. http://dx.doi.org/10.2118/179121-MS\nShen, X., Cullick, S. 2012. Numerical Modeling of Fracture Complexity with Application to Production Stimulation.\nPresented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, Texas, 6\u20138\nFebruary. SPE-151965-MS. http://dx.doi.org/10.2118/151965-MS\nShin, D.H., Sharma, M.M. 2014. Factors Controlling the Simultaneous Propagation of Multiple Competing Fractures in a\nHorizontal Well. Presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands,\nTexas, 4\u20136 February. SPE-168599-MS. http://dx.doi.org/10.2118/168599-MS.\nTuron, A., Davila, C.G., Camanho, P.P. et al. 2007. An Engineering Solution for Mesh Size Effects in The Simulation\nof Delamination Using Cohesive Zone Models. Eng. Fract. Mech. 74: 1665\u20131682. http://dx.doi.org/10.1016/\nj.engfracmech.2006.08.025"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31826-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/SPE-185649-MS Zanganeh Method.pdf", "page": 17, "chroma:document": "j.engfracmech.2006.08.025\nVan Dam, D.B., Papanastasiou, P., and de Pater, C.J. 2002. Impact of Rock Plasticity on Hydraulic Fracture Propagation\nand Closure. SPE Production & Facilities, 17(3): 149 \u2013 159. SPE-78812-PA. http://dx.doi.org/10.2118/78812-PA\nVan Den Hoek, P. 2016. A Simple Unified Pressure Transient Analysis Method for Fractured Waterflood Injectors and\nMinifracs in Hydraulic Fracture Stimulation. Presented at the SPE Annual Technical Conference and Exhibition,\nDubai, UAE, 26-28 September. SPE-181593-MS. http://dx.doi.org/10.2118/181593-MS\nYao, Y . 2012. Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in Brittle and Ductile Rocks.\nRock Mech. Rock Eng. http://dx.doi.org/10.1007/s00603-011-0211-0\nZielonka, M. G., Searles, K. H., Ning, J., and Buechler, S. R. Development and Validation of Fully-Coupled Hydraulic\nFracturing Simulation Capabilities. 2014 Simulia Community Conference, Providence RI 17."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae318b2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "Copyright 2007, Society of Petroleum Engineers \n This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & Gas Technology Symposium held in Denver, Colorado, U.S.A., 16\u201318 April 2007.  This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as \npresented, have not been reviewed by the Society of Petroleum Engineers and are subject to \ncorrection by the author(s). The material, as presented, does not necessarily reflect any"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae319ac-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than  300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435. \n \nAbstract \nSince the introduction of the G-function derivative analysis, \npre-frac diagnostic injection tests have become a valuable and \ncommonly used technique. Unfo rtunately, the technique is \nfrequently misapplied or misinterpreted leading to confusion"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31a38-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "frequently misapplied or misinterpreted leading to confusion \nand misdiagnosis of fracturing parameters. This paper presents \na consistent method of analysis of the G-function, its derivatives, and its relationship to other diagnostic techniques \nincluding square-root(time) and log( \u2206p\nwf)-log( \u2206t) plots and \ntheir appropriate diagnostic de rivatives. Actual field test \nexamples are given for the most common diagnostic curve \nsignatures. \n  \nIntroduction \nPre-frac diagnostic injection te st analysis provides critical \ninput data for fracture design models, and reservoir \ncharacterization data used to pr edict post-fracture production. \nAn accurate post-stimulation prod uction forecast is necessary  \nfor economic optimization of th e fracture treatment design. \nReliable results require an accurate and consistent interpretation of the test data. In many cases closure is \nmistakenly identified through misapplication of one or more \nanalysis techniques. In general, a single unique closure event"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31ab0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "analysis techniques. In general, a single unique closure event \nwill satisfy all diagnostic pl ots or methods. All available \nanalysis methods should be used in concert to arrive at a consistent interpretation of fracture closure.  \nRelationship of the pre-closur e analysis to after-closure \nanalysis results must also be consistent. To correctly perform \nthe after-closure analysis the tr ansient flow regime must be \ncorrectly identified. Flow regime identification has been a \nconsistent problem in many analyses. There remains no \nconsensus regarding methods to identify reservoir transient flow regimes after fracture closure. The method presented here \nis not universally accepted but appears to fit the generally assumed model for leakoff used in most fracture simulators. Four examples are presented to show the application of \nmultiple diagnostic analysis methods. The first illustrates the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31b28-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after shut-\nin. The second example shows pressure dependent leakoff \n(PDL) in a reservoir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary \nfractures or fissures. The third example shows fracture tip \nextension after shut-in. These cases generally show definable fracture closure. The fourth example shows what has been \ncommonly identified as fractur e height recession during \nclosure, but which can also indicate variable storage in a \ntransverse fracture system. \nFor each example the analysis will be demonstrated using \nthe G-function and its diagnostic derivatives, the sqrt(time) and its derivatives, and the log-log plot of pressure change \nafter shut-in and its derivatives.\n1-4 When appropriate, the after-\nclosure analysis is presented for each case, as is an empirical \ncorrelation for permeability from the identified G-function"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31bb4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "correlation for permeability from the identified G-function \nclosure time.5 A critical part of the analysis is the realization \nthat there is a common event indi cating closure that should be \nconsistently identified by all diagnostic methods. To reach a conclusion all analyses must give consistent results.  \nThe goal of this paper is to provide a method for consistent \nidentification of after-closure flow regimes, an unambiguous \nfracture closure time and stress, and a reasonable engineering estimate of reservoir flow capacity from the pressure falloff \ndata, without requiring assumptions such as a known reservoir \npressure. Other methods, based on sound transient test theory, require pressure difference curves based on the observed \nbottomhole pressure during falloff minus the \u201cknown\u201d \nreservoir pressure.\n5,8 While these methods are technically \ncorrect they can lead to confusin g results at times, especially \nin low permeability reservoirs when pore pressure is difficult"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31c2c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "in low permeability reservoirs when pore pressure is difficult \nto determine accurately prior to stimulation. \nThis is not a transient test analysis paper but is intended to \npresent a practical approach to analysis of real, and frequently marginal-quality, pre-fracture field test data. The techniques \napplied are based on some transient test theory. Some of the \nresults presented here are still under debate and development. The methods shown have been tested and, we believe, proven \nin the analysis of hundreds of  tests. Application of these \nmethods provides consistent analysis that helps to avoid \nmisinterpretation of falloff data, and give the most useful \ninformation available from diagnostic injection tests. \nStep-rate injection tests and th eir analysis are not included \nin the scope of this paper. Determination of the pressure-\ndependent leakoff coefficient is also not described here, as it \n \nSPE 107877 \nHolistic Fracture Diagnostics"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31cae-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 0, "chroma:document": "SPE 107877 \nHolistic Fracture Diagnostics \nR.D. Barree, SPE, and V.L. Barree, Barree & A ssocs. LLC, and D.P. Craig, SPE, Halliburton Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31d30-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "2  SPE 107877 \nhas been previously reported.3,4 Only the analysis of pressure \ndecline following shut-in of a fracture-rate injection test is \nconsidered. \n \nTransient Flow Regimes During and After Fracture \nClosure \nSeveral transient flow regimes may occur during a falloff test \nafter injection at fracture rate . The major flow regimes are \ngraphically illustrated in the classic paper by Cinco-Ley and \nSamaniego.6  \nImmediately after shut-in the pressure gradient along the \nlength of the fracture dissipates in a short-duration linear flow \nperiod. In a long fracture in low permeability rock the initial \nfracture linear flow can be follow ed by a bi-linear flow period \nwith the linear flow transient persisting in the fracture while \nreservoir linear flow occurs simu ltaneously.  After the fracture \ntransient dissipates the reservoir linear flow period can \ncontinue for some time, depending on the permeability of the reservoir and the volume of fluid stored in the fracture and"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31dbc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "subsequently leaked off duri ng closure. After closure the \npressure transient established around the fracture propagates \ninto the reservoir and transitions into elliptical, then \npseudoradial flow. Each of these flow regimes has a characteristic appearance on various diagnostic plots. \nFluid leakoff from a propagating fracture is normally \nmodeled assuming one-dimensional linear flow perpendicular to the fracture face. Settari has pointed out that in some cases \nof moderate reservoir permeability the linear flow regime may not occur, even during fracture extension and early leakoff.\n7 \nDuring fracture extension and shut-in the transient may \nalready be in transition to elliptical or pseudoradial flow.  In \nthis case analyses based on an  assumed pseudolinear flow \nregime will give incorrect results. In all cases an \nunderstanding of the flow regime and its relation to the \nfracture geometry is critical to arriving at a consistent interpretation of the fracture falloff test."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31e3e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "Diagnostic Derivative Examples \nFor each analysis technique various curves are used to help \ndefine closure, leakoff mechan isms, and after-closure flow \nregimes. On each plot the curves  are labeled as the primary (y \nvs. x), the first derivative ( \u2202y/\u2202x), and the semilog derivative \n(\u2202y/\u2202(lnx) or x \u2202y/\u2202x). For convenience th e primary curve is \nplotted on the left y-axis and all derivatives are plotted on the \nright y-axis for all Cartesian plots. For the log-log plot all \ncurves are shown on the same y-axis. \nFor pre-closure analysis, and consistent identification of \nfracture closure, three techniques are illustrated for each example: G-function, Square-root of shut-in time, and log-log plot of pressure change with shut-in time. All these analyses \nbegin at shut-in. The instantaneous shut-in pressure (ISIP) is taken as the incipient fracture ex tension pressure for all cases. \nWhen there is significant wellbore afterflow (fluid expansion \nor continued low-rate injection), or severe near-well pressure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31eb6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "or continued low-rate injection), or severe near-well pressure \ndrop, the ISIP can be difficult to interpret accurately and may \nbe too high to represent actual fracture extension pressure. In \nall the examples in the paper the pressures have been offset to \nan approximate ISIP of 10,000 psi to remove any relation to the original field test data. The following sections detail the \ndata and analysis for the four  major leakoff type examples.  \nNormal Leakoff Behavior \nNormal leakoff is observed when the composite reservoir \nsystem permeability is constant. The reservoir may exhibit \nonly matrix permeability or have a secondary natural fracture \nor fissure overprint in which the flow capacity of the secondary fracture system does not change with pore pressure \nor net stress. After shut-in the fracture is assumed to stop propagating and the fracture su rface area open to leakoff \nremains constant during closure. \n \nNormal Leakoff G-Function \nAs noted in previous papers, the expected signature of the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31f38-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "As noted in previous papers, the expected signature of the \nG-function semilog derivative is a straight-line through the \norigin (zero G-function and zero derivative).\n4 In all cases the \ncorrect straight line tangent to the semilog derivative of the \npressure vs. G-function curve must pass through the origin. \nFracture closure is identified by  the departure of the semi-log \nderivative of pressure with respect to G-function (G \u2202pw/\u2202G) \nfrom the straight line through the origin. During normal \nleakoff, with constant fractu re surface area and constant \npermeability, the first derivative ( \u2202pw/\u2202G) should also be \nconstant.2 The primary pw vs. G curve should follow a straight \nline.1 The example in Figure 1 shows some slight deviation \nfrom the perfect constant leakoff but is a good example of the expected curve shapes with a clear indication of closure at \nG\nc=2.31. The closure event is marked by the dashed vertical \nline [1]. \n \n0 5 10 15 20 25\nG(Time)750080008500900095001000010500"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae31fb0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "0 5 10 15 20 25\nG(Time)750080008500900095001000010500\n0200400600800100012001400160018002000\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\n \nFigure 1: Normal leakoff G-function plot \n \nNormal Leakoff Sqrt(t) Analysis \nThe sqrt( t) plot has frequently been misinterpreted when \npicking fracture closure, even  for the simplest cases. The \nprimary pw vs. sqrt( t) curve should form a straight line during \nfracture closure, as with th e G-function plot. Some users \nsuggest that the closure is identified by the departure of the \ndata from the straight line trend, similar to the way the G-\nfunction closure is picked. This is incorrect and leads to a later \nclosure and lower apparent closure pressure. The correct indication of closure is the inflection point on the p\nw vs. sqrt( t) \nplot.  \nThe best way to find the inflection point is to plot the first"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32032-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 1, "chroma:document": "The best way to find the inflection point is to plot the first \nderivative of pw vs. sqrt( t) and find the point of maximum Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
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id: 9ae320b4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 2, "chroma:document": "SPE 107877  3 \namplitude of the derivative. Many fracture-pressure analysis \nsoftware packages plot the inverse of the actual first derivative \nand show the inflection point as the minimum of the \nderivative. The plot in Figure 2, shows that the slope of the \npressure curve starts low, then increases and reaches a \nmaximum rate of decline at the inflection point, then decreases \nagain after closure. The first derivative curve in Figure 2 is \nplotted with the proper sign. The dashed vertical line [1] is the \nG-function closure pick that is synchronized in time and pressure with the sqrt( t) plot. Clearly the consistent closure \nlies at the inflection point and not at the point of departure from the straight line tangent to the pressure curve.  \nThe semilog derivative of the pressure curve is also shown \non the sqrt( t) plot. This curve is equivalent to the semilog \nderivative of the G-function for most low-perm cases. The closure pick falls at the departure from the straight line"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3212c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 2, "chroma:document": "through the origin on the semilog derivative of the P vs. sqrt( t) \ncurve. A single closure point mu st satisfy the requirement on \nboth the G-function and sqrt( t) plots. \n \n1/24/200704:00 08:00 12:00\n1/24/200716:00\nTime750080008500900095001000010500\n01002003004005006007008009001000\n1Pressure\nDerivativesP vs. \u221at\n\u221atdP/d\u221atv s .  \u221at\ndP/d\u221atv s .  \u221atFracture Closure\n \nFigure 2: Normal leakoff sqrt(t) plot \n \nNormal Leakoff Log-Log Pressure Derivative \nThe log-log plot of pressure change from ISIP versus shut-\nin time for the normal leakoff example is shown in Figure 3. \nThe heavy curve is the pressu re difference and the dashed \ncurve is its semilog derivative with respect to shut-in time. \nThe vertical dashed line is the unique closure pick from the G-\nfunction and sqrt( t) plot. It is common for the pressure \ndifference and derivative curves to be parallel immediately \nbefore closure. The slope of these parallel lines is diagnostic \nof the flow regime established during leakoff before closure."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae321ae-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 2, "chroma:document": "of the flow regime established during leakoff before closure. \nIn many cases a near-perfect \u00bd slope is observed, strongly \nsuggesting linear flow from the fracture. In this example the \nslope is greater than \u00bd sugges ting possible linear flow coupled \nwith changing fracture/wellbor e storage (See Appendix B). \nThe separation of the two parallel lines always marks fracture \nclosure and is the final confirmation of a consistent closure \nidentification. \n 2 3 456789 2 3 456789 2 3 456789 2 3 4567890.1 1 10 100 1000\nTime (0 = 8.15)2345678923456789234\n101001000\n (m = -1)  (m = 0.632)  BH ISIP = 9998 psi 1Delta-Pressure and Derivative\u2206P vs. \u2206t\n\u2206td\u2206P/d\u2206t vs. \u2206t\nFracture ClosureRadial Flow\n \nFigure 3: Normal leakoff log-log plot \n \nAfter closure the semilog derivative curve will show a \nslope of -1/2 in a fully developed reservoir pseudolinear flow \nregime and a slope of -1 in fully developed pseudoradial flow.  \nIn the example the derivative slope is -1 indicating that"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32226-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 2, "chroma:document": "In the example the derivative slope is -1 indicating that \nreservoir pseudoradial flow was observed. The late-time data shows a drop in the derivative probably caused by wellbore \neffects such as gas entry and phase segregation. The use of the \nsemilog derivative of the log-log plot for after-closure flow regime identification, as well as closure confirmation, is a \npowerful new addition to fracture  pressure decline diagnostics. \n \nAfter-Closure Analysis for Normal Leakoff Example \nThe Talley-Nolte After-Closure Analysis (ACA) flow \nregime identification plot for the normal leakoff example is \nshown as Figure 4.\n5  The heavy solid line is the observed \nbottomhole pressure during the falloff minus the initial \nreservoir pressure. The slope of the semi-log derivative of the \npressure difference function (dashed line) is 1.0 during the \nidentified pseudoradial flow period. If a linear-flow period \nexisted in this data set a derivative slope of \u00bd would exist. It is"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3229e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 2, "chroma:document": "critical to remember that the sl ope of the pressure difference \ncurve on this plot is determined solely by the guess of \nreservoir pressure used to construct the plot. The slope of the \nderivative is not affected by the input reservoir pressure value. \n \n2 3 4 5 6 789 2 3 4 5 6 789 2 3 4 5 6 7890.001 0.01 0.1 1\nSquare Linear Flow (FL^2)234567892345678923456789\n10100100010000\n (m = 1) Delta-Pressure and Derivative\u2206P vs. FL2\nFL2d\u2206P/dFL2vs. FL2\u2206P=(pw-pr)\nStart of Radial Flow\n \nFigure 4: Normal leakoff ACA log plot Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32320-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 3, "chroma:document": "4  SPE 107877 \n \nIf a pseudoradial flow regime is identified, then the \nCartesian Radial Flow plot (Figure 5) can be used to \ndetermine reservoir far- field transmissibility, kh/\u00b5. The \nviscosity used is the far-field mobile fluid viscosity and h is \nthe estimated net pay height. For the analysis of the example \ndata kh/\u00b5 = 299 md-ft/cp. For gas viscosity at reservoir \ntemperature, kh=7.9 md-ft. For the assumed net pay, the \neffective reservoir permeability is 0.097 md.  \n \n \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8\nRadial Flow Time Function72007400760078008000820084008600880090009200\n (m = 4814.2) Results\nReservoir Pressure  = 7475.68 psi\nTransmissibility, kh/\u00b5 = 298.94991 md*f t\nkh = 7.94014 md*ft\nPermeability, k = 0.0968 md\nStart of Pseudo Radial Time = 2.15 hours 1Pressure\n \nFigure 5: Normal leakoff ACA radial flow plot \n \n \nHorner Analysis for Normal Leakoff Example \nIf a pseudoradial flow period is identified, then a \nconventional Horner plot can also be used to determine"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae323de-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 3, "chroma:document": "conventional Horner plot can also be used to determine \nreservoir transmissibility. In Figure 6 the Horner slope through \nthe radial flow data is 14411 psi. Using an average pump rate \nof 18.4 bpm, kh/\u00b5 = 298 md-ft/cp. For the assumed gas \nviscosity  kh=7.9 md-ft. Using the same assumed net gives \nk=0.097 md. This result is consistent with the ACA results.  \n \n \n2 31\nHorner Time72507500775080008250850087509000925095009750\n (m = 14411) \n  (Reservoir = 7476)  1Pressure\n \nFigure 6: Normal leakoff Horner Plot \n  \nG-Function Permeability Estimate \nAn empirical correlation has also been developed to \nestimate formation permeability from the G-function closure \ntime when after-closure data is not available. The correlation \nis described in detail in the Appendix. Figure 7 shows the G-function correlation permeability estimate for the observed \nclosure time and other input parameters. The permeability \nestimate of 0.097 md is consistent with the Horner and ACA results.   \n0.0010.010.1110100"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3246a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 3, "chroma:document": "0.0010.010.1110100\n0 2 4 6 8 1 01 21 41 61 82 0\nGcData Input\nrp 1\n\u03c6 0.09 V/V\nct 7.50E-05 psi-1\nE3 . 5 M p s i\n\u00b5 1c p\nGc 2.44\nPz 966.0 psiEstimated Permeability = 0.0974 mdPermeability, md\n \nFigure 7: Normal leakoff permeability estimate \n \nPressure Dependent Leakoff \nPressure dependent leakoff (PDL) occurs when the fluid loss \nrate changes with pore pressure or net effective stress in the \nrock surrounding the fracture.  PDL is not caused by the \nnormal change in transient pressure gradient during leakoff. \nThis is part of the normal leakoff mode and is handled by the \none-dimensional linear flow solution of the diffusivity \nequation used to model fracture leakoff in a constant permeability system. The pressure dependence referred to here \nis a change in the transmissibility of the reservoir fissure or \nfracture system that dominates the fluid loss rate. PDL is only \napparent when there is substantial stress dependent \npermeability in a composite dual-permeability reservoir."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae324ec-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 3, "chroma:document": "permeability in a composite dual-permeability reservoir. \n \nG-Function for Pressure-Dependent Leakoff \nFigure 9 shows the G-function behavior expected for PDL. \nThe primary p\nw vs. G curve is concave upward and curved \nwhile PDL persists. The semilog derivative exhibits the \ncharacteristic \u201chump\u201d above the st raight line extrapolated to \nthe derivative origin. The end of PDL and the critical fissure \nopening pressure corresponds to the end of the \u201chump\u201d and \nthe beginning of the straight line representing matrix dominated leakoff. Fracture closure is still shown by the \ndeparture of the semilog derivative from the straight line \nthrough the origin.  \n Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32564-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 4, "chroma:document": "SPE 107877  5 \n0 2 4 6 8 10 12 14 16 18 20\nG(Time)8250850087509000925095009750100001025010500\n01002003004005006007008009001000\n21\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\n \nFigure 9: PDL G-function plot \n \n \nSqrt(t) Analysis for PDL \nInterpretation of the sqrt( t) plot in PDL cases has often led \nto incorrect closure picks. Figu re 10 shows an expanded view \nof the sqrt( t) plot for the example with the curves scaled for \nbetter visibility. Note that the semilog derivative is nearly \nidentical in shape and informa tion content to the G-function \nsemilog derivative. It clearly shows the PDL \u201chump\u201d and \nclosure, which has been synchronized to the G-function result.  \nIncorrect closure picks on the sqrt( t) plot will not occur if \nthe semilog derivative is used. Problems arise when the first \nderivative is used exclusively to  pick closure. In PDL cases \nthe obvious derivative maximum, or most prominent inflection \npoint, is caused by the changing leakoff associated with PDL"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae325e6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 4, "chroma:document": "point, is caused by the changing leakoff associated with PDL \nand does not indicate fracture closure. The false closure \nindication is shown on the plot. Many fracture diagnostic tests \nhave been badly misdiagnosed because the early and incorrect \nclosure was picked because of dependence on only the sqrt( t) \nplot. This example clearly illustrates why all available \ndiagnostic plots must be used in concert to arrive at a single \nconsistent closure event.  \n \n1/24/200700:20 00:40 01:00 01:20 01:40\n1/24/200702:00\nTime82508500875090009250950097501000010250\n0100200300400500\n1False ClosurePressure\nDerivativesP vs. \u221at\n\u221atdP/d\u221at vs. \u221at\ndP/d\u221at vs. \u221atFracture Closure\n \nFigure 10: PDL Sqrt plot \n \n Log-Log Pressure Derivative for PDL Example \nFigure 11 shows the log-log plot for the PDL example. The \nnormal matrix leakoff period, following the end of PDL, \nappears as a perfect \u00bd slope of the semilog derivative with a \nparallel pressure difference curve exactly 2-times the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3265e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 4, "chroma:document": "parallel pressure difference curve exactly 2-times the \nmagnitude of the derivative. The parallel trend ends at the identified closure time and pressure difference. In this \nexample a well-defined \u2212\u00bd slope, or reservoir pseudolinear \nflow period, is shown shortly after closure. The later data approach a slope of \u22121, which indicates pseudoradial flow has \nbeen established.  \n \n2 3 456789 2 3 456789 2 3 456789 2 3 4567890.1 1 10 100 1000\nTime (0 = 9.133333)23456789234567892\n101001000\n (m = 0.5) \n(m = -1) (m = -0.5)  BH ISIP = 10000 psi 1Pressure Difference and Derivative\u2206P vs. \u2206t\n\u2206td\u2206P/d\u2206tv s .  \u2206tFracture closureLinear Flow\nRadial Flow\n \nFigure 11: PDL log-log plot \n \nAfter-Closure Analysis for PDL Example \nThe ACA log-log plot (Figure 12) shows both the reservoir \nlinear and radial flow periods in their expected locations.  \n \n2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 67890.001 0.01 0.1 1\nSquare Linear Flow (FL^2)234567892345678923456789\n10100100010000\n (m = 1) \n (m = 0.5)"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae326d6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 4, "chroma:document": "10100100010000\n (m = 1) \n (m = 0.5) \n1 2 3Delta-Pressure and Derivative\u2206P vs. FL2\nFL2d\u2206P/dFL2vs. FL2\u2206P=(pw-pr)\nStart Linear Flow\nEnd Linear Flow\nStart Radial Flow\n \nFigure 12: PDL ACA log plot \n \nFigures 13 and 14 show the ACA Cartesian plots for the \nlinear and radial flow analyses. Both give consistent estimates \nof reservoir pore pressure. Th e pseudoradial flow analysis \ngives a transmissibility of 37.2 md-ft/cp and estimated \npermeability of 0.047 md.  \n Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32758-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 5, "chroma:document": "6  SPE 107877 \n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nLinear Flow Time Function8000820084008600880090009200\n (m = 1438.5) \nResults\nReservoir Pressure = 8056.66 psi\nStart of Pseudo Linear Time = 15. 9\nEnd of Pseudo Linear Time = 54.3 9 1 2PressureStart Linear Flow End Linear Flow\n \nFigure 13: PDL ACA linear flow plot \n \n0.0 0.1 0.2 0.3 0.4 0.5 0.6\nRadial Flow Time Function8000820084008600880090009200\n (m = 11373) Results\nReservoir Pressure = 8068.81 psi\nTransmissibility, kh/\u00b5 = 37.21984 m\nkh = 0.93764 md*ft\nPermeability, k = 0.0469 md\nStart of Pseudo Radial Time = 11.2 6 1Pressure\n \nFigure 14: PDL ACA radial flow plot \n \nHorner Analysis for PDL Example \nFor an average pump rate of 6.7 bpm the Horner plot gives \nkh/\u00b5=35.72 md-ft/cp. The Horner estimated permeability is \n0.046 md compared to 0.047 md from the ACA Radial Flow \nanalysis. Pore pressure estimated from the Horner plot is also"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae327e4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 5, "chroma:document": "analysis. Pore pressure estimated from the Horner plot is also \nconsistent with both the linear and radial analyses because a well-developed pseudoradial flow period does exist in this \ncase. The vertical dotted line in  Figure 15 shows the start of \npseudoradial flow. If a pseudoradial flow period does not \nexist, extrapolation of an apparent straight-line on the Horner \nplot can give extremely inaccura te estimates of pressure and \nflow capacity.  1\nHorner Time8000810082008300840085008600870088008900\n (m = 43920) \n  (Reservoir = 8064)  1Pressure\n \nFigure 15: PDL Horner plot \n \n \nG-Function Permeability Estimate for PDL Example \nThe G-function permeability correlation for the PDL \nexample is shown in figure 16. It also gives a consistent \npermeability of 0.045 md. The impact of the accelerated \nleakoff during PDL gives an estimate of the composite \nreservoir effective permeability. Note that the injected fluid \nviscosity is used for the permeability estimate based on \nclosure time.  \n \n0.0010.010.1110100"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32866-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 5, "chroma:document": "closure time.  \n \n0.0010.010.1110100\n0 2 4 6 8 1 01 21 41 61 82 0\nGcData Input\nrp 1\n\u03c6 0.08 V/V\nct 6.00E-05 psi-1\nE5 M p s i\n\u00b5 1c p\nGc 2.9\nPz 841.0 psiEstimated Permeability = 0.0453 mdPermeability, md\n \nFigure 16: PDL permeability estimate \n  \nFracture Tip Extension \nIn very low permeability reservoirs the decline in wellbore \npressure observed after shut-in may be caused by the \ndissipation of the pressure transient established in the fracture \nduring pumping. The near-well pressure decreases as the \nfracture closes, which results in a decrease of fracture width at \nthe well. The closing of the fracture volumetrically displaces \nfluid to the tip of the fracture, causing continued extension of the fracture length. Much of the pressure decline is therefore \nnot related to leakoff but to the dissipation of the linear transient along the fracture length."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae328de-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 5, "chroma:document": "Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32960-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 6, "chroma:document": "SPE 107877  7 \nG-Function Analysis for Tip Extension \nDuring fracture tip extension the G-function derivatives \nfail to develop any straight-line trends. The primary P vs. G \ncurve is concave upward, as is the first derivative. The \nsemilog derivative starts with a large positive slope and the \nslope continues to decrease w ith shut-in time, giving a \nconcave-down curvature.3,4 Figure 17 shows a typical case of \nfracture tip extension with minimal leakoff. This is another \ncase that is frequently misdiagnosed.  \n \n0 5 10 15 20 25 30\nG(Time)840086008800900092009400960098001000010200\n0255075100125150\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GPressure\nDerivatives\n \nFigure 17: Tip extension G-function plot \n \nSqrt(t) Analysis with Fracture Tip Extension \nMany times the first break in the semilog derivative curve \nhas been misinterpreted as a closure event. The mistake is \noften compounded by the use of the sqrt( t) plot. Figure 18 \nshows the sqrt( t) plot for the same data. The first derivative"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae329d8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 6, "chroma:document": "shows a large maximum very shortly after shut-in. This is \noften mistaken for closure. The semilog derivative on the \nsqrt(t) plot helps to avoid this mistaken closure pick, and \nshows the same continuously increasing trend as seen on the \nG-function semilog derivative plot. In low permeability \nsystems it is generally safe to assume that as long as the semilog derivative is still rising, the fracture has not yet \nclosed. This is not true in ve ry high permeability reservoirs \nand should always be checked using the log-log pressure difference plot.  \n \n1/25/200704:00 08:00 12:00 16:00\n1/25/200720:00\nTime840086008800900092009400960098001000010200\n0102030405060708090100\n1Incorrect ClosurePressure\nDerivativesP vs. \u221at xdP/dxvs. \u221atdP/dxvs. \u221at\n \nFigure 18: Tip extension sqrt(t) plot \n Log-Log Pressure Derivative Analysis with Tip Extension \nThe log-log plot of pressure change after shut-in is \nparticularly useful for diagnosing fracture tip extension. \nFigure 19 shows the pressure difference and pressure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32a5a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 6, "chroma:document": "Figure 19 shows the pressure difference and pressure \nderivative (semilog) for the tip extension example.  \n \n2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 6789 21 10 100 1000\nTime (0 = 33.7)23456789234567892345\n101001000\n (m = 0.25)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative\u2206P vs. \u2206t\n\u2206td\u2206P/d\u2206t vs. \u2206t\n \nFigure 19: Tip extension log-log plot \n \nIn Figure 19 the pressure derivative departs from the early \nunit-slope (storage) and establishes a \u00bc slope during fracture \ntip extension. The pressure difference curve falls on a parallel \n\u00bc slope line separated by 4-times the magnitude of the \nderivative. The \u00bc slope signature is diagnostic of bilinear flow \nrepresenting a continued dissipation of the linear pressure \ntransient along the fracture length (extension and concomitant \nfluid flow) and some linear flow driving minimal leakoff. For tip extension to occur the leakoff rate to the formation must be \nlow. As long as the parallel \u00bc slope trend continues, the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32ad2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 6, "chroma:document": "low. As long as the parallel \u00bc slope trend continues, the \nfracture has not closed and is still in the process of extending. Closure cannot be determined and no after-closure analysis \ncan be conducted. \n \nHeight Recession or Transverse Storage \nThere are two different mechanisms that can generate a \nsimilar diagnostic derivative signature during fracture closure. \nBoth are caused by an excess st ored volume of fluid in the \nfracture at shut-in relative to the expected surface area of the \nfracture for a planar, constant -height geometry model.  \nTraditionally this signature has been called \u201cfracture height \nrecession\u201d. The usual model assu mes that leakoff occurs only \nthrough a thin permeable bed and that the fracture extends in height to cover impermeable strata with no leakoff. At shut-in \nthere is a large volume of flui d stored in the fracture and the \nleakoff rate relative to the stored volume is small, hence the \nrate of pressure decline is likewise small. As the fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32b4a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 6, "chroma:document": "rate of pressure decline is likewise small. As the fracture \nempties, the rate of leakoff relative to the remaining stored \nfluid accelerates and the pressure declines more rapidly. If the \nfracture height changes duri ng leakoff, the fracture \ncompliance may also decrease, addi ng to the rate of pressure \nloss. \nHowever, the same signature is observed in many cases \nwhere fracture height growth out of zone is not observed by \ntracers, inclinometer, or micr o-seismic mapping. Some of \nthese cases show treating behavior similar to PDL cases, with Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32bc2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "8  SPE 107877 \na tendency for rapid screenout and difficulty placing high \nproppant concentration slurries. These observations suggest \nthat another mechanism may be responsible for the same \ndiagnostic derivative signature. The alternate mechanism is \ncalled \u201ctransverse fracture storage\u201d.  \nIn transverse fracture storag e a secondary fracture set is \nopened when the fluid pressure exceeds the critical fissure-\nopening pressure, just as in PD L. As the secondary fractures \ndilate they create a storage volume for fluid which is taken \nfrom the primary hydraulic fract ure. While the fracture storage \nvolume increases, leakoff can al so be accelerated so PDL and \nstorage are aspects of the same coupled mechanism of fissure dilation. The relative magnitude of the enhanced leakoff and \nstorage mechanisms determines whether the G-function \nderivatives show PDL or storage. Numerical modeling studies indicate that the storage mech anism can easily dominate even \nlarge PDL."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32c3a-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "large PDL. \nAt shut-in the secondary fractures will close before the \nprimary fracture because they ar e held open against a stress \nhigher than the minimum in-situ horizontal stress. As they close fluid will be expelled from the transverse storage volume \nback into the main fracture d ecreasing the normal rate of \npressure decline and, in effect , supporting the observed shut-in \npressure by re-injection of stor ed fluid. Accelerated leakoff \ncan still occur at the same time but if the storage and \nexpulsion mechanism exceeds the enhanced leakoff rate then \nthe only signature observed during falloff will be storage. In \nmany cases a period of linear, constant area, constant matrix \npermeability dominated leakoff will occur after the end of storage.  \n \nG-Function Analysis with Storage \nThe characteristic G-function derivative signature is a \n\u201cbelly\u201d below the straight line through the origin and tangent \nto the semilog derivative of p\nw vs. G at the point of fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32cb2-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "w vs. G at the point of fracture \nclosure. Figure 20 shows an example of slight to moderate \nstorage. In Figure 20 fracture closure, indicated by the same \ndeparture of the tangent line from the semilog derivative, \noccurs just after the end of the storage effect.  \n  \n5 10 15 20\nG(Time)750080008500900095001000010500\n0200400600800100012001400160018002000\n1\nP vs. G\nGdP/dG vs. G\ndP/dG vs. GFracture ClosurePressure\nDerivatives\n \nFigure 20: Storage G-function plot  \nSqrt(t) Analysis with Storage or Height Recession \nThe sqrt( t) plot (Figure 21) shows a clear indication of \nclosure based on both the first-derivative inflection point and \nthe semilog derivative curve. Pi cking closure in the case of \nstorage is not generally a problem. \n \n1/24/200702:00 04:00 06:00 08:00\n1/24/200710:00\nTime750080008500900095001000010500\n0100200300400\n1Pressure\nDerivativesP vs. \u221at\nxdP/dxvs. \u221at\ndP/dxvs. \u221atFracture Closure\n \nFigure 21: Storage sqrt(t) plot \n \nThe storage model, whether cau sed by height recession or"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32d2a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "The storage model, whether cau sed by height recession or \ntransverse fractures, requires that a larger volume of fluid \nmust be leaked-off to reach fr acture closure than is expected \nfor a single planar constant-heigh t fracture. In either case the \ntime to reach fracture closure is delayed by the excess fluid \nvolume that must be lost. Any estimation of reservoir \npermeability will give an incorrect result if the uncorrected \nclosure time (either G c or time-to closure in minutes, tc) is \nused. The observed closure time must be corrected by \nmultiplying by the storage ratio, rp. The magnitude of rp can \nbe determined by taking the ratio of the area under the \nG-function semilog derivative up to the closure time, divided \nby the area of the right-tria ngle formed by the tangent line \nthrough the origin at closure. For normal leakoff and PDL the \nvalue of r p is set to 1.0 even though the ratio of the areas will \nbe greater than 1 for the PDL case. It is possible that the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32dac-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "be greater than 1 for the PDL case. It is possible that the \nclosure time for PDL leakoff is proportional to the composite \nsystem permeability including both the matrix and fractures. \nFor severe cases of storage rp can be as low as 0.5 or less.  \n \nLog-Log Pressure Derivative with Storage \nFigure 22 shows the log-log plot of pressure difference and \nsemilog derivative for the storage case. Prior to closure, and \nwhile transverse storage is dominant, the semilog derivative \napproaches a unit slope, with the pressure difference curve \nnearly parallel. In some cases the two curves lie together on a \nsingle unit-slope line. In this case the curves are separated \nslightly and the slope is not exactly 1.0. After closure the \nreservoir transient signature is defined as in the previously \npresented cases. All fracture stor age effects are eliminated and \nthe reservoir pseudolinear flow period is shown by a -1/2 \nslope with a pseudoradial flow period indicated by a -1 slope \nof the semi-log derivative."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32e2e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 7, "chroma:document": "of the semi-log derivative.  \n Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32eba-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 8, "chroma:document": "SPE 107877  9 \n2 3 4 56789 2 3 4 56789 2 3 4 56789 2 3 4 560.1 1 10 100\nTime (0 = 9.416667)234567892345678923\n101001000\n (m = 0.5)  (m = -0.5)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative\u2206P vs. \u2206t\n\u2206td\u2206P/d\u2206t vs. \u2206t\n \nFigure 22: Storage log-log plot \n \n \nConclusions \nThe use of pre-frac injection/falloff diagnostic tests has \nbecome commonplace. Many impor tant decisions regarding \nfracture treatment designs and expectations of post-frac production are based on the results of these tests. In too many cases individual diagnostic plots and analysis techniques are \nmisapplied, leading to incorrect interpretations. The analyses \npresented here lead to the following conclusions: \n \n1. With consistent application of all available pressure decline diagnostics, a single unambiguous determination of fracture closure time and pressure can be made.  \n2. A single, unique closure event can be identified on all diagnostic plots.  \n3. The conventional analysis of the sqrt( t) plot, using the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32f46-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 8, "chroma:document": "3. The conventional analysis of the sqrt( t) plot, using the \ninflection point identified by the first derivative, gives incorrect indications of closure for cases of PDL and tip \nextension and should not be relied upon. \n4. A modified sqrt( t) analysis, using the semilog derivative, \nis equivalent to the G-function analysis and helps avoid incorrect closure picks in cases of PDL and tip extension. \n5. Flow regimes can be identified using the semilog pressure \nderivative on the log-log plot of \u2206p\nwf \u2212 \u2206t during the shut-\nin period following the fracture injection test. \n6. As in conventional transient test analysis, a pseudolinear \nflow period is identified by parallel \u00bd slope lines, \nseparated by 2x, on the log-log \u2206pwf \u2212 \u2206t plot up until \nfracture closure.  \n7. Bilinear flow can be identified by parallel \u00bc slope lines \nseparated by 4x on the log-log \u2206pwf \u2212 \u2206t plot prior to \nfracture closure. \n8. After closure the pseudolinear reservoir flow period is"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae32fd2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 8, "chroma:document": "8. After closure the pseudolinear reservoir flow period is \nidentified by a -1/2 slope of the semilog derivative of the \npressure difference on the log-log \u2206pwf \u2212 \u2206t plot, and a \n\u22123/2 slope of the first derivative of the pressure difference \nwith shut-in time on the same plot. \n9. Pseudoradial flow is identified by a -1 slope of the \nsemilog derivative on the log-log plot. 10. When a stable pseudolinear flow period exists, the after-\nclosure Cartesian plot of the linear flow function can be \nused to estimate reservoir pressure. \n11. When a pseudoradial flow period exists, both the \nconventional Horner analysis  and after-closure radial-\nflow analysis can be used to determine reservoir \ntransmissibility and pore pressure. \n \nAcknowledgements \nThe authors would like to thank Kumar Ramurthy, \nHalliburton, Mike Conway, Stim-Lab, and Stuart Cox, \nMarathon, for their discussion and contribution to the procedures described. Sincere thanks are also due to the many"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33054-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 8, "chroma:document": "operators whose diligence in pre-frac testing has allowed these \ndiagnostic analysis procedures to be developed and tested. \n \nNomenclature \nAf = fracture area, L2, ft2 \nB = formation volume factor, L3/L3, RB/STB \nct = total compressibility, Lt2/m, psi-1 \nCac = after-closure storage, L4t2/m, bbl/psi \nCbl = bilinear flow constant, m/Lt5/4, psi\u0387hr3/4 \nCpl = pseudolinear flow constant, m/Lt3/2, psi\u0387hr1/2  \nCpr = pseudoradial flow constant, m/Lt, psi \u0387hr  \nCfbc = before-closure fracture storage, L4t2/m, bbl/psi \nFL = linear flow time function, dimensionless  \nFR = radial flow time function, dimensionless  \ng = loss-volume function, dimensionless  \nG = G-function, dimensionless  \nh = height, L, ft  \nk = permeability, L2, md  \nLf = fracture half-length, L, ft  \nmH = slope of data on Horner plot, m/Lt2, psia  \nmL = slope of data on pseudolinear flow graph, m/Lt2, psia  \nmR = slope of data on pseudoradial flow graph, m/Lt2, psia  \np = pressure, m/Lt2, psia"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae330e0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 8, "chroma:document": "p = pressure, m/Lt2, psia  \npwf = fracture pressure measured at wellbore, m/Lt2, psia  \nq = flow rate, L3/t, bbl/D \nQt = total injection volume, L3, bbl \nrp = storage ratio, dimensionless \nSf = fracture stiffness, m/L2t2, psi/ft \nt = time, hr \nta = adjusted pseudotime, hr \n \n \nGreek \n\u03b2 = constant, dimensionless \n\u2206 = difference, dimensionless \n\u03c7 = constant, dimensionless \n\u00b5 = viscosity, m/Lt, cp \n\u03c6 = porosity, dimensionless \n \nSubscripts \na = adjusted \nc = closure \nD = dimensionless \ne = end of injection \nf = filtrate \np = pumping \n0 = end of injection  \nw = wellbore \nz = process zone \n \n Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33162-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "10  SPE 107877 \nReferences \n \n1. Nolte, K. G.: \u201cDetermination of Fracture Parameters from \nFracturing Pressure Decline, pa per SPE 3841, presented at the \nAnnual Technical Conference a nd Exhibition, Las Vegas, NV, \nSept. 23-26, 1979. \n2. Castillo, J. L.: \u201cModified Fracture Pressure Decline Analysis \nIncluding Pressure-Dependent Leakoff,\u201d paper SPE 16417, \npresented at the SPE/DOE Low Permeability Reservoirs Joint \nSymposium, Denver, CO, May 18-19, 1987. \n3. Barree, R. D., and Mukherjee, H. : \u201cDetermination of Pressure \nDependent Leakoff and Its Effect  on Fracture Geometry,\u201d paper \nSPE 36424, presented at the 71st Technical Conference and \nExhibition, Denver, CO, Oct. 6-9, 1996. \n4. Barree, R.D.: \"Applications of Pre-Frac Injection/Falloff Tests in \nFissured Reservoirs\u2014Field Examples,\" paper SPE 39932 \npresented at the SPE Rocky Mountain Regional/Low-\nPermeability Reservoirs Sympos ium, Denver, Apr. 5-8, 1998."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae331e4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "Permeability Reservoirs Sympos ium, Denver, Apr. 5-8, 1998. \n5. Talley, G. R., Swindell, T. M., Waters, G. A. and Nolte, K. G.: \u201cField Application of After- Closure Analysis of Fracture \nCalibration Tests,\u201d paper SPE 5 2220, presented at the 1999 SPE \nMid-Continent Operations Symposium, Oklahoma City, OK, \nMarch 28\u201331, 1999. \n6. Cinco-Ley, H., and Samaniego- V., F.: \u201cTransient Pressure \nAnalysis for Fractured Wells,\u201d JPT (September 1981) 1749. \n7. Settari, A.: \u201cCoupled Fracture  and Reservoir Modeling,\u201d \npresented at the Workshop on Three Dimensional and Advanced \nHydraulic Fracture Modeling, held  in conjunction with the Fourth \nNorth American Rock Mechanics Symposium, July 29, 2000, Seattle, WA. \n8. Craig, D. P. and Bl asingame, T. A.: \u201cApplication of a New \nFracture-Injection/Falloff Mode l Accounting for Propagating, \nDilated, and Closing Hydrau lic Fractures,\u201d paper SPE 1005778 \npresented at the SPE Gas Technology Symposium, Calgary, \nAlberta, Canada, May, 15-17, 2006."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3325c-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "Alberta, Canada, May, 15-17, 2006. \n9.  Hagoort, J.: \"Waterflood-Induced Hydraulic Facturing,\" PhD \nThesis, Delft Technical University, 1981. \n10.  Koning, E.J.L. and Niko, H.: \"Fractured Water-Injection Wells: \nA Pressure Falloff Test for Dete rmining Fracture Dimensions,\" \npaper SPE 14458 presented at the 1985 Annual Technical Conference and Exhibition of the Society of Petroleum \nEngineers, Las Vegas, NV,  September, 22-25, 1985. \n11.  Cinco-Ley, H., Kuchuk, F., Ay oub, J., Samaniego-V, F., and \nAyestaran, L.: \"Analysis of Pr essure Tests Through the Use of \nInstantaneous Source Response Concepts,\" paper SPE 15476 presented at the 61\nst Annual Technical Conference and Exhibition \nof the Society of Petroleum Engineers, New Orleans, LA, \nOctober, 5-8, 1986. \n  \nAppendix A - Definition of diagnostic functions \n \nThe G-Function \nThe G-function is a representati on of the elapsed time after \nshut-in normalized to the duration of fracture extension."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae332de-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "shut-in normalized to the duration of fracture extension. \nCorrections are made for the superposition of variable leakoff times while the fracture is growing. The form of the \nG-function used in this paper assumes high fluid efficiency in \nlow-permeability formations. Under that assumption the surface area of the fracture is a ssumed to vary linearly with \ntime during fracture propagation. The dimensionless pumping \ntime used in the G-function is defined as: \n( )/Dp ptt t t\u2206= \u2212 . ..........................................................(A-l) The elapsed total time from the start of fracture initiation \n(not start of pumping) is t and the total pumping time (elapsed \ntime from fracture initiation to shut-in) in consistent time units \nis tP. For the assumption of low leakoff the dimensionless time \n(\u2206tD) is used to compute an intermediate function: \n() ()[ ]5.1 5.1134\nD D D t t tg \u2206\u2212 \u2206+ = \u2206 . ............................ (A-2) \nThe G-function used in the diagnostic plots is derived from"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33360-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "The G-function used in the diagnostic plots is derived from \nthe intermediate function as follows: \n() () 04\nDD Gt gt g\u03c0\u2206 =\u23a1\u2206 \u2212 \u23a4\u23a3 \u23a6, ....................................... (A-3) \nwhere g0 is the dimensionless loss-volume function at shut-in \n(t = tp or \u2206tD = 0). All derivatives are calculated using a central \ndifference function of pressure and G-function (normalized \nshut-in time). \n \nAfter-Closure Analysis and Flow Regime Identification \nAfter-closure pressure decline analysis requires the \nidentification of fully-developed reservoir pseudolinear and \npseudoradial transient flow regimes. The flow regimes can be identified by characteristic slopes on a log-log plot of \nobserved falloff pressure minus reservoir pressure, ( p\nw(t) \u2212 pi), \nversus the square of the linear-flow time function (F L2) and the \nsemilog derivative, (X*dY/dX), of the pressure difference \ncurve.5 It is important to note that the guess of reservoir \npressure, pi, used in construction of the flow regime plot"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae333d8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "pressure, pi, used in construction of the flow regime plot \nseverely impacts the slope and magnitude of the pressure \ndifference curve. The pressure derivative, because of the \ndifference function used to generate it, is not affected by the \ninitial guess of reservoir pressure.  \nThe linear-flow time function is defined by: \n()cc\nc L ttforttttF \u2265 =\u22121sin2,\u03c0. ......................... (A-4)  \nThe linear-flow function al so requires an accurate \ndetermination of the time requ ired after shut-in to reach \nfracture closure, tc. In the pseudolinear flow period the slope \nof the derivative curve on the log-log plot should be \u00bd. For the \ncorrect estimate of reservoir pore pressure, the pressure \ndifference curve should also have a slope of \u00bd and should be \nexactly twice the magnitude of the derivative. If a stable \npseudolinear flow period is identified then a Cartesian plot of \nobserved pressure during the falloff, pw(t), versus F L should \nyield a straight line with intercept equal to the reservoir pore"}


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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 9, "chroma:document": "pressure, pi, and with a slope of mL.  \n() (),wi L L cptp m F t t\u2212= . ............................................ (A-5)  \nIf a pseudoradial flow period exists, the slope of the \nderivative and correct pressure difference curves on the log-\nlog flow regime plot should both be 1.0 and the two curves \nshould coincide. In the pseudoradial flow period, a Cartesian plot of pressure versus F\nR should also yield a straight line with \nintercept equal to pi and slope of mR.  \n Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae334dc-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "SPE 107877  11 \n() (),wi R R cptp m F t t\u2212= ............................................. (A-6) \n \nIn these equations tc is the time to fracture closure with \ntime zero set as the beginning of fracture extension, pi is the \ninitial reservoir pressure, and mR is the Cartesian slope of the \n\u201ccorrect\u201d straight line. The radial-flow function (F R) is given \nby5 \n() 6.116, 1ln41,2\u2245 =\u239f\u239f\n\u23a0\u239e\n\u239c\u239c\n\u239d\u239b\n\u2212+ =\u03c0\u03c7\u03c7\ncc\nc RtttttF . ......... (A-7)  \nIn the properly identified ps eudoradial flow period the \nreservoir far-field transmissibility can be determined from the \nslope, fracture closure time, and volume injected during the \ntest. \n \n251,000t\nRcQ kh\nmt \u00b5= . ...................................................... (A-8)  \nIn Equation A-8 the permeability, k, is in md, net pay \nthickness, h, is in feet, far-field mobile fluid viscosity,  \u00b5, \uf020in \ncp, tc in minutes, and the volume injected during the test, Qt, is \nin bbls, and the slope, mR, is in psi.  \n \nHorner analysis"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3355e-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "in bbls, and the slope, mR, is in psi.  \n \nHorner analysis \nThe conventional Horner  analysis uses a Cartesian plot of \nobserved pressure versus Horner time, ( tp + \u2206t)/\u2206t, with all \ntimes in consistent units. The fracture propagation time is tp \nand the elapsed shut-in time is \u2206t. As shut-in time approaches \ninfinity the Horner time function approaches 1. A straight-line \nextrapolation of the Horner plot to the intercept at a Horner \ntime of 1.0 gives an estimate of reservoir pressure. The slope \nof the correct straight-line extrapolation, mH, can be used to \nestimate reservoir transmissibility: \n \n()\nHmq kh 14406.162=\u00b5.....................................................(A-9) \nThe flow rate in Equation A-9 is assumed to be in barrels \nper minute and is the average rate for the time the fracture was extending. In both Equations A-8 and A-9, the viscosity is the far-field mobile fluid viscosity. The propagation of the \ntransient in pseudoradial flow oc curs at a great distance from"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae335d6-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "the fracture and is not affected by the injected fluid viscosity. \nThe major problem with the Horner analysis is that the \nresults are only valid if the da ta used to extrapolate the \napparent straight line are actually in fully developed \npseudoradial flow. There is no way to determine the validity \nof the Horner analysis or to determine the flow regime within \nthe Horner plot itself. \n \nG-function permeability estimate \nAn empirical function to  approximate formation \npermeability has been derived from numerous numerical \nsimulations of fracture closure. The correlation is based on the \nobserved G-function time at fracture closure: 1.960.0086 0.01\n0.038f z\ncp\ntp\nk\nGE rc\u00b5\n\u03c6=\n\u239b\u239e\n\u239c\u239f\u239d\u23a0. ...................................... (A-10) \nThe rate of fluid loss from the fracture before closure is \ndominated by the mobility of the injected fluid instead of the \nfar-field viscosity. The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33658-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "fluid at leakoff temperature and the relative permeability to the \nleakoff fluid. The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid \nmobility. As a general rule, an assumed injected fluid viscosity \nof 1.0 cp is used to incorp orate the effects of reservoir \ntemperature and relative permeability in the invaded zone. \n Note that the correlation gives permeability and not kh. \nThe time to closure is related to fracture volume versus created area so both the fracture height and length do not \nappear in the equation. The process zone stress, P\nz, is the net \nfracture extension pressure above closure pressure, pc, or \npz = pISIP \u2212 pc. The net extension pressure and Young\u2019s \nModulus provide a relationship between facture volume \n(width) and surface area during pumping. \nThe other parameters in Equation A-10 are defined, except \nthe storage ratio, rp. This parameter represents the amount of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae336d0-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "the storage ratio, rp. This parameter represents the amount of \nexcess fluid that must be leaked -off to reach fracture closure \nwhen the fracture geometry deviates from the normally \nassumed constant-height planar  fracture. The storage ratio, rp, \nis the ratio of the area under the G-function semilog derivative \nup until closure divided by the area of the triangle defined by \nthe straight-line (normal leakoff) tangent to the semilog \nderivative at closure.  For nor mal matrix leakoff and PDL, rp \nis therefore 1.0. For the transverse storage and height \nrecession signature, rp is some value less than 1.0. The \nobserved G-function closure time (G c) is always delayed by \nthe excess fluid volume in th e fracture for either height \nrecession or transverse storage.  Likewise the apparent time to \nclosure used in the after-closur e analysis will be delayed and \nwill cause errors in the reservoir transmissibility estimate. The \nclosure time should be corrected by multiplying by rp for all"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33798-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 10, "chroma:document": "closure time should be corrected by multiplying by rp for all \npermeability and transmissibility estimates.  \n \n \nAppendix B - Analytical Solutions and the Log-Log \nDiagnostic Graph \n \nLinear and Bilinear Flow Before-Closure \nHagoort9 developed a before-closure analytical solution \naccounting for both storage and lin ear formation flow. Koning \nand Niko10 write the solution as \n() ( )22\n12DtDf b c DDpt C ee r f c t wD\u03b2\u03c0\u03c0 \u03b2\u2206 \u239b\u239e=\u2206\u2212 \u2212 \u239c\u239f \u2206\u239d\u23a0, ................ (B-1) \nwhere \n( ) () ( )()141.2we we\nDptp t t khptwDqB\u00b5\u2212 +\u2206\u2206=, ..................................... (B-2) \n20.0002637\nD\ntfktt\ncL\u03c6\u00b5\u2206\u2206=, ............................................................ (B-3) Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33810-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "12  SPE 107877 \n2\nfbcDC\u03b2\u03c0=,  ..................................................................... (B-4) \nand \n225.615 0.8936\n2fbc fbc\nfbcD\ntf tfCC\nC\nch L ch L\u03c0\u03c6 \u03c6==. ............................................ (B-5) \nHere, Cfbc is the before-closure fracture storage constant, \nwhich is defined as8 \n2\n5.615f\nfbc\nfA\nCS=, ................................................................. (B-6) \nwhere Af  is the fracture area (one wing) and  Sf is the fracture \nstiffness. \nHagoort's solution predicts that the before-closure falloff is \na combination of fracture storag e and linear flow. As fracture \nstorage becomes small, CfbcD\u21920, linear flow will dominate the \nbefore-closure pressure falloff. In most \"hard rock\" \nenvironments, CfbcD is small, and for a fracture-\ninjection/falloff sequence with \u2206t >> te and long closure times,  \n() ( )we weptp t t t\u2212+ \u2206 \u221d , ...................................................... (B-7) \nwhere t = te + \u2206t and pw(te) is the instantaneous shut-in"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33888-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "where t = te + \u2206t and pw(te) is the instantaneous shut-in \npressure. Define a fracture-pressure difference as \n() ( )w f we wep pt pt t \u2206= \u2212 + \u2206 , ................................................... (B-8) \nand a log-log graph of \u2206pwf vs. t will exhibit a \u00bd slope during \nlinear flow before closure. An  analytical before-closure \nbilinear flow solution accounti ng for fracture tip extension \nduring shut-in does not exist, but field data suggest that during \nbefore-closure bilinear flow \n14()wfp t \u2206\u221d \u2206 , .................................................................... (B-9) \nand a log-log graph of \u2206pwf vs. t will exhibit a \u00bc slope during \nbilinear flow before closure. \n \nPseudoradial, Pseudolinear, and Bilinear Flow After-\nClosure \nCraig & Blasingame8 developed an analytical solution for a \nfracture-injection/falloff sequ ence with a propagating and \nclosing hydraulic fracture, and they derived the \"complete\" \nafter-closure impulse solutions accounting for fracture storage."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33900-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "The after-closure impulse solution for pseudoradial flow is written as  \n() 0141.2(24) 1(0) ( ) ()2ti we i\neQp C p p pt t p wsD ackh t t\u00b5+\u2212 +\u2206 \u2212 =+\u2206. ..... (B-10) \nFor after-closure pseudolinear flow, \n0141.2(24) 0.0002637 1()(0) ( ) 2t\nwe ii fQ\npt t ppC p p hL wsD ac\u03c0 \u239b\u239e+\u2206 \u2212 = \u239c\u239f+\u2212\u239d\u23a0\n121\nteckt t\u00b5\n\u03c6\u239b\u239e\u00d7\u239c\u239f+\u2206 \u239d\u23a0, ........................................... (B-11) \nand for after-closure bilinear flow, \n1434\n0141.2(24)(0.6125)(0.0002637)()()t\nwe i\ni ffQ\npt t ppp C kw ac\u00b5\u239b\u239e+\u2206 \u2212 = \u239c\u239f+\u2212\u239d\u23a0  14 3411\nteck t t\u00b5\u239b\u239e \u239b \u239e\u00d7\u239c\u239f \u239c \u239f+\u2206 \u239d\u23a0 \u239d \u23a0. ..................................... (B-12) \nFor convenience, define a constant pseudoradial flow term as \n() 0141.2(24)(0) ( )\n2ti prQp C p p C wsD ackh\u00b5+\u2212 =. ........................... (B-13) \nFor pseudolinear flow, a constant term is defined as \n12\n0141.2(24) 0.0002637 1\n(0) ( ) 2t\nplt i fQ\nCck pC p p hL wsD ac\u00b5 \u03c0\n\u03c6\u239b\u239e \u239b\u239e= \u239c\u239f \u239c\u239f+\u2212 \u239d\u23a0 \u239d\u23a0. (B-14) \nSimilarly, for bilinear flow a constant term is written as \n 14 1434\n01 141.2(24)(0.6125)(0.0002637)\n()t\nblt i ffQ"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33978-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "14 1434\n01 141.2(24)(0.6125)(0.0002637)\n()t\nblt i ffQ\nCck pp C kw ac\u00b5\u00b5\u239b\u239e \u239b\u239e= \u239c\u239f \u239c\u239f+\u2212 \u239d\u23a0 \u239d\u23a0. .. (B-15) \nWith the new definition, and defining a reservoir-pressure \ndifference, \u2206pw = pw(t) \u2212 pi, the pseudoradial-flow impulse \nsolution can be written as \n1wp rpC t\u2212\u2206= .  ................................................................... (B-16) \nSimilarly, the linear-flow impulse solution can be written as \n1/2wp lpC t\u2212\u2206= , .................................................................. (B-17) \nand the bilinear-flow impulse solution can be written as \n3/4wb lpC t\u2212\u2206= . ................................................................... (B-18) \nThe derivatives with respect to t are written for after-closure \npseudoradial flow as,  \n2\n()wprpCtt\u2212 \u2202\u2206\u2212=\u2202, ................................................................ (B-19) \nfor after-closure pseudolinear flow as, \n3/2\n() 2pl wC ptt\u2212 \u2202\u2206\u2212=\u2202, ............................................................. (B-20) \nand for after-closure bilinear flow as, \n7/4 3"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae339fa-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "and for after-closure bilinear flow as, \n7/4 3\n() 4wb lpCtt\u2212 \u2202\u2206\u2212=\u2202. ............................................................ (B-21) \nThe semilog derivatives and the impulse derivatives11 are \ncalculated by multiplying the after-closure derivatives by t and \nt2, respectively. Table B-1 show s the derivative terms for each \nafter-closure flow regime. \n \nLog-Log Diagnostic Graph \nBefore-closure linear or bilinear flow are identified by the \nrelationship between the fracture-pressure difference, \n\u2206pwf, and total time, t, but the after-closure flow regimes are \nidentified by the relationship be tween the reservoir-pressure \ndifference, \u2206pw, and t, which requires knowing the initial \nreservoir pressure, pi. However, by noting that \nwf wp p\ntt\u2202\u2206 \u2202\u2206=\u2212\u2202 \u2202, ................................................................ (B-22) \na log-log diagnostic graph can be prepared that is independent of initial reservoir pressure but can be used to identify before- \nand after-closure flow regimes. Table B-2 shows the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33a72-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 11, "chroma:document": "and after-closure flow regimes. Table B-2 shows the \ncharacteristic slopes of each flow regime for log-log graphs of \n\u2206p\nwf vs. t, wfpt\u2202\u2206\u2202 vs. t, wf tp t\u2202\u2206 \u2202 vs. t, and Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33af4-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 12, "chroma:document": "SPE 107877  13 \n2wf tp t\u2202\u2206 \u2202 vs. t .  \nIn gas reservoirs, log-log diagnostic graphs can be prepared by \nplotting \u2206pawf vs. ta, awf apt\u2202\u2206 \u2202 vs. ta, aa w fatp t\u2202\u2206 \u2202 vs. ta, \nand 2aa w f atp t\u2202\u2206 \u2202 vs. ta where adjusted pseudopressure is \ndefined as \n0pzp d ppap zi\u00b5\n\u00b5\u2320\n\u23ae\n\u23ae\u2321\u239b\u239e=\u239c\u239f\u239d\u23a0, ........................................................... (B-23) \nand adjusted pseudotime is defined as ()\n0tdttcat ict\u00b5\u00b5\u2320\n\u23ae\n\u23ae\n\u2321=. .............................................................. (B-24) \nSince adjusted pseudopressure is a function of initial \nreservoir pressure, and if initial reservoir pressure is unknown, \nit's often helpful to complete the analysis in terms of pressure \nand time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive \ngraphs plotted in terms of adjusted pseudovariables. \n \n \n  \n \n \nTable B-1. After-closure impulse solutions and derivatives. \nFlow Regime Impulse Solution Derivative Semilog \nDerivative Impulse \nDerivative"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33b76-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/spe-107877-ms.pdf", "page": 12, "chroma:document": "Derivative Impulse \nDerivative \nBilinear 3/4wb lpC t\u2212\u2206=  7/4 3\n4wb lpCtt\u2212 \u2202\u2206\u2212=\u2202 3/4 3\n4wb lpCttt\u2212 \u2202\u2206\u2212=\u2202 21 / 4 3\n4wb lpCttt\u2202\u2206\u2212=\u2202 \nPseudolinear 1/2wp lpC t\u2212\u2206=  3/2\n2pl wC ptt\u2212 \u2202\u2206\u2212=\u2202 1/2\n2pl wC pttt\u2212 \u2202\u2206\u2212=\u2202 21 / 2\n2pl wC pttt\u2202\u2206\u2212=\u2202 \nPseudoradial 1wp rpC t\u2212\u2206=  2 wprpCtt\u2212 \u2202\u2206\u2212=\u2202 1 wprptC tt\u2212 \u2202\u2206\u2212=\u2202 2 wprptCt\u2202\u2206\u2212=\u2202 \n \n \nTable B-2. Log-log graph characteristic slopes. \nBefore Closure After Closure Log-Log Graph \nBilinear Linear Bilinear Pseudolinear Pseudoradial \n vs. \n vs. wf\nawf apt\npt\u2206\n\u2206 1/4 1/2 \u23af \u23af \u23af \n vs. \n vs. wf\nawf a apt t\npt t\u2202\u2206 \u2202\n\u2202\u2206 \u2202 \u22123/4 \u22121/2 \u22127/4 \u22123/2 \u22122 \n vs. \n vs. wf\naa w fa atp t t\ntp t t\u2202\u2206 \u2202\n\u2202\u2206 \u2202 1/4 1/2 \u22123/4 \u22121/2 \u22121 \n2\n2 vs. \n vs. wf\naa w fa atp t t\ntp t t\u2202\u2206 \u2202\n\u2202\u2206 \u2202 5/4 3/2 1/4 1/2 0 \n  Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33c02-857d-11ee-bb1c-0242ac1c000c
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{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 0, "chroma:document": "CHAPTER 8\nFracture diagnostic testing\nAli Rezaei1, Mohamed Y. Soliman2and Birol Dindoruk2\n1Midstream Production Systems, Houston, TX, United States\n2Cullen College of Engineering, Petroleum Engineering, University of Houston, Houston, TX, United States\n8.1 Overview\nThis chapter summarizes several conventional and recent methods to analyze hydraulic\nfracture testing in vertical and horizontal wells. These tests are conducted for two pri-\nmary purposes: (1) to understand the reservoir properties such as minimum horizontal\nstress and permeability; (2) and to optimize the hydraulic fracturing parameters.\nFracture diagnostic techniques are critical preexecution steps toward optimum fracture\ndesign. Using these techniques, one can estimate key parameters such as the target\nzone stress regime, fracture confinement height, number of stages, treatment size,\nproppant type and amount, and optimum well direction.\n8.2 Testing types\nIn general, three different fracture diagnostics groups exist, namely direct far-field"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33c7a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 0, "chroma:document": "techniques, direct near-wellbore techniques, and indirect fracture diagnostics ( Cipolla\n& Wright 2000 ). The direct far-field techniques are suited to give the big picture\nabout fracture growth and are conducted in a separate well. The two commonly used\ntechniques for this type of fracture diagnostics are tiltmeter fracture mapping and\nmicroseismic fracture mapping. On the contrary, the second group of fracture diagnos-\ntics techniques is implemented in the same well and gives information about near-\nwellbore fracture parameters such as height and width and proppant placement.\nFinally, the most commonly used fracture diagnostics group is indirect fracture diag-\nnostics. The techniques used in this group can provide estimates about fracture con-\nductivity, dimension, and stress. This group includes fracture modeling, pressure\ntransient testing, and production data analysis ( Cipolla & Wright 2000 ). The pressure\ntransient testing (sometimes accompanied by rate transient testing) includes step"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33d10-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 0, "chroma:document": "rate test, pump in-shut in, and pump in/flow back ( Soliman, Craig, Bartko, Rahim, &\nAdams, 2005 ). These tests are most widely used because of the abundance in the\nrecorded data. Pump-in shut-in tests may be referred to as MiniFrac, or diagnostic\nfracture injection test (DFIT). There are, however, some differences between the two\ntests. Mini-Frac tests were conducted using the fluid to be used in the main fracturing\n217Unconventional Shale Gas Development\nDOI: https://doi.org/10.1016/B978-0-323-90185-7.00016-9r2022 Elsevier Inc.\nAll rights reserved."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33d92-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 1, "chroma:document": "treatment, except no proppant was added. The goal of the test was to find the leak-off\ncoefficient and efficiency of the treatment. The treatment design was consequently\nadjusted if necessary. DFIT, on the other ha nd, is conducted to determine reservoir and\nfracture properties. Other approaches such as data-based analysis and signal processing\ntechniques such as wavelet tra nsform have been developed to p erform fracture diagnostics\nanalysis in recent years. Moreover, other te chniques such as step-rates techniques have\nbeen developed. Those step rate techniques will be briefly discussed. However, the bulk\nof the chapter will be devoted to pump-in shut-in test technology.\n8.3 Step-rate test\nStep-rate test (SRT) is commonly used for estimating the pressure at which the hydrau-\nlic fracture starts to propagate (i.e., fracture propagation pressure or FPP). FPP is the\npressure at which a fracture will initiate in an unfractured well, or the existing fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33e14-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 1, "chroma:document": "in a fractured well starts to propagate. The test is primarily used for identifying the\nmaximum safe injection pressure without fracturing the reservoir rock ( Singh, Agarwal,\n& Krase, 1987 ). Although the SRT has been practiced in the industry for many years,\nthe first publication on the subject seems to be by Felsenthal (1974) . There are two\ntypes of SRT, step-up and step-down tests. In the following, a review of the two types\nis discussed. Also, it has been reported by Savitski and Dudley (2011) that the step rate\ntest is not efficient in low permeability reservoirs for identifying the closure pressure.\n8.3.1 Step-up test\nThe step-up test is performed on a well with preexisting hydraulic fracture and is used\nto estimate the fracture opening pressure. Injection starts at a low rate that is not suffi-\ncient to reopen the existing fracture. The injection is increased in a stepwise fashion\nuntil the fracture opens. At each stage, the injection continues until the pressure stabi-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33e96-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 1, "chroma:document": "lizes. It is expected that a smaller change in pressure occurs after the fracture reopens,\nresulting from the higher injectivity. Plotting the stabilized pressure versus rate yields the\npoint of change in slope, indicating the reopening pressure as illustrated in Fig. 8.1 .T h e\nfigure shows that as the fluid loss decreases (lower formation permeability), the difference\nbetween the before closure (BC) and after closure (AC) slopes get smaller, resulting in a\nharder fracture opening pressure identification. More discussions and field examples on\nthe step-up test may be found in Tan, McGowen, Lee, and Soliman (1988) .\n8.3.2 Step-down test\nIn contrary to the step up test, the step-down test is also performed by injection at\na high enough rate to propagate the fracture. As the injection rate is reduced, how-\never, it stays above the fracture opening pressure. Step down test was proposed by218 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33f18-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 2, "chroma:document": "Cleary et al. (1993) . The goal of the test is to estimate the friction- and perforation-\ndominated pressure during injection. It is hypothesized that the net pressure will be a\nfunction of the rate as expressed below. The exponent B is a strong indication of the\nstatus of the fracture, as given in Tables 8.1, 8.2, 8.3, 8.4 and 8.5 . Depending on the\nresults of the test, remedial action may be taken.\n\u0394pnet5A3qB\nlog\u0394pnet5logA1Blogq\nwhere\u0394Pnet,q, and Aand Bare net pressure, injection rate, and test constants,\nrespectively.\nHyden and Stegent (1996) presented field cases using the step-down test for find-\ning the near-wellbore restrictions.\nFigure 8.1 A typical pressure-rate cross plot of the step-up test. Courtesy Advantek.\nTable 8.1 Indication of potential problems associated with slope Bin the step-down test.\nSlope, B Potential problem\n0.25 single fracture exists\n0.5/C01.0 Presence of a tortuous path\n2.0 Insufficient perforation219 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae33f90-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 3, "chroma:document": "8.4 Pump in/shut-in test\nThe most commonly used pump-in/shut-in test is the fracture injection test. Fracture\ninjection fall-off test, also known as DFIT and minifrac test, has been a standardmethod to identify various formation properties such as closure pressure ( BminimumTable 8.2 A summary of the results for field example 1 and its comparison with G-function.\nG-function EDP\nClosure time (min) 27.47 27.95\nClosure pressure (psi) 5822 5790Closure point (#) 93 101Sampling rate 1 second\nTotal number of data points B1:2310\n3\nTable 8.3 A summary of the results for field example 2 and its comparison with G-function.\nG-function EDP\nClosure time (s) 822 761\nClosure pressure (psi) 26,756 28,756Closure point (#) 98 37Sampling rate 1 sTotal number of data points B4:6310\n3\nTable 8.4 A summary of the results for field example 3 and its comparison with G-function.\nG-function EDP\nClosure time (min) 13.6 13.5\nClosure pressure (psi) 2656 2702Closure point (#) 101 93Sampling rate 1 sTotal number of data points B10310\n3"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae343b4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 3, "chroma:document": "3\nTable 8.5 A summary of the results for field example 4 and its comparison with G-function.\nG-function EDP\nOur analysis Closure time (min) 45 46\nClosure pressure (psi) 13,539 13,521Closure point (#) 2509 2600\nSampling rate 1 s\nTotal number of data points 370 310\n3220 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34486-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 4, "chroma:document": "horizontal stress), formation permeability, and the leak-off type in the last 30 years.\nClosure pressure identification is critically important in this test because of two main\nreasons. First, it is typically considered as the minimum horizontal stress and affects\nthe subsequent hydraulic fracturing design as well as analysis of real-time fracture prop-\nagation. Second, closure pressure defines two separate classes of analysis, namely BC\nanalysis and AC analysis ( Soliman et al., 2005; Soliman, Miranda, & Wang, 2010 ).\nDFIT involves injecting water without proppant at a sufficient rate and pressure to\ncreate a fracture into a formation, followed by a shut-in period that may last for hours\nor even days depending on reservoir permeability. After starting the shut-in period,\nthe pressure will begin to fall-off due to the fracturing fluid leak-off that causes the\nfracture to close gradually, resulting from the far-field compressive stress. The closure"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34508-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 4, "chroma:document": "pressure may be determined by analyzing the decline of the recorded pressure during\nthe fall-off period.\nFig. 8.2 shows the recorded wellbore pressure during a typical minifrac test. Two\nperiods, BC (fracture dominated) and AC (reservoir dominated) are separated by a\nvertical red dashed line.\n8.4.1 Before closure analysis\nThe methods to analyze the recorded pressure fall-off data before the fracture closure\nmoment are referred to BC analysis. The main parameters that are estimated using\nthese techniques are closure pressure, leak-off coefficient, and instantaneous shut-in\npressure. The net pressure at the closure is typically considered as minimum horizontal\nstress. The most commonly used method in this category is the DFIT. Various models\nhave been proposed to analyze DFIT data and determine the fracture closure pressure.\nFigure 8.2 A Schematic of the DFIT test. After Pokalai et al. (2016) .221 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34580-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 5, "chroma:document": "Nolte (1979) introduced this technology. Further modifications of this technology\nwere introduced by further authors (e.g., Barree & Mukherjee 1996 ;Craig &\nBlasingame 2006 ;Marongiu-Porcu, Ehlig-Economides, & Economides, 2011 ;\nMohamed, Azmy, Sayed, Marongiu-Porcu, & Economides, 2011 ;Soliman, 1986 ).\nHowever, there is a significant disagreement in the calculated closure pressure from\nthese models in many cases. The primary source of error in the methods used for ana-\nlyzing DFIT is their underlying assumptions. Therefore, analyzing DFIT data using a\ndifferent methodology independent of premises such as fracture geometry is essential.\n8.4.1.1 G-function technique\nThe first and most widely used method for analyzing BC pressure fall-off was initially\npresented in Nolte \u2019s pioneering work ( Nolte, 1979, 1986 ). The technique, also\nknown as G-function, expanded by Barree and Mukherjee (1996) , is a tool for deter-\nmining closure pressure, leak-off coefficient, and fluid efficiency. This technique \u2019s cen-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34602-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 5, "chroma:document": "tral assumptions are that the leak-off is not pressure-dependent and fracture stiffness is\nconstant while the fracture is closing. Several modifications to the Nolte G-function\ntechnique have been proposed since then by addressing some of the limitations in the\nassumptions ( Castillo, 1987; Barree, Barree, & Craig, 2007; Barree & Mukherjee 1996;\nMukherjee, Larkin, & Kordziel, 1991; Barree et al., 2015 ). The primary purpose of\nthese modifications was to make the method more applicable to the new cases that\nwere encountered.\nMcClure, Blyton, Jung, and Sharma (2014) and McClure, Jung, Cramer, and\nSharma (2016) proposed a technique called the fracture compliance method. In this\ntechnique, the central assumption is that the created fracture does not reach a state of\ncomplete closure. Instead, the fracture retains a finite aperture after asperities come\ninto contact and result in an infinite fracture conductivity. Although this assumption"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3467a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 5, "chroma:document": "may be valid in some cases where the aperture of the created fracture in ultra-tight\nformations stays at \u03bcdrange, it gives contradictory results to the widely used tangent\nmethods in cases where reservoir permeability is less than one \u03bcD. Also, for such short\nfractures that are generated in DFIT, minimum to no asperities mismatch is more\nprobable, which contradicts the model \u2019s assumptions. McClure et al. (2019) further\nextended their method and recommended steps to analyze DFIT data from low per-\nmeability formations ranging from tens of nDto tens of \u03bcD.Wang and Sharma\n(2017) presented a variable compliance method based on averaging the closure point \u2019s\nG-time. Similar to the technique presented by McClure et al. (2014) andWang and\nSharma (2017) concluded that the G-function method underestimates the closure stress.\nIn another model, Liu and Ehlig-Economides (2015) and Liu, Ehlig-Economides, and\nSun (2016 ) extended the model by Marongiu-Porcu, Retnanto, Economides, and Ehlig-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae346fc-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 5, "chroma:document": "Economides (2014) to account for wellbore storage, tip extension, and pressure dependant\nleak-off, and multiple closures.222 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3477e-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 6, "chroma:document": "The definition of G-function is given below:\nG\u0394tD\u00f0\u00de54\n\u03c0g\u0394tD\u00f0\u00de2g0 \u00f0\u00de\ng\u0394tD\u00f0\u00de54\n311\u0394tD \u00f0\u00de1:52\u0394t1:5\nD/C0/C1\n;for\u03b151\ng\u0394tD\u00f0\u00de511\u0394tD \u00f0\u00de sin2111\u0394tD \u00f0\u00de1:5/C0/C1\n1\u0394t0:5\nD;for\u03b150:5\n\u0394tD5t2ti\nti\ng054\n3;for\u03b151\ng05\u03c0\n2;for\u03b150:5\nIn the above equations, \u03b1is a constant that governs the power-law propagation of\nthe fracture:\nA\nAe5t\nte/C18/C19\u03b1\n\u03b151 and 0 :5 in the above equations refer to low and high leak-off regimes,\nrespectively. A typical G-function plot is shown in Fig. 8.3 . In the plot, a straight tan-\ngent line that passes through the origin is fitted on the Gdp\ndGversus Gline. The closure\npoint is picked where the tangent line deviates from the plot. In order to read the clo-\nsure pressure, one needs to connect the deviation point to the pressure line (red line\nin the figure) using a vertical line (dashed line in the figure).\n8.4.1.2 Log /C0log analysis\nAnother widely used technique for DFIT is known as the log /C0log method\n(Marongiu-Porcu et al., 2011, 2014; Mayerhofer & Economides 1997; Mohamed"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae347f6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 6, "chroma:document": "et al., 2011; Valk\u00f3 & Economides, 1995 ). In this method, the closure time is deter-\nmined using the log /C0log diagnostic plot, which is used conventionally to analyze tran-\nsient well tests and analyze fall-off data. A typical log /C0log analysis of the pressure fall-\noff data is shown in Fig. 8.4 . In the figure, the dashed curve (pink) is the semilog\nderivative with respect to shut-in time. The slope of the derivative plot before the clo-\nsure is an identification of the flow regime before fracture closure (1/2 here indicated223 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34878-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 7, "chroma:document": "Figure 8.3 A normal leakoff G-function plot. ( Barree et al., 2007 )After Barree, R. D., Barree, V. L., &\nCraig, D. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium .\nSociety of Petroleum Engineers.\nFigure 8.4 A typical log /C0log analysis of pressure fall-off data. After Barree, R. D., Barree, V. L., &\nCraig, D. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium .\nSociety of Petroleum Engineers.224 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae348f0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 8, "chroma:document": "linear flow from fracture). In this technique, the separation of the two parallel lines is\nan identification of the fracture closure.\n8.4.1.3 Square root time method\nIn this technique, the fracture closure pressure is identified by the first peak in the\npressure plotted versus the square root of time. Fig. 8.5 shows a typical square root\ntime plot. The primary curve in this plot should form a straight line during the frac-\nture closure (similar to G-Function analysis). Unlike the G-Function analysis, where\nthe deviation from the straight line was indicating the fracture closure, in this analysis,\nthe peak of the first derivative of the pressure versus sqrt( t) indicated the fracture clo-\nsure point (pink dashed line in the figure).\n8.4.1.4 Modified technique to determine abnormalities\nUnal et al. (2019c) proposed a signal processing approach for analyzing DFIT based on\nthe signal processing approach. In the proposed technique, the noise of the recorded"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34972-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 8, "chroma:document": "pressure using wavelet transform is analyzed. The advantage of this method is that it\ndoes not depend on any assumptions of the currently available models and only relies\non the recorded pressure.\nFigure 8.5 An example of a normal Square Root time plot. After Barree, R. D., Barree, V. L., & Craig,\nD. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium . Society\nof Petroleum Engineers.225 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae349ea-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 9, "chroma:document": "Other techniques in this area include the one by Mohamed et al. (2011) and\nMarongiu-Porcu et al. (2011) , who suggested a global approach for a combined before\nand AC analysis. In these approaches, the authors used piece-wise functions. Using\nthese approach, the BC analysis can be written as:\np-\nwD512e2ted1tcD \u00f0\u00de s/C0/C1\n3q-\nD\nSK00:5OS/C2/C3\nFor the AC analysis, an enlarged equivalent wellbore radius was used. Also, the\nfluid flow in the reservoir was given as:\np-\nwD5K0\u00bdrwdOS/C138\nS1:5K1\u00f0rwD\ufb03\ufb03\ufb03\nSp\n\u00de\nThe above equation needs to be inverted numerically with the Stehfest (1970)\nalgorithm. The limitation of this technique is that it can only be used for high perme-\nability formations with short fracture.\n8.4.1.5 Transient before closure analysis\nTransient flow regimes can occur during fall off test. Cinco-Ley and Samaniego\n(1981) graphically presented the major flow regimes around a fractured well. Other\ntechniques than the conventional methods discussed above have also been developed"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34a6c-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 9, "chroma:document": "in recent years. In a similar way to the traditional G-function analysis, Siddiqui,\nSoliman, House, and Ibragimov (2016) developed a new analytical solution that could\nbe used to determine the closure pressure. In contrast to the conventional G-function\nmethod, which models the fluid leak-off into the formation based on Carter \u2019s equa-\ntion ( Howard & Fast 1957; Hubbert & Willis 1972 ), partial differential equations to\ndescribe the linear leak-off were used in the model. This method requires previous\nknowledge of fluid mobility and fracture compliance.\n8.4.2 After closure analysis\nAC analysis is referred to those analyses that are performed on the pressure fall of data\nafter the closure point (i.e., when the fracture faces are closed). The conventional well\ntest analysis can be applied with a high degree of accuracy (Soliman & Dusterhuft,\n2016). This is because that the created fracture during the pumping period of most of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34b34-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 9, "chroma:document": "the fracturing tests is small enough that ignoring it does not affect the results tremen-\ndously. However, there is one fundamental difference between the developed solu-\ntions of the traditional well testing approaches and AC analysis. The difference is that\nthe solutions of conventional well testing are derived from the constant volume, while\nthe solution of AC analysis are derived from impulse solution. In cases where the226 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34bb6-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 10, "chroma:document": "pressure fall-off is recorded long enough to reach radial flow, one can obtain reservoir\npressure and permeability using the AC analysis.\nSoliman et al. (2004, 2005) presented a technique for AC analysis to distinguish the\ndominant flow regime AC. The method could be used to analyze the data in each\nregion and determine the reservoir pressure and permeability. A summary of the tech-\nnique in the case of bilinear flow is given in the following (Soliman & Dusterhuft,\n2016):\n1.A log /C0log plot of logtinj\n\u0394t/C0/C1\nversus log/C0\nt@pf0\n@t/C1\nneeds to be plotted. Here tinj,\u0394t, and\npfoare injection time, time since shut-in, and bottom hole fall off pressure.\n2.The data will follow a straight line that indicates the flow regime.\n3.The intercept of the log tinj1\u0394t/C0/C1nversus ( pfo2pi\u00de, with n being the slope from\nthe previous step, on a cartesian plot indicates the reservoir pressure.\n4.Create a logarithmic plot oftinj\n\u0394t/C0/C1\nversus \u00f0pfo2pi\u00de."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34c38-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 10, "chroma:document": "4.Create a logarithmic plot oftinj\n\u0394t/C0/C1\nversus \u00f0pfo2pi\u00de.\n5.Assuming the bilinear flow, the intercept will be a function of permeability and\nreservoir conductivity.\nbr5264 :6V\nh\u03bc\u00f0\u00de0:75 1\n\u03c6ctk/C18/C190:251\nOkfwf !\nAlso, the end of bilinear flow is a function of both fracture conductivity and\npermeability as the following:\ntDebf54:55\nCfD22:5/C18/C1924\nCfD5kfw\nkLf\nCombining these equations by assuming that the fracture conductivity is low,\nthe formation permeability can be calculated as:\nk5264 :6V\nh3\u03bc\nbr1\n2:637tef/C0/C10:25\nThe fluid properties used in these equations are reservoir fluid properties.\n6.If the shut in time gets sufficiently large, the pseudo-radial flow will probably be\nobserved. For this regime to happen, the dimensionless time from the following\nequation should be greater than 1:\nt$3:7923103\u03c6\u03bcCtL2\nf\nkh\n7.A similar approach can be followed for the linear flow.227 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34cba-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 11, "chroma:document": "8.5 Pump-in/flow-back test (micro-frac)\nThe conventional DFIT has several limitations. For example, one needs to wait so\nlong (days) for the closure to happen. Also, the test gives inaccurate results in the case\nof the temperature difference between the test fluid and rocks or when the reservoirs\nare naturally fractured. Pump-in flow-back test is designed to address some of the lim-\nitations of the conventional pump-in shut-in approaches. During the test, usually, less\nthan four barrels of fluids are injected into the well (Soliman & Dusterhoft, 2016). A\nfollow back starts at the end of the test (rather than the shut-in). During the test,\nthe flow back period is analyzed to find the closure pressure. A typical recorded pres-\nsure for the flow back test is shown in Fig. 8.6 . As can be seen, the pressure plot in\nthe flow back region is concave with two almost constant slopes before and after\npoints A and B. The slope of the curve above point A represents the stiffness of the"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34d32-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 11, "chroma:document": "system, and the slope below point C represents the stiffness of the fracture alone\n(Savitski & Dudley 2011 ). Note that the return rate is very low and should stay con-\nstant during the flow back period, as shown in Fig. 8.6B . Different interpretations\nhave been proposed for picking up the closure pressure at points A, B, or C in this\ntest (Nolte,1990; Plahn et al., 1997; Soliman & Daneshy, 1991 ).\nSavitski and Dudley (2011) showed that in low permeability formations, the con-\ntinuous flow back at a constant rate, as shown in Fig. 8.6A , does not result in an\ninterpretable trend. Hence, an incremental flow back test with pressure stabilization\nprotocol was suggested. In the suggested approach, instead of a continuous fluid flow,\nrequired by the conventional pump in flow back test, a small incremental volumetric\nflow back followed by the pressure stabilization is used. The pressure stabilization\nallows for the reaching an equilibrium condition between the wellbore and fracture"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34db4-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 11, "chroma:document": "pressures. The advantage of this approach is that it removes the uncertainty about the\nFigure 8.6 Examples of recorded pressure during the pump-in flow-back test. (A) Schematic of the\npressure (B) a field case reported by Savitski and Dudley (2011) .228 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34e36-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 12, "chroma:document": "transient flow between the fracture and wellbore and results in a quasiequilibrium situ-\nation. An example of a field case from Savitski and Dudley (2011) using this approach\nis shown in Fig. 8.7 .\n8.6 New analysis technique\nSoliman and Kabir (2012) andSoliman and Gamadi (2012) revisited applications of\nsome of the mentioned techniques to unconventional and tight reservoirs, respectively.\nThey concluded that, in general, these methods are not practical for these reservoirs.\nTo address these limitations, new approaches have emerged in recent years. These\ntechniques are mainly based on the recorded data from the tests and have limited\nassumptions. They include signal processing approaches and data analysis approaches.\nIn the following, we review the signal processing approach in detail.\n8.6.1 Wavelet analysis\nSignal processing is a branch of electrical engineering that models and analyzes differ-\nent data types and their representations of physical events. Traditionally, signal proces-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34eb8-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 12, "chroma:document": "sing involves using filters to separate the signal of interest or even remove the\nunwanted frequency. The main objective of signal processing is to extract relevant\ninformation from a signal by transforming it. Fourier and wavelet transforms\nFigure 8.7 The quasiequilibrium pump in flow back, suggested by Savitski and Dudley (2011) .229 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34f30-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 13, "chroma:document": "are conventional techniques for signal analysis. Fourier transform is applied for\nstationary signals, signals whose properties do not evolve in time (e.g., sinewaves).\nFourier transform is not recommended for nonstationary signals because any abrupt\nchange in time would spread out the signal over the whole frequency domain ( Rioul\n& Vetterli, 1991 ). Also, the events cannot be located in time using the Fourier\ntransform.\nWavelet transform is of interest for the analysis of nonstationary signals. Wavelet is\na small wave that grows and decays primarily in a limited time period; on the contrary,\nFourier transform decomposes signal into sin waves, which keep on oscillating up and\ndown to infinity. The advantage of wavelet transform over Fourier transform in DFIT\nis the one can identify the events and their frequencies in time using this technique\n(Unal et al., 2019a, 2019b ). Wavelet is a function characterizing by the following\nproperties ( Percival & Walden, 2000 )"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae34fb2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 13, "chroma:document": "properties ( Percival & Walden, 2000 )\n\u0081The integral of real-valued function \u03c8is equal to zero:\n\u00f01N\n2N\u03c8u\u00f0\u00dedu50: \u00f08:1\u00de\n\u0081The square of \u03c8integrates to unity:\n\u00f01N\n2N\u03c82u\u00f0\u00dedu51: \u00f08:2\u00de\nBased on orthogonality, there are two types of wavelet transforms, continuous and\ndiscrete wavelet transform.\n1.Continuous Wavelet Transform (CWT): This transform can be used to obtain a simul-\ntaneous time-frequency analysis of a signal. CWT is best suited for time-frequency\nanalysis as these wavelets do not have negative frequency components. Examples\nof analytical wavelets are: Morse wavelet, Bump wavelet, and analytical Morlet\nwavelet. Continuous wavelet is now widely used in various disciplines, from x-ray\nimages in the medical field to well-logging in geophysics.\n2.Discrete Wavelet Transform (DWT): It is an essential tool needed for studying\ntime series via wavelet by decomposing the signal into a mutually orthogonal set of\nwavelets. The critical usage for discrete wavelet analysis is de-noising and compres-"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae3502a-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 13, "chroma:document": "sion of signals and images. DWT has numerous applications in science, engineer-\ning, mathematics, and computer science. For instance, DWT improved the analysis\nof dynamic fracture behavior in unconventional reservoirs. In reservoir engineer-\ning, DWT provided means of inferring inter-well connectivity in water flooding\noperations ( Unal et al., 2019c ).230 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae350a2-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 14, "chroma:document": "The base scale in DWT is set to 2. Different scales can be obtained by raising the\nbase scale to integer values represented as follows\n2j\u00f0j51;2;3;4...\u00de\nThe translation occurs at integer multiples represented as\n2jm\u00f0m51;2;3;4...\u00de\nThe DWT analyzes the signal at different frequency bands with different resolutions\nby decomposing the signal into coarse approximation and detail information. As shown\ninFigure 8.8 , the original signal passes through a high pass filter D1and low pass filter\nA1. Half of the samples are eliminated after filtering according to Nyquist \u2019s rule, which\nstates that in order to reproduce a signal adequately, it should be periodically sampled\nat a rate, that is 2 X, the highest frequency has been recorded ( Cramer & Grenader,\n1959). Filtering a signal corresponds to the mathematical operation of convolution of\nthe signal with an impulse response of the filter. The convolution equation is given by\nxn\u00bd/C1383hn\u00bd/C1385XN\nk52Nxk\u00bd/C138:h\u00bdn2k/C138\u00f0 8:3\u00de"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae35110-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 14, "chroma:document": "xn\u00bd/C1383hn\u00bd/C1385XN\nk52Nxk\u00bd/C138:h\u00bdn2k/C138\u00f0 8:3\u00de\nFigure 8.8 Discrete wavelet multilevel decomposition. The signal is separated to high pass (noise)\nand a low pass (approximation) at each level.231 Fracture diagnostic testing"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae35192-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 15, "chroma:document": "The low-pass filter halves the resolution but keeps the scale intact. The procedure\ncan mathematically be expressed as\nyn\u00bd/C1385XN\nk52Nhk\u00bd/C138:x\u00bd2n2k/C138\u00f0 8:4\u00de\nThe signal is then subsampled by two by discarding every other sample. This pro-\ncess continues one level of decomposition and mathematically can be expressed as\nyhighk\u00bd/C1385X\nnxn\u00bd/C138:g\u00bd2k2n/C138\u00f0 8:5\u00de\nylowk\u00bd/C1385X\nnxn\u00bd/C138:g\u00bd2k2n/C138\u00f0 8:6\u00de\nwhere yhighk\u00bd/C138andylowk\u00bd/C138are the outputs of the high-pass and low-pass filters, respec-\ntively. For the next level of decomposition, the low-pass sub-band A1is iteratively fil-\ntered by the same technique into narrower sub-bands AjandDj. With this technique,\nthe signal of interest can be captured with a few large magnitudes of discrete wavelet\ndecompositions ( Polikar, 2001 )(Fig. 8.8 ).\nIn DWT, the total energy of the signal is decomposed into components (i.e., dif-\nferent decomposition levels). Therefore DWT preserves the energy of the signal."}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae35200-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 15, "chroma:document": "Parseval \u2019s theorem relates the energy of the total signal to the energy of each compo-\nnent and its wavelet coefficients ( Burrus, Gopinath, & Guo, 1998 ). For a discrete sig-\nnal with the number of data points, the detail coefficients energy of the signal can be\ncalculated as follows:\nEd\ni5dijj2;i51;2;3;... ;l\nwhere51, 2, 3, ...,lis the wavelet decomposition level up to l.\n8.6.1.1 Field example 1\nFigure 8.9 shows the analysis of the MiniFrac by G-function of the first field\nexample case presented in this section ( Eltaleb et al., 2020 ). In this test, an average of\n16.8 bpm of treated water was injected into the formation for 15 minutes. Pressure\nfall-off was recorded after shut-in for 20 minutes, with a sampling rate of 1 seconds.\nNormal leak-off (i.e., fracture area is constant during shut-in) can be recognized from\nthe derivative shape. The closure can be identified evidently when G3dp/dGdeviates\nfrom the straight-line, reporting a closure pressure of 5822 psi and closure time of"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae35278-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 15, "chroma:document": "27.8 minutes. The fracture closure occurred at the 93rd data point in the analysis data\nset (Fig. 8.9 ).232 Unconventional Shale Gas Development"}


created_at: 2023-11-17 19:17:23
topic: persistent://default/default/8e5da3e9-8391-4ad8-a826-4d5919dbf381
id: 9ae352f0-857d-11ee-bb1c-0242ac1c000c
encoding: FLOAT32
metadata:
{"source": "drive/MyDrive/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf", "page": 16, "chroma:document": "This example is analyzed using our proposed technique. Figure 8.10 shows the\nenergy distribution plot of the noise at the decomposition level D 5. Two different dis-\ntinct regions (A and B) can be identified based on the energy magnitude. It can be\nnoticed that the energy magnitude is declining, then leveling off, and remains constant\nat a minimum level until the end of the job. Note that there are almost six orders of\nmagnitude reduction in the energy of the noise until the minimum. We hypothesize\nthat after the closure point, the energy of the noise should be minimum and stay con-\nstant. Also, it can be inferred from the energy trend that there is a change in system\nproperties with respect to time. This change is attributed to the decrease in fracture\nvolume over time@V\n@t/C0/C1\n. The closure point is identified from the intersection between\nlines connecting the energy levels of regions A and B, as indicated by the dashed line"}


