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SPE-171175-MS
Examples of Mini-Frac Test Data Interpretation in Low-Permeability
Reservoir
N. Makhota, A. Davletbaev, and A. Fedorov, RN-UfaNIPIneft LLC; R. Asmandiyarov, RN-Yuganskneftegas LLC;
I. Afanasiev, A. Sergeychev, and I. Yamalov, Rosneft Oil Company
Copyright 2014, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Russian Oil and Gas Exploration and Production Technical Conference and Exhibition held in Moscow , Russia,
14–16 October 2014.
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 have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessar ily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this p aper without the written


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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; i llustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Summary
When dealing with ultra low-permeability reservoirs, one comes across a certain problem of choosing low
cost way of getting reservoir pressure data. Classical well test implies long-lasting shut down ofproduction wells (up to several months) due to low permeability and presence of extended hydraulicfractures. The paper contains examples of surveys carried out using hydrodynamic logging. These testsshow significant restrictions of the method applied in low permeability resevoirs.
Well pressure based on minifrac test data was evaluated here for ultra low permeability reservoirs.


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Well pressure based on minifrac test data was evaluated here for ultra low permeability reservoirs.
Then the data was then compared with the results of average pressure at hydrodynamic logging. Fractureclosure pressure data was used to enrich correlation dependencies for fracture closure pressure (sponta-neous growth of fracture) in injection wells using well pressure fall-off curve interpretation methodsduring minifrac test. Each minifrac test in horizontal wells with multi-stage fracturing was analysed, theresults of injection efficiency ratio were gained that are in agreement with production profile accordingto PLT and allow to test the most productive well influx intervals.
Introduction
When dealing with ultra-low permeability reservoir almost all step-out wells are subjected to hydraulicfracturing. As a rule the main frac is preceded by minifrac test which is following stop to monitor wellpressure fall-off curve [


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G. LeBlanc et. al., 2010 ;V.A. Baikov et.al., 2012 ]. Minifrac test data is used for
setting geomechanical models and revising program sheet for the main frac.
Large volume fracs carried out in all wells, fractures being as long as 150 meters. Classical well tests
by means of well pressure build-up curve methods are complicated by the necessity of shutting down thewells (for more than 60 days). As a result it leads to significant production loss and large expenditures forobtaining well data in the regions of drilling new wells. Before shutting off the performance of a wellshould have a significant period of time (up to 6 months) until it has steady state flow. As a result withthe dynamic drilling in new zones being used reservoir parameters and pressure data can be received bymeans of standard well tests in a significant period of time by the moment when data received is not upto date. Thus, there is certain problem of getting low cost and up to date well pressure data in ultra


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low-permeability fields. The authors convinced that the necessary reservoir data should be received before
starting up the producer well.
This work contains detailed analyses of initial data for bottom hole pressure change during minifrac
test, interpretation in Ecrin, estimation of fracture closure and reservoir pressures. The gained reservoirpressure results through minifrac test were compared with the ones gained through hydrodynamic logging.The results compared showed satisfactory repeatability of measurement data on level of reservoirpressure. The interpretation was carried out during minifrac test with the help of deep pressure gauges.Minifrac tests are commonly used that is why in new or infill drilling zones the reservoir pressure can bereceived without substantial costs and with sufficient precision for all wells.
Determination of reservoir pressure according to hydrodynamic logging


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Determination of reservoir pressure according to hydrodynamic logging
Determination of reservoir pressure according to hydrodynamic logging in the zones of drilling new wellsas a rule is carried out in open-hole wells by means of test tools dropped down on cable (XPT«Schlumberger», FMT «Weatherford» packages are usually used for logging of reservoir pressure).During the measurement with the packages in each test interval sampler container collects a small amountof liquid (several cubic centimeters) with pressure curve recording. As a rule, there are two samplingcycles and two pressure buildup curves are recorded. The results achieved are further compared. If thevalues are more or less fit (with sufficient precision), then the sampling is of high integrity. Otherwise,depending on the difference in pressure values the sampling is of low or satisfactory integrity.
On
fig. 1 there is a diagram showing pressure change and fluid influx when testing one of the intervals


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On
fig. 1 there is a diagram showing pressure change and fluid influx when testing one of the intervals
on well XX22. Due to the limit volume of sampler container the cycling time for influx is short: 17 sec
- for the first cycle, 13 sec- for the second. After sampling cycles well pressure buildup curves are
recorded: 11 minfor the first, 5 minfor the second. It is seen on the diagram (for both ones) that after
wellbore storage effect logarithmic derivative tend to zero ( fig. 1b ). This can be explained by a short
period of sampling before pressure build-up curve. The pressure on build-up cycle quickly aligns andtends to constant. In this case the difference between the observed values is: ~1.4 atm.
When testing one well up to 30 reservoir pressure values can be achieved through different time
intervals. Experience of studying highly-compartmentalized low-permeability reservoirs shows that in theinterval of one geological object pressure values can differ (100 atmand even more).


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Fig. 2 shows test
report for well XX39. In object interval C formation pressure values during hydrodynamic logging aredifferent by the value of 190 atm, the depths of the testing intervals is as much as 14 m. The difficulty
is in data averaging for the object. Only one value should be given to an object when forming intervalpressure map, reservoir models.
Currently the engineers from “Rosneft” and “Schlumberger” use the following expressions (1) to
evaluate pressure average value. This approach takes into account the thickness of beds according to
Figure 1—Dynamics of bottom hole pressure and liquid rate (a) and its logarithmic derivative of build-up cycles (b) on well XX222 SPE-171175-MS


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geophysical logging interpretation and study intervals as well as actuality of each sampling. An average
pressure for the object is expressed as follows:
(1)
Where Priis a reservoir pressure of i-th bed ( i) in which several measurement tests can be carried out,
(k//H9262)i–mobility of i-th bed. In the first expression the summing up is carried out for all thin beds Lwith
informative measurement test. Second and third expressions the summing up is carried out for allmeasurement tests Min each i-th thin bed. It is widely accepted that weight factors take the following
values according to test reliability ( w
ij),wij/H110051 – for measuring high integrity, wij/H110050.5 – for measuring
satisfactory integrity, wij/H110050.1 - for measuring low integrity.
Amid the deteorating filtration properties of the wells drilled there is a tendency for low integrity study


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Amid the deteorating filtration properties of the wells drilled there is a tendency for low integrity study
number and number of dry measuring growing. In this case the absence of influx during the test can beexplained by the limitations of the measuring instrument (the model doesn’t support obtaining influx fromultra low permeability reservoirs). For example, when measuring well XX39 in the interval of Achimovskireservoir (with average permeability less than 1 mD) 12 measurements were carried out - 8 of them weredry sampling (with no influx from reservoir), 3 samplings were of low integrity and 1 with satisfactoryintegrity. In other case, during XX44 well surveying for object US0 measurements were carried out in 26intervals. All cases showed dry samplings. After hydraulic fracturing the well was put on operation. Thewell is being productive for 3 years.
Thus, tests with the help of hydrodynamic logging have restrictions in low-permeability reservoirs,


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Thus, tests with the help of hydrodynamic logging have restrictions in low-permeability reservoirs,
measuring pressure using well pressure buildup curve interpretation method for producer after hydraulicfracturing leads to significant oil production losses.
The examples of reservoir pressure determining according to minifrac test
data
Example 1 . Measuring pressure change curve using bottomhole pressure gauge and with the help of
injection fluid flow at the stage of minfra /slovotest during hydraulic fracture for XX45 well is shown on fig.
Figure 2—Interpretation results of testing by using hydrodynamic logging (a) /izhepressure profile according to the depth (b) for well XX39SPE-171175-MS 3


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3a. During the injection for 4.3 min. 12.8 m3of fluid was injected. Well pressure fall-off curve
interpretation method was recorded after injection stop. Initial data interpretation (for the well pressurefall-off curve interpretation method) was carried out in Ecrin (Kappa Eng). According to pressurederivative diagram using index with G-function [
Barree R. et. al., 2009 ] fracture closure time (~3.1 min)
and closure pressure (361.6 atm) were determined ( fig.3b ). By the shape of the curve GdP/dG fluid loss
type from the fracture to reservoir and closure time point were determined [ Craig D. et. al., 2000 ].
Reservoir pressure (using minifrac test) was determined by means of pressure analyses after the
closure. Pressure derivative diagram shows flow patterns according to line flow function square ( fig. 3 ).
Figure 3—Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square


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Figure 3—Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square
linear flow pressure derivative chartFL(/slovo), radial flow function FR(d) during minifrac test for well XX45
Figure 4—Data on bottomhole pressure and flow rate of the liquid injected (a), pressure and pressure derivative chart with G-function (b), square
linear flow pressure derivative chartFL(/slovo), radial flow function FR (d) during minifrac test for well XX714 SPE-171175-MS


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So, the ½ slope corresponds to pseudolinear flow pattern, singular slope-to pseudoradial. In this case,
pseudoradial flow pattern can be observed for well XX45 in ~30 minfrom stopping time on well pressure
fall-off curve interpretation method that is why reservoir pressure is determined according to linear curveapproximation from the last point measured to the beginning of pseudoradial flow pattern on radial flowfunction pressure diagram F
R(fig.3d ). After minifrac test data analyses was carried out the pressure for
well XX45 amounts to 278.9 atm.
Example 2 . Pressure change curve view during minifrac test for well XX71 is shown on fig.4a . Liquid
injection was carried out for 4.2 min, 12.8 m3were injected. At this formation fracturing pressure is 464.7
atm. However, in this case it is seen on pressure derivative diagram where pseudoradial flow pattern can


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atm. However, in this case it is seen on pressure derivative diagram where pseudoradial flow pattern can
not be detected on square linear flow, that is why reservoir pressure was measured on pseudolinear flowpattern with ½ slope. Estimated value of reservoir pressure is 294, 6 atm(
fig. 4b ), fracture closure pressure
– 374.4 atm.(fig. 4b ).
Example 3 . Minifrac test review with stop at well pressure fall-off curve interpretation method is shown
atfig. 5a . Formation fracturing pressure is 464.7 atm, fracture closure pressure – 321.1 atm(fig.5b ).
Pseudoradial flow pattern is not tested as it was in the previous case ( fig.5c ), thus, reservoir pressure
assessment is done through pseudolinear flow pattern – 246.4 atm(fig.5d ).
Before hydraulic fracturing the measurements were carried with the help of hydrodynamic logging for


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Before hydraulic fracturing the measurements were carried with the help of hydrodynamic logging for
observed in the examples above wells. Consequently, after averaging the pressure with expressions (1)reservoir pressures at minifrac test and hydrodynamic logging were compared. Well XX45 at pseudoradialflow pattern has the best congruity: reservoir pressure on minifrac test is 278.9 atm, hydrodynamical
logging pressure is 272.5 atm. In wells in which pressure estimation was carried out on pseudolinear flow
pattern pressure variation is greater: in well XX71 pressure on minifrac test is 294.6 atm. On hydrody-
namic logging - 275.0 atm. In well XX82 at minifrac test 246.4 atm, at hydradynamic logging – 262.7 atm.
Thus, when analyzing bottomhole gauge data on injection and fall-off cycles at minifrac tests one can
get estimated values of reservoir pressure. The tests carried out showed that the difference betweenpressures at mininfrac test and hydrodynamic logging can reach 20 atm.


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Figure 5—Bottomhole pressure and fluid flow rate data (a), pressure and pressure derivative chart on G-function (b), pressure derivative chart with
square linear flow function FL(/slovo), pressure change chart at radial flow function FR(d) at minifrac test for well XX82SPE-171175-MS 5


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Correlation relationship of fracture closure pressure vs. reservoir pressure
In the paper [ Davletbaev et. al., 2010 and2014 ] correlation relationship between fracture closure pressure
and reservoir pressure is given. This generalised relationship has sufficient statistics for reservoir pressurerange for more than 350 atm, for the range with less pressure testing points used are not enough. The


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reason for this is that well candidates for the study in steady flow regime are chosen with substantialinjection volumes and. as a rule, waterflooding radius in these wells are comparable to welltestinvestigation radius. When analyzing minifrac tests data using downhole gauge one can also get reservoirpressure and fracture closure pressure values. Fracturing is carried out in step-out wells in undisturbedzones with initial reservoir pressure, in infill wells in zones with pressure less than initial reservoirpressure. Consequently, correlation relationship can be complemented with data from the range of missingpressures for the relationship observed. Correlation relationship of fracture closure pressure with reservoirpressure for reservoirs T10, T11, T12 according to study on steady state injection test which is added withthe results of analyses of minifrac tests with bottomhole pressure gauges for the reservoirs.
Data comparison of minifrac test and production profile of PLT for


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Data comparison of minifrac test and production profile of PLT for
horizontal well with multi-stage fracturing
At this time the number of horizontal wells drilled in low-permeability collectors (at multi-stage
fracturing) is growing. However, according to reports on fracturing the planned production is notcorrelated to the volumes of proppant agent. The most difficult question is - which fractures in horizontalhole provide influx for the well as the injection of the proppant agent in the fracture do not guarantee it’sinflux to the well. For these purposes in a number of wells PLT logging is carried out with productionprofile record for horizontal well. This is done for determining the influx from intervals with fixedhyradulic fractures.
Influx/production distribution to frac sleeves recoreded for production profile at PLT logging is shown
for well XX46 on
fig.7 The study helped to learn that the main influx/prodcution (60%) is provided by


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for well XX46 on
fig.7 The study helped to learn that the main influx/prodcution (60%) is provided by
hydraulic fracture in the interval with frac sleeve P4, low level influx comes from other 4 fractures. Inaddition to this 25% of liquid comes from nonhermetic linear coupling.
In all five cases before the main fracturing minifrac test were carried out. The results were analized as
in
fig.3–5. Apart from fracture closure pressure estimation and etc. fluid injection efficiency kiewas also
Figure 6—The connection of fracture closure pressure with reservoir pressure for reservoirs T10, T11, T12 according to study results at steady
injection patterns and minifrac test data6 SPE-171175-MS


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received. Injection efficiency is a parameter showing the difference between the initial and maximum
fracture with the efflux absence. So is determined by efflux efficiency at hydraulic fracture and reservoircharacteriscics. The difference k
ieat different stages of hydraulic fracturing along horizontal well at equal
fluid properties indicates the difference between reservoir intervals. The less leakage efficiency is thebetter reservoir characterisctics are and more fluid moves to reservoir at hydraulic fracturing. Injectionefficiency for 4 minifrac tests are shown on
fig. 7 . There is no data for the sleeve P3. The estimation is


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fig. 7 . There is no data for the sleeve P3. The estimation is
unsuccessful. Comparison of injection efficiency at hydraulic fracturing and according to productiondistribution at PLT logging shows that interval with themain influx (60%) has the lowest injectionefficiency at minifrac tests. Thus, analysis of minifrac tests can reveal the most productive intervals athydraulic fracturing in horizontal wells. This helps to save expenditures for proppant agent volumeinjected to non-commercial intervals and reveal the most productive ones for planning of proppant agentinjection volume.
Conclusion


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Recording and interpretation of minifrac test data give the possibility of estimating reservoir pressure withsufficient precision in ultra low-permeability reservoir rocks where it is impossible to get informativemeasurements with the help of hydradynamic logging and there’s no possibility to test operating wellsafter reservoir fracturing due to significant production loss. It should be noted that more or less precisereservoir pressure measurements can be received using well pressure fall-off curve interpretation methodduring minifrac test at pseudoradial flow pattern. Apart from that realization of hydradynamic logging inopen well bore connected with additional technological risks and well construction time increment due tostudy (average time is 60 hours). Along with that minifrac tests are usually the integral part of hydraulicfracturing and additional expenditures will be connected with wireline gage run. Consequently, themethod applicated is cost and expenditures saving connected


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and additional expenditures will be connected with wireline gage run. Consequently, themethod applicated is cost and expenditures saving connected with reservoir pressure detection. Hydro-dynamic logging is much more expensive.


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Minifrac test analyses provide data on fracture closure pressure and can complement correlation
relationship on steady state injection test in injection wells with spontaneous growth of fracture.Evaluation of injection efficiency during minifrac test provides an opportunity to receive the mostproductive intervals in horizontal wells with multi-stage fracturing and is corresponding with theinflux/production profile on PLT logging.
Acknowledgments
The authors express thanks to the colleagues Latipov Alfred, Kazakevitch Eduard (Schlumberger) for theparticipation, help, advice and realization of averaging reservoir pressure method according to hydrody-namic logging study, to Serdyuk Alexander Nikolaevich, Valeev Sergei Valerievich (LLC “RN-
Figure 7—Influx profile comparison with the results of minifrac test for well XX46SPE-171175-MS 7


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Yuganskneftegaz”), Yudin Aleksei (Schlumberger) for the help and coordination at planning the mini-frac
measurement, to Ivan Tichonov and Askar Valiullin (LLC “PKF Gis-Nefteservis”) for help and analysesof PLT logging results for horizontal well XX46 with multi-stage fracturing.
Nomenclature
F
L – linear flow time function
FR – radial flow time function
hi – bed thickness, m
k – reservoir permeability, mD
(k//H9262)i – mobility of i-th bed, mD/cP
(k//H9262)HL_ij – mobility of reservoir fluid according to j-th measurement with hydrodynamic logging in
i-th bed
L – number of beds with measurement results after hydrodynamic loggingM – measurement number with hydrodynamic logging in one bedP – pressure, atm
tdP/dt – pressure derivatives, /az/tverdo/myslite
P
ra v – average reservoir pressure, atm
Pri – reservoir pressure for i-th bed, atm
PrH Li j – reservoir presuure according to j-th measurement with hydrodynamic logging in i-th bed,
atm


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Pri – reservoir pressure for i-th bed, atm
PrH Li j – reservoir presuure according to j-th measurement with hydrodynamic logging in i-th bed,
atm
wij – measurement accuracy at hydrodynamic logging, unit fraction
oPij – hydrostatic pressure diferential between the depth of the measurement and reservoir
pressure calculation, atm
/H9262 – reservoir fluid fiscosity, cP
Pi – reservoir pressure, atm
Pff – fracture closure pressure, atm
kie – injection efficiency at minifrac test
SSIT – steady state injection test
References
Baikov V.A., Burakov I.M., Latypov I.D., Yakovlev A.A., Asmandiyarov R.N. 2012. Waterflood
induced hydraulic fracturing control under reservoir pressure maintenance conditions on RN-Yugan-skneftegas oilfields. Neftyanoe khozyaystvo – Oil Industry . No 11. P. 30–33 (Russian)
Barree R., Mukherjee H. 1996. Determination of Pressure-dependent Leakoff and Its Effects on
Fracture Geometry . SPE 36424,


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Barree R., Mukherjee H. 1996. Determination of Pressure-dependent Leakoff and Its Effects on
Fracture Geometry . SPE 36424,
Barree R., Barree V., Craig D. 2009. Holistic Fracture Diagnostics: Consistent Interpretation of
Prefrac Injection Tests Using Multiple Analysis Methods . SPE 107877,
Craig D., Eberhard M., Barree R. 2000. Adapting High Permeability Sands for Estimating Reservoir
Engineering Parameters . SPE 60291.
Davletbaev A., Baikov V., Bikbulatova G., Asmandiyarov R., Nazargalin E., Slabetskiy A., Sergey-
chev A., Nuriev R. 2014. Field Studies of Spontaneous Growth of Induced Fractures in Injection Wells .
SPE-171232.
Davletbaev A., Baikov V., Ozkan E., Garipov T., Usmanov T., Asmandiyarov R., Slabetskiy A.,
Nazargalin E. 2010. Multi-Layer Steady-State Injection Test with Higher Bottomhole Pressure than the
Formation Fracturing Pressure . SPE 136199.
LeBlanc G., Gunasan E., Boutaud de la Combe J-L., Jaffrezic V. 2010. Reservoir Characterization


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Formation Fracturing Pressure . SPE 136199.
LeBlanc G., Gunasan E., Boutaud de la Combe J-L., Jaffrezic V. 2010. Reservoir Characterization
Using Injection Test After-Closure Analysis: Field Case History in a Depleted Oil Reservoir . SPE 128052.8 SPE-171175-MS


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SPE-170956-MS
The Effect of Changing Fracture Compliance on Pressure Transient
Behavior During Diagnostic Fracture Injection Tests
Mark W. McClure, Christopher A. J. Blyton, Hojung Jung, and Mukul M. Sharma, The University of Texas at
Austin, Department of Petroleum and Geosystems Engineering
Copyright 2014, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Amsterdam, The Netherlands, 27–29 October 201 4.
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 have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessar ily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this p aper without the written


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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; i llustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
Diagnostic Fracture Injection Tests (DFIT) are commonly interpreted with G-function derivative plots. In


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these plots, an upward deviation from linearity is generally attributed to fracture height recession ortransverse storage. This interpretation neglects changing fracture compliance during closure. Resultspresented in this study show that in low permeability formations, an upward deviation of the G-functionplot G*dP/dG is caused by a sharp reduction in fracture compliance due to closure. It is not necessary toinvoke more complicated explanations such as height recession or transverse storage. The specific shapeof the G*dP/dG curve is controlled by factors that affect fracture compliance before and after closure.These factors include the fracture height, fracture stiffness, and the residual aperture (which is affected bythe microscopic and macroscopic roughness of the created hydraulic fracture). Wellbore storage broadensthe post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that constantly increases during theshut-in period may be caused by wellbore storage, rather than


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peak in the G*dP/dG curve. A curve of G*dP/dG that constantly increases during theshut-in period may be caused by wellbore storage, rather than fracture tip extension. In higher permea-bility formations (with non-zero wellbore storage), the effect of fracture compliance is less important. Asa result, changing fracture compliance during closure does not have a significant effect on the pressuretransient plots, and closure decreases the rate of pressure decline because it reduces the rate at which fluidcan leak off into the matrix. In all cases, as long as a single planar fracture has formed, the best pick forthe minimum principal stress is at the deviation from linearity on the G*dP/dG plot, whether or not thisdeparture is concave up or concave down. Closure picks via a tangent to near the peak in G*dP/dG (asis commonly performed) may significantly underestimate the minimum principal stress. These picksshould be confirmed with other measurements, such as a subsequent injection period


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underestimate the minimum principal stress. These picksshould be confirmed with other measurements, such as a subsequent injection period or a pump-inflow-back test. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed inDFIT tests suggests that the actual geometry of the created fracture is rougher and more complex than istypically assumed by standard models.


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Introduction
OverviewDiagnostic fracture injection tests (DFIT) are a type of minifrac test performed in low permeabilityformations. The procedure includes pumping a small volume of water into a short wellbore interval over


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a few minutes, most often out of a perforated cased wellbore. The pressure transient after shut-in is used
to infer the closure pressure (assumed equal to the minimum principal stress), the formation permeability,the leak-off coefficient, and the initial formation pressure (
Barree et al., 2007 ;Soliman and Kabir, 2012 ;
Cramer and Nguyen, 2013 ).
Conventionally, closure pressure is estimated from the shut-in pressure transient of a minifrac test by
plotting pressure versus a function of time called G-function time (or alternatively, versus the square rootof time) and identifying the point where a deviation from linearity occurs (
Nolte, 1979 ;Castillo, 1987 ).
Equivalently, closure may be identified by plotting G*dP/dG and identifying the departure from linearity(
Barree et al., 2007 ).
Minifrac tests in very low permeability formations have proven difficult to interpret because plots of


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Barree et al., 2007 ).
Minifrac tests in very low permeability formations have proven difficult to interpret because plots of
G*dP/dG may appear concave up or down ( Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). In some
cases the trend is concave down with a persistent gradual increase apparent. This makes the departurefrom linearity difficult to identify. These trends are usually attributed to a variety of different non-idealprocesses. In such cases the interpretation is difficult given the non-unique nature of the problem.
In this study, we simulated DFIT tests in low permeability formations assuming a single planar
fracture. We used CFRAC (Complex Fracturing ReseArch Code), a hydraulic fracturing simulator thatperforms a full numerical simulation for fluid flow in the fracture and the matrix, calculates the stressesinduced by fracture opening, includes wellbore storage, and uses a nonlinear joint closure model todescribe the evolution of fracture compliance after closure (


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McClure and Horne, 2013 ). The simulations
indicated that a concave up trend on a G*dP/dG plot can be explained as a natural consequence of thedecreasing fracture compliance that occurs during closure. A concave downward G*dP/dG trend can becaused by wellbore storage. The simulations also indicated why these behaviors are more likely to beobserved in low permeability formations.
Our simulations suggest that closure pressure should be picked according to the classical method of
identifying the point of departure from linearity on the pressure versus G-function time (or square root oftime) plot or the G*dP/dG versus G-function time plot (
Castillo, 1987 ), whether or not the G*dP/dG curve
is concave up or concave down. In the presence of curving G*dP/dG plots, widespread industry practiceis to pick closure much later (at a much lower pressure), near the peak of the G*dP/dG curve (
Barree et


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Barree et
al., 2007 ;Soliman and Kabir, 2012 ;Padmakar, 2013 ;Cramer and Nguyen, 2013 ;Wallace et al., 2014 ).
Our results suggest that in low permeability formations, closure actually occurs near the bottom of thepeak in the G*dP/dG curve where a subtle departure from linearity is often present.
Literature review
Nolte (1979) published the first analysis technique for post fracturing pressure decline. The method
allowed the overall fluid loss coefficient, efficiency, geometry, and time for the fracture to close to becalculated. The assumptions made were consistent with PKN fracture geometry, presented by
Perkins and
Kern (1961) and extended by Nordgren (1972) . These include constant fracture height, elastic rock


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Perkins and
Kern (1961) and extended by Nordgren (1972) . These include constant fracture height, elastic rock
deformation with no slip at bedding planes, a bi-wing fracture created by a constant fluid injection rateand continuous propagation during pumping. Further, it was assumed that fluid loss is a function of theinverse square root of exposure time. This assumption is often termed Carter leakoff (Howard and Fast,1970). Because of the greater exposure time closer to the wellbore, the leakoff rate from the fracture isgreatest furthest from the wellbore.
An analytical approach was used by
Nolte (1979) to develop a function of shut-in time and pumping
time that produces a straight line when plotted against pressure. This time function is commonly referredto as G-function time. The overall fluid loss coefficient can be determined from a lumped parameter equalto the gradient of the pressure versus G-function time. The only reference to closure stress by
Nolte (1979)


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Nolte (1979)
was an estimate taken from a pump-in and flow-back test and not from analysis of post fracturing pressuredecline.2 SPE-170956-MS


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Nolte (1986) later extended the earlier work to other geometries, including both KGD ( Khristianovich
and Zheltov, 1955 ;Geertsma and de Klerk, 1969 ) and radial fracture geometry. Reference to closure stress
was only made with regard to corrections needed in the case of its alteration due to the injection processitself and not a determination from the post-fracturing pressure decline.
Martins and Harper (1985)
presented a similar analysis for elliptical fracture geometry. They also included a discussion of pressuredecline analysis, noting that the rate of decrease of average pressure depends on both the rate of fluid lossto the formation and the fracture compliance. They noted that for a given fracture length and height, orradius in the case of radial fracture geometry, the fracture compliance remains constant while the fractureis open. The focus of the work was primarily on determination of the overall leakoff coefficient.
Subsequent work by


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Subsequent work by
Nolte (1988) covered pressure transient analysis of fracture treatments both during
pumping and after shut-in. An observable increase in the rate of pressure decline upon closure could beidentified from data plotted against time, allowing an inference of closure stress. The author noted that thiswas due to the increasing stiffness of the fracture as it closed on proppant. A departure from linearity ona plot versus the square root of time was also presented. The author suggested that this closure stressestimate could be compared with the results of a pump-in and flow-back test.
Castillo (1987) suggested an extension in application of the Nolte (1979) G-function time to the


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Castillo (1987) suggested an extension in application of the Nolte (1979) G-function time to the
identification of fracture closure and hence its use to infer closure stress. This was based on identifyinga departure from linearity of the pressure versus G-function time plot, after a period of post shut-infracture extension, if present. The first derivative of pressure with respect to G-function time was used asan identification aid.
Plots of G-time multiplied by the derivative of pressure with the respect to G-time are commonly used
as a diagnostic tool.
Barree and Mukherjee (1996) used a numerical fracture simulation code to produce


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as a diagnostic tool.
Barree and Mukherjee (1996) used a numerical fracture simulation code to produce
pressure transients of non-ideal fracture cases, potentially allowing for their identification in field data.The simulations included an ideal base case, a case with post shut-in fracture extension, a case withpressure dependent leakoff, a case with both pressure dependent leakoff and pressure dependent fracturecompliance, and a case with fracture height recession from bounding layers. Ad-hoc functions were usedto relate leakoff with Young’s modulus to pressure, and so the results should be considered qualitative.
In later research
Barree et al. (2007) used other diagnostic tools, in addition to G-function time, to
examine simulated pressure transients for a number of cases with departures from the basic G-functiontime assumptions of
Nolte (1979) . The phenomena studied included an ideal base case, fracture extension


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Nolte (1979) . The phenomena studied included an ideal base case, fracture extension
following shut-in, pressure dependent leakoff, and fluid storage from transverse or dilated naturalfractures.
Barree et al. (2007) picked the fracture closure pressure by drawing a straight line from the
origin tangent to the G*dP/dG curve where it begins to curve down. Barree et al. (2007) interpreted a
concave up G*dP/dG curve as representing either fracture height recession or opening of transversefractures. A concave down G*dP/dG curve was interpreted as fracture tip extension.
Barree et al. (2007)
considered fracture tip extension to be a process that could occur for an extended time after shut-in, oreven for the entire duration of the shut-in period.
Padmakar (2013) presented results of a coupled geomechanics and reservoir flow simulation study


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Padmakar (2013) presented results of a coupled geomechanics and reservoir flow simulation study
conducted using commercially available software. An advantage of this approach was that it enabled studyof both pre and post fracture closure pressure transient behavior. A Barton-Bandis model was used tomodel the fracture permeability, which included both hysteresis in loading and retention of residualaperture on closure. There was only one simulated pressure transient presented, which showed asignificant departure from the ideal response expected.
Padmakar (2013) noted that the pick made for
closure stress was lower than that input to the simulation.
Wallace et al. (2014) modeled a single fracture case and complex discrete fracture network cases with


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Wallace et al. (2014) modeled a single fracture case and complex discrete fracture network cases with
an assumed fracture density and aperture. The complex discrete fracture network was upscaled to acontinuum model for simulation. The effect of fracture closure was captured in a simple and non-hysteretic manner by use of non-linear transmissibility multiplier versus pressure curves. The simulatedSPE-170956-MS 3


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pressure transients produced by their simulations were analyzed by means of the pressure and derivative
versus G-function time approach. The apparent closure pick made for the single fracture case matched thatinput to the simulator. The complex discrete fracture case showed a pressure transient signature that wasinterpreted to be the result of transverse storage or natural fracture dilation.
Methodology
Simulation methodologySimulations were performed with CFRAC (Complex Fracturing ReseArch Code). CFRAC is a two-dimensional discrete fracture network simulator that simultaneously calculates fluid flow and stressesinduced by fracture deformation. Fluid flow in the matrix is also simulated. The full details of CFRACwere summarized by
McClure and Horne (2013) , and only a brief overview is given in this section. Some
modifications have been made to CFRAC since the version described by McClure and Horne (2013) , and
those modifications are described here in detail.
Fluid flow


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those modifications are described here in detail.
Fluid flow
Fluid flow is single-phase liquid water (no proppant) and isothermal. CFRAC solves the unsteady statemass balance equation with Darcy’s law:
(1)
(2)
where mis the mass density at a location (kg/m3),tis time (s), qfluxis the mass flux vector (kg/(s-m2)),
sis a source term for a well (kg/(m3-s)),/H9267is fluid density (kg/m3),kis permeability (m2),/H9262is fluid
viscosity (MPa-s), Pis pressure (MPa), and xiis location in the direction i(m).
The mass flux is converted to mass flow rate, q(kg/s), by multiplying by the cross sectional area for
flow. The cross-sectional area for flow is equal to the height (the out-of-plane dimension in a two-dimensional simulation), h, multiplied by width available for flow. For flow in the fractures, the width for
flow is defined equal to the hydraulic aperture, e. The permeability of the fracture is defined as equal to:
(3)


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flow is defined equal to the hydraulic aperture, e. The permeability of the fracture is defined as equal to:
(3)
When permeability is multiplied by hydraulic aperture, the result is the transmissivity (units of m3),T,
and the classical cubic law results ( Witherspoon et al., 1980 ):
(4)
We define two different types of fracture aperture. The “void” aperture, E, is the volume of fluid stored
per surface area of fracture. The “hydrauli” aperture, e, is related through the cubic law to the
transmissivity. In actual practice, the hydraulic aperture may be lower than the mechanical aperture (thephysical apparent thickness of the fracture) or E, the void aperture (the volume of fluid stored per area).


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Therefore, while transmissivity is a physically meaningful parameter because of its role in Darcy’s law,hydraulic aperture is an “effective” parameter that is only meaningful in terms of its relationship totransmissivity. Void aperture, E, is an important physical parameter because it describes the fluid stored
in the fracture.
The equations are solved fully implicitly (backward Euler in time) using the finite volume method
described by
Karimi-Fard et al. (2004) . Time step duration is chosen adaptively during the simulation.
Following the method of Karimi-Fard et al. (2004) , fractures are meshed as one-dimensional line
segments, and the surrounding matrix is meshed with triangles. Karimi-Fard et al. (2004) described the
method for calculating geometric transmissibility factors for calculating flow between elements. Thesegeometric transmissibility factors calculate an effective “area available for flow between two elements”4 SPE-170956-MS


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and “distance for flow between two elements” in such a way that the method is convergent to the exact
solution.
For a matrix element, the local mass density is defined as the product of fluid density and porosity. For
a finite volume element, this mass density is converted to mass contained in the element (kg) bymultiplying by element area and formation height. Fluid density and matrix porosity are given by theconstitutive relationships:
(5)
(6)
where /H9267initand/H9278initare the fluid density and porosity at the initial pressure Pinit, and cfandc/H9278are the
compressibility of the fluid and matrix (assumed constant).
In fractures, the local mass density can be defined as simply the fluid density (since the “porosity”
within the fracture can be considered to be unity). To calculate the mass content of a finite volume fractureelement (kg), the fluid density is multiplied by element length, formation height, and the void aperture,
E.


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E.
Nonlinear relationships are used to relate pressure, stress, and deformation to void aperture and
hydraulic aperture. Before describing these relations, it is necessary to define some terminology. Wedefine a “closed” fracture as a fracture where the fluid pressure is less than the normal stress and the wallsof the fracture are in contact. Roughness of the fracture walls allows closed fractures to retain hydraulicand void aperture. We define an “open” fracture as a fracture where the fluid pressure has reached thenormal stress and the walls are no longer in contact.
For calculating the aperture of closed fracture elements, CFRAC uses the following joint closure
relation, which was used by
Willis-Richards et al. (1996) and was based on the work of Barton et al.
(1985) :
(7)
where E0and/H9268n,Eref are specified constants that characterize the joint. E0is the residual void aperture


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(1985) :
(7)
where E0and/H9268n,Eref are specified constants that characterize the joint. E0is the residual void aperture
and/H9268n,Eref is the 90% closure stress for void aperture. /H9268n=is the effective normal stress, equal to the
normal stress minus fluid pressure.
We use the same relation with different constants to calculate hydraulic aperture:
(8)
If a fracture element is open, then its aperture is equal to its residual aperture (aperture at normal stress
of zero) plus the mechanical opening that has occurred. The aperture at zero normal stress is e0andE0.
In this case, aperture is defined as:
(9)
(10)
where Eopenis mechanical opening, the amount of physical separation between the fracture walls that
has occurred during opening. For hydraulic fracture elements, the value of E0is permitted to evolve during
the simulation (and is different for each element), as described in Section 2.1.3.


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the simulation (and is different for each element), as described in Section 2.1.3.
The problem is solved using a conforming mesh created using the publicly available program Triangle
(Shewchuk, 1996 ). Fracture elements consist of line segments and matrix elements consist of triangles.
The mesh is created so that the fracture element length is constant, ensuring the accuracy of themechanical calculations.SPE-170956-MS 5


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For the simulations in this study, there was a single hydraulic fracture and a wellbore, represented as
a line segment. The problem domain was 70 m long in the direction of fracture propagation and 15 meterswide in the direction perpendicular to fracture propagation. The domain was meshed into roughly 68,000matrix elements and roughly 250 fracture elements. The path of the propagating hydraulic fracture wasassumed in advance, linear and perpendicular to the minimum principal stress. By specifying thepropagation path in advance, the entire problem domain could be meshed prior to the simulation andre-meshing during fracture propagation was unnecessary.
At the beginning of a simulation, the hydraulic fracture elements do not physically exist, but have


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At the beginning of a simulation, the hydraulic fracture elements do not physically exist, but have
already been meshed and are treated as “inactive.” As the fracture initiates and propagates along thepredetermined path, elements are converted from inactive to active. Inactive elements are treated asnon-existent and omitted from the system of equations.
The edges of the domain were assumed to be no-flow boundaries. The simulations were run for a short
enough duration that the edges of the model did not have an effect on the results. The well was a lineoriented perpendicular to the direction of fracture propagation with a length of four meters. The fracturewas assumed to initiate at the center of the well (which is also the center of the problem domain).
Figure
1shows a zoomed in image of the center of the problem domain (the figure shows only a part of the
problem domain).
Mechanical calculations


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Figure
1shows a zoomed in image of the center of the problem domain (the figure shows only a part of the
problem domain).
Mechanical calculations
In addition to solving the fluid flow equations, CFRAC calculates the stresses induced by fracturedeformation - both opening and sliding. In the simulations for this study, there was no fracture slidingbecause there was only a single hydraulic fracture, which was linear and perpendicular to the minimumprincipal stress. Therefore, the treatment of fracture sliding will not be described in this section.
The stresses are calculated using the boundary element method of
Shou and Crouch (1995) , which uses


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The stresses are calculated using the boundary element method of
Shou and Crouch (1995) , which uses
the displacement discontinuity (boundary element) method using quadratic basis functions assuming anelastically homogeneous and isotropic formation, linear elastic deformation, and small strain. Themechanical boundary condition is that the fracture(s) are embedded in an infinite whole space. Thetechnique satisfies the equations of classical continuum mechanics: quasistatic stress equilibrium, thecompatibility equations, and the linear elastic constitutive relation. Calculation results are fully convergentto analytical solutions for crack deformation (such as
Sneddon, 1946 ). The calculations are fully


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Sneddon, 1946 ). The calculations are fully
numerical, enabling complex problems to be solved (for example, arbitrarily varying fluid pressure alongthe fracture length). An advantage of the displacement discontinuity method is that it is only necessary tomesh the fractures themselves, not the area around the fractures, to perform stress calculations.
Figure 1—An image of the center of the problem domain (the full problem domain is larger). The blue triangles are the matrix elements. The
horizontal red line is the path traveled by the propagating hydraulic fracture and the vertical red line is the wellbore.6 SPE-170956-MS


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For an open element, the stress boundary condition is that effective normal stress must be equal to zero:
(11)
where
 is the normal stress on the fracture due to remote loading and /H9004/H9268nis the change in normal
stress due to backstress created by the deformation of all the fracture elements in the system.
In addition to calculating the stresses induced by the opening deformation of open elements, CFRAC
has the capability to calculate the stresses induced due to normal deformations (changes of E) of closed
elements (the cstress option in McClure and Horne, 2013 ). This capability was used in the simulations
performed for this study. For brevity, the numerical details are omitted here, and they can be found in
McClure and Horne (2013) . Normal deformations of closed elements are small and the resulting induced
stresses have only a minor effect on simulation results.
The compliance (derivative of aperture with respect to pressure) of open fractures is not explicitly


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The compliance (derivative of aperture with respect to pressure) of open fractures is not explicitly
defined. Instead, the compliance behavior of the fracture is a property that arises emergently from thecoupling of fluid pressure and deformation in the mechanical network. When fractures are closed, theircompliance is primarily controlled by
Equation 7 , but also affected by the stresses caused by normal
displacement of the fracture.
Poroelastic behavior in the matrix is not included, which means that stresses induced by fracture
deformation are only calculated on other fractures, not throughout the matrix and that changes in fluidpressure of the matrix do not cause stress changes in the fracture.
A code called Hmmvp (
Bradley, 2011 ) is used to very accurately and efficiently approximate the


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A code called Hmmvp (
Bradley, 2011 ) is used to very accurately and efficiently approximate the
matrices of interaction coefficients arising from the displacement discontinuity method. This improves theefficiency and scaling of the method and enables much larger fracture networks to be simulated (thoughin this study, only a single, relatively small fracture was simulated).
The
Shou and Crouch (1995) method assumes plane strain deformation, implying infinite fracture size
in the out-of-plane dimension. In this study, the Olson (2004) adjustment was used to adjust the stress
calculations to account for the finite fracture height.
The nonlinear system of equations is solved using a specially developed iterative method with iterative
coupling between (1) opening and fluid flow and (2) sliding displacements. Within the nonlinear solver,the sparse matrix linear solver UMFPACK is used (
Davis, 2004 ), which also uses BLAS ( Lawson et al.,


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Davis, 2004 ), which also uses BLAS ( Lawson et al.,
1979 ;Dongarra et al., 1988 ), AMD ( Amestoy et al., 2004 ), and LAPACK ( Anderson et al., 1999 ).
Fracture propagation is determined using linear elastic fracture mechanics. The stress intensity factor
at each crack tip, KI,is numerically evaluated using the method of Olson (2007) and if it exceeds the
fracture toughness, KIc(this is specified material property), the fracture extends by a single element (per
timestep).
The simulations in this study used arithmetic averaging of transmissivity between adjacent elements.
For fluid properties such as viscosity and density, upwinding is used when calculating properties for flowbetween elements.
Residual aperture of a hydraulic fracture
There are unique challenges for defining the aperture of newly forming hydraulic fractures. For this study,different constitutive relations for the aperture of newly formed fractures were used than in past versionsof CFRAC (


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McClure and Horne, 2013 ). When an element first forms at the tip of the fracture, it is given
a very small residual void aperture, 10 microns. We define “residual aperture” as the aperture wheneffective normal stress is equal to zero, which is E
0. Because the elements are initiated with a very small
residual aperture, the algorithm does not perfectly conserve mass. However, because the initial apertureis very small, the error is slight.
A special algorithm is applied to allow residual aperture to increase as a hydraulic fracture opens.
Therefore, each element may have a different E
0, and it can evolve during the simulation. As shownSPE-170956-MS 7


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below, if the residual aperture of the elements is kept constant at 10 microns, the simulation results during
closure are very unrealistic. But the elements cannot be initiated with a larger residual aperture becausethis would create a significant mass balance error. So an algorithm is needed that allows elements to havevery low residual aperture initially, but for their residual aperture to increaes as they open. This algorithmis an attempt to mimic the natural formation of fracture roughness as a crack forms and opens. Thisprocess does not appear to be investigated in detail in the literature, and so we are using an approach thatis somewhat ad-hoc, but physically plausible and necessary for realistic model behavior.
The percentage of total opening retained as residual aperture is defined as the “residual aperture


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The percentage of total opening retained as residual aperture is defined as the “residual aperture
percentage.” It is not clear what value to use for the residual aperture percentage. For most simulations,we used 50%, but this was treated as a sensitivity analysis parameter and turned out to have a very strongeffect on simulation results. Presumably, as a fracture opens more and more, there will be some limitingvalue for which further opening does not increase residual aperture further. We have not defined a limitingresidual aperture, but this was not necessary because the maximum opening in any simulation performedin this study was only 0.6 mm, which is the same order of magnitude as a plausible residual aperture fora fracture.
Hydraulic aperture of fracture elements was treated differently than void aperture. For all hydraulic
fracture elements, regardless of the residual void aperture, the residual hydraulic aperture, e
0, was set to


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fracture elements, regardless of the residual void aperture, the residual hydraulic aperture, e
0, was set to
be constant at 20 microns. When fracture elements opened, their hydraulic aperture could grow higherthan e
0due to the separation of the fracture walls, as defined by Equation 9 .
The fluid flow calculations were validated by comparing against analytical solutions to the diffusivity
equation for one-dimensional flow and radial flow. Also, the DFIT simulations shown in this paper wereperformed to very late time (using a larger problem domain than used for the simulations in this study)so that late time radial flow could be identified in the pressure transient. Plots of pressure versus theinverse of time were used to calculate permeability (using the method of
Gu et al., 1993 ) and to confirm
that the calculated permeability was consistent with the permeability specified as a model input.Validation of the mechanical calculations was described by
McClure and Horne (2013) .


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McClure and Horne (2013) .
Wellbore storage
Wellbore storage was newly implemented in CFRAC for this study. During injection, a constant injectionrate was imposed in the simulation. This was accomplished by adding an additional equation to the systemof equations, specifying that the sum of the flow rates between the well and the adjacent fracture elementsmust sum to the specific injection rate. To include wellbore storage, the equation was modified slightly:
(12)
where qis the specified mass injection rate, fwiis the mass flow rate between the well and an adjacent
fracture element i, Cwis the wellbore storage coefficient (m3/MPa), /H9267wis the density of the injection fluid,
andNis the number of elements connected to the wellbore. The equation is expressed in units of


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andNis the number of elements connected to the wellbore. The equation is expressed in units of
kilograms per second. The wellbore storage term accounts for the fluid stored in the wellbore due to fluidcompressibility. The wellbore storage coefficient is equal to the product of wellbore volume and fluidcompressibility. If a surface shut-in is used during the DFIT test, the wellbore storage may be significantenough to affect the pressure transient, even if the well is full of only water, which has low compress-ibility. This is especially true because the volume of fluid injected during a DFIT test is so small, only afew barrels. Wellbore storage may be minimized if a downhole shut-in tool is used (
Cramer and Nguyen,
2013 ).
At the beginning of the simulation, the wellbore fluid pressure is equal to the initial fluid pressure. The


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Cramer and Nguyen,
2013 ).
At the beginning of the simulation, the wellbore fluid pressure is equal to the initial fluid pressure. The
hydraulic fracture cannot initiate until the fluid pressure has exceeded the minimum principal stress. Ifthere is no wellbore storage, the fluid pressure will rise instantaneously to the minimum principal stress,forcing the hydraulic fracture to initiate. But with wellbore storage, the initial injection into the wellbore8 SPE-170956-MS


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will be taken up by wellbore storage and fluid pressure will not rise instantaneously. As long as Cwis
nonzero, Equation 12 can be satisfied even if the hydraulic fractures have not yet formed.
In all simulations except one, it was assumed that there is no hydraulic connection between the
wellbore and the matrix. This setting was intended to mimic a DFIT test in a perforated, cased wellbore.In one of the simulations, fluid flow was allowed directly between the wellbore and the adjacent matrix(as shown in
Figure 1 ).
Details of the simulations
Results of eleven simulations are presented in this paper. The simulations were designed to mimic a DFITtest. Injection was performed for five minutes at 1 kg/s, and then the well was shut-in. The total simulationduration was ten hours.
The simulation parameters used in all the simulations are given in
Table 1 . The specific differences
between the simulations are given in Table 2 . The simulations are named BC(base case), LP(low


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Table 1 . The specific differences
between the simulations are given in Table 2 . The simulations are named BC(base case), LP(low
permeability), HP(high permeability), LRAP (low residual aperture percentage), HRAP (high residual
aperture percentage), LCS (low closure stress), HCS (high closure stress), LH(low height), HWS (high
wellbore storage), HRAP-HWS (high residual aperture percentage and high wellbore storage), and
CMT (conventional minifrac test).
Simulation BCwas designed to be representative of a DFIT test in a low permeability formation. As
discussed in Section 3, the shut-in response of Simulation BCdoes not quite appear realistic, and several
modifications were investigated to make the result more realistic. The permeability was 10 /H9262D, the initial
fluid pressure was 18 MPa, and the minimum principal stress was 28 MPa. Estimates of fracture closurepressure should therefore be close to 28 MPa.


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fluid pressure was 18 MPa, and the minimum principal stress was 28 MPa. Estimates of fracture closurepressure should therefore be close to 28 MPa.
Except for Simulations CMT andLH, all simulations were the same as Simulation BCexcept for
varying a single parameter. Simulation CMT had three parameters varied from the base case: much higher
permeability, a direct connection between the well and the matrix, and a modest wellbore storagecoefficient. These parameters were chosen to represent a minifrac test in a conventional reservoirTable 1—Simulation settings used in all simulations.
/H9271p(Poisson’s ratio) 0.25 G(shear modulus) 15 GPa
Pinit(initial fluid pressure) 18 MPa /H9268yy(initial y-direction principal stress) 28 MPa
h(fracture/formation height) 15 m KIc(fracture toughness) 3 MPa-m1/2
e0(reference hydraulic aperture, Equation 8 )2 0 /H9262m /H9262(fluid viscosity) 1 cp
/H9267init(initial fluid density) 1000 kg/m3cf(fluid compressibility) 0.000458 MPa-1


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/H9267init(initial fluid density) 1000 kg/m3cf(fluid compressibility) 0.000458 MPa-1
/H9278init(initial porosity) 0.03 c/H9278(pore volume compressibility) 0.00145 MPa-1
Ehfresid,min (minimum residual aperture) 10 /H9262m
Table 2—Simulation settings specific to the different simulations.
Simulation name BC LP HP LRAP HRAP LCS HCS LH HWS HRAP-HWS CMT
/H9268n,Eref (90% closure stress for void
aperture) (MPa)20 20 20 20 20 5 90 20 20 20 20
/H9268n,eref (90% closure stress for
hydraulic aperture) (MPa)20 20 20 20 20 5 90 20 20 20 20
Residual aperture percentage 50% 50% 50% 0% 90% 50% 50% 50% 50% 90% 50%
Cw(wellbore storage coefficient)
(m3/MPa)00000000 0.01 0.01 0.003
k(permeability) ( /H9262D) 10 1 100 10 10 10 10 10 10 10 1000
h(fracture/formation height) (m) 15 15 15 15 15 15 15 2.5 15 15 15
Hydraulic connection from well to
matrixNo No No No No No No No No No Yes
Injection rate (kg/s) 1111111 0 . 5 111SPE-170956-MS 9


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(relatively high permeability) and with an open-hole
completion. Simulation LHhad a lower fracture
height, and also had a lower injection rate (for thesame injection duration).
Results and Discussion
Simulation BC
Simulation BC before closure Figure 3 shows a
close view of the G*dP/dG and pressure curvesprior to closure. Following convention, in all G-function plots, the derivative terms dP/dG andG*dP/dG are plotted as positive. At the beginningof the transient, there was an initial slight upwardmovement of G*dP/dG, which was caused by asmall amount of tip extension that occurred afterclosure. Then the curve formed a straight line, asexpected for a DFIT test before closure (
Castillo,
1987 ). The total length of the hydraulic fracture that
formed was 28.75 m.
Based on the slope of the plot of pressure versus
G-time prior to closure, it should be possible toestimate the leakoff coefficient, C
L(Equation 9 -58
from Economides and Nolte, 2000 ):
(13)


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G-time prior to closure, it should be possible toestimate the leakoff coefficient, C
L(Equation 9 -58
from Economides and Nolte, 2000 ):
(13)
where his the fracture height, p*is the slope of
pressure versus G-time, E=is equal to the Young’s
modulus divided by one minus the square of Pois-son’s ratio (equal to 40 GPa), and t
pis the time of
production (in our case, representing the duration ofinjection). We have used the form of the equationappropriate for the PKN model, a Newtonian injection fluid, and a fracture that has uniform leakoff alongits face. From
Figure 3 , the value of p*was equal to roughly 0.1 MPa/G-time. Using this value and the
values from Table 1 , we calculate that the leakoff coefficient to be 2.2 /H1100310-6m3/(m2-s).
The simulation was not strictly a PKN model. It used the Olson (2004) adjustment to account for


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The simulation was not strictly a PKN model. It used the Olson (2004) adjustment to account for
formation height, meaning that if fracture length is much greater than height, it will behave like the PKNmodel and if length is much less than height it will behave like the KGD model. The half-length wasroughly equal to the formation height, so neither the KGD nor PKN model would be expected to perfectlyapply.
Because the formation properties are known, we can calculate the leakoff coefficient from the
definition, for a fracture with no filter cake (
Howard and Fast, 1957 ):
(14)
where/H9004Pis the increase in fluid pressure above the formation fluid pressure. The values in Equation
14can be read from Table 1 andTable 2 , except /H9004P, which was taken to be 10.4 MPa. Using Equation
14, the leakoff coefficient can be calculated as 4.4 /H1100310-6m3/(m2-s). This value is in reasonable agreement
Figure 2—G-function plot from Simulation BC.


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Figure 2—G-function plot from Simulation BC.
Figure 3—Zoomed in plot of G*dP/dG and pressure at early time from
Simulation BC.10 SPE-170956-MS


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with the estimate from the pressure transient using Equation 13 , considering the variety of approximations
used in deriving and applying Equation 13 .
Simulation BC after closure A correct pick of the closure stress should be roughly around the minimum
principal stress, 28 MPa. Figure 2 andFigure 3 show that in Simulation BC, pressure was equal to 28 MPa
at a G-function time around 2. The most striking feature in Figure 3 is that when closure occurs, the
pressure begins to drop much more rapidly, causing a large spike in G*dP/dG. In field data, the trend inthe decline of pressure with time may accelerate or decelerate following closure (
Nolte, 1988 ). In low
permeability formations, it is very common to see a concave up G*dP/dG plot, indicating the decline inpressure during shut-in is accelerating (
Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). The
common interpretation of this data is that closure occurs later, near the top of the peak in G*dP/dG ( Barree


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common interpretation of this data is that closure occurs later, near the top of the peak in G*dP/dG ( Barree
et al., 2007 ;Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ), but as discussed below, this would not
be the correct interpretation of the data generated in our simulations.
In Simulation BC, the G*dP/dG plot increased more sharply than is typically observed in the field. In
Sections 3.3, 3.4, and 3.6, we discuss how residual aperture, closure stress, and fracture height affect thesharpness of this transition. Despite the rather unrealistically sharp increase in derivative, the result inSimulation BCdemonstrates an important physical process, and later simulations show how the settings
could be modified to create a more realistic looking simulation result.
Figure 3 shows a close view of the preclosure plot of pressure and G*dP/dG in Simulation BC.A s
predicted by Castillo (1987) based on the solution of Nolte (1979) , pressure makes a straight line (and so


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predicted by Castillo (1987) based on the solution of Nolte (1979) , pressure makes a straight line (and so
therefore G*dP/dG should also form a straight line).
The sharp increase in G*dP/dG occurred because the fracture compliance dramatically dropped when
closure occurred. The effect of changes in fracture compliance on the pressure derivative can be bestexplained through an application of the chain rule:
(15)
where Vis the volume of fluid in the fracture,
 is the average pressure in the fracture, and tis time.
For simplicity, we are assuming that the average fracture pressure is equal to the pressure measured at thewellbore. The first term,
, is proportional to the reciprocal of the compliance of the fracture. The second
term,
 , is the leakoff rate from the fracture. The leakoff rate is proportional to the inverse of the square
root of time at early time, but is also a function of pressure, decreasing as fracture pressure decreases. As


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root of time at early time, but is also a function of pressure, decreasing as fracture pressure decreases. As
fracture closure occurs, the compliance drops abruptly, causing a rapid increase in the magnitude of
and
 . Eventually, the leakoff rate decreases sufficiently (encouraged by the lowering fluid pressure
in the fracture), and the magnitude of the pressure derivative drops back down. An inflection point in
pressure occurs at the maximum value of the magnitude of the pressure derivative, which occurs when thesecond derivative of pressure is zero, implying:
(16)
According to classical analytical solutions, the compliance of an open fracture (walls out of contact)
is very high and proportional to the fracture size. For example, for an open, circular crack with constantfluid pressure, the relationship between maximum aperture and pressure is (
Sneddon, 1946 ):
(17)
where Ris radius, Pis pressure, /H9271pis Poisson’s ratio, and EYoung’s Modulus is the Young’s modulus. For


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Sneddon, 1946 ):
(17)
where Ris radius, Pis pressure, /H9271pis Poisson’s ratio, and EYoung’s Modulus is the Young’s modulus. For
a crack with radius of 20 m, Poisson’s ratio equal to 0.25, and Young’s modulus equal to 37.5 GPa(equivalent to shear modulus of 15 GPa, as used in our simulations), the compliance at fracture center,
,
is equal to 0.64 mm/MPa.SPE-170956-MS 11


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Figure 4 shows plots of aperture and compliance
(the derivative of aperture with respect to pressure)versus effective normal stress based on
Equation 7
forE0equal to 0.5 mm and a value of /H9268n,Eref equal
to 5 MPa and 20 MPa. Figure 4 makes clear that
fracture closure results in a sharp drop in fracturecompliance. If
/H9268n,Eref is low enough, compliance
may initially remain high as closure begins, butclosure will drop rapidly after effective normalstress has reached several MPa. If
/H9268n,Eref is higher,
compliance will drop less rapidly as effective nor-mal stress increases, but there will be a more abruptdrop in compliance when closure occurs.
In Simulation BC,
/H9268n,Eref was equal to 20 MPa.
Figure 4 shows that for this value, the derivative of


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In Simulation BC,
/H9268n,Eref was equal to 20 MPa.
Figure 4 shows that for this value, the derivative of
aperture with respect to pressure will be 0.2 mm/MPa, sharply lower than the compliance of the openfracture and continuing to decline rapidly. This ex-plains why the pressure derivative increases sharplywhen fracture closure occurs. As discussed below,the fact that such a sharp increase in pressure de-rivative is not usually observed suggests either thepost-closure compliance must be higher than inSimulation BCor the pre-closure compliance must
be lower than in Simulation BC.
Barree et al. (2007) suggested that a concave up
G*dP/dG plot could be caused by fracture heightrecession or opening of transverse fractures. Ouranalysis suggests that the upward curving figure is anatural result of the decreasing fracture complianceas the fracture closes, and does not necessarily in-dicate fracture height recession or opening of trans-verse fractures.
Effect of matrix permeability


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Effect of matrix permeability
Decreasing matrix permeability delays closure, ex-tending the duration of the pre-closure straight lineon the G*dP/dG curve. However, in both Simula-tionHPandLP, a sharp increase in G*dP/dG oc-
curred only after closure (
Figure 5 andFigure 6 ).
Effect of residual aperture percentage
The residual aperture percentage has a huge effecton the shape of the G*dP/dG curve (
Figure 7 and
Figure 8 ). A large spike occurred in the G*dP/dG
plot for Simulation LRAP. If residual aperture per-
centage is zero, then the residual aperture whenclosure occurs is very tiny, 10
/H9262m (the default initial
value used by the model). Equation 7 implies that
Figure 4—Aperture (blue lines) and dE/d /H9268n=(red lines) versus effective
normal stress for E0equal to 0.5 mm and /H9268n,Eref equal to 5 MPa (solid
lines) and 20 MPa (dashed lines).
Figure 5—G-function plot from Simulation LP.
Figure 6—G-function plot from Simulation HP.12 SPE-170956-MS


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the derivative of aperture with respect to pressure (for a closed fracture) is directly proportional to the
residual aperture. Therefore, if residual aperture is very small, the contrast in compliance that occurs withclosure is huge, causing an extremely sharp spike in the G*dP/dG curve. This behavior is not realistic, butthe result from Simulation LRAP is instructive to illustrate the effect of the compliance contrast during
closure.
Simulation HRAP had a high residual aperture percentage, 90%. The maximum opening was around
0.64 mm, tapering to around 0.2 mm near the tip, and so the residual aperture in the fracture was 90% ofthose values. Apparently, a residual aperture of 90% was high enough to prevent an abrupt drop incompliance when closure occurred.
Figure 8 shows that the result from this simulation looked like a
typical DFIT curve from a low permeability formation, with the plot of G*dP/dG gently curving up.
Effect of closure stress


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typical DFIT curve from a low permeability formation, with the plot of G*dP/dG gently curving up.
Effect of closure stress
Figure 9 andFigure 10 show the effect of changes in closure stress on the G-function plot. As shown in
Figure 4 , a lower closure stress means that the fracture has a higher compliance when closure begins. As
a result, when closure starts, the G*dP/dG curve increases less abruptly when closure stress is low. InSimulation HCS, the closure stress was 90 MPa, and so compliance at closure was very low, and the
contrast in compliance at closure was extreme, causing a very abrupt increase in the G*dP/dG curve.
Figure 7—G-function plot from Simulation LRAP.
Figure 8—G-function plot from Simulation HRAP.
Figure 9—G-function plot from Simulation LCS.
Figure 10—G-function plot from Simulation HCS.SPE-170956-MS 13


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Effect of wellbore storage
The effect of wellbore storage on the pumping pe-riod can be seen in
Figure 11 , which shows the
injection pressure during pumping for SimulationBCandHWS . In Simulation BC, injection pressure
instantaneously rose above the minimum principalstress and initiated a hydraulic fracture. In Simula-tion HWS , injection for nearly two minutes was
required before the fluid pressure has reached theminimum principal stress. For the first two minutesof pumping, the hydraulic fracture had not yetformed and all injected fluid was being stored in thewellbore.
In both simulations, a breakdown pressure was


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In both simulations, a breakdown pressure was
observed modestly above the minimum principalstress. As a model assumption, the fracture initiatedwhen the fluid pressure reached the minimum prin-cipal stress (to avoid mesh dependence, the fracturewas initiated at a half-length of 1 meter). But inorder for the fracture to propagate, the fluid pressureneeded to rise high enough that the stress intensityfactor at the tips reached the fracture toughness.Once this occurred, injection pressure dropped be-cause as the fracture grew, a lower fluid pressurewas required for the stress intensity factor to reachthe fracture toughness.
The effect of wellbore storage was to decrease
the rate at which pressure declined (
Figure 12 and
Figure 13 ). Other than wellbore storage, Simulation
HWS was identical to Simulation BC. In Simulation
BC, there was very little drop in pressure prior to


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HWS was identical to Simulation BC. In Simulation
BC, there was very little drop in pressure prior to
closure, and then an abrupt drop in pressure afterclosure. Wellbore storage has an effect when thepressure in the wellbore is changing (
Equation 12 ),
so in Simulation HWS , the effect of wellbore storage
was felt primarily after closure. It is interesting tonote that in conventional pressure transients, pres-sure changes most rapidly at the beginning of thetransient, and so wellbore storage is only importantat early time. But in a DFIT test, pressure maychange most rapidly after closure, and also the totalvolume of fluid in the formation may not be muchlarger than the volume expelled due to wellborestorage. As a result, wellbore storage can affect theentire pressure transient response, not just the be-ginning.
Wellbore storage causes fluid to flow out of the well into the formation as the pressure draws down.
A wellbore storage coefficient of 0.01 m


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Wellbore storage causes fluid to flow out of the well into the formation as the pressure draws down.
A wellbore storage coefficient of 0.01 m
3/MPa is equivalent to 10 kg/MPa. The total mass of fluid injected
Figure 11—Injection pressure versus time during Simulation BC (blue)
and Simulation HWS (red).
Figure 12—G-function plot from Simulation HWS.
Figure 13—G-function plot from Simulation HRAP-HWS.14 SPE-170956-MS


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was 300 kg, and the increase in pressure from in-
jection was a bit over 10 MPa, so in SimulationHWS , roughly 100 kg (one third of the total) was
consumed by wellbore storage. Therefore, at thetime of shut-in, only 200 kg had been injected intothe formation. After shut-in, as the pressure contin-ued to decline back to 18 MPa, most of the remain-ing 100 kg flowed out of the well into the formation.Unsurprisingly, the constant resupply of fluid intothe formation from the wellbore (due to storage)during shut-in slowed the rate of pressure decline,causing the G*dP/dG curve to have a graduallyincreasing trend for a long time after closure (
Figure
12).
After closure, Simulation HRAP-HWS behaved
very similarly to Simulation HWS (Figure 13 ). However, prior to closure, the high residual aperture


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12).
After closure, Simulation HRAP-HWS behaved
very similarly to Simulation HWS (Figure 13 ). However, prior to closure, the high residual aperture
percentage caused the fracture compliance to be much lower, preventing a very flat G*dP/dG curve. Thecombination of high residual aperture percentage and wellbore storage resulted in only a very subtle effectin the G*dP/dG curve when closure occurred (at around 29 MPa, a bit higher than the true minimumprincipal stress, due to the high residual aperture, as discussed in Section 3.3).
As an example comparison to field data, Simulations HRAP andHRAP-HWS appear similar to the
DFIT field data in
Figure 8 from Cramer and Nguyen (2013) . In their Figure 8 , “Test 1” was performed
with a surface shut-in (larger wellbore storage). The G*dP/dG curve shows a gradually increasing andthen eventually gradually decreasing trend, just as in Simulation HRAP-HWS (
Figure 13 ). Their “Test 2”


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Figure 13 ). Their “Test 2”
was performed with a downhole shut-in (lower wellbore storage). For this test, the plot of G*dP/dG wasconcave upward and the more sharply peaked, followed by a steady decline. This test was qualitativelysimilar to Simulation HRAP (
Figure 8 ).
Effect of fracture height
In Simulation LH, the fracture height was reduced from 15 to 2.5 m. To maintain a comparable fracture
length, injection rate was reduced to 0.5 kg/s. The compliance of an open fracture fracture scales with thesmallest dimension, which for most simulations in this study (including Simulation LH) was height. Note
that for plane strain models, the fracture compliance would scale with length, but we used the
Olson
(2004) adjustment, which enabled the 2D model to behave more like a PKN model, and so compliance
scaled with height. Therefore, because Simulation LHhad a lower height, it had a lower pre-closure


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scaled with height. Therefore, because Simulation LHhad a lower height, it had a lower pre-closure
compliance. This resulted in a less flat G*dP/dG curve prior to closure ( Figure 14 ). The post-closure
compliance was low enough that there was still an abrupt increase in derivative after closure. However,from prior results, we can infer that if the closure stress had been lower (as in Simulation LC)o ri ft h e
residual aperture percentage had been higher (as in Simulation HRAP ), the transition would have been less
abrupt or non-existent.
Conventional minifrac test


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abrupt or non-existent.
Conventional minifrac test
As discussed above, in DFIT tests in low permeability formations, an upward curving G*dP/dG plot iscommonly observed, indicating that pressure begins to drop more rapidly after closure. As discussed inSection 3.1, this occurs because the system compliance drops rapidly after shut-in. However, in manyminifrac tests, especially in permeable rocks, pressure drops less rapidly, not more rapidly, after closure.An example is given in Figure 7.3 of
Zoback (2007) .Nolte (1988) noted that after closure, the pressure
trend may begin to drop more or less rapidly, depending on conditions.
Figure 14—G-function plot from Simulation LH.SPE-170956-MS 15


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Simulation CMT was performed to demonstrate
how and why pressure may drop less rapidly afterclosure. Simulation CMT was different from the
base case because it had: (1) much higher formationpermeability, 1 mD, (2) a small but nonzero well-bore storage, and (3) a direct hydraulic connectionbetween the well and the formation (as if the wellwas completed open-hole).
Figure 15 shows that for
Simulation CMT , dP/dG was monotonically de-
creasing, with closure occurring rapidly, and afterclosure (which must have occurred around 28 MPabecause of the model setup), pressure dropped lessrapidly than before closure. In the other simulationsin our study, dP/dG increased post closure and thendecreased again, with sometimes a very short periodof high derivative at the beginning of closure due totip extension.
The different behavior of Simulation CMT can be understood as being caused by the competition


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The different behavior of Simulation CMT can be understood as being caused by the competition
between storage and leakoff. In Simulation CMT , the much higher permeability caused the fracture that
formed during pumping to be much smaller than in the other simulations in our study, with a total lengthof 3 m (as opposed to 28.75 m in Simulation BC). The compliance of an open fracture scales with the
smallest dimension (length in Simulation CMT ), and so not only was the fracture much smaller, it was also
much less compliant.
Thus, in Simulation CMT , the hydraulic fracture was much smaller and much less compliant than in
the other simulations, and as a result the fracture had very little ability to store fluid, even before closure.Even with a rather small value of wellbore storage coefficient, 0.001 m
3/MPa, the wellbore had greater


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3/MPa, the wellbore had greater
fluid storage capacity than the fracture that formed. As a result, when closure occurred, the overall systemcompliance (which includes the fracture and the wellbore) was not much affected.
Another effect of closure is to reduce the transmissivity of the fracture. In a formation with very low
permeability, even a closed fracture may have high transmissivity relative to the matrix. But in a higherpermeability formation, the transmissivity of the fracture after closure may readily become negligiblerelative to the matrix. As a result, the closure of the hydraulic fracture reduces the rate at which fluid canleak out of the wellbore/hydraulic fracture system.
Overall, in Simulation CMT , closure had a minor effect of the compliance of the wellbore/hydraulic


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Overall, in Simulation CMT , closure had a minor effect of the compliance of the wellbore/hydraulic
fracture system but reduced the rate at which leakoff could occur from the fluid stored in the wellbore.The total effect was that closure reduced the rate at which pressure declined, unlike in the othersimulations.
Estimating the minimum principal stress in low permeability formations
Barree et al. (2007) described a variety of methods for interpreting reservoir behavior from G-function
plots. He suggested that an upward curving G-function plot indicates “storage” due to opening oftransverse fractures or fracture height recession. Our results indicate that the upward curving of thederivative curve is a natural consequence of the decreasing compliance of the fracture as it closes (thoughthis effect should only be apparent in low permeability formations).
Barree et al. (2007) also suggested that when an upward curving G-function plot is apparent, the timing


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Barree et al. (2007) also suggested that when an upward curving G-function plot is apparent, the timing
of fracture closure should be identified by drawing a tangent straight line from the origin to near the peakof the G*dP/dG curve. Then the pressure at that time should be evaluated to estimate the closure pressure.This method of interpretation would significantly underestimate the closure stress in nearly every
Figure 15—G-function plot from Simulation CMT. Note that the x-axis
shows only G-time of 0 to 6, unlike the other figures.16 SPE-170956-MS


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simulation we performed. In most simulations, this
method would lead to an interpretation of the min-imum principal stress equal to around 24-25 MPa,lower than the true value of 28 MPa.
Our simulations indicate that closure would
properly be picked at the bottom of the peak in theG*dP/dG curve. Specifically, in the early transient,the G*dP/dG curve makes a straight line (except fora very early feature indicating fracture tip exten-sion). When the curve deviates upward from thestraight line, closure has occurred.
The G-function plots from the simulations with
high residual aperture percentage are consistentwith the shape of G-function plots from field data(
Figure 8 andFigure 13 ). However, in these cases,
it is less clear where to pick closure, because theupward deviation caused by closure is less sharp.Because of the residual aperture, the upward deviation appears to begin when the fluid pressure is a bitabove the true minimum principal stress (


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Figure 13 ). Because the walls begin to come in contact while
there is still a significant amount of residual aperture, the backstress on the fracture has not returnedentirely to zero (which would only entirely occur with full closure if the fracture aperture returned to zero).As a result, closure begins with the pressure modestly above the minimum principal stress.
Figure 16 demonstrates the different ways to pick closure from the curve, either at the first deviation
from the straight line in the G*dP/dG curve or at the intersection of the G*dP/dG curve with a tangent linedrawn from the origin. The actual minimum principal stress is 28 MPa.
In practice, we suspect that the backstress is unlikely to cause closure to begin at a pressure more than


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In practice, we suspect that the backstress is unlikely to cause closure to begin at a pressure more than
0-2 MPa greater than the minimum principal stress. A more severe difficulty may arise if it is ambiguouswhere to identify where the “upward deviation” of the G*dP/dG curve occurs (as perhaps was the casein Simulation HRAP ). However, a relatively sharp increase in G*dP/dG (or alternatively, an abruptly more
rapid drop in pressure) is often apparent in field data in unconventional formations.
Castillo (1987) suggested picking the closure stress by plotting pressure versus the square root of time
and then identifying the deviation from linearity. This method has been widely adopted, sometimesreplacing the G-function plot with a simpler plot of pressure versus the square root of time (
Zoback, 2007 ).


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Zoback, 2007 ).
In the absence of tip extension, this method is essentially equivalent to what we have suggested (becausein most cases, G-function plots will give results that are reasonably similar to a square root of time plot).However, care would need to be taken to avoid picking closure at the end of the tip-extension period. Forexample, in
Figure 3 , fracture closure would best be picked at G-time of 2, not at the end of the very early
time linear feature at G-time of roughly 0.1.
A simple method for checking picks of the closure stress would be to perform a second, rather low rate


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time linear feature at G-time of roughly 0.1.
A simple method for checking picks of the closure stress would be to perform a second, rather low rate
injection test at the end of the shut-in period. Because of hoop stress and the tensile strength of theformation, the fracture may not initiate at exactly the minimum principal stress during the first injectionperiod, making the leakoff point unreliable for estimating the minimum principal stress. However, oncethe fracture has formed, it should be relatively easy to reopen it, and fracture initiation effects will not beapparent. Even in the second test, hoop stress effects could theoretically cause a slight breakdown pressureto be apparent, but shortly after that breakdown, the fluid pressure should stabilize at a fracturepropagation pressure close to the minimum principal stress. The value of repeated injection periods forestimating the minimum principal stress has been emphasized by other authors (
Hickman and Zoback,
1983 ).


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Hickman and Zoback,
1983 ).
Figure 16—Comparison of two methods for identifying closure stress.
Drawing a line tangent to G*dP/dG gives a closure stress around 25MPa. Identifying closure at the beginning of the upward deviation in theG*dP/dG curve gives a closure a bit less than 29 MPa.SPE-170956-MS 17


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Estimating the minimum principal stress in high permeability formations
Simulation CMT was an example of a conventional minifrac test in a relatively higher permeability
formation. In this test, the G*dP/dG curve was initially straight and then began to bend downward whenclosure occurred (indicating a more slowly decreasing fluid pressure). For this curve, the methodsuggested by
Castillo (1987) , to pick closure at the deviation of linearity of the plot of pressure versus
G-time, would give the correct closure, though it may be ambiguous exactly where to identify the“deviation from linearity.”
Barree et al. (2007) interprets a concave down G*dP/dG curve as indicating tip extension. Our results
are consistent with that interpretation, as the very early time deviation in the G-function plot in Figure 3
did correspond to a limited degree of tip extension. However, our results suggest that care should be takenin interpreting tip-extension based on a concave down plot of G*dP/dG.


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Figure 12 ,Figure 13 , and Figure
15show downward curving G*dP/dG curves. This behavior was caused by wellbore storage in the first
two cases and the effect of closure in a higher permeability formation in the latter case. In the case offracture tip extension,
Barree et al. (2007) cautions against picking the closure stress at the first break in
linearity of the G*dP/dG curve, suggesting this may lead to an early pick of the closure stress (see hisFigures 17 and 18). However, in Simulations HWS (
Figure 12 ),HRAP-HWS (Figure 13 ), and CMT (Figure
15), the correct pick for closure stress would be at the break in linearity of the G*dP/dG curve, and a later
pick would yield a substantial underestimate. Simulations HWS andHRAP-HWS show that wellbore
storage can cause the G*dP/dG curve to slowly trend upward for long after closure has occurred.
Implications for actual fracture geometry
It is notable that in the base case simulation, Simulation BC, and in several of the other simulations,


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Implications for actual fracture geometry
It is notable that in the base case simulation, Simulation BC, and in several of the other simulations,
closure caused an extremely abrupt upward deviation in the G*dP/dG curve. While an upward trendingG*dP/dG curve is common in low permeability formations, such an abrupt deviation seems unrealistic andartificial. In some of our simulations, abrupt deviations did not occur, or the deviations were less abrupt.It is worth considering what this behavior suggests about the geometry of the fractures that are actuallycreated during DFIT tests in low permeability formations.
As described in Section 3.1, the analytical solution for the relation between pressure and aperture for
an open crack suggests that cracks on the scale of meters or larger have very significant compliance. Thisis why such an abrupt increase in G*dP/dG is observed during closure, such as in Simulation BC. Because


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this feature is not typically observed in field data, an implication may be that open hydraulic fractures maynot be as compliant as classical fracture mechanics models suggest.
Fracture mineback (
Warpinski and Teufel, 1987 ;Jeffrey et al., 2009 ), coring ( Warpinski et al., 1993 ),
and laboratory experiments ( Blanton, 1982 ;Renshaw and Pollard, 1995 ;Bahorich et al., 2012 ;Suarez-
Rivera et al., 2013 ) suggest that real hydraulic fractures are complicated features with branching,
non-planarity, and termination against natural fractures ( Mahrer, 1999 ; Cipolla et al., 2008). All of these


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features would tend to decrease fracture compliance. If fluid is actually distributed into a larger numberof smaller features, the overall system compliance will be lower because compliance scales with fracturesize. The back stress on each fracture would tend to increase normal stress on adjacent fractures, furtherdecreasing system compliance. Non-planar fracture formation may result in the normal stress not beinghomogeneous along the newly formed fracture(s). As a result, closure could occur more gradually,because the fracture(s) must close against a distribution of different closure stresses. Stress heterogeneitymay also exist because of either the prior geological loading or heterogeneity in elastic properties in theformation. Stress heterogeneity would tend to smear out closure in the pressure transient response bycausing closure to occur across a distribution of normal stresses along the fracture faces. All of thesebehaviors may manifest themselves as an effectively “rougher”


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to occur across a distribution of normal stresses along the fracture faces. All of thesebehaviors may manifest themselves as an effectively “rougher” fracture, which may explain why ourmodel using a high residual aperture percentage appeared quite realistic.18 SPE-170956-MS


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In short, the DFIT transient pressure response suggests a more complicated geometry than is classically
assumed in hydraulic fracture modeling (including in this study). The idealized, homogeneous problemgeometry of our base case simulation created an unrealistically abrupt transition at closure, but introducingfeatures associated with heterogeneity and complexity into the model smooths the transition and causesit to appear more realistic. From the perspective of hydraulic fracture modeling, simplification isnecessary to make the problem tractable, but it is important to consider to what degree simplificationactually changes the model predictions and behaviors in a meaningful way.
Other potential complications
There are several other potentially complicating issues that could cloud DFIT interpretation and deservegreater study in future work.
DFITs involve injection of water and are frequently done in hydrocarbon bearing formations. The


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DFITs involve injection of water and are frequently done in hydrocarbon bearing formations. The
G-function analysis is dependent on the assumption that leakoff scales with the square root of time. Thisscaling relationship is based on a single-phase, one-dimensional solution to the diffusivity equation.Nonlinear multiphase flow effects may have an unpredictable effect on the scaling of leakoff rate withtime, affecting the shut-in transient, especially in shale formations, where leakoff behavior is notwell-understood.
As noted in Section 3.10, direct hydraulic fracture observations have generally indicated complex
fracture geometry. If multiple fractures form, if propagating fractures terminate and branch, or if otherforms of complex geometry develop, this could complicate DFIT interpretation significantly, potentiallycausing the minimum principal stress to be overestimated (
McClure, 2014 ).
Conclusion
Our analysis leads to the following conclusions:


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McClure, 2014 ).
Conclusion
Our analysis leads to the following conclusions:
1. In low permeability formations, a concave up deviation of the G*dP/dG curve from a linear trend
can be explained by decreasing fracture compliance due to fracture closure. It is not necessary toinvoke more complicated explanations such as height recession or transverse storage.
2. Since we have shown that the concave up deviation is caused by a drop in fracture compliance,
the specific shape of the post fracture closure curve is affected by factors that affect fracturecompliance before and after shut-in, including the fracture height, the fracture 90% closure stress,and the residual aperture percentage (equivalent to the microscopic or macroscopic roughness ofthe created hydraulic fracture).
3. Wellbore storage broadens the post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that


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3. Wellbore storage broadens the post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that
constantly increases for a long time during the shut-in period may be caused by wellbore storage,rather than fracture tip extension.
4. In higher permeability formations (with nonzero wellbore storage), the effect of fracture compli-
ance is less important. Closure decreases the rate of pressure decline because it reduces the rateat which fluid can leak off into the matrix from the wellbore.
5. In all cases, the best pick for the minimum principal stress is at the deviation from linearity on the
G*dP/dG plot. Closure picks based near the peak in G*dP/dG (as is commonly performed) maysignificantly underestimate the minimum principal stress. These picks should be confirmed withother measurements, such as a subsequent injection period.
6. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed in DFIT tests


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6. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed in DFIT tests
suggests that the actual geometry of the created fracture is rougher and more complex than istypically assumed by standard models such as PKN and KGD.SPE-170956-MS 19


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Acknowledgements
The authors would like to acknowledge the support provided by the corporations supporting our research
though the Hydraulic Fracturing and Sand Control JIP at the University of Texas at Austin.
Nomenclature
A : matrix of interaction coefficients (MPa/m)
CL : leakoff coefficient (m3/(m2-s))
Cw : wellbore storage coefficient (m3/MPa)
cf : fluid compressibility (MPa-1)
c/H9278 : pore volume compressibility (MPa-1)
D : vector of displacement discontinuities (m)
E : void aperture (m)
E= : Young’s moduls divided by one minus the square of Poisson’s ratio (MPa)
Ehfresid,min : minimum residual aperture (m)
EYoung’s Modulus : Young’s modulus (MPa)
E0 : residual void aperture (m)
Eopen : mechanical separation between walls for an open element (m)
e : hydraulic aperture (m)
e0 : residual hydrualic aperture (m)
fwi : mass flow rate between the well and an adjacent element (kg/s)
G : shear modulus (MPa)
h : fracture/formation height (m)
KI : stress intensity factor (MPa-m1/2)


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G : shear modulus (MPa)
h : fracture/formation height (m)
KI : stress intensity factor (MPa-m1/2)
KIc : fracture toughness (MPa-m1/2)
k : permeability (m2)
m : mass density at a location (kg/m3)
N : number of elements connected to the wellbore
P : pressure (MPa)
: average pressure (MPa)
Pinit : initial pressure (MPa)
p*: slope of the pressure versus G-time curve (MPa/G-time)
q : mass flow rate (kg/s)
qflux : mass flux (kg/(m2-s))
R : radius of a circular crack (m)
S : matrix of induced stresses (MPa)
s : mass source term (kg/(m2-s))
T : transmissivity (m3)
t : time (s)
tp : time of production (s)
x : location (m)
V : volume of fluid in the fracture (m)
/H9004P : change in pressure above initial pressure (MPa)
/H9004/H9268n : induced normal stress (MPa)
/H9262 : fluid viscosity (MPa-s)
/H9267 : fluid density (kg/m3)
/H9267init : initial fluid density (kg/m3)
/H9267w : density of fluid in the wellbore (kg/m3)20 SPE-170956-MS


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/H9268n : normal stress (MPa)
/H9268n= : effective normal stress (MPa)
/H9268n,Eref : 90% closure stress for void aperture (MPa)
/H9268n,eref : 90% closure stress for hydraulic aperture (MPa)
: remote loading normal stress (MPa)
/H9268yy : initial principal stress in the y-direction (MPa)
/H9271p : Poisson’s ratio (unitless)
/H9278 : porosity (unitless)
/H9278init : initial porosity (unitless)
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drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-187348-MS Variable compliance.pdf


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SPE-187348-MS
New Variable Compliance Method for Estimating In-Situ Stress and Leak-Off
from DFIT Data
HanYi Wang and Mukul M. SharmaThe University of Texas at Austin
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.
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 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
any 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


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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 SPE copyright.
Abstract
Over the past two decades, Diagnostic Fracture Injection Tests (DFIT) or Injection-Fall-off Fracture
Calibration Tests have evolved into a commonly used and reliable technique to evaluate reservoir properties,
fracturing parameters and obtain in-situ stresses. Since the introduction of DFIT analysis based on G-
function and its derivative, this method has become standard practice for quantifying minimum in-situ stress
and leak-off coefficient. However, the pressure decline model that underlies the G-function plot makes two
distinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or


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distinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or
compliance) is assumed to be constant during fracture closure. Fracture closure is a gradual process that
starts when asperities at the fracture tip first come into contact with each other. As pressure declines due
to leak-off, more and more of the fracture wall comes into contact. It is important to model this process
quantitatively to obtain good estimates of in-situ stresses and leak-off. In this paper, we present a model that
accounts for changes in fracture stiffness/compliance as the fracture closes, with leak-off that is dependent
on fracture pressure. The model is, therefore, capable of analyzing DFIT data from the end of pumping to
days or even weeks after shut-in.
We first review Nolte's original G-function model and examine the assumptions inherent in the model.
We then present a new global pressure transient model for pressure decline after shut-in which not only


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We then present a new global pressure transient model for pressure decline after shut-in which not only
preserves the physics of unsteady-state reservoir flow behavior, elastic fracture mechanics and material
balance, but also incorporates the gradual changes of fracture stiffness/compliance due to the contact of
rough fracture walls during closure. This global model allows us to analyze the whole spectrum of DFIT data
by bridging before closure and after closure data seamlessly. Analysis of synthetic cases, along with field
data are presented to demonstrate how the coupled effects of fracture geometry, fracture surface asperities,
formation properties, pore pressure and wellbore storage can impact fracturing pressure decline and the
estimation of minimum in-situ stress.
It is shown that so-called "normal leak-off" behavior that is modeled using Carter leak-off is an
oversimplification that leads to significant errors in the interpretation of the data. All the before


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oversimplification that leads to significant errors in the interpretation of the data. All the before
closure analysis conducted under a "normal leak-off" assumption should be reexamined cautiously. Most
importantly, this article reveals that previous methods of estimating minimum in-situ stress often lead to
significant over or underestimates, because of their failure to account for changes in fracture stiffness/


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2 SPE-187348-MS
compliance correctly as the fracture closes progressively from the edge to the center. Based on our modeling
and simulation results, we propose a much more accurate and reliable method to estimate the minimum in-
situ stress, fracture pressure dependent leak-off rate and evaluate the compliance of the un-propped fracture.
Introduction
Diagnostic fracture injection tests (DFIT), which have also been referred to as Injection-Falloff Tests,
Fracture Calibration Tests, Mini-Frac Tests in the literature, involve pumping a fluid (typically water), at
a constant rate for a short period of time, creating a relatively small hydraulic fracture before the well
is shut in. The pressure transient data after shut-in is analyzed to obtain in-situ stresses and reservoir
properties. A typical pressure trend is qualitatively shown in Fig. 1. Conventionally, DFIT analysis has
focused on acquiring fracturing treatment design parameters, such as fluid efficiencies, leak-off coefficient


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focused on acquiring fracturing treatment design parameters, such as fluid efficiencies, leak-off coefficient
and fracture closure pressure (which is normally interpreted as minimum in-situ stress). However, in recent
years, DFIT analysis has been extended to obtain reservoir properties such as reservoir pore pressure, and
permeability in unconventional reservoirs, where traditional pressure transient tests are impractical. The
reservoir properties determined by DFIT are representative, because the created fracture can pass through
near-wellbore damage zone and provide a large volume of investigation for true formation properties. In
addition, the net pressure trends in a DFIT can be also used to infer the induced fracture complexity in
different geological settings (Potocki 2012). Thoese valuable information obtained from DFITs provides
key input parameters for modeling hydraulic fracture propagation (Wang 2015; Wang 2016; Wang et al.


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key input parameters for modeling hydraulic fracture propagation (Wang 2015; Wang 2016; Wang et al.
2016), stimulation design (Ramurthy et al. 2011), development of reservoir models (Mirani et al. 2016;
Loughry et al. 2015; Wang 2017) and post-fracture analysis (Fu et al. 2017). Without a realistic estimation
of the fracturing parameters and reservoir properties, it would not be possible to optimize hydraulic fracture
design and evaluate the economic viability of producing hydrocarbons from unconventional reservoirs.
Figure 1—Diagram showing sequence of events observed in a DFIT
The pressure decline data after shut-in can be divided into two distinct regions for analysis: before closure
pressure data and after closure pressure data (separated by the fracture closure pressure). Analysis models are
developed differently for these regimes. Before closure analysis (BCA) assumes normal leak-off behavior


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developed differently for these regimes. Before closure analysis (BCA) assumes normal leak-off behavior
or uses the portion of normal leak-off data (a straight line on G function and square root of time plots) to
estimate fracture geometry, leak-off coefficient or formation permeability (Mayerhofer et al. 1995; Nolte
1979 and 1986; Valko’ and Economides 1999). A departure from the straight line indicates fracture closure
(Castillo1987). After closure analysis (ACA) models have been developed by several authors (Benelkadi
and Tiab 2004; Chipperfield 2006; Nolte et al. 1997; Soliman et al. 2005; Soliman et al. 2010) to estimate


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SPE-187348-MS 3
pore pressure and infer formation permeability. Because both BCA and ACA only use some portion of the
DFIT data, global consistency of interpretation can't be guaranteed, even if the individual analyses match
a certain portion of the data well. Liu et al. (2016) and Marongiu-Porcu et al. (2014) attempted to bridge
BCA and ACA on a log-log plot and the closure pressure is picked at the end of 3/2 slope of the pressure
derivative. However, their work is still based on G-function model's assumptions and the 3/2 slope just
arises from a spatial integration of Carter's leak-off (McClure 2017). Currently, no model is available to
simulate DFIT pressure decline under a single, coherent mathematic framework, and correctly capture the
dynamic evolution of fracture stiffness/compliance during closure. In the following section, we are going to
review and examine the fundamental limitations of the G-function model, show that closure does not occur


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review and examine the fundamental limitations of the G-function model, show that closure does not occur
at a single point in time but is a continuous and gradual process, and derive a fully coupled global model
the fits the entire pressure curve during the entire fracture closure process.
The advent of fracturing pressure decline analysis was pioneered by the work of Nolte (1979 and 1986).
With the assumptions of power law fracture growth, negligible spurt loss, constant fracture surface area after
shut-in and Carter's leak-off model (one-dimensional leak-off of fluid from a constant pressure boundary,
the solution to the diffusivity equation predicts that the leak-off rate will scale with the inverse of the square
root of time), a remarkably simple and useful equation for the pressure decline can be obtained:
(1)
Here, is the instantaneous shut-in pressure at the end of pumping, Pf is the fracture pressure at


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(1)
Here, is the instantaneous shut-in pressure at the end of pumping, Pf is the fracture pressure at
dimensionless time Δt D.tp is the total pump time.r p is the productive fracture ratio, which is the ratio of
fracture surface area that is subject to leak-off to the total fracture surface area. For low permeability,
unconventional reservoirs, r p ≈1. To increase the readability of this article, r p will be assumed to be 1 for the
rest of the discussion. CL is Carter's leak-off coefficient which is a constant. Sf is the fracture stiffness, which
can be calculated using Table 1 for different fracture geometries. It is assumed that Sf is a constant until the
fracture closes instantaneously when the fluid pressure in the fracture reaches the closure pressure. Because
fracture stiffness is the reciprocal of fracture compliance, they will be used interchangeably throughout this
article.
Table 1—Fracture stiffness expressions for different fracture geometry models


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article.
Table 1—Fracture stiffness expressions for different fracture geometry models
E′ is the plane strain Young's modulus and can be calculated using Young's Modulus, E, and Poisson's
Ratio, v:
(2)
The dimensionless time Δt D is defined by:
(3)
Where 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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4 SPE-187348-MS
(4)
Where the g-function of time is approximated by,
(5)
From Eq. (1), we can infer that for normal leak-off behavior (constant Carter's leak-off coefficient,
constant leak-off area and constant fracture stiffness during fracture closure), the pressure declines
linearly with G(Δt D). Castillo (1987) usedNolte's G-function for modeling the pressure declinebehavior
and developed the straight-line plotof the G-function vs. pressure. The slope of this curve is used for the
computation of the leak-off coefficient that is independent of pressure. Any departure from this straight-
line is interpretedasclosure of the fracture.
Unfortunately, plots of pressure versus G-function often yield curves with multiple points of inflectionthat
have been attributed to abnormal leak-off behavior (such as pressure dependent leak-off, fracture height
recession, closing of secondary transverse fractures, and fracture tip extension), which makes it difficult


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recession, closing of secondary transverse fractures, and fracture tip extension), which makes it difficult
to interpretthe changes in slope and identify fracture closure. So identification of fracture closure pressure
and non-ideal behavior is usually done using plots of pressure and GdP/dG versus G-function (Barree and
Mukherjee, 1996; Barree, 1998, Barree et al., 2014), where the closure is picked at the tangential point
between a straight line that passes the origin and the GdP/dG curve. This prevailing method of determining
minimum in-situ stress (although has been widely accepted, but has never been theoretically proved) will be
classified and discussed as "tangent line method" in the following text. The fracturing pressure decline model
that underlying G-function plot suffers from two distinct and important issues: (1) leak-off is not pressure-
dependent, i.e. a constant pressure boundary is assumed (2) fracture compliance/stiffness is assumed to be


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dependent, i.e. a constant pressure boundary is assumed (2) fracture compliance/stiffness is assumed to be
constant during fracture closure. This is why G-function based models are only used for before closure
analysis and are not capable of analyzing DFIT data from the end of pumping to days or even weeks after
shut-in, which requires bridging both before and after closure data seamlessly.
Model Formulation
There are two essential components of the general DFIT model formulated in this paper. In this section, we
will present a generalization of the pressure transient model that couples the fluid pressure in the fracture
with the leak-off rate without making any assumptions about the pressure in the fracture that acts as a
boundary condition for leak-off. In addition, special cases will be demonstrated to link previous models and
our fully coupled approach.
Global DFIT Model
In unconventional reservoirs, fluid leak-off during pumping is very small and no filter-cake is formed on the


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Global DFIT Model
In unconventional reservoirs, fluid leak-off during pumping is very small and no filter-cake is formed on the
fracture surface because only water is injected. In addition, the duration of injection is very short (typically
3-5 minutes) while the total shut-in time may be days or even weeks. In order to correctly capture fracturing
pressure response during a DFIT in unconventional reservoirs, the pressure dependent leak-off at the fracture
surface and the dynamic changes of fracture compliance have to be accounted for. Fig. 2 illustrates one-
dimensional leak-off into a semi-infinite formation.


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SPE-187348-MS 5
Figure 2—Illustration of one-dimensional leak-off
Assuming linear Darcy flow and a slightly compressible, single phase fluid in the reservoir, the
differential form of the mass balance can be written as:
(6)
where P is the pressure, k is formation permeability, μ f is fluid viscosity, ϕ is formation porosity and ct is
total formation compressibility. Inside the fracture, the average fracture width 
  and fracturing net pressure
Pnet is related by the fracture stiffness:
(7)
From a material balance perspective (fluid compressibility is negligible compared to that of the fracture),
the rate of fluid leak-off(one wing of the fracture) into the formation, qf, after shut-in equals the rate of
shrinkage of fracture volume (one wing of the fracture), V f, as pressure declines:
(8)
And
(9)
where Af is the fracture surface area of one face of one wing of the fracture. So Eq. (8) can be re-written as
(10)


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(8)
And
(9)
where Af is the fracture surface area of one face of one wing of the fracture. So Eq. (8) can be re-written as
(10)
At the boundary of the fracture surface, both Darcy's flow and material balance have to be honored
(11)
Rearranging Eq. (11), we get the boundary condition at the fracture surface
(12)
In the above derivation, it is assumed that the whole fracture surface area is subject to leak-off, which is
the norm in low permeabity reservoirs. If only a portion of fracture surface is considered permeable, then


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6 SPE-187348-MS
one just need to multiply a leak-off surface area ratio (leak-off surface area devided by total fracture surface
area) on the left side of Eq. (12).
With initial condition (disregarding the pressure disturbance during the short injection period)
(13)
(14)
where P 0 is the initial pore pressure.
The governing Eq. (6), plus the initial condition of Eq. (13)-(14) and boundary condition of Eq. (11)
uniquely describes the pressure transient behavior during DFITs. In the above derivations, the influence of
wellbore storage (WBS) is not included. For typical DFITs, untreated water is normally used as injection
fluid. Since water is slightly compressible, the wellbore fluid will expand after shut-in with the declining
pressure. Ignoring the early time fluid transient behavior along the wellbore after shut-in and assuming
fracturing pressure and wellbore pressure have reached quasi-equilibrium, leak-off volume into formation


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fracturing pressure and wellbore pressure have reached quasi-equilibrium, leak-off volume into formation
should equal the shrinkage of fracture volume plus the expansion of wellbore fluid. We can modify the leak-
off rate from Eq. (10) to account for wellbore fluid expansion as follows:
(15)
Where V W is half the wellbore volume (only one wing of the fracture needs to be modeled) and C W
is the compressibility of water. Using Eq. (15) with Eq. (10), we can define the fracture-wellbore system
stiffness as:
(16)
Under extreme circumstances, when wellbore storage is negligible (V w Cw ≈ 0), fracture stiffness
dominates the system stiffness, and when Vw CW are large and Af is small, WBS dominates the system
stiffness. If WBS has to be included in the model, one just need simply replace Sf with SS in Eq. (12). It should
be emphasized there are two stages of fracture closure: mechanical closure, where the fracture surfaces come


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be emphasized there are two stages of fracture closure: mechanical closure, where the fracture surfaces come
into contact at asperities, and the other is hydraulic closure, where fracture pressure in the closed section
is disconnected from the open section or wellbore pressure. In the above derivations, we assume that the
mechanically closed fracture still retains hydraulic conductivity because of its residual fracture width.
Special Case 1: Constant Fracture Compliance and Carter's Leak-off.   If we assume that a constant
shut-in pressure (ISIP) is applied at the fracture surface, the solution of Eq. (6) to obtain the leak-off or
the velocity Uf across fracture surface into an intact, homogenous formation can be found as ( Economides
and Nolte 2000):
(17)
If we take Δt out of Eq. (17), then the R.H.S of Eq. (17) essentially becomes Carter's leak-off coefficient
CL, which is a constant parameter. The total leak-off rate qf from one wing of the fracture starting from
shut-in is:
(18)


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SPE-187348-MS 7
Combining Eq. (18) and Eq. (10), we have:
(19)
Eq. (19) is a simple first order differential equation, with the initial condition of Pf (Δt = 0) = ISIP, the
solution of Eq. (19) leads to the pressure declining proportionally to the square root of time:
(20)
Eq. (20) is the fundamental basis for the square root of time plot. From Eq. (1) and (20), we can infer that
for normal-leak-off behavior with constant fracture stiffness and assuming Carter's leak-off, the pressure
and its derivative will be straight lines on both G-function and the square root of time plots. The fracture
pressure continues declining even when it reaches the initial reservoir pore pressure. This non-physical
prediction indicates that the assumption of Carter's leak-off and constant stiffness made in the derivation
prevent Notle's G-function model and the square root of time model from capturing the actual pressure
response during fracture closure and late time pressure transient behavior.


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response during fracture closure and late time pressure transient behavior.
Special Case 2: Constant Fracture Compliance and Fracture Pressure Dependent Leak-off.   When the
leak-off rate is fracture pressure dependent, a closed analytic form of pressure decline is not available,
but the solution can be obtained using a superposition time. A detailed derivation of the pressure decline
solution is presented in Appendix A. And from Eq. (A.4), the fracture pressure at the nth time interval, Pf,n,
can be calculated explicitly:
(21)
where Pf,0 = P0, Pf,1 = ISIP, Δt 1 = 0. It is important to note that in this article, the term "fracture pressure
dependent leak-off" (FPDL) specifically refers to a leak-off rate that depends on the declining fluid pressure
inside the fracture, not to how the leak-off rate changes with net stress in the rock surrounding the fracture,
which is often denoted as pressure dependent leak-off or PDL ( Barree et al. 2009). In some cases, excessive


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which is often denoted as pressure dependent leak-off or PDL ( Barree et al. 2009). In some cases, excessive
pressure drop can be observed during early-time of shut-in in low permeability reservoirs, but it is the result
of high apparent ISIP, wellbore storage and additional friction away from the wellbore, and not caused by
PDL, which is unlikely to happen in nanodarcy permeability systems.
Variable Fracture Compliance Due to Closure on Rough Surfaces and Asperities
As pressure declines inside the fracture after shut-in, the fracture will gradually close and the fracture
aperture will approach the scale of the surface roughness. If the fracture faces are perfectly parallel and
smooth, they will come into contact all at once when the fluid pressure inside the fracture declines to the
far field stress, and the fracture is then mechanically and hydraulically closed. However, there is abundant


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far field stress, and the fracture is then mechanically and hydraulically closed. However, there is abundant
evidence to suggest that fractures retain their conductivity after the walls have come into contact (mechanical
closure). Fractures retain a finite aperture after mechanical closure due to a mismatch of asperities on the
fracture walls. van Dam et al. (2000) presented scaled laboratory experiments on hydraulic fracture closure
behavior. They observed up to a 15% residual aperture (compared to the maximum aperture during fracture
propagation) long after shut-in. Fredd et al. (2000) demonstrated fracture surface asperities can provide
residual fracture width and sufficient conductivity in the absence of proppants. Using sandstone cores from
the East Texas Cotton Valley formation, sheared fracture surface asperities that had an average height of


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about 2.286 mm were observed. Warpinski et al. (1993) reported hydraulic fracture surface asperities of
about 1.016 mm and 4.064 mm for nearly homogeneous sandstones and sandstones with coal and clay-
rich bedding planes, respectively. Sakaguchi et al. (2008) created tensile fractures in large rock blocks and
measured the asperity height and distribution. Their work shows that the fracture surfaces can be assumed
to be a fractal object and most of the asperities fall within 1 to 2 mm in height. Wells and Davatzes (2015)
conducted topographic measurement on dilated fractures from core samples and found the asperity height
ranges from hundreds to thousands of micrometers. Bhide et al. (2014) created X-ray microtomographic
images from shear induced fractures and the roughness values obtained varied from 1.8 to 1.95 mm along
the length of the rock samples. Zou et al. (2015) conducted experiments on 20 fractured shale samples and


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the length of the rock samples. Zou et al. (2015) conducted experiments on 20 fractured shale samples and
found the average asperity height to be 1.88 mm. An experimental study ( Zhang et al. 2014) on Barnett Shale
samples reveals that the surface topography of the displaced fracture can be altered because of rock failure,
and the fracture surface exhibited parallel strips of crushed asperities. Field measurements ( Warpinski et al.
2002) using a down-hole tiltmeter array indicated that the fracture closure process is a smooth, continuous
one which often leaves 20%-30% residual fracture width, regardless of whether the injection fluid is
water, linear-gel or cross-linked-gel. The microscopic measurement and modeling of surface roughness and
mechanical properties of asperities can often be up-scaled to macroscopic contact laws that relate fracture
width and the associated contact stress. Willis-Richards et al. (1996) proposed contact law to relate fracture


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width and the associated contact stress. Willis-Richards et al. (1996) proposed contact law to relate fracture
width and the net closure stress for fractured rocks, based on the work of Barton et al. (1985):
(22)
where wf is the fracture aperture and, w0 is the contact width, which represents the fracture aperture when
the contact normal stress is equal to zero, σ c is the contact normal stress on the fracture, and σ ref is a contact
reference stress, which denotes the effective normal stress at which the aperture is reduced by 90%. it should
be emphasized that the contact width w0 is determined by the tallest asperities, and the strength, spatial and
height distribution of asperities are reflected by the contact reference stress σ ref (e.g., if the tallest asperities
on two fracture samples are the same, then they should have the same W0, but the one with a higher median
asperity height will have higher value of σ ref, provided other properties are the same).


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asperity height will have higher value of σ ref, provided other properties are the same).
With known fracture geometry, rcok properties and surface roughness, the question now is how to
estimate fracture stiffness/compliance as a function of pressure, if it continuously changes during fracture
closure. Wang and Sharma (2017) presented a method and general algorithms to model the dynamic behavior
of hydraulic fracture closure on rough fracture surfaces and asperities, using linear elastic solutions that
coupled with contact law for three different fracture models (PKN, KGD and radial fracture geometry).
Given the fracture geometry, rock properties, contact parameters and minimum principal stress, their
approach can predict the evolution of fracture aperture profile, total fracture volume and fracture stiffness
as fracturing pressure declines. A brief discussion of modeling non-local dynamic fracture closure behavior
can be found in the Appendix B.


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as fracturing pressure declines. A brief discussion of modeling non-local dynamic fracture closure behavior
can be found in the Appendix B.
Assuming a Young's modulus of 20 GPa, Poisson's ratio of 0.25, W0 of 2 mm, σ ref of 5 MPa for a
PKN fracture geometry with 10 m fracture height, 35 MPa minimum in-situ stress, the evolution of the
fracture width profile and contact stress distribution can be determined, as the fluid pressure inside the
fracture gradually declines. The results are shown in Fig. 3 and Fig. 4. To demonstrate the impact of fracture
roughness and surface asperities on fracture closure behavior, the case without surface asperities (fracture
surface is completely planar and smooth) is also included. The result shows that at relative high fracturing
fluid pressure, the fracture asperities have negligible impact on fracture width distribution, and the contact
stress is always concentrated at the tip of the fracture, where the contact stress is much higher than in the


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stress is always concentrated at the tip of the fracture, where the contact stress is much higher than in the
middle of the fracture. We can also see that the fracture surfaces do not contact each other like parallel
plates. In fact, the fracture closes on rough surfaces starting from the tip, and closes progressively all the


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SPE-187348-MS 9
way from the edges to the center of fracture. As fluid pressure continues to decline, more and more of the
fracture surfaces come into contact and these changes the subsequent fracture closure behavior. At lower
fluid pressures, contact stresses start to counter-balance the in-situ stress and the fracture becomes stiffer
and less compliant. If the fracture faces were perfectly parallel and smooth, the fracture width would have
collapsed to zero when the fluid pressure dropped to 35 MPa. The moment when all fracture surfaces come
into contact on asperities and the contact stress becomes non-zero on the entire fracture surface, the fracture
is mechanically closed and this mechanical closure stress is higher than the minimum in-situ stress.
Figure 3—Fracture width evolution with and without asperities at different fluid pressure for a PKN geometry
Figure 4—Fracture width and the corresponding contact stress distribution at different fluid pressure for a PKN geometry


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Figure 4—Fracture width and the corresponding contact stress distribution at different fluid pressure for a PKN geometry
The simulated fracture volume evolution as a function of fluid pressure is shown in Fig. 5. As can be
seen, when the fluid pressure inside the fracture is relatively high, the fracture volume declines linearly
with pressure (indicates roughly constant fracture compliance/stiffness). However, as the pressure declines
to a certain level (2 MPa higher than the input minimum in-situ stress of 35 MPa), the fracture volume
and pressure departs from a linear relationship. The traditional method of estimating fracture stiffness using
Table 1 can only be used when the fracture pressure is still relatively high and the asperities are only in
contact at the edges of the fracture. However, as pressure declines further and more and more fracture surface
area comes into contact, the effective fracture stiffness becomes pressure dependent:
(23)


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10 SPE-187348-MS
Figure 5—Fracture volume and fracture stiffness evolution as fluid pressure declines for a PKN geometry
The relationship between Pf and Vf can be used to calculate the pressure dependent fracture stiffness
and the result is also shown in Fig. 5. This pressure dependent fracture compliance/stiffness obtained
from non-local fracture closure modeling (Wang and Sharma 2017) can be incorporated into Eq. (12) to
make Sf pressure dependent. The "tangent line method" for picking up minimum in-situ stress is based
on the underlying assumptions of G-function model that assumes perfectly smooth fracture surfaces, and
the stiffness remains constant until fluid pressure drops to minimum in-situ stress. This leads to a sudden
increase in fracture stiffness when all fracture surfaces get into contact simultaneously. Fracture stiffness
only remains constant at high fracture pressures. It starts to increase long before the fracturing pressure


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only remains constant at high fracture pressures. It starts to increase long before the fracturing pressure
declines to the minimum in-situ stress (35 MPa in this case). McClure et al. (2016) proposed a "compliance
method" for picking closure pressure on the G-function plot based on numerical simulations, where closure
pressure is picked at the point where the fracture stiffness starts to increase. Their method can overestimate
minimum in-situ stress if the mechanical closure pressure is interpreted as the minimum in-situ stress.
There is compelling field evidence to indicate that this occurs in the field. Fig. 6 shows the normalized
tiltmeter response plotted against wellbore pressure during the shut-in period of 2B well at three different
stations from the GRI/DOE M-site. Each tiltmeter response is normalized by dividing by the maximum
tilt (displacement, also an indication of fracture width) measured at that instrument during the test. The


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tilt (displacement, also an indication of fracture width) measured at that instrument during the test. The
tiltmeter demonstrates that soon after shut-in, the measured displacement declines linearly with pressure.
If the fracture surface area remains constant during shut-in, then the fracture volume is proportional to the
average displacement and should also decline linearly with the pressure within this period. After the wellbore
pressure declines to a certain level, the measured displacement vs pressure departs from a linear relationship.
This field measurement is consistent with the general trend shown in Fig. 5. As pressure continues declining,
more and more fracture surface area comes into contact with the rough fracture surfaces and asperities, and
the fracture stiffness increases gradually. Even though these data were measured from the end of injection to
weeks of shut-in, we can still observe the existence of a residual fracture width that is supported by surface


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weeks of shut-in, we can still observe the existence of a residual fracture width that is supported by surface
asperities and mismatches even after the fracturing fluid pressure drops below the minimum in-situ stress.
So considering the nature of mineralogical heterogeneity and the existence of thin laminated rock layers,
the formation of asperities and tortuous fracture walls are inevitable during hydraulic fracture propagation.


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SPE-187348-MS 11
Figure 6—Normalized tiltmeter data from shut-in of 2B from the GRI/DOE M-site. Data Courtesy of Norm Warpinski
Simulation Results and Analysis
The pressure decline behavior during DFITs in low permeability formations is governed by Eq. (6) (a PDE)
with Eq. (12) (an ODE) acting as the boundary condition. To solve this system of PDE with an ODE
boundary condition, the method of lines (MOL) is used, by replacing the spatial derivative in the PDE with
algebraic approximations. The PDE can then be transformed into a system of ODEs, which can be solved
simultaneously and efficiently by well-established numerical methods (Schiesser and Griffiths 2009). Since
the MOL essentially replaces the problem PDEs with systems of approximating ODEs, the addition of other
ODEs is easily accomplished. In this section, our fully-coupled global DFIT model is used to investigate how
different fracture geometry, contact parameters and reservoir properties impact fracturing pressure response


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different fracture geometry, contact parameters and reservoir properties impact fracturing pressure response
and our interpretation. It should be mentioned that the discussion presented in this section is restricted to
low permeability formations where the leak-off rate and poroelastic effects are small.
Normal leak-off behavior
Normal leak-off behavior is a special case of our model that satisfies all the assumptions of Nolte's G-
function model, and it can be clearly identified with straight lines of pressure and its derivatives on G-
function and the square root of time plots. All previous BCA models assume that these straight lines indicate
normal leak-off behavior and hence, leak-off coefficient or formation permeability can be determined based
on this portion of the data. However, one key assumption in all BCA is Carter's leak-off. To examine the
impact of incorrectly assuming Carter's leak-off for the duration of the DFIT, we extend the concept of


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impact of incorrectly assuming Carter's leak-off for the duration of the DFIT, we extend the concept of
"normal leak-off behavior" to account for different fracture boundary conditions, while other assumptions
remain the same (e.g., constant fracture compliance/stiffness and constant surface area during closure). The
simulation results from three different leak-off models are compared using input parameters from Table
2. Fig. 7 and Fig. 8 show the simulated pressure and its derivatives on G-function and square root of
time plots for different formation permeability. As expected, Carter's leak-off (using Eq. (1) or Eq. (20),
they are interchangeable) always produce straight lines on diagnostic plots. But when pressure dependent
leak-off at the fracturing surface is considered (using Eq. (21) and our fully coupled simulation), we no
longer obtain straight lines. This is because as fracturing pressure declines, leak-off rate departs from the
linear relationship with


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longer obtain straight lines. This is because as fracturing pressure declines, leak-off rate departs from the
linear relationship with 
 , so the assumption of Carter's leak-off is violated. Carter's leak-off model
overestimates the leak-off rate and pressure drop by assuming a constant ISIP at the fracturing surface and
leads to larger errors as shut-in time gets longer.


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12 SPE-187348-MS
Figure 7—Comparison of "normal leak-off" behavior for different models, k=0.0001 md
Figure 8—Comparison of "normal leak-off" behavior for different models, k=0.001 md
Table 2—Input parameters for DFIT simulation with constant fracture compliance
The most important observation here is that the so-called "normal leak-off behavior" with straight lines
on G-function and the square root of time plots is not "normal" at all. For example, the closure time for
typical DFIT in a formation with 0.0001 md permeability is around 2~4 days. And in the above case shown
in Fig. 7, the fracturing pressure and its derivative depart from the straight line after 4 hours of shut-in,
even with constant fracture compliance/stiffness, surface area and reservoir properties. Many field data sets
do show "normal leak-off behavior" with straight lines that extend all the way to the point of downward


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do show "normal leak-off behavior" with straight lines that extend all the way to the point of downward
deflection. This occurs if some mechanisms that accelerate pressure drop (e.g., the gradual increase of
fracture stiffness) happen to offset the effects of declining pressure at the fracture surface, so the pressure
data fits the straight lines predicted by Carter's leak-off model coincidentally. So any DFIT analysis that is
based on the assumption of Carter's leak-off and straight lines on G-function and the square root of time
plots should be re-examined.


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SPE-187348-MS 13
Variable Fracture Compliance on Pressure Decline and In-Situ stress Estimation
As discussed earlier, G-function and previous BCA models assume Carter's leak-off behavior as if the
pressure in the fracture is a constant value (i.e., ISIP). In addition, these models also assume fracture
compliance/stiffness is constant and only changes abruptly at the moment of fracture closure, when
fracturing pressure drops to the minimum in-situ stress. The failure to account for the coupled mechanisms
of pressure dependent leak-off at the fracture surface and the gradual increase of fracture stiffness during
closure explains why these models are only useful on a small portion of the before closure data, and are
not able to simulate the fracturing response for the entire duration of the DFIT test. These simplifying
assumptions also make their interpretation of DFIT data and estimation of minimum in-situ stress inaccurate.


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assumptions also make their interpretation of DFIT data and estimation of minimum in-situ stress inaccurate.
In this section, our global DFIT model is used to investigate how each factor impacts the pressure decline
trend and our estimation of minimum in-situ stress and fracture compliance.
PKN Fracture Geometry.   We first examine a PKN geometry using parameters listed in Table 2 and
assuming w0 is 2 mm, σ ref is 5 MPa and reservoir permeability is 0.0005 md as the Base Case. Fig. 9 shows
the contact stress at different fracture width. As expected, when the fracture width is larger than the contact
width w0, the contact stress is zero. However, when the fracture width is smaller than the contact width,
the contact stress and fracture width follows a hyperbolic relationship. Fig. 10 shows the corresponding
fracture stiffness evolution for different W0 with the given fracture geometry and rock properties. The results


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fracture stiffness evolution for different W0 with the given fracture geometry and rock properties. The results
indicate that as the fracture width decreases, the rough fracture walls will come into contact sooner if the
contact width is larger, so the noticeable changes of fracture stiffness occur earlier when the contact width
is larger. We can also observe that when the contact width is larger, the increase in fracture stiffness is more
gradual and smooth. In the extreme case of W0 close to zero, as would be the case for relative flat fracture
surfaces, we can infer that fracture stiffness will increase abruptly as soon as the fluid pressure drops to the
minimum in-situ stress (35 MPa in this case). This pressure dependent fracture stiffness obtained from non-
local fracture modeling (Wang and Sharma 2017) can be directly put into the boundary ODE of Eq. (12) to
predict fracture pressure decline for certain reservoir properties.


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predict fracture pressure decline for certain reservoir properties.
Figure 9—The relationship between contact stress and fracture width for different W 0


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14 SPE-187348-MS
Figure 10—Fracture stiffness evolution for different W 0 with a PKN geometry
Fig. 11 shows the fracturing pressure and its derivatives for different W0 on G-function and square root
of time plots. Relate with Fig. 9 and Fig. 10, we can see that the contact width impacts pressure decline
response significantly, because it alters the evolution of fracture stiffness. Large contact width leads to a
smooth pressure decline trend while small contact width leads to steep changes in pressure decline rate and
pressure derivatives. We can also infer that if the contact width is close to zero and all fracture walls come
into contact simultaneously, then a sudden change in pressure decline rate and the pressure derivative spikes
on both G-function and square root of time plots are inevitable, which is unrealistic and never observed in
field cases. So the conventional assumption that a fracture closes on flat, smooth fracture surfaces where W0


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field cases. So the conventional assumption that a fracture closes on flat, smooth fracture surfaces where W0
= 0 does not reflect reality. In addition, different contact widths lead to different derivative slopes, so any
BCA models that attempt to match this portion of data and estimate reservoir properties without considering
the variable fracture compliance is questionable. Fig. 11 also shows G-function and square root of time plots
give the same quantitative information only slightly different in scales. More importantly, the estimated
minimum in-situ stress from pressure derivatives using the "convention method" and "compliance method"
give inconsistent results for different contact widths, with the same input parameters of fracture surface
area, minimum in-situ stress and reservoir properties. A detailed comparison and a new approach to estimate
minimum in-situ stress will be discussed later.
Figure 11—Pressure decline response for different W 0 with a PKN geometry


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minimum in-situ stress will be discussed later.
Figure 11—Pressure decline response for different W 0 with a PKN geometry
Next, we examine how contact reference stress affects the pressure decline response. Fig. 12 and Fig. 13
show the relationship between contact stress and fracture width for different contact reference stress and the
corresponding fracture stiffness evolution at different fracturing pressure. For the same contact width, the
higher the contact reference stress, the more rapid the increase of contact stress as the fracture width shrinks.
Physically, the contact reference stress represents how hard and strong the fracture surface asperities are.
The lower the contact reference stress, the more gradual the change in fracture stiffness as pressure declines.


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SPE-187348-MS 15
Even though the contact reference stress does not have much impact on the pressure at which the fracture
stiffness starts to changes noticeably, it does impact the fracture stiffness evolution, as shown in Fig. 13.
Fig. 14 shows the fracturing pressure and its derivatives for different contact reference stress on G-function
and the square root of time plots. It can be observed that the contact reference stress has negligible influence
on early time pressure decline when fracturing pressure is still high. This is because only the fracture edges
start to come into contact during this period and the fracture stiffness remains roughly constant. However,
after this period as the fracture pressure declines further, more and more fracture surfaces come into contact
and the contact reference stress begins to affect the pressure decline trend and the peak value of the pressure
derivatives.
Figure 12—The relationship between contact stress and fracture width for different σ ref


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derivatives.
Figure 12—The relationship between contact stress and fracture width for different σ ref
Figure 13—Fracture stiffness evolution for different σ ref with a PKN geometry
Figure 14—Pressure decline response for different σ ref with a PKN geometry
Besides the contact parameters, fracture geometry also affects fracture stiffness and its evolution during
closure. Fig. 15 shows fracture stiffness evolution for different fracture height while all the other parameters


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16 SPE-187348-MS
remain the same as the Base Case. Similar to what Table 1 implies, for a PKN fracture geometry, the smaller
the fracture height, the higher the initial fracture stiffness. We can also observe that smaller fracture height
leads to noticeable changes in fracture stiffness at higher fracturing pressure. This is because smaller fracture
height results in smaller fracture width at the same net pressure, so the more fracture surfaces can come into
contact at a higher fracture pressure. When fracture height is 5m, the effective fracture stiffness is already
affected by surface contact when fluid pressure is 39 MPa.
Figure 15—Fracture stiffness evolution for different fracture height with a PKN geometry
Fig. 16 shows the fracturing pressure and its derivatives for different fracture height on G-function and
the square root of time plots. The results indicate that fracture height impacts the pressure decline trend


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the square root of time plots. The results indicate that fracture height impacts the pressure decline trend
significantly. Larger fracture height leads to later occurrence of the peak of the pressure derivative and this
also increases the peak value of the pressure derivatives.
Figure 16—Pressure decline response for different fracture height with a PKN geometry
Fig. 17 shows the fracturing pressure and its derivatives for different reservoir permeability on G-function
and the square root of time plots. As expected, the pressure declines more rapidly when the reservoir
permeability is large and the decline rate slows down as the difference between fracturing pressure and
initial reservoir pressure becomes smaller and smaller.


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SPE-187348-MS 17
Figure 17—Pressure decline response for different reservoir permeability with a PKN geometry
Next, we examine the impact of wellbore storage on the fully coupled pressure decline trend. The
water compressibility is assumed to be 4.35e-4 MPa−1. Fig. 18 shows the fracture-wellbore system stiffness
evolution for different wellbore volume. As can be seen, when fracturing pressure is high, fracture stiffness
dominates the system stiffness. However, as fracturing pressure continues declining, the fracture become
less and less compressible and the role of wellbore storage becomes apparent. In general, the larger the
wellbore volume, the more gradual and slower the increase in system stiffness will be.
Figure 18—Fracture-wellbore system stiffness evolution for different wellbore volume with a PKN geometry
Fig. 19 shows the corresponding fracturing pressure and its derivatives for different wellbore volumes


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Fig. 19 shows the corresponding fracturing pressure and its derivatives for different wellbore volumes
on G-function and square root of time plots. It can be observed that larger wellbore volume leads to more
gradual pressure decline trends. A larger wellbore volume also delays the occurrence of fracture closure
and lowers the peak of the pressure derivative curve. It can be seen that WBS has a small impact during
early time of shut-in when the system stiffness is still dominated by fracture stiffness, however, as more
and more of the fracture surface comes into contact and the fracture becomes stiffer, WBS effects become
apparent, and the after-flow of fluid from wellbore to fracture long after shut-in decelerates the pressure
decline rate and extends the tail of the pressure derivative after its reaches the peak. So the WBS effect on
pressure decline response must be accounted for when interpreting and analyzing DFIT data.


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Figure 19—Pressure decline response for different wellbore volume with a PKN geometry
Fig. 20 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-
function and square root of time plots. It can be observed when the initial reservoir pressure is low; the
pressure declines more rapidly. This is also reflected by Eq. (18), a lower reservoir pressure leads to a higher
leak-off rate. However, when the reservoir pressure is over pressurized with high initial pore pressure, the
pressure decline trend resembles a "normal-leak-off behavior". In this case (P 0=28MPa), one can still notice
that there exist a subtle "bump" in the pressure derivatives that indicates an increase in fracture stiffness as
reflected in Fig. 10. For small values of σ ref, this increase can be too gradual to be noticeable on G-function
and square root of time plots. This example deomonstrates that long closure time not only can be attributed


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and square root of time plots. This example deomonstrates that long closure time not only can be attributed
to low formation permeability, but also can be the result of small pressure difference between ISIP and
initial pore pressure.
Figure 20—Pressure decline response for different initial reservoir pressure with a PKN geometry
In all the above synthetic simulations, the input minimum in-situ stress remains 35 MPa while other
parameters are modified from the Base Case. The pressure decline trend varies substantially for different
fracture geometry, contact parameters, reservoir permeability, wellbore volume and reservoir pore pressure.
Any interpretation of the pressure data to determine the minimum in-situ stress must properly account for
all these parameters. Choosing minimum in-situ stress where the pressure or its derivative departs from a
straight line (Nolte 1979 and 1986) is not reliable, because it ignores the changes in fracture compliance/


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straight line (Nolte 1979 and 1986) is not reliable, because it ignores the changes in fracture compliance/
stiffness during closure and the pressure dependent leak-off at the fracture surface. In addition, pressure does
not decline linearly with G-function or 
  in most cases. The most widely used "tangent line method" to
pick up minimum in-situ stress using a tangential line with pressure derivatives ( Barree et al., 2009; Barree
et al., 2014) always underestimates minimum in-situ stress, especially in under-pressure reservoirs. Even
though this method has been the common practice in the industry, it is an emprical method and is subject to
the assumptions discussed earlier. The "compliance method" identifies closure pressure as the point at which
fracture stiffness starts to change (McClure et al. 2016). As we have discussed earlier, the fracture closes


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SPE-187348-MS 19
progressively from the edges to the center, As a result the fracture stiffness begins to increase long before
the fracture pressure drops to the minimum in-situ stress. A comparison of different methods of estimating
minimum in-situ stress (closure pressure is interpreted as minimum in-situ stress) for all previous synthetic
cases is presented in Table 3.
Table 3—Comparison of different methods of estimating minimum in-situ stress for PKN fracture geometry
Examining Eq. (6)~Eq. (16) closely, it is evident that the pressure decline is governed by a linear PDE
system in the formation that is coupled with an ODE at the fracture surface. The minimum in-situ stress is
just one of many implicit factors that shape the evolution of the fracture pressure. Previous DFIT models
simplified this PDE-ODE system assuming Carter's leak-off and constant fracture stiffness during fracture


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simplified this PDE-ODE system assuming Carter's leak-off and constant fracture stiffness during fracture
closure, so that a closed form of an analytic solution could be obtained. Since our model shows minimum
in-situ stress implicitly coupled with pressure, it is not possible to analytically estimate the minimum in-situ
stress from a single diagnostic plot. Based on the analysis of our synthetic cases, it is clear that the "tangent
line method" underestimates minimum in-situ stress and the "compliance method" overestimates closure in
all cases if the closure pressure is interpreted as the minimum in-situ stress.
Our analysis shows that the answer lies between these two limiting cases, regardless of parameters
used in simulation. To keep the analysis pratical and not resorting to a complex inverse problem, we
propose a "variable compliance method," where the minimum in-situ stress can be obtained by averaging


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propose a "variable compliance method," where the minimum in-situ stress can be obtained by averaging
the dimensionless G-time or the square root of time of these two methods and extrapolating back to the
pressure curve that corresponds to the averaged G-time or the square root of time. An exception is when
pressure derivative plot resembles the so-called "normal leak-off behavior," the changes in fracture fracture/
compliance may not be detectable on diagnostic plots and the "tangent line method" gives an estimation of
minimum in-situ stress within an acceptable error.
The estimated minimum in-situ stress using "variable compliance method" is also included in Table 3.
When we compare the estimated minimum in-situ stress obtained from the different methods with the input
minimum in-situ stress, Table 3 shows that the "variable compliance method" gives significantly lower
errors. The maximum error does not exceed a few 100 KPa, while the "tangent line method" yield errors


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errors. The maximum error does not exceed a few 100 KPa, while the "tangent line method" yield errors
that are an order of magnitude larger.
Even though we have focused on the PKN fracture geometry in this section, these conclusions are also
applicable to a KGD fracture geometry. We can obtain the results for a KGD geometry if we replace fracture
height with fracture length in calculating the evolution of fracture stiffness, as discussed in the Appendix B.


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20 SPE-187348-MS
PKN Fracture Geometry with Multiple Layers.   In previous section, we have analyzed pressure decline
response of DFIT in a single layer; however, it is possible that fracture can penetrate into adjacent layers
where the minimum in-situ stresses are different. Fig. 21 illustrates the examples where fracture geometry
span across multiple layers. In a single layer case, fracture gradually closes from edges to center on rough
fracture wall, while in multilayer cases; fracture can close first within the interval where the minimum in-
situ stress is the largest.
Figure 21—Illustration of multilayer cases with different minimum in-situ stresses
Assuming the input parameters are the same as the base case in the previous section (provided in Table
2), and fracture geometry and minimum in-situ stresses are illustrated in Fig. 21, the fracture stiffness
evolution can be calculated using non-local fracture modeling (Wang and Sharma 2017), as shown in Fig.


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evolution can be calculated using non-local fracture modeling (Wang and Sharma 2017), as shown in Fig.
22. "Multilayer Case 1" is the most common scenario where height recession can happen during fracture
closure process where fracture closes first within formation barriers (e.g., Layer 1 and Layer 3) where
stresses are higher. Compared to the single layer case in Fig. 22, height recession (i.e., Multilayer Case 1)
leads to earlier and steeper increase in fracture stiffness. The fracture stiffness evolution of the single layer
case and "Multilayer Case 2" are similar, because they have the same average stress.
Figure 22—Stiffness evolution for single and multilayer cases
Fig. 23 shows the fracturing pressure and its derivatives for single and multilayer cases on G-function
and square root of time plots. It can be observed that the trend of pressure derivatives generally reflects


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and square root of time plots. It can be observed that the trend of pressure derivatives generally reflects
the evolution of fracture stiffness. The case of height recession (i.e., Multilayer Case 1) leads to earlier and
steeper increase in pressure derivatives on both G-function and square root of time plots.


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SPE-187348-MS 21
Figure 23—Pressure decline response for single and multilayer cases
A comparison of different methods of estimating minimum in-situ stress for above single and multilayer
cases is presented in Table 4. Again, we can oberve that the tangent line method under estimates while
compliance method overestimates minimum in-situ stress (i.e., 35 MPa in the target layer), and our "variable
compliance method" gives the least errors for all cases.
Table 4—Comparison of different methods of estimating minimum in-situ stress for single and multilayer cases
Based on the disscussion by McClure et al. (2016), the "compliance method" gives an estimate of the
mecanical closure pressure, and this closure pressure can be 1-2 MPa above the actual minimum in-situ
stress, depending on fracture geometry and contact width. Our anlaysis in Table 3 shows that the differrence
between the infered closure pressure and input minimum in-situ stress that is constrained within 1-2 MPa


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between the infered closure pressure and input minimum in-situ stress that is constrained within 1-2 MPa
is only limited to a few cases where fracture height is large or contact width is small. For a typical DFIT,
the injeciton volume is normally only a few cubic meters and the resulting fracture height or radius is often
limited. In addition, depending on rock mineralogy and heterogeneity, the height of asperities can range
from hundreds to thousands of micrometers (Bhide et al. 2014; Fredd et al. 2000; Sakaguchi et al. 2006;
Wells and Davatzes 2015; Warpinski et al. 1993; Zou et al. 2015), and, as shown by the field tiltimeter
survey, the residual fracture width can be as large as 20-30% of the maximum fracture width during pumping
(Warpinski et al. 2002). Remember is determined by the tallest asperities, so it is challenging to calibrate
the minimum in-situ stress using the "compliance method". In addtion, if the fracture has penetetrated


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the minimum in-situ stress using the "compliance method". In addtion, if the fracture has penetetrated
into barrier layers with higher minimum pricinpal stress (e.g., Multilayer Case1 in Table 4), the fracture
walls will come into contact at a higher pressure within the barrier layers and increase the overall fracture
stiffness, so the "compliance method" can furthur overstimate the minimun in-situ stress. However, in all
simulated cases the presented "variable compliance method" provides a much more consistent estimation
of minimum in-situ stress. Most importantly, the results are insensitive to input parameters such as fracture
surface roughness and fracture geometry, which significantly reduces the uncertainties associated with the
"compliance method".
Radial Fracture Geometry.   Next, we investigate the pressure decline response for radial fracture geometry,
using parameters from Table 2 and assuming W0 is 2 mm, σ ref is 5 MPa, fracture radius is 10 m and reservoir


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22 SPE-187348-MS
permeability is 0.0005 md as the Base Case. Fracturing pressure decline response for different W0 and are
shown in Fig. 24. The results and trends are similar to those obtained for the PKN geometry, a small contact
width leads to steeper changes in pressure decline rate and the slope of pressure derivative, and increases
the peak value of pressure derivative.
Figure 24—Pressure decline response for different W 0 with a radial geometry
Fig. 25 shows the pressure decline response on G-function and the square root of time plots for different
contact reference stress. As in the PKN cases, a larger value of contact reference stress increases the pressure
decline rate after the fracturing pressure drops to a certain level and reduces the peak value of pressure
derivatives.
Figure 25—Pressure decline response for different σ ref with a radial geometry
Fig. 26 shows the corresponding pressure decline response on G-function and the square root of time


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Fig. 26 shows the corresponding pressure decline response on G-function and the square root of time
plots. A larger fracture radius leads to lower fracture stiffness and this, in turn, results in a lower pressure
decline rate.
Figure 26—Pressure decline response for different fracture radiiwith a radial geometry.


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SPE-187348-MS 23
Fig. 27 shows the fracturing pressure and its derivatives for different reservoir permeability on G-
function and the square root of time plots. As expected, pressure declines more rapidly when the reservoir
permeability is large. At early time, the pressure decline rate increases because an increase in the fracture
stiffness dominates changes in leak-off rate. At late time when the fracture becomes less compliant, the
pressure decline rate decreases due to smaller and smaller differences between the fracture pressure and the
initial reservoir pressure. Eventually, the leak-off rate will drop to zero as the fracture pressure becomes
equal to the reservoir pressure.
Figure 27—Pressure decline response for different reservoir permeability with a radial geometry
Fig. 28 shows the pressure decline response for different wellbore volumes on G-function and the square
root of time plots. Compared to Fig. 19, it can be observed that the WBS effect has stronger influence on


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root of time plots. Compared to Fig. 19, it can be observed that the WBS effect has stronger influence on
pressure decline trend in our radial fracture geometry for the same wellbore volume. This is because the
fracture volume and fracture surface are smaller than that in previous case of PKN fracture geometry, which
intensifies the impact of WBS effect.
Figure 28—Pressure decline response for different wellbore volume with a radial geometry
Fig. 29 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-
function and square root of time plots. Similar to Fig. 20, pressure declines rapidly in depleted reservoirs and
declines gradually with much lower peak values of pressure derivatives in over-pressurized reservoirs. Also
the pressure decline trend resembles a "normal-leak-off behavior" when pore pressure is abnormally high.


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24 SPE-187348-MS
Figure 29—Pressure decline response for different initial reservoir pressure with a radial geometry
Table 5 shows a comparison of the different methods of estimating minimum in-situ stress (closure
pressure is interpreted as the minimum in-situ stress) for the above synthetic cases for a radial fracture
geometry. Again, the results indicate that the "tangent line method" always underestimates minimum in-
situ stress while the "compliance method" overestimates the minimum in-situ stress (if the closure stress
is assumed to be equal to the minimum in-situ stress) if no correction is applied (except for the so-
called "normal leak-off behavior" where the changes in fracture stiffness/compliance are undetectable). The
estimated minimum in-situ stress using our proposed "variable compliance method" give the smallest errors
in all simulated cases.
Table 5—Comparison of different methods of estimating minimum in-situ stress for radial fracture geometry.


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in all simulated cases.
Table 5—Comparison of different methods of estimating minimum in-situ stress for radial fracture geometry.
From our synthetic analysis presented in this section, we can conclude that the pressure decline response
during DFIT is affected by many parameters, such as fracture geometry, reservoir properties, fracture surface
roughness, etc. So history match DFIT data can be non-unique if we don't have good knowledge (from other
independent analysis and measurements) of the constraints on these parameters. Nevertheless, our proposed
"variable compliance method" gives the most reliable estimation of minimum in-situ stress regardless of
these parameters, so the uncertainties that associated with fracture geometry, surface roughness and reservoir
properties do not impact our estimation of minimum in-situ stress.


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SPE-187348-MS 25
Field Data Interpretation
Field Case 1
The first field case to be analyzed comes from a horizontal well-A drilled through a shale formation. The
measured depth is around 5500 m and a diagnostic fracture injection test is conducted at the toe of the
horizontalwellbore, with 2.35 m3 of water injection in 3 minutes, then the well was shut-in for 27 days. Fig.
30 shows the pressure decline response on G-function and the square root of time plots. As in the previous
synthetic cases, even though G-function and the square root of time plots give exactly the same quantitative
information, it is more intuitive to infer the shut-in time on the square root of time plot, while the shut-in
time is implicit in the value of G-function, as reflected by Eq. (3)~Eq. (5). This field case also exhibits linear
pressure derivatives and low peak value of pressure derivative with a long extended tail after it reaches the


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pressure derivatives and low peak value of pressure derivative with a long extended tail after it reaches the
peak, we can infer that the impact of wellbore storage should be substantial and this reservoir is potentially
over-pressurized.
Figure 30—Pressure decline response from Well-A on G-function and square root of time plots
From the signature of pressure derivatives, the changes in fracture stiffness/compliance are undetectable,
so the minimum in-situ stress is picked at 31.8 MPa as the "tangent line method" suggests. A closer look at
the square root of time plot, shows that the pressure drops significantly during the first one hour of shut-in.
By examining the pressure derivative curve, whose extrapolated value is not zero when Δt=0, we can identify
that the excessive pressure drop at the beginning is caused by tip extension or near-wellbore tortuosity. The
apparent ISIP is 45.5 MPa while the true ISIP is 34.5MPa. Remember net pressure is the pressuredifference


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apparent ISIP is 45.5 MPa while the true ISIP is 34.5MPa. Remember net pressure is the pressuredifference
between thefluid pressure inside the fracture and closure pressure. In reality, the pressure measured at the
wellhead can be much higher than expected during fracture propagationduring theearly time of shut-in
period, because significant pressure drop can happen in the near-wellbore region and along the horizontal
wellbore, due to friction and tortuous/complex fracture path that initiated from perforation clusters. So
pressure data from this period cannot be used to infer fracturing parameters and reservoir properties. If a
rate step down test can be performed before complete shut-in, the relationship between early-time excessive
pressure drop and flow rate can be quantified.
Log-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis


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Log-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis
and assist the interpretation of closure stress (Barree et al. 2019; Hawkes et al. 2012; Soliman et al. 2015).
Even though Well-A was shut-in for nearly four weeks, pseudo-radial flow is still absent, as shown in
Fig. 31. This is a clear evidence of extremely low formation permeability. Without pseudo-radial flow,
formation pore pressure and permeability cannot be determined independently with enough confidence
using just after closure analysis alone. One has to analyze the whole spectrum of DFIT data to reach a
consistent interpretation. Using late time data from the linear flow regime, and plotting Pf vs 
  on
Cartesian coordinates, the intercept on the y-axis when Δt → ∞ is the estimated initial pore pressure, which
is 29.1 MPa. This indicates it is indeed an over-pressurized reservoir. From geological and petrophysical


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26 SPE-187348-MS
studies, it is known that the thickness of the target formation is 24.4 m with an average porosity of 0.07
and in-situ fluid viscosity of 0.257 cp, The Young's modulus is 38.9 GPa, the Poisson's ratio is 0.2 and
formation total compressibility is 3e-3 MPa−1. Hydraulic fracture modeling indicates that the fracture is
well contained within the target formation with penny-shaped fracture geometry, and the fracture radius is
roughly 12m. Based on this information, the pressure decline response can be matched globally using our
fully coupled DFIT model (Eq. (6)-Eq. (16) with pressure dependent fracture stiffness obtained from our
non-local fracture closure model (Wang and Sharma 2017)), from the end of tip extension/near-wellbore
tortuosity dominated period to the end of the test.
Figure 31—Log-log pressure derivative plot of Well-A
Fig. 32 shows the pressure decline response predicted by our fully coupled DFIT model and field data


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Fig. 32 shows the pressure decline response predicted by our fully coupled DFIT model and field data
on G-function and the square root of time plots. The results indicate that our simulated pressure matches
extremely well with the field data for the entire duration of the test, excluding the first hour of shut-in. Our
matched reservoir permeability is 220 nd, which is within the range of other petrophysical measurements.
The matched contact width and contact reference stress are 0.7 mm and 3 MPa, respectively, and Fig. 33
shows the corresponding fracture and fracture-wellbore system stiffness, based on the matched fracture
geometry, wellbore volume and contact parameters. Because the DFIT was conducted at the toe of a
horizontal well, it is no surprise that the wellbore storage effect is significant, considering such a large
contrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts


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contrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts
to increase noticeably when pressure drops to 34 MPa, over 2 MPa above the minimum in-situ stress. It
is apparent that the so-called "normal leak-off behavior" is just a coincidence when the effect of pressure
dependent leak-off at the fracture surface and variable fracture compliance counterbalance each other during
the fracture closure process.
Figure 32—Matched pressure decline response for Well-A on G-function and square root of time plots


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SPE-187348-MS 27
Figure 33—Matched fracture and fracture-wellbore system stiffness for Well-A
Field Case 2
The second field case to be analyzed comes from a vertical well-B drilled through a shale formation. The
total wellbore length is around 2000 m and a diagnostic fracture injection test is conducted with 4.7 m3 of
water injection in6 minutes, then the well was shut-in for 11 days. Fig. 34 shows the pressure decline response
on G-function and the square root of time plots. If we pick the fracture closure pressure using the "tangent
line method," both G-function and the square root of time plots give the same minimum in-situ stress of
50.4 MPa. One can also notice that there is a slight "bump" in the pressure derivatives soon after shut-in. For
shale formations, this early-time accelerated pressure drop is not likely to be caused by PDL, but attributed
to tip extension and near wellbore tortuosity (continued after-flow of fluid from wellbore to fracture after


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to tip extension and near wellbore tortuosity (continued after-flow of fluid from wellbore to fracture after
shut-in). It should be observed that in the early-time pressure transient response, the pressure derivative
starts to deviate upwards at around 55 MPa. Using "variable compliance method" through averaging the G-
function or the square root of time that corresponding to a pressure of 50.4 MPa and 55 MPa, respectively,
we can obtain the estimated minimum in-situ stress of 52.8 MPa by picking the pressure that corresponds
to the averaged G-function or square root of time.
Figure 34—Pressure decline response from Well-B on G-function and square root of time plots
From Fig. 35, we can observe that only linear flow emerged after closure. Using late time data from the
linear flow regime, the estimated initial pore pressure is 33.7MPa. Geological and petrophysical studies
indicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid


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indicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid
viscosity of 0.28 cp. The Young's modulus is 39.5 GPa, the Poisson's ratio is 0.25 and formation total
compressibility is 1.9e-3 MPa−1. Hydraulic fracture modeling shows that the fracture is well contained within
the target formation with roughly 34 m fracture half length. Based on this information and assuming PKN
geometry, the pressure decline response can be matched globally using our global DFIT model by adjusting
reservoir permeability and contact parameters.


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28 SPE-187348-MS
Figure 35—Log-log pressure derivative plot of Well-B
Fig. 36 shows the predicted pressure decline response and field data on G-function and the square root of
time plots. The results indicate that our simulated pressure agrees well with field data and it even captures
the downward curve of pressure derivatives before closure. Traditionally, this phenomenon is interpreted
as "height recession" or "transverse storage," where the top and bottom barrier formations close first and
squeeze fluid into the target formation during the closure process. This can be regarded as a special case of
"variable compliance," as demonstrated in Fig. 23. Because fractures always close from their edges to the
center on rough walls, this constantly alters fracture compliance/stiffness. This variable fracture compliance
coupled with fracture pressure dependent leak-off dictates our pressure decline trend for a given formation.


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coupled with fracture pressure dependent leak-off dictates our pressure decline trend for a given formation.
In this field case, our matched reservoir permeability is 210 nd, and the matched contact width and contact
reference stress are 1.2 mm and 1.1 MPa, respectively.
Figure 36—Matched pressure decline response for Well-B on G-function and square root of time plots
Fig. 37 shows the corresponding fracture and fracture-wellbore system stiffness as a function of fluid
pressure inside fracture. It can be observed that both fracture and system stiffness increase gradually as
pressure declines. Even though this test was conducted in a vertical well with relative moderate wellbore
volume, the wellbore storage effect on the system stiffness is still significant. Re-examining Eq. (16), we
can see that the relative influence of wellbore storage not only depends upon the ratio of wellbore to fracture


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can see that the relative influence of wellbore storage not only depends upon the ratio of wellbore to fracture
volume, but it also depends on fracture stiffness itself. For the same wellbore volume, the higher the fracture
stiffness, the less compressible the fracture is, so the compressed fluid in the wellbore plays a bigger role to
compensate leak-off and pressure drop. In this field case, with fracture height contained in 5 m, the fracture
stiffness should be at least 5.36 MPa/mm (From Table 1) even without considering the contact of fracture
surfaces. If the fracture geometry and reservoir properties can be constrained with enough confidence, the
estimated contact parameters and fracture stiffness evolution can be used to reconstruct the dynamic fracture


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SPE-187348-MS 29
closure process and the evolution of fracture width distribution, thus, the un-propped fracture conductivity
and its pressure/stress dependence can be inferred.
Figure 37—Matched fracture and fracture-wellbore system stiffness for Well-B
Conclusions and Discussion
Since the introduction of Nolte's pioneering work, diagnostic fracture injection test (DFIT) has evolved into
a commonly used technique and are generally accepted as a reliable way to obtain fracturing parameters
(e.g., minimum in-situ stress, leak-off behavior) and reservoir properties (e.g., initial pore pressure and
representative formation permeability). These key parametersare needed to run fracture models, post-
fracture productionprediction and economical evaluations. In this article, we first reviewed Nolte's G-
function model and the associated assumptions (adopted by many subsequent works), and derived the


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function model and the associated assumptions (adopted by many subsequent works), and derived the
equations that underpin the interpretation of the square root of time plot in low permeability reservoirs. We
demonstrated that the square root of time plot gives the same quantitative information obtained from the
G-function plot.
A new global DFIT model is proposed, which for the first time, has the capability to predict pressure
decline response across the entire duration of a DFIT test using a single, coherent mathematic framework in
low permeability formations. We show how each factor impacts the pressure decline response and the effects
of previous overlooked coupled mechanisms are examined and discussed, along with synthetic and field
case studies. The content of this article promotes our conceptual understanding of pressure transient and
coupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions


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coupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions
reached from the analysis presented in this paper include the following.
1.Carter's leak-off, (leak-off rate is proportional to 
 ), is applicable during stable fracture
propagation, when net pressure is roughly constant and injection time is limited. However, during the
fracture closure peroid, the fluid pressure in the fracture declines significantly, and the leak-off rate
becomes fracture pressure dependent and ceases to be proportional to 
 .
2. The so-called "normal leak-off," with a straight line of pressure and its derivatives on G-function and
the square root of time plots, is not "normal" at all. This is sometimes observed in data sets when
mechanisms that accelerate pressure decline (e.g., increase in fracture stiffness during closure) offset
those that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and


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those that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and
wellbore storage effects). So any "before closure" analysis based on this conventional "normal leak-
off" assumption to estimate fracture geometry and formation permeability should be re-examined.
3. Hydraulic fractures do not close as two smooth parallel plates, but rather close progressively from
edges to the center on asperities and roughness surfaces, as confirmed by our modeling results and
field tiltmeter measurements. The fracture stiffness increases (compliance decreases) gradually as


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30 SPE-187348-MS
more and more fracture surfaces come into contact, even before the fracture pressure drops to the
minimum in-situ stress.
4. It is shown that the pressure decline response during fracture closure is governed by a PDE-ODE
system, where the pressure at the fracture surface should be solved simultaneously with the pressure
inside the formation. Previous DFIT models assumed a constant fracture compliance/stiffness and
Carter's leak-off oversimplified and decoupled this system, which prevents them from predicting
pressure decline across the entire duration of the DFIT test.
5. The conventional "tangent line method" underestimates the minimum in-situ stress, especially
in depleted reservoirs. The "compliance method" picks the mechanical closure pressure but
overestimates the minimum in-situ stress (if the two are equated), especially when height recession
occurs. Height recession (or transverse storage) is a just special type of variable compliance
phenomenon.


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occurs. Height recession (or transverse storage) is a just special type of variable compliance
phenomenon.
6. The "variable compliance method" proposed in this study suggests that picking an average of the
dimensionless G-time (or the square root of time) of the "tangent line" and "compliance" methods
provides a much more robust and reliable way of estimating the minimum in-situ stress when concave
pressure derivative exist. The uncertainties that associated with fracture geometry, surface roughness
and reservoir properties do not impact our estimation of minimum in-situ stress using "variable
compliance method". Results using this method match well with field data.
7. Fracture closure on asperities and rough fracture surfaces is a gradual process starting at the fracture
tip. The model presented here can be used to define the relationship between fracture width and
contact stress using DFIT pressure data. If other matching parameters can be constrained with enough


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contact stress using DFIT pressure data. If other matching parameters can be constrained with enough
certainty, this information can be used to infer unpropped fracture conductivity as a function of
effective stress. This adds tremendous value to DFIT analysis for completion design and estimates
of well productivity and EUR.
Nomenclature
a = Half fracture height (PKN model) or half fracture length (KGD model), L, m
Af =Fracture surface area (one face of one wing),L2, m2
ct =Formation total compressibility, Lt2/m, 1/Pa
cW =Water compressibility, Lt 2/m, 1/Pa
CL =Carter's leak-off coefficient, 
 , 
E = Young's modulus, m/Lt 2, Pa
E′ = Plane strain Young's modulus, m/Lt 2, Pa
g(Δt D) =Dimensionless g-function of time
G(Δt D) =Dimensionless G-function of time
hf =Fracture height, L, m
ISIP = Instant shut-in pressure, m/Lt2, Pa
k = Formation permeability, L2, m2
P = Pressure, m/Lt2, Pa
Pf =Fracturing pressure, m/Lt2, Pa
Pnet =Fracturing net pressure, m/Lt 2, Pa


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k = Formation permeability, L2, m2
P = Pressure, m/Lt2, Pa
Pf =Fracturing pressure, m/Lt2, Pa
Pnet =Fracturing net pressure, m/Lt 2, Pa
P0 =Initial reservoir pressure, m/Lt2, Pa
qf =Leak-off rate (one wing), L3/t, m3/s
rD =Normalized radius
rp =Productive surface ratio
rf =Local fracture radius, L, m
Rf =Fracture radius, L, m


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SPE-187348-MS 31
Sf =Fracture stiffness, which is the reciprocal of fracture compliance, m/L2t2, Pa/m
SS =Fracture-wellbore system stiffness, m/L2t2, Pa/m
t =Generic time,t, s
tD =Dimensionless time
te =Generic time at the end of pumping, t, s
tp =Pumping time, t, s
Δt = Total shut-in time, t, s
ΔtD =Dimensionless shut-in time, t, s
xf =Fracture half-length, L, m
Uf =Flux velocity through fracture surface with constant fracturing pressure, L/t, m/s
Vf =Fracture volume (one wing), L3, m3
VW =Half of wellbore volume, L3, m
W0 =Contact width, L,m
Wf =Local fracture width, L, m
 =Averagefracture width, L, m
μf =Fluid viscosity, m/Lt, Pa·s
v = Poisson's ratio
σc =Contact stress, m/Lt2, Pa
σhmin =Minimum in-situ stress, m/Lt2, Pa
σref =Contact reference stress, m/Lt2, Pa
ϕ = Formation porosity
Subscript
n = Variables at nth time interval
j =Variables at jth time interval
i =Variables at jth time interval
References


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ϕ = Formation porosity
Subscript
n = Variables at nth time interval
j =Variables at jth time interval
i =Variables at jth time interval
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34 SPE-187348-MS
Appendix A
Solutions of Pressure Decline under Fracture Pressure Dependent Leak-off
In order to overcome the limitations of Nolte's G-function model and circumvent the use of a constant leak-
off coefficient, Mayerhofer et al. (1995) presented a model that describes unsteady-state fluid flow from
fractures of varying area into the formation, with the filter cake considered as a time and rate dependent
skin effect, allowing for superposition of the injection history, filter cake deposition and associated rate
convolution. Valko’ and Economides (1999) simplified Mayerhofer et al. (1995) model by assuming
constant leak-off rate during fracture propagation and constant fracture area during closure. A special before
closure analysis (BCA) plot with a straight line can be constructed to determine reservoir permeability and
fracture face resistance, without acquiring the initial guess of permeability and fracture area. However, the


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fracture face resistance, without acquiring the initial guess of permeability and fracture area. However, the
process to generate the special BCA plot and predict fall-off pressure is complicated and cumbersome,
prone to mistakes that can lead to dimension inconsistency ( Soliman et al. 2013). In addition, their approach
used a constant-rate solution of an infinite-conductivity well introduced by Gringarten et al. (1974). The
results obtained by use of the constant-rate solution are questionable to account for the fracture pressure
dependent leak-off behavior. In this appendix, we derive the solutions of pressure decline under fracture
pressure dependent leak-off using superposition based on a constant-pressure solution.
The basic principle of superposition is that the response of a system to a number of perturbations is equal
to the sum of the responses to each of the perturbation as if they were present by themselves if the responses


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to the sum of the responses to each of the perturbation as if they were present by themselves if the responses
are linearly related to the perturbations. From Eq. (18), we can calculate the leak-off rate under a constant
fracture pressure P f:
(A.1)
Divide the shut-in time Δt into n time steps, and the leak-off rate at the nth time step can be determined
based on superposition:
(A.2)
The pressure difference P f,j - P f,j-1 is the pressure in the fracture at time step j minus the pressure in the
formation at the fracture-formation interface, which equals to the pressure in the fracture at the previous
time step. Combine Eq. (A.2) with Eq. (10) via material balance, we can get:
(A.3)
let P f,0 = P 0, Pf,1 = ISIP, Δt 1 = 0 then for n ≥ 2, integrate Eq. (A.3) we can obtain:
(A.4)
Eq. (A.4) can be solved explicitly with marching time interval, for example, when n = 2
(A.5)


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On closer observation of Eq. (A.5), we can realize that it is exactly Carter's leak-off solution to Eq. (19),
because during the first time step, no superposition is needed. The fracture pressure dependent leak-off
coefficient can be recovered as:
(A.6)
It should be mentioned that Eq. (A.4) is only valid when Sf is constant, where the linear response system
is the prerequisite of using superposition principal.


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36 SPE-187348-MS
Appendix B:
Model Non-Local Fracture Closure on Rough Surfaces and Asperities
England and Green (1963) showed that the width of a 2D fracture with any arbitrary pressure/stress profile
is given by:
(B.1)
where γ is a dummy variable, y is the distance to the middle of the fracture, a is the half fracture height
(PKN model) or half fracture length (KGD model). Let PNet(y) be the net pressure/stress distribution along
the y-direction, which is the total pressure/stress acting on the fracture surfaces minus the far field stress
in the fracture opening direction. To obtain F(γ) and G(γ), PNet(y) has to be divided into even and odd
functions, such that PNet(y) = -f(y) - g(y), with f(y) being even and g(y) being odd. Then F(γ) and G(γ) are
determined by:
(B.2)
(B.3)
where β is a dummy variable. If the P Net(y) is symmetric, then g(y) and G(γ) = 0. For example, assuming a
uniform pressure P Net inside the fracture, f(y) =-P Net, g(y) = 0, From Eq. (B.2):
(B.4)


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uniform pressure P Net inside the fracture, f(y) =-P Net, g(y) = 0, From Eq. (B.2):
(B.4)
Substituting Eq. (B.4) into Equation (B.1), we have,
(B.5)
Equation (B.5) is the well-known equation for a 2D static pressurized crack. If a is the fracture half-
height, then Eq. (B.5) describes the fracture width distribution forthe PKN fracture geometry, and if a is
the fracture half-length, Eq. (B.5) describes the fracture width distribution for the KGD fracture geometry.
Whenan analytic form of P Net(y) is not available a priori (e.g., coupled with other physics) or P Net(y) is
not acontinuous function, numerical integration to calculate the fracture width has to be implemented in a
discretized manner along the fracture height or length. Fig. B1 shows a fracture that is discretized into a
number of n segments. In each segment, there exists a uniform net pressure/stress that acts on the fracture


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number of n segments. In each segment, there exists a uniform net pressure/stress that acts on the fracture
surfaces. Any net pressure/stress distribution can be approximated by representing it in this manner given
enough number of n.


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SPE-187348-MS 37
Figure B1—DiscretizePKN/KGD fracture into n segments with different net pressure/stress
Define integration operator I:
(B.6)
the corresponding fracture width profile at the mth segment can be obtained by integrating Eq. (B.1) across
each segment to account for non-local elastic behavior (Wang and Sharma 2017):
(B.7)
Repeat this calculation from the 1st segment to the nth segment, the entire fracture width profile under
arbitrary normal loading can be obtained. Fig. B2 shows an example where with a fracture half-height
divided into three segments with equal length, and a certain net pressure/stress is imposed on each segment
individually. As we can see, when the net pressure/stress is the same on all segments, an elliptical fracture
width profile is obtained, which is also validated by Eq. (B.5). When the net pressure/stress varies along
different segments, the fracture width profile will depart from the smooth, elliptical curve. And P Netn has


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different segments, the fracture width profile will depart from the smooth, elliptical curve. And P Netn has
more impact on the overall width distribution when it is imposed close to fracture center.


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38 SPE-187348-MS
Figure B2—Fracture width distribution along fracture half-height across three segments: a = 30 m, E=20 GPa, v =0.25
The radial crack opening due to an excess pressure distribution in a linearly elastic material of infinite
extent can be expressed as (Sneddon and Lowengrub 1969):
(B.8)
where Rf is the length of fracture radius, u and s are dummy variables and r D is the normalized radius, which
is defined as
(B.9)
where rf is the local fracture radius. For a radius crack with a uniform pressure distribution P Net, Eq. (B.8)
can be integrated directly to get an analyticalsolutionfor the fracture width profile:
(B.10)
Barr (1991) proved that the double integral of Eq. (B.8) can be transformed into a single integral with
the aid of elliptic integrals:
(B.11)
F(φ, m) is the elliptic integral of the first kind, which is defined as:
(B.12)
where φ are input parameters for elliptic integral, and θ is a dummy variable.


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(B.12)
where φ are input parameters for elliptic integral, and θ is a dummy variable.
Similar to Fig. B1, a radial fracture can be discretized into n segments, as shown in Fig. B3. In each
segment, there exists a uniform net pressure/stress that acts on the opposite fracture surface. If n is large
enough, the net pressure/stress distribution can be approximated to any distribution of P Net(r).


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SPE-187348-MS 39
Figure B3—Discretized radial fracture (n segments) with different net pressure/stress acting along the radial direction
Define integration operators R1 and R2 as:
(B.13)
(B.14)
and the corresponding fracture width profile at the mth segment under arbitrary P Net(r) can be obtained by
integrating Eq. (B.11) across each segment (Wang and Sharma 2017):
(B.15)
where
(B.16)
(B.17)
Repeat this calculation from the 1st segment to the nth segment, the entire fracture width profile can be
obtained. Fig. B4 shows an example where with a fracture radius divided into three segments with equal
length, and a certain net pressure/stress is imposed on each circular segment individually. When the net
pressure/stress is the same on all segments, it reproduces the same fracture width distribution as predicted
by Eq. (B.10). And when the net pressure/stress varies along different segments, the fracture width within


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by Eq. (B.10). And when the net pressure/stress varies along different segments, the fracture width within
a certain segment can be substantially impacted by the load imposed on other segments.


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40 SPE-187348-MS
Figure B4—Fracture width distribution along fracture radius across three segments: R f = 30 m, E=20 GPa, v =0.25
In the above discussion, we have presented the method and associated algorithms to determine the
fracture width profile for PKN, KGD and Radial fracture geometries, given an arbitrary distribution of net
pressure/stress. Now, if the fracture closes on rough surfaces and asperities, then the net pressure/stressis
impacted by the contact stress, which itself is a function of fracture width distribution. Therefore, the net
pressure/stressdistribution imposed on fracture faces and the entire fracture width distribution have to be
solved simultaneously. Because both the net pressure/stress distribution and fracture width distribution are
discretized, they can be only solved iteratively. Fig. B5 shows the iterative scheme to solve the net pressure/
pressureand fracture width distribution.


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pressureand fracture width distribution.
Figure B5—Solution flow diagram for solving coupled linear fracture mechanics with contact problem
In this Picard iteration scheme, α r is a relaxation factor that is less than or equal to 1.0. It controls how
much the net pressure/stress should be corrected within iterations. Smaller α r can avoid over-correction
and improve convergence stability, but more iterations may be required with small value of α r. When the
changes in net pressure/stress between iterations is less than the tolerance value, the estimation of contact
stress across the entire fracture faces is converged and the final net pressure/stress distributionwith the
corresponding fracture width profile can be obtained. So given initial fracture geometry, rock properties
and contact law parameters, the fracture width profile can be uniquely determined under certain fracturing


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and contact law parameters, the fracture width profile can be uniquely determined under certain fracturing
pressure. By discretizing fracturing pressure within a certain range, the relationship between fracture volume
vs. pressure and fracture stiffness vs. pressure can also be obtained.


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SPE-185649-MS
Reinterpretation of Fracture Closure Dynamics During Diagnostic Fracture
Injection Tests
B. Zanganeh and C. R. Clarkson, University of Calgary; R. V. Hawkes, Trican Well Service Ltd.
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April 2017.
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 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
any 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


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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 SPE copyright.
Abstract
A fit-for-purpose, fully coupled stress-pore pressure simulation model (Abaqus®) is used to simulate
diagnostic fracture injection tests (DFITs) and generate before closure pressure responses. The simulated
responses are used to explain field observations, and to propose a new concept: progressive fracture closure.
The cohesive zone model (CZM) is used to model fracture propagation and closure associated with
DFITs. The customized model in Abaqus® is capable of modeling all the physical processes involved
in a typical DFIT including: porous media deformation? fluid flow inside the reservoir? hydraulic
fracture initiation, propagation and closure? compliance change before and after closure; residual fracture


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fracture initiation, propagation and closure? compliance change before and after closure; residual fracture
conductivity? and fluid flow inside the fracture and fluid interaction between the fracture and reservoir
(leakoff).
A key result obtained is that the previously-introduced fracture compliance method is demonstrated to be
the most reliable approach to identify fracture closure. Depending on the pressure distribution around the
fracture, the closure pressure is usually found to be higher than the minimum principal stress. Based on the
continuity equation, the compliance method is expanded to include the progressive fracture closure (PFC)
concept. PFC refers to the scenario where fracture closure occurs gradually along the length of fracture, from
the tip of the fracture to near the wellbore. Different estimates of closure pressure will be obtained early and
late in this process. Several field cases are presented which exhibit progressive fracture closure. A consistent


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late in this process. Several field cases are presented which exhibit progressive fracture closure. A consistent
closure signature can be identified for these cases using the primary pressure derivative. This study further
suggests that wellbore storage can mask the closure signature. Therefore, reducing the wellbore storage
effect by using downhole shut-in makes it easier to identify fracture closure, in addition to accelerating the
test.
A common DFIT fracture closure referred to as "fracture height recession" is reinterpreted to be caused
by the PFC phenomenon. This finding has tremendous implications for interpretation of before closure flow
regimes and associated reservoir behavior. This study also confirms that the compliance method, which
corresponds to fracture tip closure, provides more of a true measure of closure pressure than conventional
approaches which correspond to fracture closure near the well.


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2 SPE-185649-MS
Introduction
The DFIT, also known as a "minifrac," has become the standard pre-stimulation method for determining
fracturing treatment properties, effective reservoir permeability and initial reservoir pressure in
unconventional resources. A DFIT is an injection-falloff test conducted before the main hydraulic fracturing
treatment. The DFIT procedure involves breakdown of the rock and creation of a short fracture during the
injection of high pressure fluid (usually water), and recording of the pressure response during the shut-
in period. The pressure response is analyzed to obtain fracturing treatment parameters such as breakdown
pressure, fracture closure pressure and leakoff characteristics. If the falloff period is sufficient, good
estimates of formation permeability and reservoir pressure can be obtained.
Before-closure analysis (BCA) was pioneered by Nolte (1979), who introduced the G-function based


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Before-closure analysis (BCA) was pioneered by Nolte (1979), who introduced the G-function based
on material balance and the Carter leakoff model (Howard and Fast 1957); Nolte noted that the shut-in
pressure has a linear relationship with G-function during closure. Barree and Mukherjee (1996) and Barree
(1998) recommended using the G-function and its diagnostic derivatives (
  and 
 ) to identify fracture
closure. Using these diagnostics, closure can be picked as the deviation from a straight line on a plot of
pressure, and its derivatives, versus the G-function. Barree (1998) and Barree et al. (2009) also presented
signatures for non-ideal leakoff behaviors such as tip extension, pressure-dependent leakoff and height
recession/transverse storage.
Mohamed et al. (2011) and Marongiu-Porcu et al. (2011) presented a model to predict the falloff pressure
trend of an idealized DFIT (normal leakoff), and identified fracture closure using standard pressure transient


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trend of an idealized DFIT (normal leakoff), and identified fracture closure using standard pressure transient
(PTA) log-log diagnostic plots. They identified normal leakoff as a 
  slope on the Bourdet-derivative (with
respect to superposition time), and picked closure as the deviation from the 
  slope. Bachman et al. (2012)
presented a systematic PTA approach, and a combination of diagnostic plots, to identify various flow
regimes before and after fracture closure. Van Den Hoek (2016) and McClure et al. (2016) demonstrated
limitations of superposition time and Agarwal's time (1980) for analyzing DFITs where the pumping time
is small. They stated that the Bourdet-derivative (with respect to superposition time and Agarwal's time)
starts to exceed one at shut-in times equal to roughly one-tenth of the pump time and that the 
  slope is
not related to fracture closure.
Van Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments


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slope is
not related to fracture closure.
Van Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments
on different rock types. They observed that fracture closure happens at the tip first, and then towards the
wellbore. They reported a break on a plot of pressure versus G-function due to the decrease of fracture
compliance at closure, and argued that, instead of a deviation from a straight line on the pressure derivative
(with regard to G-function), the local minimum should be selected as closure pressure. Further, they stated
that plastic deformation can cause a fracture to remain open in the vicinity of the wellbore. McClure et
al. (2014) and McClure et al. (2016) investigated the effect of fracture compliance on pressure behavior
using a numerical simulator. Their modeling of after-closure compliance behavior was based on the Barton-
Bandis (1985) model, which is an empirical relationship. They demonstrated that the previous "fracture


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Bandis (1985) model, which is an empirical relationship. They demonstrated that the previous "fracture
height recession" signature presented by Barree et al. (2009) is the natural behavior of closure caused by
a change in fracture compliance. Based the "fracture compliance method," closure is picked as the start of
upward deviation on the 
  curve. McClure et al. (2016) argued that previously-presented methods tend
to underestimate the closure pressure.
Van Den Hoek (2016) presented a PTA approach for modeling pressure behavior in DFITs and
waterflood-induced fractures based on simplified numerical and analytical solutions. In his work, fracture
growth rate was a predefined input into the simulator, the Carter model was used as the leakoff model for
the DFIT, and closure was modeled as a gradual decline of fracture compliance representing a combination
of "hinge" (constant length) and "zipper" (length recession) closure.


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SPE-185649-MS 3
The cohesive zone model (CZM) for modeling crack propagation was originally proposed by Dugdale
(1960) and Barenblatt (1962). Recently, this method has been successfully used for modeling hydraulic
fracture initiation and propagation during fracturing treatment (Yao 2012; Shen and Cullick 2012; Shin and
Sharma 2012; Chen 2012; Zielonka et al. 2014; Haddad and Sepehrnoori 2015). Some of the advantages
of the CZM compared to conventional linear elastic fracture modeling are that it avoids a singularity at the
crack tip, the location of crack tip is not an input and is a direct output of the solution, and it is capable
of modeling fracture tip material softening in quasi-brittle rocks such as shale ( Chen 2012; Haddad and
Sepehrnoori 2015). The CZM has been implemented in the Abaqus® finite element program and validated
analytically and experimentally by Zielonka et al. (2014) and Searless et al. (2016).


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analytically and experimentally by Zielonka et al. (2014) and Searless et al. (2016).
In this study, the CZM implemented in the Abaqus® finite element program is used to model fracture
initiation, propagation and closure for DFITs. A pressure-dependent leakoff model is coupled with the
program through user subroutines. The model is used to generate synthetic pressure responses, which in
turn are compared to field observations and used to study fracture closure dynamics.
Theory and Methods
In the following, key components of the simulation model used in this study are described, as is the model
setup used for generating synthetic DFITs.
Cohesive Zone Model (CZM)
With the CZM, the fracture is modeled as a gradual separation between two material (rock) surfaces. This
separation is modeled as a progressive degradation of cohesive strength along the cohesive layer, which is
a pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's


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a pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's
Guide 2016; Chen 2012). With traction-separation behavior (Fig. 1(a)), prior to damage initiation, cohesive
elements exhibit a reversible linear elastic response (characterized by the cohesive layer stiffness, Kn) until
the traction reaches the maximum tensile strength (T n0) with a separation of δ0. After damage initiation,
the cohesive traction evolves from the maximum strength (T n0) down to zero where the element is fully
damaged with the separation of δf. The damage evolution is governed by the critical energy release rate ( Gc)
that is equal to the area under traction-separation curve. Fig. 1(b) is a schematic of fracture propagation
in a cohesive layer, demonstrating the fully damaged cohesive zone (filled with fracturing fluid) and the
fracture process zone (damage initiation and evolution).


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fracture process zone (damage initiation and evolution).
Figure 1—(a) Typical traction-separation behavior; (b) Damage evolution in cohesive elements.
Different guidelines have been presented for selecting the cohesive element mesh sizes and stiffness.
Turon et al. (2007) suggested the following equation for the cohesive layer stiffness:


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4 SPE-185649-MS
(1)
where Kn is the stiffness of the cohesive layer, E is the Young's modulus of the material and tadjacent  is
thickness of the adjacent sub-laminate. They recommended α values much larger than 1 (α>>1). Zielonka
et al. (2014) and Searless (2016) used α=100, and presented close matches with analytical solutions and
experimental results. Haddad and Sepehrnoori (2015) derived an optimum α value of 60 (assuming tadjacent  =
1) by conducting a sensitivity analysis. As will be shown in the next section, we have observed that cohesive
layer stiffness is a major parameter affecting fracture compliance behavior before and after closure.
Fluid Flow within the Fracture
Fig. 2 is a plan view schematic of a wing of a fracture showing the components of fluid flow in the fracture.
There are two components of fluid flow inside the fracture: tangential flow within the fracture gap (q frac)


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There are two components of fluid flow inside the fracture: tangential flow within the fracture gap (q frac)
and normal flow (leakoff) from the fracture to the surrounding rock (q leak). The continuity equation for fluid
flow inside the fracture is:
(2)
where w is the fracture opening, and q inj is the injection rate into the fracture.
Figure 2—Plan view schematic of one wing of the fracture.
Tangential Flow
The tangential flow is formulated based on Poiseuille's law:
(3)
where w is the fracture opening, μ is the fluid viscosity and Pf is the fluid pressure along the fracture length.
Leakoff Model
The Carter leak-off model (Howard and Fast 1957) has been used extensively for modeling of hydraulic
fracture propagation and in the analytical solutions for DFIT analysis. It is a one dimensional pressure-
independent leak-off model, applicable to high-viscosity fracturing fluids causing the formation of a low
permeability filter cake on the fracture walls and is described by:
(4)


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permeability filter cake on the fracture walls and is described by:
(4)
where qleak(x) is leakoff rate at point x, Ccarter is the Carter leakoff coefficient, t is the time elapsed since the
start of injection and τ(x) is the time when the fracture is created at point x (exposure time). The Carter leakoff
coefficient depends on filter cake created on fracture walls, and it is not related to formation properties such
as reservoir permeability and pressure.


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SPE-185649-MS 5
In this study, a rigorous leakoff model based on the solution of the 1-D diffusion equation in a half-space
with an imposed pressure history at the boundary is used (Sarvaramin and Garagash 2015):
(5)
where qleak(x) is leakoff rate at point x, S is the storage coefficient and αl is the hydraulic diffusivity. Unlike the
Carter model, this model (Sarvaramin and Garagash 2015) is pressure-dependent, and it is directly related
to formation properties such as permeability, porosity and total compressibility. The final approximation
of this model is similar to Howard and Fast (1957)'s solution for leakoff of low viscosity fracturing fluids,
except that the pressure difference term is not constant, and it is a function of time and location.
Model Setup and Description
In this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure


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Model Setup and Description
In this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure
in a DFIT. Fig. 3 provides a schematic of the simulation model. Minimum and maximum horizontal stresses
are acting in the Y and X directions, respectively. In the Base Case model, wellbore storage is neglected,
and the fluid is injected directly at the injection point (perforation). In a more complex scenario, a block
representing the wellbore storage is considered. The fluid is injected into the wellbore block, which is
connected to the model using pipe elements. The pipe elements also model possible pressure drops during
the test.
Figure 3—Plan view schematic of the simulation model setup.
In order to consider the appropriate far-field boundary conditions and model propagation of the fracture in
an infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where


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an infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where
the modeled region is small in size compared to the surrounding medium (formation), and they provide
quiet boundaries in the simulation (Abaqus Analysis User's Guide 2016). Chen (2012) and Haddad and
Sepehrnoori (2015) have discussed the advantages of using infinite elements in improving the accuracy
and runtime of the simulation. For well-testing applications, we have observed improvements in the
representation of pressure transient behaviors, particularly in after-closure flow regimes and removal of
unrealistic boundary effects, using infinite elements.
The cohesive elements are embedded in the formation rock, and act as the potential pathway for hydraulic
fracture growth. During the injection of high pressure fluid, once the required criteria (as discussed in the
previous section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-


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previous section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-
in, the fracture aperture reduces to closure aperture (w closure), which is a predefined input into the simulator.
After closure, the aperture can further decrease below the wclosure, but it will always be a positive value,


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6 SPE-185649-MS
meaning the fracture retains a finite aperture and conductivity even after closure. In low permeability
formations, even a few microns of residual aperture results in an infinite conductive fracture relative to
matrix permeability.
Table 1 lists properties of the Base Case simulation model. In order to compare simulation results with
field observations, four other models with the same reservoir and rock mechanical properties, but different
settings, are considered (Table 2). In Model 2, the cohesive layer stiffness is reduced to observe its effect on
fracture compliance and pressure behavior. In Model 3, the first cohesive element connected to the injection
point remains open during shut-in and closure. In Model 4, wellbore storage is included that is comparable
with real field data.
Table 1—Base Case model simulation properties
Input parameter Value Input parameter Value
Permeability, md 0.025 Initial porosity, % 10


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Table 1—Base Case model simulation properties
Input parameter Value Input parameter Value
Permeability, md 0.025 Initial porosity, % 10
Young's modulus, GPa 20.7 Fracture fluid viscosity, cp 1
Poisson's ratio 0.25 Shmin, MPa 38.6
Total compressibility, KPa 4.68×10-6 Shmax, MPa 44.1
Pumping time, sec 120 Initial pore pressure, MPa 24.8
Pumping rate (per unit thickness),
m3.sec-11.5×10-4 Maximum tensile strength, MPa 1.25
Cohesive layer stiffness, GPa.m-1 2070
Table 2—Differences in model setup and simulation settings
Base Case Model 2 Model 3 Model 4
Cohesive layer stiffness,
GPa.m-12070 20.7 2070 2070
Displacement boundary
condition applied to the first
cohesive elementNo No Yes No
Wellbore storage and pipe
elementsNo No No Yes
Results and Discussion
In the following, synthetic before-closure signatures are generated for the Base Case and Models 2-4 and
interpreted for various physical phenomena. Field cases are then analyzed in the context of the simulation


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interpreted for various physical phenomena. Field cases are then analyzed in the context of the simulation
model results to understand the signatures that commonly occur in the field.
Base Case
Fig. 4(a) illustrates the pressure behavior during injection for the Base Case simulation model. Because no
wellbore is included in the Base Case model, pressure increases quickly during injection, until it reaches the
breakdown point pressure. At this point, damage initiates (fracture initiation), and then propagates along
the cohesive element (fracture propagation). The smooth pressure behavior during propagation indicates
proper mesh size of the model and cohesive elements. Fig. 4(b) illustrates the fracture aperture profile during
propagation and early shut-in time. Fracture tip extension occurs for approximately 1 meter after shut-in.


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SPE-185649-MS 7
Figure 4—Base Case model: (a) pressure profile during injection and fracture propagation; (b)
aperture profile during injection and illustration of tip extension during the early shut-in period.
Fig. 5(a) illustrates fracture opening at a typical element along the length of the fracture (e.g. x=1)
during pressure falloff. As the fluid inside the fracture leaks off to the surrounding formation and pressure
inside the fracture decreases, the fracture opening gradually reduces to the closure aperture. The fracture
aperture further reduces after the closure, but will always be a positive value, meaning the fracture retains
a finite aperture and conductivity even after closure. As demonstrated in Fig. 5(a), the rate of aperture
change with time before closure, 
 , is significantly different from after closure behavior,
. The simulated response is similar to the experimental results of Van Dam et al. (2002).


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, is significantly different from after closure behavior,
. The simulated response is similar to the experimental results of Van Dam et al. (2002).
Based on the continuity equation of fluid flow in the fracture ( Eq. 2), the change of 
  at the time of closure
causes an abrupt change in pressure. The same behavior is observed for change of the aperture with pressure
(Fig. 5 (b)) in our simulated case. This is in agreement with the concept of compliance contrast at closure
presented previously by Van Dam et al. (2002) and McClure et al. (2014).
Figure 5—Base Case model: fracture aperture behavior at x=1 during falloff with (a) time and (b) pressure.
Progressive Fracture Closure.   Fig. 6 illustrates the accumulated leakoff volume from the fracture face
along the fracture length for a typical time during falloff (e.g. 15 minutes). The accumulated leakoff volume
is higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this


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is higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this
results in a non-uniform pressure profile and effective stress (on fracture walls) along the fracture length
(Fig. 6). The effective stress is maximum at the fracture tip (x=11) and gradually decreases along the fracture


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8 SPE-185649-MS
(up to x=5) where it remains nearly constant up to the perforation (x=0). Therefore, fracture closure first
occurs at the tip and then moves toward the wellbore, which is referred to herein as "progressive fracture
closure". Fig. 7 illustrates the fracture aperture over time during progressive closure. As expected, closure
first occurs at the tip (x=11) moving toward x=5, where fracture suddenly closes up to perforation (x=0).
Figure 6—Accumulated leakoff volume and effective stress profile at fracture face along the fracture.
Figure 7—Progressive fracture closure from the tip of the fracture to the perforation.
The progressive closure causes an increase in the pressure derivative. Fig. 8 illustrates the pressure falloff
trend, and its semi-log derivative 
 , on a G-function plot. Using the Carter leakoff model, we would
expect a linear trend on the 
  curve; however, due to the pressure-dependency included in the leakoff
model used herein (Eq. 5), the


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expect a linear trend on the 
  curve; however, due to the pressure-dependency included in the leakoff
model used herein (Eq. 5), the 
  curve slightly deviates from the straight line with a concave downward
trend. The start of the upward deviation (derivative spike) on the 
  plot is correlated with tip closure, and
the end of derivate spike indicates a fully closed fracture (closure at the wellbore).


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SPE-185649-MS 9
Figure 8—Base Case model: before-closure analysis using G-function and derivative plots.
The PTA plot in Fig. 9 includes the Primary Pressure Derivative (PPD), Bourdet-derivative with respect
to Agarwal's time (
 ) and Bourdet-derivative with respect to shut-in time (
 ). The early time
fluctuations on the derivative plots are caused by fracture tip extension. According to Mattar and Zaoral
(1992), the PPD for any type of reservoir effect or flow regime will always either be a constant or decreasing,
and any non-reservoir effect will appear as an increase in the PPD. The non-reservoir effects in the Base
Case model are fracture tip extension and fracture closure. After the early time tip extension signature
(fluctuations), the start of PPD violation (upward trend) is caused by fracture tip closure. The end of PPD
violation corresponds to a fully closed fracture. Closure pressure may be estimated at the start and end


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violation corresponds to a fully closed fracture. Closure pressure may be estimated at the start and end
of the PPD violation as 38.7 MPa and 38.3 MPa, respectively. In real field tests, it is likely that stress
conditions near wellbore area are significantly affected during drilling and completion. Therefore, the
difference between closure pressure at the perforation and fracture tip can be larger.
Figure 9—Base Case model: before-closure analysis using PTA plot.
Consistent with previous claims (Van Den Hoek 2016; McClure et al. 2016) it is observed that 
starts to deviate from 
  at Δt = 0.1 tp, and the closure time is not long enough for the 
  slope to establish.


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10 SPE-185649-MS
Model 2
The main difference between this model and the Base Case is that the 
  ratio used in the cohesive layer
stiffness calculation (Eq. 1) is reduced from 100 to 1. As shown in Fig. 10, even after closure, the fracture
aperture continues to decrease from the closure aperture to a residual aperture. The difference between
 and 
  is also less than for the Base Case.
Figure 10—Model 2: fracture aperture behavior during falloff with (a) time and (b) pressure.
Similar to the Base Case, progressive closure from the tip of the fracture to near the wellbore causes a
clear break in pressure, and a spike in the pressure derivative on the G-function and PTA plot; however, its
magnitude is smaller than for the Base Case model (see Fig. 11). Fracture tip closure starts at 39.4 MPa,
wellbore closure occurs at 38.4 MPa, and the progressive closure period lasts for 5 minutes.
Figure 11—Model 2: before-closure analysis using (a) G-function (b) PTA plot.


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Figure 11—Model 2: before-closure analysis using (a) G-function (b) PTA plot.
Low values of cohesive layer stiffness are not realistic. But even for low cohesive layer stiffness, a small
change in aperture behavior along the length of fracture caused an abrupt change in pressure behavior. While
the progressive fracture closure signature (or previously known as height recession/transverse storage) is
usually observed in field data, a very sharp increase in derivative for a fully closed fracture is not common.
With the next two models, a few scenarios are presented that can affect this sudden pressure change.


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SPE-185649-MS 11
Model 3
In this model, the first cohesive element connected to the injection point remains open during closure.
This was achieved by applying appropriate boundary conditions to the first element. Fig. 12 presents
BCA analysis plots for Model 3. The overall pressure behavior (Fig. 12(a)) is smooth and similar to field
observations. Also, the progressive closure signature (upward deviation in derivative plots) is smooth.
Figure 12—Model 3: before-closure analysis using (a) G-function (b) PTA plot.
Model 4
In this case, wellbore storage is included in the model by addition of a well block and pipe elements. Fig.
13 provides pressure transient analysis for this model, without any smoothing applied to the derivative.
As demonstrated by the schematics included in the figure, at early time, the afterflow caused by wellbore
storage increases both the fracture aperture and length identified by a unit slope on Bourdet-derivative.


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storage increases both the fracture aperture and length identified by a unit slope on Bourdet-derivative.
The transition from the unit slope occurs when the fracture walls start to close (the so-called hinge closure
suggested by Van Den Hoek 2016). Due to low leakoff, this can cause additional tip extension during the
early closure (abrupt changes and fluctuations) as fluid is pushed toward the tip of fracture. After this
transition, leakoff becomes the dominant factor affecting the pressure transient and similar flow regimes
to previous models appear.


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12 SPE-185649-MS
Figure 13—Model 4: before-closure analysis using PTA plots.
In the presence of wellbore storage, the first fracture element connected to the wellbore can stay open
without applying any boundary conditions (Model 3). This can be explained based on the continuity equation
(Eq. 1). Initially it was assumed that the qinj term is zero during shut-in. In the presence of afterflow caused by
wellbore storage, the qinj term is no longer zero and can act to reduce the abrupt change in the 
  term during
closure. Therefore, progressive fracture closure stops at the wellbore without causing an abrupt change in
pressure and derivatives (as with the Base Case model and Model 2). Also, as mentioned earlier, Van Dam
et al. (2002) demonstrated that plastic deformation can cause the fracture to remain open in vicinity of
the wellbore. While plastic deformation is not modeled in this study, it can be another reason why a very


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the wellbore. While plastic deformation is not modeled in this study, it can be another reason why a very
sharp spike in derivative plots is not observed in field data. Nonetheless, the start of progressive closure (tip
closure) can be used to estimate closure pressure and minimum horizontal stress.
As suggested by Cramer and Nguyen (2013), using downhole shut-in accelerates fracture closure and
time to reach radial flow after closure. These advantages can be explained with Model 4. In the presence
of wellbore storage, fracture length and volume increase significantly during the early falloff period. With
larger fracture length, a much longer time is required for radial flow to be established in the formation.
Further, re-supply of fluid into the fracture due to afterflow, coupled with a larger volume, delays fracture
closure significantly. In addition to the benefits suggested by Cramer and Nguyen (2013), reducing afterflow


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closure significantly. In addition to the benefits suggested by Cramer and Nguyen (2013), reducing afterflow
by using downhole shut-in results in a clearer signature of fracture closure (abrupt change in pressure and
derivatives).
In many field cases, a unit slope on the Bourdet-derivative is observed. Hawkes et al. (2013) introduced
a composite permeability concept to explain after-closure behavior. Bachman et al. (2015) expanded this
idea and stated this behavior can happen before or after closure. A simpler explanation for unit slope after
closure is the effect of wellbore storage. Depending on the re-supply rate of fluid from the wellbore, and its
magnitude relative to leakoff, a unit slope after closure can exist.
Field Example 1
The analysis for this dataset is provided in Fig. 14 and Fig. 15. The pressure values are recorded using
surface gauges. No smoothing is used for the purpose of comparison with Model 4. The overall trend and


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SPE-185649-MS 13
signatures are very similar to Model 4 except that the progressive closure period is longer. The early unit
slope (Fig. 14) and fluctuations indicate fracture volume increase and tip extension due to afterflow. The
transition from a unit slope represents the start of hinge closure. Similar to Model 4, tip extension occurs
during this period as hinge closure pushes the fluid to the tip of the fracture. Progressive fracture closure
starts with PPD violation (or upward deviation on derivative plots) at 22.7 MPa and ends at 19.5 MPa. The
progressive closure period lasts around 5 hours and the difference between closure pressure estimates is
significant.
Figure 14—Field Example 1: before-closure analysis using PTA plot.
Figure 15—Field Example 1: before-closure analysis using the G-function plot.
Field Example 2
This well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very


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Field Example 2
This well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very
similar to Model 4 (Fig. 16). The upward deviation at the end of the 
  slope was of particular interest in the
previous analysis, and no explanation was provided at the time for this feature.


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14 SPE-185649-MS
Figure 16—Field Example 2: before-closure analysis using PTA plot (modified from Hawkes et al., 2013; Well 058L).
The early time unit slope indicates fracture expansion and tip extension due to wellbore storage. At the
end of the unit slope, hinge closure starts and is accompanied by tip extension. After this period (considering
that falloff time in this case is large enough), the 
  slope appears on the Bourdet-derivative. Consistent
with our modeling results, the upward deviation at the end of the 
  slope, that is consistent with the start
of the PPD violation, corresponds to the progressive fracture closure period. The start of the PPD violation
indicates tip closure (52.8 MPa) and the end of PPD violation represents a fully closed fracture (50.9 MPa).
The progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.
Field Example 3


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The progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.
Field Example 3
In this test, pumping time was relatively long (19 minutes). Combined with the low permeability of the
formation, this probably resulted in a long fracture requiring more time to close. The pressure values
are recorded using surface gauges and significant ambient temperature variations between day and night
occurred during the falloff period. As shown in Fig. 17(a), there are severe fluctuations in the G-function
derivative plot. These functions can result in misinterpretation of the falloff period (e.g. near wellbore
complexities, tip extension, pressure dependent leakoff, etc.); however, when ambient temperature is
plotted, it is observed that pressure fluctuations are a direct effect of ambient temperature variation. To
remove the temperature effect from the surface pressure data, the following equation is used to account for


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remove the temperature effect from the surface pressure data, the following equation is used to account for
fluid volume change with pressure and temperature:
(6)
where Preading is the recorded pressure at surface, T is the current ambient temperature, T0 is the ambient
temperature at time of shut-in, β is the thermal expansion coefficient of wellbore fluid and Cf is the
compressibility of wellbore fluid. In this equation, temperature dependency of β and Cf is neglected which
can further increase its accuracy.


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SPE-185649-MS 15
Figure 17—Field Example 3: (a) effect of ambient temperature variation on derivative plot (b) removing
the effect of ambient temperature using Eq. 6 and BCA based on height recession signature.
As demonstrated in Fig. 17(b), applying Eq. 6 to recorded pressure values reduces the ambient
temperature effect significantly without using a larger derivative window. Closure pressure is estimated to
be 21.5 MPa, based on conventional interpretation of fracture height recession/transverse storage.
Fig. 18 provides before-closure analysis based on PTA plot and using the progressive closure concept.
Although closure time is much longer compared to pumping time, a 
  slope is not clear in this test. After the
tip extension period, the start of the PPD violation at 25.2 MPa represents tip closure. Due to the very low
permeability of the formation, and large fracture created during pumping, the progressive closure period


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permeability of the formation, and large fracture created during pumping, the progressive closure period
is very long (around 6 days). The end of the PPD violation at 23.3 MPa indicates a fully closed fracture.
The closure pressure estimate based on the conventional interpretation of height recession (Fig. 17(b))
underestimates the pressure by 3.7 MPa.
Figure 18—Field Example 3: before-closure analysis using PTA plots.
Conclusions
The concept of compliance change at the time of closure is validated in this study. We demonstrated that
even for a soft material with low cohesive layer stiffness (not realistic) this feature exists. Therefore, the
fracture compliance method (McClure et al. 2016) is a reliable approach to identify fracture closure.


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16 SPE-185649-MS
For planar fractures, closure is a transient process, starting from the tip of the fracture to the vicinity of
the wellbore. We refer to this process as "progressive fracture closure". The common signature referred to as
"fracture height recession/transverse storage" is reinterpreted to be caused by this phenomenon. The start of
progressive closure (tip closure) is picked as the closure pressure, which is consistent with the compliance
method. Progressive fracture closure explains the increase of Bourdet-derivative after the 3/2 slope. It can
also provide information about the leakoff characteristics, fracture dimensions and stress distribution around
the fracture. The application of progressive fracture closure to evaluate these properties will be discussed
in a follow up paper.
The primary pressure derivative (PPD) is a powerful tool for identifying progressive fracture closure.


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in a follow up paper.
The primary pressure derivative (PPD) is a powerful tool for identifying progressive fracture closure.
The start of a PPD violation corresponds to tip closure, and the end of the violation indicates full closure.
Other PTA derivative plots are useful in the identification of flow regimes before the start of progressive
closure and explaining other PPD violations. For instance, PPD violations happening before closure during
the unit slope on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are
caused by fracture expansion and tip extension due to wellbore storage.
In previous studies, the magnitude and duration of the tip extension period has been underestimated.
The use of the cohesive zone model in this study enabled us to capture this process accurately and the
simulations are consistent with field observations. The model results illustrate that tip extension can occur


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simulations are consistent with field observations. The model results illustrate that tip extension can occur
during early fracture expansion period due to wellbore storage and during the hinge closure (and can last
for several hours).
The afterflow caused by wellbore storage can mask the 
  change at closure, and its transition from
the tip of the fracture to near the wellbore. Further, it may result in excessive tip extension and a larger
fracture which delays after closure radial flow. Therefore, removing wellbore storage using downhole shut-
in not only accelerates closure and radial flow, but also provides a clear signature of compliance change
and progressive closure.
A simple method is presented for removing the effect of ambient temperature change on pressure values
without applying large derivative windows for smoothing. The temperature effect may be interpreted
incorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results


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incorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results
in loss of accuracy and removal of signatures such as tip extension and PPD violation.
Acknowledgements
The authors would like to thank Dassault Systemes Simulia Corporation for providing Abaqus® program
and Seven Generations Energy Corporation for providing field data. The authors would also like to
acknowledge Dr. Jack R. Jones for his valuable advice and comments. The sponsors of Tight Oil Consortium
are acknowledged for their support. Partial funding for this project was obtained through an NSERC
Collaborative Research and Development grant (CRDPJ 452752-13) led by Clarkson.
Nomenclature
Ccarter Carte leakoff coefficient, m3/sec0.5
CfWellbore fluid compressibility, Pa-1
EYoung's modulus, Pa
GG-function time, dimensionless
GcCritical energy release rate, Pa.m
KnStiffness of the cohesive layer, Pa/m
mSlope of straight line, dimensionless
PPressure, Pa


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GcCritical energy release rate, Pa.m
KnStiffness of the cohesive layer, Pa/m
mSlope of straight line, dimensionless
PPressure, Pa
PfFluid pressure in the cohesive element, Pa


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SPE-185649-MS 17
qfracGap flow rate within the cohesive element, m3/sec
qinjInjection rate into the cohesive element, m3/sec
qleakLeakoff rate from cohesive element, m3/sec
SStorage coefficient, Pa-1
ShmaxMaximum horizontal stress, Pa
ShminMinimum horizontal stress, Pa
taAgarwal's time, dimensionless
tadjacent Thickness of the adjacent sub-laminate, m
Tn0Maximum tensile strength, MPa
tpPumping time, sec
wFracture opening (aperture), m
wclosure Closure aperture, m
Greek Variables
αCoefficient for calculation of cohesive layer stiffness, dimensionless
αlHydraulic diffusivity, m2/sec
βThermal expansion coefficient, °C-1
δ0Separation at damage initiation, m
δfSeparation at failure, m
ΔtShut-in time, sec
τ(x) Exposure time, sec
μFracturing fluid viscosity, cp
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The Fracture-Compliance Method for
Picking Closure Pressure From Diagnostic
Fracture-Injection Tests
Mark W. McClure and Hojung Jung, University of Texas at Austin; Dave D. Cramer, ConocoPhillips; and
Mukul M. Sharma, University of Texas at Austin
Summary
In this paper, we present the fracture-compliance method, a tech-
nique for estimating the closure pressure from diagnostic fracture-
injection tests (DFITs). The method is based on the observation
that fractures retain a ﬁnite aperture after asperities come into
contact (mechanical closure). An empirical, nonlinear joint-clo-sure law is used to relate the after-closure fracture aperture and
stiffness (the reciprocal of compliance) to effective normal stress.
Fracture closure increases fracture stiffness, which, in low-perme-
ability formations, causes an increase in the pressure derivative.
On the basis of these insights, we propose the fracture-compliancemethod, which consists of picking closure at the ﬁrst point of


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On the basis of these insights, we propose the fracture-compliancemethod, which consists of picking closure at the ﬁrst point of
deviation from linearity on a plot of pressure or G/C2dP/dGvs.G-
time (after the end of the very-early-time transient associated withwellbore and near-wellbore friction and fracture tip-extension).
The contribution of this paper is to provide theoretical justiﬁca-
tion for why closure is best picked with the fracture-compliance
method, and not with other widely used methods. We provide a
series of numerical DFIT simulations to demonstrate the sensitiv-ity of the pressure transient to input parameters. Governing equa-
tions are derived and used to demonstrate the effect of changing
fracture aperture after closure. A ﬁeld DFIT data set is analyzed
with the new method. Finally, a ﬁeld example is presented in
which downhole tiltmeter measurements provide an independentestimate of the minimum principal stress.
Introduction


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which downhole tiltmeter measurements provide an independentestimate of the minimum principal stress.
Introduction
Background. DFITs are used for a variety of purposes, including
estimating the magnitude of the minimum principal stress in a for-
mation. Fluid, usually water, is injected at a low-to-moderate rate
for a short period of time, creating a relatively small hydraulic
fracture, and then the well is shut in. The pressure transient fol-
lowing shut-in is analyzed to identify the closure pressure, deﬁnedas the ﬂuid pressure at which the fracture walls come into contact.
The closure pressure is taken to be an estimate of the minimum
principal stress. In recent years, there has been a growing interestin using DFITs to evaluate very-low permeability formations,
including shale reservoirs. DFITs belong to a family of fracture-
injection/falloff tests that include minifracture and extended leak-
off tests. These fracturing tests were interpreted with a variety of


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injection/falloff tests that include minifracture and extended leak-
off tests. These fracturing tests were interpreted with a variety of
different methods and used for many decades (Hubbert and Wil-lis 1957; Godbey and Hodges 1958; Haimson and Fairhurst
1967). Nevertheless, there is still not universal agreement on
best practices for interpreting DFITs to estimate the minimum
principal stress.
Fig. 1 shows a ﬁeld example of pressure data from a DFIT,
with different stages of the test numbered. This test was original
presented by Cramer and Nguyen (2013). Initially, pressure rises
linearly with time, because relatively little ﬂuid leaks out into the
formation and the transient is dominated by wellbore storage (1).
At the leakoff point, the transient deviates from linearity, indicat-ing that a hydraulic fracture has formed (or reopened) at the well-bore (2). The pressure ﬂattens out to the fracture-propagation
pressure (3). When the injection rate increases between minutes 4


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pressure (3). When the injection rate increases between minutes 4
and 5, the fracture-propagation pressure increases, primarily
because of increased wellbore and near-wellbore pressure drop,
and then ﬂattens out again. When injection is stopped, there is animmediate drop in pressure to the initial shut-in pressure (ISIP) as
wellbore and near-wellbore frictional pressure drop reduces (4).
Subsequently, the pressure continues to fall off, and the fracture-
closure pressure is reached when the walls of the fracture come
into contact (5). Further discussion is provided in the Field DFIT#1 subsection.
Methods for Interpreting DFITs. Fig. 2 shows plots of the
shut-in pressure transient from the DFIT shown in Fig. 1 in four
different types of plots: a G-function plot, a Bourdet plot with de-
rivative taken with respect to shut-in time, a Bourdet plot with de-
rivative taken with respect to superposition time, and a Bourdet
plot with time-integrated pressure (Bourdet et al. 1989; Barree


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rivative taken with respect to superposition time, and a Bourdet
plot with time-integrated pressure (Bourdet et al. 1989; Barree
et al. 2007; Marongiu-Porcu et al. 2011; Craig 2014). Conven-
tional applications of these plots for diagnosing the closure pres-sure are described in this section. Fig. 2 shows both ﬁeld data and
a numerical-simulation match to the data, as described in the Field
DFIT #1 section.
Fracture-closure pressure is most-often diagnosed from plots
of pressure vs. the square root of time or a specialized function of
time called the G-function (Nolte 1979; Nolte and Smith 1981).
For 1D leakoff of ﬂuid from a constant-pressure boundary, the so-
lution to the diffusivity equation predicts that the leakoff rate willscale with the inverse of the square root of time (Howard and Fast
1957). When coupled with a wellbore/fracture system of constant
compliance, the result is pressure decay scaling with the square


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1957). When coupled with a wellbore/fracture system of constant
compliance, the result is pressure decay scaling with the square
root of time. The G-function is a generalization of this concept
that includes the effect of fracture propagation, which causes theduration of leakoff to be different at each point along the fracture.
Nolte (1979) postulated that fracture-growth scales in time
according to a power law given by an exponent a. The G
-function
is a complex function of a, but it is only weakly dependent on a
between the reasonable bounds of [1/2] to 1.0. If ais assumed to
be 1.0, Gcan be written in a particularly convenient form:
GðDtD;aÞ¼4
pgðDtD;aÞ/C0gð0;aÞ ½/C138 ; ð1Þ
where DtD¼Dt=te,Dtis time because shut-in teis duration of
injection, and
gðDtD;a¼1Þ¼4
3ð1þDtDÞ1:5/C0Dt1:5
Dhi
: ð2Þ
AllG-function plots in this paper use the convention of assum-
ing that aequals 1.0, and use the deﬁnition of Ggiven in Eqs. 1
and 2. The speciﬁc choice of adoes not affect the method of pick-


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ing that aequals 1.0, and use the deﬁnition of Ggiven in Eqs. 1
and 2. The speciﬁc choice of adoes not affect the method of pick-
ing closure from the transient.
On either plot, pressure is expected to form a straight line
before closure and deviate from the straight line when closureoccurs. Therefore, the deviation of pressure from the straight line
can be used to identify the fracture-closure pressure, an estimate
of the minimum principal stress (Castillo 1987; Zoback 2007). In..............
..........
Copyright VC2016 Society of Petroleum Engineers
Original SPE manuscript received for review 26 August 2014. Revised manuscript received
for review 17 November 2015. Paper (SPE 179725) peer approved 8 December 2015.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1321 Total Pages: 19
ID:jaganm Time: 18:38 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1321


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addition, G/C2dP/dGvs.Gis often plotted {equivalent to d P/
d[ln(G)] vs. G}, where Pis pressure. As long as pressure forms a
straight line, G/C2dP/dGshould also form a straight line, and so
closure can be picked at the deviation from the straight line (Bar-
ree and Mukherjee 1996). In many cases, the Gfunction is
approximately equal to Dt0:5, and the plots can be made with Dt0:5
substituted for G.
Unfortunately, plots of pressure vs. G-time often yield curving
lines with multiple points of inﬂection. Efforts were made to inter-
pret and explain these nonideal observations. Wellbore and near-
wellbore pressure drop, pressure-dependent leakoff, fracture-
height recession, closing of secondary transverse fractures, andfracture tip-extension are some of the most commonly cited sour-
ces of nonideal behavior (Nolte 1991; Nolte et al. 1993; Barree
and Mukherjee 1996; Barree 1998; Barree et al. 2007). In high-


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ces of nonideal behavior (Nolte 1991; Nolte et al. 1993; Barree
and Mukherjee 1996; Barree 1998; Barree et al. 2007). In high-
permeability formations, leakoff limits fracture length, and clo-sure occurs rapidly. But in very-low permeability formations, clo-
sure can occur hours or days after shut-in, and nonideal transientsare nearly always observed.
G-function plots of DFITs are conventionally interpreted
according to the method described by Barree and Mukherjee
(1996) and Barree et al. (2007), which we refer to as the “tangent
method.” According to the tangent method, an ideal DFIT has aG/C2dP/dGcurve that is straight until closure, at which time the
curve bends, decreasing in slope; eventually, the curve crests,
then turns downward. A concave upward G/C2dP/dGcurve is inter-
preted as being caused by fracture-height recession or transverse-
fracture storage. A concave downward G/C2dP/dGis interpreted as
being caused by pressure-dependent leakoff or fracture tip-exten-


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fracture storage. A concave downward G/C2dP/dGis interpreted as
being caused by pressure-dependent leakoff or fracture tip-exten-
sion. These interpretations are based on numerical simulations
(Nolte 1991; Barree and Mukherjee 1996; Barree 1998) and are
used widely (Economides and Nolte 2000; Barree et al. 2007;
01 2345678 9 1 0
Elapsed Time (minutes)–1.40
–1.20
–1.00
–0.80
–0.60–0.40
–0.20
0.0065
60
55
50
45
40
35
30Wellbore-storage trendline
Pressure rollover
indicates fracture reopeningat 55.3 MPaISIP = 57.1 MPa
Water hammer
event at shut in
pdata
qwaterCalculated Bottomhole Pressure (MPa)Water-Injection Rate (m3/min) 123
4
5?
Fig. 1—Example of rate and pressure measurements from a DFIT. This test was original presented by Cramer and Nguyen (2013).
0 50 100 150 200
G-time
10–210–110–1
100100
100100100
102102104
101101101
10–2100102
102102
Δt (hours) Δt (hours)Δt (hours)I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)


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102102104
101101101
10–2100102
102102
Δt (hours) Δt (hours)Δt (hours)I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)
Field data (red)Field data (red), simulation data (blue) Field data (red), simulation data (blue)
Field data (red), simulation data (blue)60
55
50
45
40
35Pressure (solid) (MPa)ΔP (solid), τ*dP/dτ (dashed) (MPa)
G*dP/dG (dashed) (MPa)10
8
6
4
20
Fig. 2—Field DFIT #1. Field data are plotted in in red. A simulation match to the data is plotted in blue (described in the Field DFIT
#1 subsection). The transient was originally shown in Fig. 4 of Cramer and Nguyen (2013). Upper left: G-function plot. Upper right:
Bourdet plot with derivative with respect to time. Lower left: Bourdet plot with derivative with respect to superposition time. Lowerright: Bourdet plot with integrated pressure (zoomed in). Arrows indicate the identiﬁcation of closure from the tangent method


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(blue), from the fracture-compliance method (orange), from the Mohamed et al. (2011) method (green), and from the Craig and Bla-
singame (2006) method (red).J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1322 Total Pages: 19
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Soliman and Gamadi 2012; Soliman and Kabir 2012; Cramer
and Nguyen 2013; Wallace et al. 2014; Meng et al. 2014; Barree
et al. 2014).
When the G/C2dP/dGcurve is concave upward, closure is
picked according to the tangent method by drawing a straight line
from the origin, tangent to the G/C2dP/dGcurve (Barree et al.
2007). The point of tangency is the time of closure, and the frac-
ture-closure pressure is identiﬁed as the pressure at that time. Thisis typically equivalent to picking closure at the end of the second
straight line on a plot of pressure vs. G-time. When the G/C2dP/dG
curve is concave downward, closure is picked at approximately
the time the G/C2dP/dGcurve stops increasing. The closure pick
from the tangent method is labeled with the blue arrow in theupper-left panel of Fig. 2.
Bourdet plots are log-log plots that show the change in pres-
sure since the start of the transient, DP, and d P/d(ln(Dt)) (which
can be equivalently written as Dt/C2dP/dDt) vs. the time elapsed


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sure since the start of the transient, DP, and d P/d(ln(Dt)) (which
can be equivalently written as Dt/C2dP/dDt) vs. the time elapsed
since the start of the transient, Dt(Bourdet et al. 1989). Mohamed
et al. (2011) and Marongiu-Porcu et al. (2011; 2014) advocate
interpreting DFITs with Bourdet plots in which the derivative is
taken with respect to superposition time, s. For the method pro-
posed by Mohamed et al. (2011), superposition time is deﬁned as
s¼
teþDt
Dt; ð3Þ
where teis duration of injection. If Dt<<te, taking the derivative
with respect to superposition time has no effect, compared to tak-
ing the derivative with respect to Dt.I fDt>>te, taking the deriva-
tive with respect to superposition time adds 1.0 to the slope of thederivative curve. These authors advocate picking closure at the
beginning of the [1/2] slope on the derivative plot (labeled with a
green arrow in the lower-left panel of Fig. 2).
Craig and Blasingame (2006) propose interpreting DFITs by


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green arrow in the lower-left panel of Fig. 2).
Craig and Blasingame (2006) propose interpreting DFITs by
integrating the pressure data after shut-in to ﬁnd the effectivepressure response, I. This can be written as
IðDPÞ¼ð
teþDt
0½PðtÞ/C0Pinit/C138dt; ð4Þ
where Pinitis the initial ﬂuid pressure. This transformed pressure
response can then be plotted on a Bourdet plot. For application to
a DFIT test, the “wellbore storage” region on the Bourdet plot ofintegrated pressure has a very long duration because matrix per-
meability is very low and the storage includes both the wellbore
and the fracture. Craig and Blasingame (2006) suggest identifying
closure at the point where the fracture/wellbore storage changes
(caused by the changing fracture stiffness at closure). This corre-sponds to the time when a deﬂection can be identiﬁed during the
wellbore storage period in the Bourdet plot with respect to inte-
grated pressure. This is labeled with a red arrow in the lower-rightpanel of Fig. 2.


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grated pressure. This is labeled with a red arrow in the lower-rightpanel of Fig. 2.
We advocate picking closure from a plot of pressure vs. G(or
the square root of time), but in contrast to the tangent method
(Barree and Mukherjee 1996; Barree et al. 2007), we recommend
picking closure at the initial deviation from linearity (after the endof the very early-time transient associated with wellbore and near-
wellbore friction and fracture tip-extension). We refer to this
method as the “fracture compliance method” for picking closure.
The discussion in this paper elaborates on concepts presented by
McClure et al. (2014).
The fracture compliance can be deﬁned in several equivalent
ways: the derivative of fracture closure (fracture closure is equal
to negative change in aperture) with respect to effective normal
stress, the negative of the derivative of aperture with respect to


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to negative change in aperture) with respect to effective normal
stress, the negative of the derivative of aperture with respect to
effective normal stress, or the derivative of aperture with respectto pressure [Chapter 12 from Jaeger et al. (2007)]. The fracture
stiffness is equal to the reciprocal of the fracture compliance. The
name “fracture compliance method” is chosen because the
method is based on understanding how fracture compliance (and
equivalently, stiffness) change because of fracture closure.At very early time, there is often a sharp drop in pressure
caused by the disappearance of friction in the wellbore and near-
wellbore. This period should not be considered when picking clo-sure pressure. A linear period on the plot of Pvs.Gshould be
visible after the end of this initial period. When closure occurs in
low-permeability formations, increasing fracture stiffness causes
pressure to deﬂect downward and G/C2dP/dGto deﬂect upward, as


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low-permeability formations, increasing fracture stiffness causes
pressure to deﬂect downward and G/C2dP/dGto deﬂect upward, as
discussed in the Stiffness Contrast at Closure section. A detailedderivation of the equations describing this process are provided in
Appendix A.
In high-permeability formations, closure reduces the leakoff
rate, and closure can cause pressure to deﬂect upward and G/C2dP/
dGto deﬂect downward (though continue increasing), as dis-
cussed in Section 3.1.6. The G/C2dP/dGplot is useful for identify-
ing closure because it makes it easy to identify these deﬂections.
The key difference between our work and most previous work
on this topic is that we consider the effect of residual aperture on
ﬂuid storage after mechanical closure. We use a 3D hydraulic-fracture simulator that rigorously solves the hydromechanically
coupled problem of fracture initiation and closure. When fracture


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coupled problem of fracture initiation and closure. When fracture
walls come into contact, the simulator switches to a nonlinearjoint-closure law on the basis of the work of Barton et al. (1985),
which allows the fracture to remain hydraulically conductive
and continue to store ﬂuid after mechanical closure has occurred.
Similar approaches were used by other authors (Koning and
Niko 1985; Plahn et al. 1997; van den Hoek 2002; Ribeiro andHorne 2013).
There is abundant evidence demonstrating that fractures retain
storativity and conductivity after the walls have come into contact
(mechanical closure). Fractures retain aperture after mechanical
closure because of mismatch of asperities in the fracture walls.An extensive literature exists on the relationship between fracture
aperture, conductivity, ﬂuid pressure, and stress (Gangi 1978;
Witherspoon et al. 1980; Walsh 1981; Barton et al. 1985; Brown
and Bruhn 1998; Pyrak-Nolte and Morris 2000). The –1/2 slope


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Witherspoon et al. 1980; Walsh 1981; Barton et al. 1985; Brown
and Bruhn 1998; Pyrak-Nolte and Morris 2000). The –1/2 slope
pressure transients frequently observed in Bourdet plots of DFITdata indicate the presence of a post-closure inﬁnite conductivity
fracture, demonstrating that the fracture has retained signiﬁcant
aperture after closure. In high-permeability formations, the ﬂuidstorage and conductivity-retained post-closure quickly becomes
negligible. But in very-low permeability formations, leakoff into
the matrix is very slow, and an extremely narrow fracture aperture
can have a dominant effect on the pressure transient.
The dimensionless fracture conductivity (or equivalently,
dimensionless fracture transmissivity) is deﬁned as:
C
fD¼T
kLf; ð5Þ
where Tis the fracture transmissivity, kis the matrix permeability,
andLfis the fracture half-length. A fracture will behave similar to
an inﬁnite-conductivity fracture as long as Cfd>300. From the


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andLfis the fracture half-length. A fracture will behave similar to
an inﬁnite-conductivity fracture as long as Cfd>300. From the
cubic law (Witherspoon et al. 1980), the fracture transmissivity isdeﬁned as
T¼e3
12; ð6Þ
where eis the hydraulic aperture.
For a fracture with half-length of 50 m and a matrix permeabil-
ity of 10/C019m2(/C25100 nd), the fracture will be inﬁnitely conduc-
tive as long as the fracture aperture is greater than 26 mm.
Methods
The numerical simulations are performed with CFRAC (Complex
Fracturing ReseArch Code), which fully couples ﬂuid ﬂow with
the stresses induced by fracture deformation in discrete-fracture
networks (McClure and Horne 2013). All simulations described in
this paper assume a single planar fracture perpendicular to theminimum principal stress. The simulations are fully 3D, with the
fractures discretized into square elements in the vertical and hori-
zontal directions. Complete descriptions of CFRAC are provided..............................


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zontal directions. Complete descriptions of CFRAC are provided..............................
...................
...............................
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by McClure and Horne (2013) and McClure et al. (2015). Rele-
vant details are summarized in this section.
The ﬂuid is single phase, single component, constant compres-
sibility ( cf), constant viscosity ( l), and has initial density of qinit.
The simulations are isothermal, and the effect of gravity is
neglected. The simulations are initialized with homogeneous
stress and ﬂuid pressure.
Nonlinear relationships are used to relate pressure, stress, and
deformation to aperture. We deﬁne a “closed” fracture as a frac-
ture in which the ﬂuid pressure is less than the normal stress and
the walls of the fracture are in contact. Roughness of the fracture
walls allows closed fractures to retain hydraulic and void aperture.
We deﬁne an “open” fracture as a fracture where the ﬂuid pres-sure has reached the normal stress and the walls are no longer
in contact.
The aperture of closed-fracture elements is estimating with the
following joint-closure relation, which was written in this form by


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in contact.
The aperture of closed-fracture elements is estimating with the
following joint-closure relation, which was written in this form by
Willis-Richards et al. (1996), on the basis of the work of Bartonet al. (1985):
E¼
E0
1þ9r0
n=rn;ref; ð7Þ
where Eis the void aperture (ﬂuid volume stored in the fracture
per surface area), E0is the residual aperture (the aperture at effec-
tive normal stress equal to zero), and rn,refis the 90% closure
stress (i.e., the effective normal stress that reduces Eto 10% of
E0). The fracture-effective normal stress, r0
n, is equal to the nor-
mal stress, rn,minus ﬂuid pressure, P. In the simulations for this
study, the void aperture and hydraulic aperture are assumed to
be equal.
For open elements, the aperture is deﬁned as:
E¼E0þEopen; ð8Þ
where Eopenis the mechanical separation between the fracture walls.
A special algorithm is used to deﬁne the value of E0for hy-
draulic fractures. When a hydraulic-fracture element ﬁrst starts, it


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A special algorithm is used to deﬁne the value of E0for hy-
draulic fractures. When a hydraulic-fracture element ﬁrst starts, it
is given a very small aperture, 0.1 mm. But the residual aperture of
a fully open fracture should be on the order of ten or hundred of
microns. To handle this issue, an algorithm is used that allows E0
to increase as the fracture opens. The algorithm mimics the natu-
ral process of fracture-roughness development as the crack starts
and opens. The algorithm sets E0of each element to be equal to
90% of the maximum aperture reached by the element, allowingE
0to increase, but never decrease. A maximum value for E0is
speciﬁed, Ehf,max,resid .
Fluid leakoff is calculated, assuming that the reservoir is
slightly compressible with compressibility of c/,initial porosity
of/init, and contains the same ﬂuid as the injection ﬂuid. Leakoff
is calculated with the semianalytical method of Vinsome and
Westerveld (1980), which assumes 1D leakoff, but accounts for


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is calculated with the semianalytical method of Vinsome and
Westerveld (1980), which assumes 1D leakoff, but accounts for
changing pressure in the fracture over time.
Poroelastic-stress changes in the fracture caused by ﬂow in the
matrix are neglected. Also, poroelastic-pressure response in thematrix caused by fracture deformation is neglected. Poroelastic-
stress effects on the fracture are not typically considered for DFIT
analysis (Nolte 1979; Barree and Mukherjee 1996). In a typical
DFIT (very-low permeability), the region of pressure increase
around the fracture is very narrow, and so the poroelastic effect ofstress on the fracture is slight (Perkins and Gonzalez 1985; Wal-
lace et al. 2014).
The mechanical calculations are performed, assuming that the
formation is mechanically homogeneous, isotropic, and linearly
elastic. The calculations are performed with the boundary-element
method of Okada (1992), which uses constant displacement, rec-


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elastic. The calculations are performed with the boundary-element
method of Okada (1992), which uses constant displacement, rec-
tangular elements in an elastic half-space. This method satisﬁes
the equations of quasistatic-stress equilibrium, Hooke’s law,and compatibility, giving results that are convergent with
mesh reﬁnement to exact analytical solutions (Babazadeh and
McClure 2015).For a mechanically open element, the condition is enforced
that (Crouch and Starﬁeld 1983)
r
r
n/C0PþDrn¼0; ð9Þ
where rr
nis the normal stress on the fracture caused by remote
loading (calculated from the initial conditions) and Drnis the
change in normal stress caused by stress induced by the cumula-
tive deformation of all the fracture elements in the system, as cal-
culated from the Okada (1992) method.
Closed elements experience small normal displacements in
response to changes in effective normal stress (Eq. 7). The stress
changes caused by these aperture changes are included in the sim-


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response to changes in effective normal stress (Eq. 7). The stress
changes caused by these aperture changes are included in the sim-
ulations with the method described in Sections 2.2.3 and 2.2.4 of
McClure and Horne (2013).
Fracture initiation occurs at the wellbore when the ﬂuid pres-
sure reaches the minimum principal stress. Propagation is handled
with linear elastic-fracture mechanics. The stress-intensity factor
at the crack tips (vertically and horizontally) is calculated numeri-
cally, and propagation occurs when the stress-intensity factorreaches the fracture toughness.
In the simulations for this paper, the fracture height is assumed
to be limited to no more than 7 m. The only exception is Simula-tion G1, in which there is no height conﬁnement and the fracture
propagates radially.
The wellbore and near-wellbore pressure drop is modeled as
proportional to the square of the ﬂow rate from the wellbore to
the formation,
DP
nw¼CnwQ2; ð10Þ


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proportional to the square of the ﬂow rate from the wellbore to
the formation,
DP
nw¼CnwQ2; ð10Þ
where Cnwis a constant and Qis the volumetric ﬂow rate from the
wellbore to the formation. This term combines the effect of fric-
tion in the wellbore, the perforations, and the near-wellbore
region. The perforation pressure drop is a function of the numberof ﬂowing perforations, the perforation diameter, and the perfora-
tion shape (Cramer 1987). Friction in the wellbore is related to
wellbore diameter and the wellbore roughness (Moody 1944).
Typically, the pressure drop associated with injection through
well tubulars and perforations during DFIT treatments is verysmall, especially when the injection rate is 2 bbl/min or less. The
most-important pressure drop is typically downstream of the per-
forations: (1) along a tortuous ﬂow path from the perforations tothe primary fracture and/or (2) caused by the interaction of multi-


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forations: (1) along a tortuous ﬂow path from the perforations tothe primary fracture and/or (2) caused by the interaction of multi-
ple (short) fractures of different orientations (longitudinal, multi-
ple transverse orientations, horizontal) which is set up as the
result of near-wellbore stress concentration (e.g., hoop stress).
Results and Discussion
In the following sections, various simulation results are presented
and discussed. In the Generic Simulations subsection, a series ofgeneric simulations are performed to investigate the effect of differ-
ent physical processes on pressure-transient behavior. In the Field
DFIT #1 subsection, a simulation result is provided that matches
Field DFIT #1. Field DFIT #1 is a straight-forward, uncomplicated
case, and so an almost perfect match to the data is possible. In theGas Research Institute/Department of Energy (GRI/DOE) M-site
Tests subsection, data are provided from the GRI/DOE M-site ﬁeld-


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Tests subsection, data are provided from the GRI/DOE M-site ﬁeld-
demonstration project. At this site, hydraulic-fracture injectionswere performed in conjunction with downhole tiltmeter measure-
ments in an offset well that provide an independent estimate of the
closure pressure. Table 1 provides the settings used in all simula-
tions. Table 2 provides settings that are varied between simulations.
In all ﬁgures showing pressure-transient results, arrows indi-
cate the identiﬁcation of closure from the tangent method (blue),
the fracture compliance method (orange), the Mohamed et al.
(2011) method (green), and the Craig and Blasingame (2006)
method (red).
Generic Simulations. In this section, a series of generic DFIT
simulations is presented. Superposition time, s,andG-time are........................
.....................................................
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calculated as functions of time of shut-in, Dt,and the duration of
injection, te. Before the leakoff point, injected ﬂuid is almost
entirely stored in the wellbore, and leakoff into the formation isnegligible. To account for this, t
eis deﬁned to be equal to the du-ration of injection after the leakoff point is reached, rather than
the entire duration of injection.
Simple Baseline Simulations. Simulation G1 assumes radial-
fracture geometry (no height containment) with no wellbore stor-
age, very-small residual aperture, and fairly high fracture tough-ness. Simulation G2 is similar to Simulation G1, but the hydraulic
fracture is assumed to be conﬁned in height to 7 meters. Figs. 3 and
4show the results from Simulations G1 and G2. For calculating t
e,
the “leakoff point” in Simulations G1 and G2 occurs immediately at
the start of injection because there is not any wellbore storage.
Simulations G1 and G2 have a very sharp (unrealistic) spike in


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the start of injection because there is not any wellbore storage.
Simulations G1 and G2 have a very sharp (unrealistic) spike in
derivative when closure occurred. Closure clearly and unambigu-
ously occurs at a ﬂuid pressure equal to 55.1 MPa, precisely the
same as the minimum principal stress assumed in the simulation.
Closure can be identiﬁed consistently in all four plots.
(a) Stiffness Contrast at Closure. In Appendix A, a detailed
derivation is provided for the equations governing a DFIT pres-sure transient. After shut-in and the period after shut-in when
ﬂow-rate-dependent pressure gradients dissipate, it is shown thatTable 1 —Simulation settings used in all simulations. Variables are
deﬁned in the Nomenclature section.
Table 2 —Simulation settings varied between simulations. Variables are deﬁned in the Nomenclature section.
60
57
5154
48
45
42Pressure (solid) (MPa)
0 20 40 60 80 100 12030
25
20
15
10
5
0
G-time
Δt (hours) Δt (hours)Δt (hours)ΔP (solid), τ*dP/dτ (dashed) (MPa)


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5154
48
45
42Pressure (solid) (MPa)
0 20 40 60 80 100 12030
25
20
15
10
5
0
G-time
Δt (hours) Δt (hours)Δt (hours)ΔP (solid), τ*dP/dτ (dashed) (MPa)
I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)ΔP (solid), Δt*dP/dΔt ((dashed)
(MPa)G*dP/dG (dashed) (MPa)
10–210–2
100100
102
10–2100100100100
102102103
102105
101101
101
Fig. 3—Results from Simulation G1. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure
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dP
dt¼/C0qleakoff
cfVwlþA
Sf; ð11Þ
where cfis the compressibility of the ﬂuid, Vwlis the volume of
the well, Ais the surface area of the fracture, qleakoff is the volu-
metric leakoff rate from the fracture, and Sfis the fracture stiff-
ness. Eq. 11 neglects the effect of the compressibility of the ﬂuid
in the fracture and the effect of the changing wellbore volume
because both terms are of negligible magnitude in typical DFITapplications (Appendix A).
For a radial fracture, the stiffness of an open fracture (deﬁned
asS
f,o) can be written as (Sneddon 1946)
Sf;r¼3pGm
8ð1/C0/C23ÞRf; ð12Þ
where Gmis the shear modulus, /C23is Poisson’s ratio, and Rfis the
radius. For a fracture with length much longer than its height, the
stiffness can be estimated with the Perkins-Kern-Nordgren (PKN)
assumption of plane strain in vertical cross section (Sneddon1946; Perkins and Kern 1961; Nordgren 1972):
S
f;PKN¼4Gm
pð1/C0/C23Þhf; ð13Þ
where hfis the fracture height.


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S
f;PKN¼4Gm
pð1/C0/C23Þhf; ð13Þ
where hfis the fracture height.
Before closure, the leakoff rate scales with time according to
theG-function (Nolte 1979), and the denominator of Eq. 11 is
constant, which is why a plot of pressure vs. G-time forms a
straight line.
In Appendix A, the stiffness of a closed fracture is derived [in
analogy to the discussion in Chapter 12 from Jaeger et al. (2007)]:
dr0
n
dE¼Sf;c¼Sf;oþSf;cont; ð14Þ
where Sf,cis the stiffness of a closed fracture and Sf,contis the con-
tribution to the stiffness from the contacting of the fracture walls.
Sf,contcan be calculated from a constitutive law that relates aper-
ture to effective normal stress (such as Eq. 7) by solving for effec-
tive normal stress and then taking the derivative with respect
to aperture:Sf;cont¼rn;ref
9E0
E2¼rn;ref
9E01þ9ðrn/C0PÞ
rn;ref/C20/C212
: ð15Þ
In Simulation G1, the fracture is circular. The radius at shut-in


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to aperture:Sf;cont¼rn;ref
9E0
E2¼rn;ref
9E01þ9ðrn/C0PÞ
rn;ref/C20/C212
: ð15Þ
In Simulation G1, the fracture is circular. The radius at shut-in
is 21 m, but the fracture extends slightly after shut-in and reachesa ﬁnal radius of 21.6 m. From Eq. 12, the preclosure stiffness is
1.09 MPa/mm. From Eq. 13, the preclosure stiffness in Simulation
G2 (which has fracture height of 7 m) is 3.64 MPa/mm.
Eq. 15 shows that two parameters, E
0andrn;ref, affect the con-
tact stiffness at the moment of closure. To avoid a sharp disconti-nuity in stiffness and the pressure derivative, the stiffness at
mechanical closure cannot be much greater than the stiffness
before closure. In Simulations G1 and G2, the value of E
0at clo-
sure is very small, causing Sf,contto be very large (5.7 /C2106MPa/
mm when Pis equal to rn), which is why there is a very sharp
change in the pressure derivative with closure.
(b) Comparison to Analytical Solutions. As summarized by


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change in the pressure derivative with closure.
(b) Comparison to Analytical Solutions. As summarized by
Marongiu-Porcu et al. (2011), the pressure trend before closure
can be used to estimate the leakoff coefﬁcient and the fracture size
(Nolte 1979; Shlyapobersky et al. 1988; Valko and Economides1999). The technique involves drawing a straight line through the
plot of pressure vs. G-time during the preclosure period.
In Simulation G1, the slope of the line is –0.0166 MPa/ Gwith
an intercept of 56.51 MPa, and the fracture has radial geometry.
This yields an estimate of fracture radius of 20.4. In the simula-tion, the actual radius at shut-in is 21 m, and a bit of additional
propagation occurs so that the ﬁnal radius is 21.6 m. With the ﬁnal
value of radius in the simulation, 21.6 m, the equations yield an
estimate of leakoff coefﬁcient of 6.95 /C210
/C07m/s. From Howard
and Fast (1957), CL¼DPﬃﬃﬃﬃﬃﬃﬃﬃﬃk/ct
plr
, where ctis total compressibil-


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estimate of leakoff coefﬁcient of 6.95 /C210
/C07m/s. From Howard
and Fast (1957), CL¼DPﬃﬃﬃﬃﬃﬃﬃﬃﬃk/ct
plr
, where ctis total compressibil-
ity, the sum of the ﬂuid compressibility and porosity compressi-
bility. Assuming a value of DPequal to 56.51 – 35 ¼21.51 MPa,
the true value of CLis 9.18 /C210/C07m/s, close to the estimated
value, though somewhat larger. The modest discrepancy is per-
haps not surprising because the derivation of these G-function-
based methods relies on several assumptions, such as Carter
leakoff and power-law growth of the fracture, that are not exactlysatisﬁed in our more-general numerical calculations. With the true
values of radius at shut-in (21 m) and leakoff coefﬁcient, the ana-
lytical equation predicts an average fracture aperture of 1.12 mm,.........................
........................
.......................
..........................
60
57
5154
48
45
42Pressure (solid) (MPa)
0 20 40 60 80 100 12030
25
20
15
10
5
0
G-time


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.......................
..........................
60
57
5154
48
45
42Pressure (solid) (MPa)
0 20 40 60 80 100 12030
25
20
15
10
5
0
G-time
Δt (hours) Δt (hours) Δt (hours) ΔP (solid), τ*dP/dτ (dashed) (MPa)
I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)
10–210–2
100100
102
10–2100100100100
102102103
102105
104
102
101101
101
Fig. 4—Results from Simulation G2. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure
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compared to the actual value from the simulation value of
1.11 mm. With the actual simulation values of average aperture
and radius at shut-in, the analytical equation predicts efﬁciency
(volume of ﬂuid in the fracture divided by volume injected) at
shut-in of 97%, the same as the actual efﬁciency at shut-in during
the simulation.
In Simulation G2, the slope of the line is -0.0496 MPa/ Gwith
an intercept of 58.19 MPa. With the equations for a PKN fracture,
this yields an estimate of fracture half-length of 133.7 m. At shut-in, the actual fracture half-length is 108.6 m. The fracture propa-
gates a bit further after shut-in, reaching 122.6 m. Withthe actual
value of ﬁnal fracture half-length, the analytical equation predicts
leakoff coefﬁcient of 6.95 /C210/C07m/s. With actual values of half-
length at shut-in and leakoff coefﬁcient, the analytical expressionspredict average aperture of 1.02 mm, compared to the actual simu-


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length at shut-in and leakoff coefﬁcient, the analytical expressionspredict average aperture of 1.02 mm, compared to the actual simu-
lation value of 1.01 mm. With the actual value of average aperture
and fracture half-length at shut-in, the analytical expression pre-
dicts efﬁciency at shut-in of 97%, the same as the actual efﬁ-
ciency at shut-in during the simulation.
Effect of Increasing Residual Aperture. Simulation G3 is a
PKN-like simulation (height conﬁned to 7 m), identical to Simula-tion G2, except that the fracture is given a maximum residual
aperture of 500 mm. The results are shown in Fig. 5.
A discontinuity in the derivative of pressure still occurs when
closure occurs, but it is much less pronounced than in Simulations
G1 and G2. The 90% closure stress is 51 MPa, and so the stiffnessat closure caused by contact is 11.3 MPa/mm, a factor of 3 greater
than the stiffness of the fracture preclosure.
As indicated by the orange arrows, the fracture compliance


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than the stiffness of the fracture preclosure.
As indicated by the orange arrows, the fracture compliance
method indicates closure at the upward deviation of the G/C2dP/dG
curve, at G-time of approximately 20 (2.4 hours) and a pressure
equal to 57 MPa, or at G-time of 25.5 (3.8 hours) and a pressure
equal to 56.6 MPa.
As indicated by the blue arrow, the tangent method indicates
closure from the G-function plot at G-time equal to approximately
46, equivalent to approximately 12 hours. This yields an estimate
of closure pressure equal to approximately 51.8 MPa. In the Bour-
det plot with superposition derivative, there appears to be a 1.5
slope, followed by a disturbance in the derivative at around thetime of closure, followed by what appears to be a resumption of the
1.5 slope until approximately 12 hours. Picking closure at the end
of the second 1.5 slope (solid green arrow) yields a closure estimatesimilar to the tangent method, approximately 51.8 MPa. Picking


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of the second 1.5 slope (solid green arrow) yields a closure estimatesimilar to the tangent method, approximately 51.8 MPa. Picking
closure at the end of the ﬁrst 1.5 slope (dashed green arrow) yields
a result similar to the fracture compliance method. In the Bourdet
plot with integrated pressure, an upward deﬂection of the deriva-
tive plot can be seen as early as 1.7 hours after shut-in (red arrow),
equivalent to G-time of 16.7, giving a closure pick of 57.2 MPa.
From the simulation results, it is possible to identify when and
how mechanical closure occurs. Fig. 6 shows the distribution of
aperture and residual aperture in the fracture shortly after shut-in.The red lines represent the residual aperture, and the blue lines
represent the total aperture. The upper part of the ﬁgure shows the
aperture distribution along the fracture length, down the midpoint.
The lower part of the ﬁgure shows the aperture distribution along
a vertical line running through the center of the fracture.


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The lower part of the ﬁgure shows the aperture distribution along
a vertical line running through the center of the fracture.
Fig. 7 shows the aperture distribution after 2.32 hours of shut-
in. Here, one can see that although the fracture has not closed
along its length, a fracture-height recession process has begun to
occur. The fracture has closed to the residual aperture along the
upper and bottom 1.5 m of the fracture. The beginning of this frac-ture-height recession process is what causes the initial slight dis-
continuity and increase in the derivative at G-time equal to
approximately 20.
The height recession is not caused by stress layering in the for-
mation (stress is homogeneous). Instead, height recession iscaused by the residual aperture. As predicted by the PKN approxi-
mation, the aperture distribution in the vertical direction is ellipti-
cal, tapering to zero at the edges (though this is imperfectlycaptured in the simulation because of mesh coarseness). The max-


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cal, tapering to zero at the edges (though this is imperfectlycaptured in the simulation because of mesh coarseness). The max-
imum residual aperture is speciﬁed to be a constant value,
500mm. As the ﬂuid pressure decreases, aperture gradually
decreases, maintaining an elliptical shape and continuing to taper
to zero at the top and bottom. As aperture decreases, regions de-velop where the aperture has reached the residual aperture (me-
chanical closure). These regions extend inward from the top and
bottom of the fracture.
Fig. 8 shows that at approximately 4 hours, the aperture has
reached the residual aperture everywhere along the fracture. Thisis complete mechanical closure and coincides with the sharp
increase in the derivative.
Even though fracture-height recession occurs in this simula-
tion (and occurs in all the simulations with non-negligible residual
aperture), there is no ambiguity between its effect on the pressure
60
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tion (and occurs in all the simulations with non-negligible residual
aperture), there is no ambiguity between its effect on the pressure
60
54
48
42Pressure (solid) (MPa) ΔP (solid), τ*dP/dτ (dashed) (MPa)
Δt (hours) Δt (hours)Δt (hours)I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)
100100102
102102
102104103
102101101
101G-time0 20 40 60 80 100 12015
10
5
0
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)100
10010010010–1
10–210–2101
Fig. 5—Results from Simulation G3. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure
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transient and the effect of closure on the pressure transient. Fig. 8
shows that the fracture closes fully by G-time of 28, coinciding
with the time when G/C2dP/dGbegins to increase rapidly (Fig. 5).
The fracture-height recession occurring before complete closure(seen by comparison between Fig. 7 and Fig. 8) during the period
of approximately 20 G-time to 28 G-time causes a visible upward
deﬂection of G/C2dP/dG, but this deﬂection is slight compared to
the much-larger deﬂection that occurs after complete closure.
The true value of the minimum principal stress is 55.1 MPa,
but the ﬂuid pressure at the time of mechanical closure is slightly
larger, approximately 57 MPa. This occurs because there is stillsome stress shadow on the fracture at closure as a result of the re-
sidual aperture. This stress shadow can be estimated by multiply-
ing the residual aperture (500 mm) by the fracture stiffness causedby elastic deformation of the formation, S
f,o(calculated from Eq.


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ing the residual aperture (500 mm) by the fracture stiffness causedby elastic deformation of the formation, S
f,o(calculated from Eq.
13). This yields an estimate of 1.82 MPa, close to the overesti-
mate that is observed.
Adding Wellbore Storage. In Simulation G3, several aspects
of the simulation remain unrealistic. Wellbore storage isneglected, there is not near-wellbore pressure loss, and there is
still a rather sharp discontinuity in derivative with closure. In Sim-
ulation G4, these issues are resolved. Simulation G4 is similar toSimulation G3 but includes wellbore storage, wellbore/near-well-
bore pressure drop, and uses a much lower value of r
n;ref, 5 MPa,
to prevent a sharp increase in derivative with closure. From these
values, we can calculate Sf,contto be 1.1 MPa/mm at Pequal to rn
(Eq. 15). This is less than the stiffness caused by the elastic com-
pliance of the surrounding formation, Sf,o(3.64 MPa/mm). Thus,
Time since shut-in: 0.0013386 hr, G-time: 0.08962


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pliance of the surrounding formation, Sf,o(3.64 MPa/mm). Thus,
Time since shut-in: 0.0013386 hr, G-time: 0.08962
Distance (m)Distance (m)
–3 –2 –1 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5
1.51
10.5
0.50
0–150 –100 –50 0 50 100 150
Fig. 6—Distribution of aperture during Simulation G3 soon after shut-in. The upper ﬁgure shows aperture along the length of the
fracture, and the lower ﬁgure shows aperture along a vertical cross section at the center of the crack.
Time since shut-in: 2.3205 hr, G-time: 19.6252
Distance (m)Distance (m)
–3 –2 –1 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5
1.51
10.5
0.50
0–150 –100 –50 0 50 100 150
Fig. 7—Distribution of aperture during Simulation G3 at 2.3205 hours after shut-in (19.6252 G-time). The upper ﬁgure shows aper-


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Fig. 7—Distribution of aperture during Simulation G3 at 2.3205 hours after shut-in (19.6252 G-time). The upper ﬁgure shows aper-
ture along the length of the fracture, and the lower ﬁgure shows aperture along a vertical cross section at the center of the crack.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1328 Total Pages: 19
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when mechanical closure occurs in Simulation G4, the stiffness
increases only fractionally, and a sharp discontinuity in pressure
derivative does not occur.
The fracture forms after 36.6 seconds of injection, at a break-
down pressure of 61.15 MPa, before dropping down to a fracture-propagation pressure approximately 60 MPa. The fracture-propa-
gation pressure is considerably higher than the ISIP because of
the wellbore and near-wellbore pressure drop (Fig. 9).
The pressure transient in this simulation appears realistic and
is similar to the match to ﬁeld data described in the Field DFIT #1subsection. The fracture-compliance method gives a closure pick
atG-time equal to 33, pressure of 56.7 MPa, and 4.3 hours after
shut-in (orange arrow). The closure pick from the tangent methodis at G-time equal to 77, pressure of 52.75 MPa, and 22 hours af-
ter shut-in (blue arrow).
The method of Mohamed et al. (2011) gives a closure pick


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ter shut-in (blue arrow).
The method of Mohamed et al. (2011) gives a closure pick
similar to that of the tangent method (green arrow). The Bourdet
plot with integrated pressure gives a rather ambiguous pick forclosure, with the ﬁrst deﬂection of the curve at 1 hour (15.1 G-
time; dashed red arrow), earlier than the interpretation from any
other method. An inﬂection is also apparent at approximately 20
hours (73 G-time), close to the timing of the closure pick from the
tangent method (solid red arrow).
As in Simulation G3, fracture-height recession occurs. How-
ever, this causes only the very-slight concave upward behavior
seen in the G/C2dP/dGcurve from approximately G-time equal to
Time since shut-in: 4.4788 hr, G-time: 27.8749
Distance (m)Distance (m)
–3 –2 –1 0 1 2 3Aperture distribution in vertical cross section at the wellboreAperture at Crack Center (mm) Aperture Along Crack Center (mm)1.5
1.51
10.5
0.50
0–150 –100 –50 0 50 100 150


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1.51
10.5
0.50
0–150 –100 –50 0 50 100 150
Fig. 8—Distribution of aperture during Simulation G3 at 4.4788 hours after shut-in (27.8749 G-time). The upper ﬁgure shows aper-
ture along the length of the fracture, and the lower ﬁgure shows aperture along a vertical cross section at the center of the crack.
60
54
48
42Pressure (solid) (MPa)ΔP (solid), τ*dP/dτ (dashed) (MPa)
Δt (hours) Δt (hours) Δt (hours) I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)
100100102
102102
102104
101G-time0 20 40 60 80 100 12015
10
5
0
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)100
10010010010–2
10–210–2
Fig. 9—Results from Simulation G4. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure
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20 to G-time equal to 38. There is no ambiguity between this
modest effect and the sharply increasing G/C2dP/dGcurve from 38
G-time to 80 G-time, which is interpreted as fracture-height reces-
sion by the tangent method. As we have shown here, the increas-
ingG/C2dP/dGcurve is caused solely by the increasing fracture
stiffness after mechanical closure.
Reduction in the wellbore and near-wellbore pressure drop
causes the very rapid, brief period of pressure decrease immedi-ately after shut-in.
Fracture Tip-Extension. Simulation G5 is performed with a
lower value of fracture toughness, 2 MPa /C1m1/2(instead of
10 MPa /C1m1/2, used in Simulations G1-G4). Simulation G5 is oth-
erwise identical to Simulation G4. With this lower value of frac-
ture toughness, a large amount of fracture tip-extension occurs
after shut-in. As in Simulation G4, the fracture forms after 36.6seconds of injection, at a breakdown pressure of 61.15 MPa,


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after shut-in. As in Simulation G4, the fracture forms after 36.6seconds of injection, at a breakdown pressure of 61.15 MPa,
before dropping down to a fracture-propagation pressure approxi-
mately 60 MPa (Fig. 10).
The fracture tip-extension is visible on the G-function plot. As
described by Nolte (1991) and Barree and Mukherjee (1996), tip-extension causes the G/C2dP/dGcurve to increase sharply and then
curve concave down. At shut-in, the fracture half-length is 110 m.
However, after shut-in the fracture reaches a half-length of 204 m.
Fracture growth does not end until the fracture closes. The tip-
extension accelerates closure because it increases surface area andleakoff. Also, as the fracture grows, the average aperture along
the fracture is forced to decrease to accommodate mass balance,
causing aperture to reach the residual aperture sooner.
The high fracture length from this simulation may seem unre-
alistic for such a relatively modest volume of ﬂuid pumped. Simu-


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The high fracture length from this simulation may seem unre-
alistic for such a relatively modest volume of ﬂuid pumped. Simu-
lations (not shown in this paper) show that with multiple
propagating-fracture strands, the transient looks similar, but the
overall fracture length is proportionally decreased, according tothe number of fractures. Thus, the transient from a single fracture
with a half-length of 200 m will be similar to the transient from
multiple-fracture strands, each with length much less than 200 m.
Stress shadowing tends to cause only one or two fractures to prop-
agate, but near-wellbore pressure drop has the opposite effect,encouraging propagation of a larger number of fractures.
Investigation of the simulation results indicates that closure
occurs at G-time of approximately 3.9 (0.1 hour) and a pressureof 56.5 MPa. The fracture-compliance method indicates that clo-
sure occurs at a G-time of 12.7 (0.7 hour) and a pressure of 55.5


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sure occurs at a G-time of 12.7 (0.7 hour) and a pressure of 55.5
MPa (orange arrow). The tangent method indicates that closure
occurs at G-time of 32.6 (4.2 hours) and pressure of 52.35 MPa
(blue arrow). The Bourdet plot with superposition-derivativemethod suggests that closure occurs at 3.2 hours (28.3 G-time)
and a pressure of 53.3 MPa (green arrow). The integrated-pressure
plot suggests that closure occurs at approximately 2.5 hours (24.7G-time) and a pressure of 54 MPa (red arrow).
Higher Permeability. Simulation G6 is performed with a
permeability of 10
/C015m2(/C241 millidarcy), much higher than the
other simulations. As in Simulations G4 and G5, the fracture
forms after 36.6 seconds of injection, at a breakdown pressure
of 61.15 MPa, before dropping down to a fracture-propagation
pressure approximately 60 MPa. The other simulations in thisstudy are performed with a 1D leakoff model. But this simulation
is performed with a fully numerical treatment of ﬂow in the


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is performed with a fully numerical treatment of ﬂow in the
matrix around the fracture, with the method described by
McClure (2014).
Fig. 11 shows that before mechanical closure, a straight line
can be seen on the G/C2dP/dGplot, as expected. When mechanical
closure occurs, at approximately 0.85 G-time (0.012 hours), the
rate of pressure drop decreases (orange arrow). This behavior is
different from the previous cases, when closure causes the rate of
pressure drop to increase. The different behavior can be under-stood by considering Eq. 11, which shows that the rate of pressure
decrease is equal to the product of system stiffness (the reciprocal
of total storage coefﬁcient, deﬁned in Appendix A) and leakoffrate. In a very-low permeability formation, the fracture remains
inﬁnitely conductive (dimensionless fracture conductivity greater
than 300) even after closure. But with higher permeability, the
dimensionless fracture conductivity rapidly drops below 300 after


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than 300) even after closure. But with higher permeability, the
dimensionless fracture conductivity rapidly drops below 300 after
closure. As a result, the rate at which ﬂuid can ﬂow from the well(where ﬂuid is stored because of wellbore storage) into the frac-
ture is reduced by closure. If leakoff rate decreases enough to
counteract the effect of increasing stiffness, the rate of pressure
decline will decrease.
In addition, in cases of high leakoff, the fracture length is
much shorter. Eq. 11 shows that with lower fracture area, closure
has a relatively smaller impact on the system stiffness, and well-
bore storage has a relatively larger effect. Thus, closure has less
60
51
42Pressure (solid) (MPa)ΔP (solid), τ*dP/dτ (dashed) (MPa)
Δt (hours) Δt (hours) Δt (hours) I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)
100100102
102102
102104
101G-time0 20 40 60 80 100 12015
10
5
0
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)100101
10010010010–1
10–210–2


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102102
102104
101G-time0 20 40 60 80 100 12015
10
5
0
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)100101
10010010010–1
10–210–2
Fig. 10—Results from Simulation G5. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure
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effect on the overall stiffness, which lessens the tendency of clo-
sure to increase the derivative.
The linear ﬂow period that corresponds to the straight line on
theG/C2dP/dGcurve can be seen from approximately 0.001 hour
to approximately 0.01 hours on the Bourdet plot (upper right ofFig. 11). After the linear ﬂow period, the Bourdet derivative-plot
ﬂattens out to a slope of approximately zero (with some gradual
increase), corresponding to a radial ﬂow period (which begins
rapidly because of the relatively high permeability and short frac-
ture), before ﬁnally reaching an impulse radial ﬂow period with anegative one slope, starting at approximately 1.00 hour ( G-time of
15). Comparison of the Bourdet plot with the G-function plot indi-
cates that the G/C2dP/dGcurve continues increasing after closure,
though more slowly.
In the Bourdet plot with superposition derivative, the late-time
impulse radial appears as a ﬂat line, as expected. The intermediate


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In the Bourdet plot with superposition derivative, the late-time
impulse radial appears as a ﬂat line, as expected. The intermediate
time, approximately radial, period shows a slope of 1.0, which is
expected because taking the derivative with respect to superposi-tion time increases the log-log slope by one at shut-in times much
greater than the period of injection (Section 1.2.2). Careful
inspection of the Bourdet plot with respect to integrated pressure
shows no indication of closure.
As in the previous cases, closure should be picked at the devia-
tion from linearity of the G/C2dP/dGcurve (orange arrow). How-
ever, in this relatively high-permeability case, the slope of the
G/C2dP/dGcurve decreases after closure, rather than increasing.
The tangent method provides a closure pick at approximately 8
G-time, at a pressure far below the minimum principal stress
(blue line).
Field DFIT #1. In this subsection, we review Field DFIT #1.


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G-time, at a pressure far below the minimum principal stress
(blue line).
Field DFIT #1. In this subsection, we review Field DFIT #1.
This DFIT was previously shown on the left side of Figs. 4 and 5
from Cramer and Nguyen (2013). The simulation that we perform
to match the data is called Simulation F1 in Table 2. The injection
rate over time is shown in Fig. 1.
There are several reasons that this is a good example of a
DFIT transient.
•Cement integrity is excellent, as evidenced by the cement-
bond log, assuring adequate interzonal isolation throughout
the injection and falloff periods.
•The test interval is mechanically strong (low-clay content)
and sufﬁciently thick (7 m) for establishing a dominant verti-cal hydraulic fracture. These mechanical characteristics are
indicated by openhole log signatures: neutron density, resis-
tivity, and sonic log responses (Fig. 12).
•As shown in Fig. 12, the stress contrast to higher-stress
bounding rock layers is sufﬁcient for containing a DFIT-


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•As shown in Fig. 12, the stress contrast to higher-stress
bounding rock layers is sufﬁcient for containing a DFIT-
scale fracture to the test interval.
•A conductive drilling-induced tensile fracture is exhibited
in the center of the test interval by an openhole image log(Fig. 13). This pre-existing fracture facilitates noncomplex,
planar fracture initiation and propagation.
•The vertical orientation of the wellbore ensures that the axial
starter fracture will be in line with the plane normal to the
minimum principal/horizontal stress, further reducing the
potential for fracture complexity (Cramer and Nguyen
2013).
•The pressure rollover event observed near the beginning of
injection and at a relatively low bottomhole pressure is a
strong indication that the preexisting fracture is reopened
during the DFIT (Fig. 1). Reopening of an existing fracture
ensures that the inﬂuence of wellbore hoop-stress zone is
mitigated or eliminated.


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during the DFIT (Fig. 1). Reopening of an existing fracture
ensures that the inﬂuence of wellbore hoop-stress zone is
mitigated or eliminated.
•The onset of the pressure rollover event, in which the pres-
sure plot departs from the wellbore-storage trend line (at
55.3 MPa), is, in effect, the fracture-reopening pressure.
This fracture-reopening pressure should be slightly above
the fracture-closure pressure.
•There is a water-hammer event at the end of injection, con-
ﬁrming that an excellent connection existed between the
wellbore and primary, far-ﬁeld fracture. This is assurancethat there was no distortion of the shut-in pressure response
caused by near-wellbore fracture complexity or tortuosity.
•Borehole breakout analysis of adjacent intervals indicates
the existence of a large difference between minimum and
maximum horizontal stresses. Consequently, the probabilityof a primary hydraulic fracture interacting with and reopen-


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maximum horizontal stresses. Consequently, the probabilityof a primary hydraulic fracture interacting with and reopen-
ing crossed natural fractures is very-low; transverse fracture
storage is extremely unlikely.
•Permeability in the tested interval is very low, in the range
of 50 to 150 nd. Consequently, dimensionless fracture con-ductivity should be very high following mechanical-fracture
closure, resulting in a stable ﬂuid leakoff rate during closure.
Thus, the change in fracture compliance/stiffness caused by
60
40Pressure (solid) (MPa)ΔP (solid), τ*dP/dτ (dashed) (MPa)
Δt (hours) Δt (hours) Δt (hours) I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)102G-time0 246 8 10 125
0
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)
100100100100
105
100100
10–310–110–210–210–210–2
Fig. 11—Results from Simulation G6. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.


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Fig. 11—Results from Simulation G6. Upper left: G-function plot. Upper right: Bourdet plot with derivative with respect to time.
Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot with integrated pressure.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1331 Total Pages: 19
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fracture closure should dominate the pressure falloff
response when fracture closure occurs.
The simulation very closely matches the ﬁeld data. The simu-
lation match is accomplished with a minimum principal stress
equal to 54.3 MPa. This compares well with the fracture-reopen-
ing pressure of 55.3 MPa, which is derived in analyzing Fig. 2.
The corresponding net pressure at the ISIP (57.1 MPa) is equal to
2.8 MPa, which is a reasonable value with the small-injection vol-ume and compatible with simulations performed with commercial
hydraulic-fracture models.Fig. 2 shows there is a very-brief period of rapid-pressure drop
at the start, as wellbore friction and near-wellbore friction
decrease. Subsequently, the G/C2dP/dGcurve appears to be contin-
uously concave upward during the shut-in period. However, a
close examination of the data indicates that it forms a relativelyﬂat straight line until approximately G-time of 7 (0.48 hours),


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close examination of the data indicates that it forms a relativelyﬂat straight line until approximately G-time of 7 (0.48 hours),
when it begins to increase sharply. Thus, according to fracture-
compliance method, the closure estimate is at G-time of 7, for a
pressure of 55.2 MPa (orange arrow). Plots of aperture vs. posi-
tion from the simulation results (similar to those shown in theTVD
m
3025.0
3030.03035.03040.0
3045.0
HCGR GAPI0 200 100
HSGR GAPI0 200 100AT90 Ω.m0.10 100
AT30 Ω.m
AT10 Ω.mPDMC dec0.30 –0.10 0.1000
TNPH_LIM V/V0.30 –0.10 0.1000DTCO US/F50.00 150 100
DTSM_SL US/F
DTSM_FA US/FCorr_Stress MPa40.00            65.00 52.50
Log_Stress MPa40.00               65.00 52.50Dyn_PR unitless0.15              0.30 0.22
Static_E GPa20.00                50.00 35.00
Fig. 12—Openhole log data for Field DFIT #1.
Conductive Fracture
Perforations: 3035–3040 m
Fracture noted from 3037 to 3039 m
From Lane/CoP303930383037


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Conductive Fracture
Perforations: 3035–3040 m
Fracture noted from 3037 to 3039 m
From Lane/CoP303930383037
Fig. 13—Wellbore imaging log for Field DFIT #1.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1332 Total Pages: 19
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Effect of Increasing Residual Aperture subsection) conﬁrm that
closure occurs at G-time of 7 in the simulation. Fracture closure
occurs at a pressure approximately 0.9 MPa greater than rhmin
because of the stress shadow caused by the residual aperture.
The tangent method yields an estimate of closure pressure
approximately G-time of 36 (approximately 9.4 hours), which is
at a pressure approximately 49.3 MPa. The net pressure corre-sponding to this estimate is 7.8 MPa, which is unrealistically high
with the small injection volume because it implies exceptionally
high fracture toughness. The Mohamed et al. (2011) method picksclosure at approximately 8 hours ( G-time of 33.2), for a pressure
of approximately 50 MPa. There is a slight deviation in the Bour-
det plot with integrated pressure curve at approximately 4 hours
(G-time of 23) and at pressure of 52.5 MPa.
The GRI/DOE M-Site Tests. The GRI/DOE M-Site project
involved a series of hydraulic-fracturing injections into two inter-


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The GRI/DOE M-Site Tests. The GRI/DOE M-Site project
involved a series of hydraulic-fracturing injections into two inter-
vals - the “B-sand” and the “C-sand”. In this section, we consider
the tests in the B-sand. The B-sand consists of a complexsequence of sandstone, mudstone, siltstone, and shale. The site
was unusually well-instrumented, involving a microseismic arrayand downhole tiltmeters in an offset well (Branagan et al. 1996).
The tiltmeters provide an independent estimate of the mini-
mum principal stress. In the B-sand, seven different injection tests
were performed, some involving clear water, others involving
gelled or cross-linked ﬂuid, and one involving proppant. Pump-inﬂuid volume during these tests ranges from 3.0 to 95 m
3(19 to
600 bbl). Pressure measurements are available for an extended
time after shut-in for two of the tests: the initial breakdown injec-tion, which involved 3.0 m3(19 bbl) of crosslinked gel pumped at


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time after shut-in for two of the tests: the initial breakdown injec-tion, which involved 3.0 m3(19 bbl) of crosslinked gel pumped at
13.25 L/s (5 bbl/min), and injection 6B, which involved 63.6 m3
(400 bbl) of linear gel pumped at 58.3 L/s (22 bbl/min). The ma-trix permeability is approximately 0.1 md ( /C2510/C016m2), signiﬁ-
cantly higher than Field DFIT #1 (see the Field DFIT #1subsection).
Fig. 14 shows the result from the initial breakdown injection
test. Breakdown of the formation occurred 2.7 minutes before
shut-in at 30.77 MPa before dropping to a fracture-propagation
pressure of approximately 27 MPa.
The 6B injection shut-in transient is unusual (showing no clear
indication of closure at any pressure), was the sixth injection into
the interval, and used a much larger volume than is typically used
in a DFIT. Therefore, the results from that test are not presented.
TheG-function plot shows a period of fracture tip-extension at


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in a DFIT. Therefore, the results from that test are not presented.
TheG-function plot shows a period of fracture tip-extension at
early-time, which terminates at G-time /C241. Afterward, the G/C2dP/
dGcurve ﬂattens out before beginning to increase sharply after
approximately G-time of 3.5 (0.16 hours; indicated by orange
arrow). According to the fracture-compliance method, this
increase in the G/C2dP/dGcurve represents closure, and so the
minimum principal stress should be picked from the pressure cor-
responding to G-time of 3.5, which is 24.3 MPa.
The tangent method and the Mohamed et al. (2011) method
yield a closure pick at approximately 0.85 hours of shut-in ( G-
time of 9.5), corresponding to a pressure of 22.75 MPa (blue and
green arrows). In the Bourdet with integrated pressure plot, there
is no apparent deﬂection indicating closure.
Fig. 15 shows the normalized tiltmeter response plotted
against wellbore pressure during injection 2B, the test immedi-


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Fig. 15 shows the normalized tiltmeter response plotted
against wellbore pressure during injection 2B, the test immedi-
ately after the breakdown test shown in Fig. 14. A smoothed ver-
sion of this ﬁgure is shown in Fig. 9A-4 of Economides and Nolte
30
25
20Pressure (solid) (MPa) ΔP (solid), τ*dP/dτ (dashed) (MPa)
Δt (hours) Δt (hours)Δt (hours)I (solid), Δt*dI/dΔt (dashed)
(MPa-hr)G-time0 246 8 100
ΔP (solid), Δt*dP/dΔt (dashed)
(MPa)G*dP/dG (dashed) (MPa)
101101
100100100100
10–110–110–1
10–310–3
10–2100100
10–110–1
10–310–210–210–2
Fig. 14—Field data from the M-site, B-sand breakdown injection. Upper left: G-function plot. Upper right: Bourdet plot with deriva-
tive with respect to time. Lower left: Bourdet plot with derivative with respect to superposition time. Lower right: Bourdet plot withintegrated pressure.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20 22 26
Pressure (MPa)4628
4558
4487
44184348
4274Normalized Tilt
Fig. 15—Normalized tiltmeter data from Injection 2B from the


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0.5
0.4
0.3
0.2
0.1
0
20 22 26
Pressure (MPa)4628
4558
4487
44184348
4274Normalized Tilt
Fig. 15—Normalized tiltmeter data from Injection 2B from the
GRI/DOE M-site. Data courtesy of Norm Warpinski.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1333 Total Pages: 19
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(2000). Each tiltmeter response is normalized by dividing by the
maximum tilt measured at that instrument during the test. The tilt-
meters demonstrate that the fracture began opening at approxi-mately 23.9 MPa, very close to the closure pressure estimated by
the fracture-compliance method from the shut-in transient after
the breakdown injection (which immediately preceded the 2B
injection). At 23.9 MPa, the derivative of tilt with respect to pres-
sure becomes much greater, indicating an abrupt drop in fracturestiffness at that pressure corresponding to reopening.
Fig. 15 shows only the data during injection, not shut-in. The
shut-in tiltmeter response is much smoother and shows a more-
gradual relationship between tilt and wellbore pressure. This dif-
ference is caused by the contribution to the tiltmeter responsefrom the poroelastic effect of the ﬂuid leaked off into the forma-
tion. During injection, the amount of leakoff is relatively minimal,


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tion. During injection, the amount of leakoff is relatively minimal,
which is why the tiltmeter response during injection (reopening)
is more appropriate for estimating the minimum principal stress.
Branagan et al. (1996) report substantial residual deformation
(20% of the maximum deﬂection) many weeks after the ﬁrst four
injections into the B-sand, which they interpret as suggesting that
the fractures retained signiﬁcant residual aperture after closure.
Nonuniqueness in Simulation Results. The simulation matches
to the ﬁeld data are subject to nonuniqueness. Lower shear modu-
lus, higher fracture toughness, and greater fracture height (or ra-
dial geometry) all result in lower fracture surface area, which can
be compensated by using a higher permeability or lower ﬂuid vis-
cosity. The selection of ﬂuid viscosity is complicated because inmost cases clear water is being injected into a gas- or oil-saturated
formation. Leakoff will be affected by multiphase effects, and the


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formation. Leakoff will be affected by multiphase effects, and the
injection ﬂuid has a different viscosity and compressibility than
the reservoir ﬂuid. In the simulations in this paper, we have not
made an attempt to choose a realistic viscosity and have merelyset the ﬂuid viscosity to 1 cp. Adjustments to viscosity require
proportional adjustments to the permeability to achieve a simula-
tion match.
Because of this nonuniqueness, many of the values used in the
simulation matches should not necessarily be interpreted as beingactual estimates for those parameters in the formation. However,
there are certain features of the transients that allow unique esti-
mation of formation parameters. For example, the wellbore-stor-age coefﬁcient can be estimated with high conﬁdence from the
slope of a plot of pressure vs. volume injected before the leakoff
point or breakdown pressure. Additional data sources can be used
to constrain parameters. For example, the shear moduli can be


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point or breakdown pressure. Additional data sources can be used
to constrain parameters. For example, the shear moduli can be
estimated from core or geophysical logs.
Regardless of these uncertainties, the estimates for closure
pressure that are based on the simulations matches are always
unique. In every case, closure causes the G/C2dP/dGcurve to
deﬂect upward (except the high-permeability case, when it
deﬂected downward). Therefore, the location of the deﬂection intheG/C2dP/dGcurve can be taken as a unique estimate for the clo-
sure pressure. Because the residual aperture creates a stress
shadow, the closure-pressure estimate could be up to 1 to 2 MPa
greater than the actual value of r
hmin(depending on the magni-
tude of the residual aperture and the geometry of the fracture),which creates some uncertainty (see the Effect of Increasing Re-
sidual Aperture subsection).
The net pressures implied by the tangent method of G-function


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sidual Aperture subsection).
The net pressures implied by the tangent method of G-function
interpretation and the Mohamed et al. (2011) method are much
higher than the values derived from our proposed method. For
example, for Field DFIT #1, the net pressure at the ISIP is 2.8
MPa according to the pick from the fracture-compliance method,
and 7.8 MPa according to the pick from the tangent method.Matching these very-high net pressures requires very-high values
of fracture toughness or poorly constrained adjustments for a
ﬂuid-lag zone or inelastic deformation ahead of the crack tip (Jef-
frey 1989; Yew and Liu 1993; Mack and Warpinski 2000). This is
not inconceivable, but in the context of the results from this study,the more-likely explanation is that net pressure is overestimated
because of the method of DFIT interpretation. The lower net pres-
sures implied by the fracture-compliance method are consistentwith values of fracture toughness derived from laboratory
measurements.


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sures implied by the fracture-compliance method are consistentwith values of fracture toughness derived from laboratory
measurements.
Conclusions
1. Using a hydraulic-fracturing simulator that includes the effect
of residual-fracture aperture, simulations of DFITs can be per-
formed that closely match ﬁeld data, including the preclosure,
closure, and post-closure periods.
2. The results suggest that the tangent method for identifying clo-
sure from G-function plots (Barree and Mukherjee 1996; Bar-
ree et al. 2007) tends to underestimate the minimum principal
stress in low-permeability formations because it neglects the
effect of changing residual fracture aperture after closure.
3. The method proposed by Mohamed et al. (2011) for picking
closure from a Bourdet plot with superposition time derivativetends to give results similar to the tangent method, underesti-
mating closure pressure in low-permeability intervals.
4. The method of Craig and Blasingame (2006) for identifying


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mating closure pressure in low-permeability intervals.
4. The method of Craig and Blasingame (2006) for identifying
closure from a Bourdet plot with integrated pressure is based
on picking closure from a change in storage and thus identiﬁesthe correct closure pressure in some cases. However, the
deﬂection on the plot is often ambiguous or impossible to
identify.
5. Closure can best be picked by identifying the departure from
linearity on the plot of pressure vs. G-time or the square root of
time (after the very-early time period after shut-in when dissi-
pating wellbore and near-wellbore friction causes the pressure
to drop rapidly). Identiﬁcation of this departure from linearitycan be assisted by the G/C2dP/dGcurve. In low-permeability
formations, the G/C2dP/dGcurve should deﬂect upward at clo-
sure. We refer to this method as the “fracture compliance
method” for picking closure.
6. In high-permeability cases, the G/C2dP/dGcurve may deﬂect


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sure. We refer to this method as the “fracture compliance
method” for picking closure.
6. In high-permeability cases, the G/C2dP/dGcurve may deﬂect
downward (with pressure deﬂecting upward) when closure
occurs because closure can signiﬁcantly reduce the rate of
leakoff.
7. The residual fracture aperture upon closure can cause a stress
shadow of up to 1 to 2 MPa. As a result, the closure pressurecan be as much as 1 to 2 MPa above the value of the minimum
principal stress.
8. In a fracture-injection test from the GRI/DOE M-site project,
the closure pick can be independently estimated from down-
hole tiltmeters and is consistent with the pick from the frac-ture-compliance method presented in this paper.
Nomenclature
A¼fracture area, m
2
cf¼ﬂuid compressibility, MPa/C01
c/¼porosity compressibility, MPa/C01
Cac¼after-closure storage coefﬁcient, m3/MPa
CfD¼dimensionless fracture conductivity, unitless
CL¼Carter leakoff model leakoff coefﬁcient, m3/(m2/C1s)


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CfD¼dimensionless fracture conductivity, unitless
CL¼Carter leakoff model leakoff coefﬁcient, m3/(m2/C1s)
Cnw¼constant for wellbore and near-wellbore pressure
drop, MPa/(m3/s)2
Cw¼wellbore-storage coefﬁcient, m3/MPa
Ct¼total storage coefﬁcient for fracture and wellbore
system, m3/MPa
e¼hydraulic aperture, related to the transmissivity by
the cubic law, m
E¼void aperture, volume of ﬂuid stored per fracture
surface area, m
E0¼residual fracture aperture, m
Ehf,max,resid ¼maximum residual aperture, m
Eopen¼mechanical fracture opening, m
g¼dimensionless variable used in deﬁning G,deﬁned
in Eq. 2, unitless
G¼G-time, deﬁned in Eq. 1, unitless
Gm¼shear modulus, MPa
hf¼fracture height, m
I¼integrated pressure, MPa /C1s
k¼permeability, m2J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1334 Total Pages: 19
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KIc¼fracture toughness, MPa /C1m1/2
Lf¼fracture half-length, m
m¼mass of ﬂuid in the wellbore and fracture, kg
P¼ﬂuid pressure, MPa
P0¼reference pressure used to calculate ﬂuid density,
MPa
Pinit¼initial ﬂuid pressure, MPa
qinj¼volumetric surface-injection rate, m3/s
qm,inj¼surface mass injection rate, kg/s
qleakoff ¼total volumetric leakoff rate across entire fracture,
m3/s
qm,leakoff ¼total-mass leakoff rate across entire fracture, kg/s
Q¼volumetric ﬂow rate, m3/s
Rf¼fracture radius, m
Sf¼fracture stiffness, MPa/m
Sf,c¼stiffness of a closed fracture, MPa/m
Sf,cont ¼contribution to stiffness of a closed fracture from
contact stress, MPa/m
Sf,o¼stiffness of open fracture, or equivalently, contribu-
tion to stiffness of a closed fracture from the defor-
mation of the surrounding formation, MPa/m
Sf,PKN¼stiffness of an open PKN fracture, MPa/m
Sf,r¼stiffness of an open radial fracture, MPa/m
t¼time, seconds
te¼duration of injection, seconds
T¼transmissivity, m3


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Sf,r¼stiffness of an open radial fracture, MPa/m
t¼time, seconds
te¼duration of injection, seconds
T¼transmissivity, m3
Vfr¼volume of ﬂuid in fracture, m3
Vwl¼volume of ﬂuid in wellbore, m3
a¼power-law exponent for fracture growth used in the
G-time derivation, unitless
DP¼change in pressure since the start of the transient,
MPa
DPnw¼wellbore and near-wellbore pressure drop, MPa
Dt¼time from start of transient, seconds
DtD¼dimensionless time from start of transient, unitless
Drn¼deformation-induced change in normal stress, MPa
l¼ﬂuid viscosity, MPa /C1s
/C23¼Poisson’s ratio, unitless
q¼ﬂuid density, kg/m3
q0¼ﬂuid density at the reference pressure, kg/m3
qfr¼average density of ﬂuid in fracture, kg/m3
qinit¼initial ﬂuid density, kg/m3
qinj¼density of injected ﬂuid, kg/m3
qleakoff ¼average density of ﬂuid leaking off from the frac-
ture, kg/m3
qwl¼average density of ﬂuid in wellbore, kg/m3
rhmin¼minimum principal stress, MPa
r0
n¼effective normal stress, MPa
rn¼normal stress, MPa
rr


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qwl¼average density of ﬂuid in wellbore, kg/m3
rhmin¼minimum principal stress, MPa
r0
n¼effective normal stress, MPa
rn¼normal stress, MPa
rr
n¼remote-loading normal stress, MPa
rn,ref¼reference normal stress, MPa
/init¼initial porosity, unitless
/¼porosity, unitless
s¼superposition time, unitless
Acknowledgments
The ﬁnancial support provided by the Fracturing and Sand Con-
trol JIP at the University of Texas at Austin is gratefully acknowl-
edged. The authors would like to thank Norm Warpinski for
supplying data from the GRI/DOE M-site project used in the pa-per and Priscila Ribeiro and Bertrand Rohmer for their useful
comments. Thank you to our friends and colleagues who encour-
aged us in pursuing this research.
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Appendix A: Derivation of the Equations
Governing Pressure-Transient BehaviorDuring a DFIT
Here, we derive the equations governing pressure-transient behav-
ior during a DFIT.
We describe the entire fracture and wellbore system with a
“lumped parameter model,” in which we assume thatdP
dtis uni-
form throughout the entire system. This approach is isothermal
and implicitly neglects pressure gradient caused by ﬂow in thewell, perforation, near-wellbore region, and along the fracture.


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and implicitly neglects pressure gradient caused by ﬂow in thewell, perforation, near-wellbore region, and along the fracture.
Considering these effects would not fundamentally change the
conclusions drawn from the analysis. Related approaches are used
by Raaen et al. (2001) and Craig and Blasingame (2006), though
the derivation we provide here has important differences.
The pressure derivative with time can be decomposed with the
chain rule as
dP
dt¼dP
dmdm
dt; ðA-1Þ
where mis the total mass of ﬂuid in the wellbore and in the frac-
ture and Pis the pressure that is either measured at the bottom of
the well or calculated from surface-pressure measurements.dm
dtis
equal to the surface mass-injection rate, qm,inj, minus the rate at
which ﬂuid mass ﬂows out of the well and fracture into the ma-
trix,qm,leakoff . We can expand these terms by saying that
dm
dt¼qm;inj/C0qm;leakoff ¼qinjqinj/C0qleakoff qleakoff ; ðA-2Þ
where qinjis the volumetric surface injection rate, qinjis the den-


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dm
dt¼qm;inj/C0qm;leakoff ¼qinjqinj/C0qleakoff qleakoff ; ðA-2Þ
where qinjis the volumetric surface injection rate, qinjis the den-
sity of the ﬂuid injected, qleakoff is the volumetric rate of leakoff
into the matrix, and qleakoff is the average density of the ﬂuid leak-
ing off into the matrix.dP
dmcan be expanded by starting with
the deﬁnition
m¼qwlVwlþqfrVfr; ðA-3Þ
where qwlis the average density of the ﬂuid in the wellbore, Vwlis
the volume of the wellbore, qfris the average density of the ﬂuid
in the fracture, and Vfris the volume of the fracture.
To simplify further, we assume that the ﬂuid is slightly com-
pressible and has constant compressibility, cf. In this case, we
can write
q¼q0exp½ðP/C0P0Þcf/C138; ðA-4Þ
where P0is a reference pressure and q0is the density of the ﬂuid
at that pressure. Taking the Taylor-series expansion and discard-
ing higher-order terms provides the approximation
q¼q0½1þcfðP/C0P0Þ/C138: ðA-5Þ
We can take the derivative of this linearized equation with


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ing higher-order terms provides the approximation
q¼q0½1þcfðP/C0P0Þ/C138: ðA-5Þ
We can take the derivative of this linearized equation with
respect to pressure and determine that
dq
dP¼q0cf: ðA-6Þ
Now, we take the derivative of Eq. A-3 with respect to
pressure:dm
dP¼dqwl
dPVwlþqwldVwl
dPþdqfr
dPVfrþqfrdVfr
dP: ðA-7Þ
Combining Eqs. A-1, A-2, and A-7, we arrive at
dP
dt¼qinjqinj/C0qleakoff qleakoff
dqwl
dPVwlþqwldVwl
dPþdqfr
dPVfrþqfrdVfr
dP: ðA-8Þ
Eq. A-8 can be simpliﬁed by relying on the assumption that
the ﬂuid is slightly compressible. If we assume that qinj,qleakoff ,
qwl, and qfrare all approximately equal (for isothermal fresh
water, they would be within a few percentages), and these quanti-
ties are also equal to the q0value used to calculate the derivatives
of ﬂuid density with respect to pressure in the wellbore and the
fracture, then we can cancel them out of the numerator and the de-nominator of Eq. A-8 and arrive at
dP
dt¼qinj/C0qleakoff
cfVwlþdVwl
dPþcfVfrþdVfr


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fracture, then we can cancel them out of the numerator and the de-nominator of Eq. A-8 and arrive at
dP
dt¼qinj/C0qleakoff
cfVwlþdVwl
dPþcfVfrþdVfr
dP¼qinj/C0qleakoff
Ct;
/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1/C1 ð A-9Þ
where Ctis the total storage coefﬁcient of the entire system. The
deﬁnition of total-storage coefﬁcient used in Eq. A-9 is more gen-
eral than the expressions used by Raaen et al. (2001) and by Craig
and Blasingame (2006).
Craig and Blasingame (2006) deﬁne the after-closure storage
coefﬁcient as (with slightly different variable deﬁnitions)
Cac¼cfVwlþcfVfr; ðA-10 Þ
where Vfris assumed constant. Comparing this expression with
Eq. A-9, we can see that Craig and Blasingame (2006) implicitly
assume that qwldVwl
dPandqfrdVfr
dPare zero. The former term, the
changing wellbore volume, is routinely assumed negligible. How-
ever, the latter term, for the changing fracture volume after clo-


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changing wellbore volume, is routinely assumed negligible. How-
ever, the latter term, for the changing fracture volume after clo-
sure, is large, as shown next. Craig and Blasingame (2006) permitthat the ﬂuid in the wellbore and fracture may have different com-
pressibility, whereas we assume that their compressibility is the
same. The distinction should have a negligible effect in a DFIT,where clear water is injected.
Raaen et al. (2001) use an approach similar to that shown in
Eq. A-9. However, they assume that “when the fracture is closed,
there are three main contributions to the system stiffness: from
the compliance of the wellbore ﬂuid, from the deformation of thecasing and top-side tubes, and from the deformation of the open-
hole below the casing shoe.” Therefore, they implicitly assume
that the
dqfr
dPVfrandqfrdVfr
dPterms in Eq. A-9 are zero.
With the assumptions we describe, Eq. A-9 applies during all


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that the
dqfr
dPVfrandqfrdVfr
dPterms in Eq. A-9 are zero.
With the assumptions we describe, Eq. A-9 applies during all
phases of a fracturing test: during injection, the before-closureshut-in period, during closure, and the after-closure shut-in period.
In reality, during injection, pressure gradient caused by ﬂow
through the well, perforations, near-wellbore region, and fracture
causedP
dtto vary spatially throughout the system, but after the
early shut-in period, these pressure gradients should be minimal.
The total storage coefﬁcient has four parts: compressibility of
ﬂuid in the wellbore, changes in the wellbore volume, compressi-
bility of the ﬂuid in the fracture, and changes in fracture volume.Conventionally, changes in wellbore volume are assumed negligi-
ble relative to c
fVwland thedVwl
dPterm is neglected (Horne 1995).
dVfr
dPcan be written asA
Sf, where Ais the surface area of the
fracture and Sfis the fracture stiffness, the derivative of effective


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dVfr
dPcan be written asA
Sf, where Ais the surface area of the
fracture and Sfis the fracture stiffness, the derivative of effective
normal stress with respect to aperture (Jaeger et al. 2007).dVfr
dP
depends on fracture geometry and mechanical characteristics of
the formation. In practice, the fracture geometry is usually not...........................
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well-constrained. Two limiting cases for fracture geometry are
circular and PKN (Perkins and Kern 1961; Nordgren 1972). If
there is no fracture-height restraint caused by layering, then thefracture will be approximately circular (though not completely cir-
cular because the trends in stress with depth will cause it to prefer-
entially propagate upward). If the fracture is conﬁned between
higher stress intervals, then the fracture length will be much
greater than its height, and stiffness can be approximated from a2D plane-strain solution, the key assumption of the PKN model.
For a radial fracture, the stiffness of the fracture (when open),
S
f,o, can be written as (Sneddon 1946):
Sf;r¼3pGm
8ð1/C0/C23ÞRf; ðA-11 Þ
where Gmis the shear modulus, /C23is Poisson’s ratio, and Rfis the
radius. For a fracture with length much longer than its height, the
stiffness can be estimated as (Sneddon 1946; Perkins and Kern
1961; Nordgren 1972):
Sf;PKN¼4Gm
pð1/C0/C23Þhf: ðA-12 Þ
The magnitudes of theA


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stiffness can be estimated as (Sneddon 1946; Perkins and Kern
1961; Nordgren 1972):
Sf;PKN¼4Gm
pð1/C0/C23Þhf: ðA-12 Þ
The magnitudes of theA
Sf,cfVwl, and cfVfrterms can be
roughly assessed (for example, in the fracture simulation shown
in the Effect of Increasing Residual Aperture subsection). The
ﬁnal fracture length is approximately 250 m, with a height of 7 m,
for a surface area of 1750 m2. Fig. 6 shows that the average frac-
ture aperture is roughly 1 mm, which means that Vfris roughly
1.75 m3andcfVfris roughly (with the value of water compressi-
bility given in Table 1) 0.0008 m3/MPa. Vwldepends on wellbore
length and diameter. A typical value is approximately 100 m3, and
socfVfrcan be estimated as 0.0458 m3/MPa. The fracture shown
in Fig. 6 has length much greater than height, 7 m. Therefore, its
preclosure stiffness can be assessed from Eq. A-12 and is equal to
3.638 MPa/mm. TheA
Sfterm is equal to 0.48 m3/MPa. This analy-


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preclosure stiffness can be assessed from Eq. A-12 and is equal to
3.638 MPa/mm. TheA
Sfterm is equal to 0.48 m3/MPa. This analy-
sis indicates that the dominant effect in the total-storage coefﬁ-
cient is likely to be the fracture stiffness, the wellbore storage has
minor but nonnegligible effect, and the compressibility of the
water in the fracture is negligible.
After shut-in, the pressure derivative can be written as (assum-
ing constant wellbore volume and that the cfVfrterm is negligible)
dP
dt¼/C0qleakoff
cfVwlþA
Sf: ðA-13 Þ
If the fracture continues propagating after closure, Amay
change signiﬁcantly and Sfmay change (depending on the geome-
try). If the fracture is no longer propagating, Sfwill be constant
before fracture closure, but will change when the fracture closes.
Using the Carter model to calculate leakoff rate yields a value
ofqleakoff that scales with the inverse of the square root of time


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Using the Carter model to calculate leakoff rate yields a value
ofqleakoff that scales with the inverse of the square root of time
(Howard and Fast 1957). This is why plots of pressure vs. thesquare root of time during shut-in yield a straight line. If leakoff
rate scales with time according to the G-function introduced by
Nolte (1979), then a plot of pressure vs. G-time will yield a
straight line. The straight line will only occur if the denominator
in Eq. A-13 is constant, which is why closure causes a deviationfrom the straight line. Closure also decreases the fracture trans-
missivity. If this causes the fracture to no longer be effectively in-
ﬁnite conductivity, then, pressure gradients can develop along the
fracture, and this can cause the lumped parameter approach used
in this section to no longer be applicable.
To understand the pressure trend after closure, it is necessary
to investigate how stiffness is affected by closure. The stiffness
after closure, S


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To understand the pressure trend after closure, it is necessary
to investigate how stiffness is affected by closure. The stiffness
after closure, S
f,c, includes the effect of the elastic response of the
formation (as shown in Eqs. A-11 and A-12), but also includes a
contribution from the contacting fracture walls (Section 12.3 fromJaeger et al. 2007).As a simplifying assumption, we assume that both the stress
and aperture are constant along the fracture. This is not strictly
correct, but is sufﬁciently accurate for providing a conceptual ex-planation. Under this simplifying assumption, we can write
r
0
n¼rr
nþESf;o/C0P: ðA-14 Þ
Because the initial aperture of a hydraulic fracture is zero
(because it does not initially exist), backstress is created by its
entire aperture, E.
To ﬁnd the fracture stiffness, Eq. A-14 can be solved for Pand
differentiated with respect to E:
Sf;c¼dE
dP/C18/C19/C01
¼dP
dE¼Sf;o/C0dr0
dE¼Sf;o/C0dE
dr0n/C18/C19/C01
¼Sf;oþdE
dP/C18/C19/C01


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differentiated with respect to E:
Sf;c¼dE
dP/C18/C19/C01
¼dP
dE¼Sf;o/C0dr0
dE¼Sf;o/C0dE
dr0n/C18/C19/C01
¼Sf;oþdE
dP/C18/C19/C01
¼Sf;oþSf;cont;/C1/C1/C1/C1/C1/C1/C1/C1/C1 ð A-15 Þ
where Sf,contis the contribution to the stiffness from the contact
stress. Sf,contcan be calculated from any constitutive equation that
speciﬁes Eas a function of effective normal stress. For example,
Eq. 12 yields
Sf;cont¼/C0dE
dr0
n/C18/C19/C01
¼9E0
rn;ref1þ9r0
n
rn;ref/C18/C19/C02"#/C01
¼rn;ref
9E0
E2: ðA-16 Þ
Thus, with Eq. 12, the stiffness of a closed fracture is
Sf;c¼Sf;oþrn;ref
9E0
E2: ðA-17 Þ
The value of Sf,contincreases as the fracture closes. At closure
Sclosure
f;cont¼rn;ref
9E0: ðA-18 Þ
Referring to the simulation described in the Effect of Increas-
ing Residual Aperture subsection, we can calculate Sclosure
f;cont to be
11.333 MPa/mm (for a total Sfof 14.971 MPa/mm), signiﬁcantly
greater than the value of Sf,oof 3.638 MPa/mm. Closure causes a


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f;cont to be
11.333 MPa/mm (for a total Sfof 14.971 MPa/mm), signiﬁcantly
greater than the value of Sf,oof 3.638 MPa/mm. Closure causes a
sharp increase in Sffrom Sf,otoSf,c, which (from Eq. A-13) causes
an increase in the pressure derivative. The derivative increasesvery sharply because stiffness increases so much with closure,
increasing by roughly a factor of 4, and continuing to increase fur-
ther as aperture drops further below E
0. In the Adding Wellbore
Storage subsection, a simulation is shown with a smaller value of
rn;ref, 5 MPa instead of 51 MPa, and so Sclosure
f;contis 1.111 MPa/mm
(for a total Sfof 4.749 MPa/mm), causing only a modest increase
compared to the stiffness before closure. As a result, the deriva-
tive increases less abruptly.
Mark McClure is the founder and chief executive officer of


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tive increases less abruptly.
Mark McClure is the founder and chief executive officer of
McClure Geomechanics and is a consulting professor at Stan-ford University. He was an assistant professor in the Departmentof Petroleum and Geosystems Engineering at the University of
Texas at Austin from 2013 to 2015. McClure’s research focuses
on modeling and characterization of hydraulic fracturing,induced seismicity, and enhanced geothermal systems. Hewas honored with the 2015 SPE Regional Completions Optimi-
zation and Technology Award from the Southwest North Amer-
ican Region, Outstanding Service as a Technical Editor for SPE
Reservoir Evaluation & Engineering in 2014, the Hank Ramey
Award for Outstanding Research and Service to the Depart-
ment from the Department of Energy Resources Engineering at
Stanford University in 2012, and Best Paper in Geophysics from
the Society of Exploration Geophysicists (SEG) in 2011. McClureholds a BS degree in chemical engineering, an MS degree in


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the Society of Exploration Geophysicists (SEG) in 2011. McClureholds a BS degree in chemical engineering, an MS degree in
petroleum engineering, and a PhD degree in energy-resources
engineering, all from Stanford University.
Hojung Jung is a PhD degree student at the University of Texas
at Austin. Her research interests include experiments and.......................
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simulations of geomechanical problems in the context of hy-
draulic fracturing and carbon dioxide storage. Jung holds an
MS degree in petroleum engineering from the University ofTexas at Austin and a BS degree (cum laude) in energy-resour-ces engineering from Seoul National University, South Korea.
She is a member of SPE.
Dave Cramer is a senior engineering fellow on the ConocoPhil-
lips Global Completions Engineering staff in Houston and spe-
cializes in hydraulic-fracturing applications. He has published
48 technical papers and holds two US patents. Cramer is aregistered professional engineer in Colorado. Industry recogni-tions include the Henry Mattson Technical Achievement
Award by the SPE Denver Chapter in 1993 and the SPE Interna-
tional Completions Optimization and Technologies Award in2011. Cramer was an SPE Distinguished Lecturer during2003–2004 and the SPE Region Director for the US and Canada


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Rocky Mountain Region from 2004 to 2007.Mukul M. Sharma is a professor and holds the “Tex” Moncrief
Chair in the Department of Petroleum and Geosystems Engineer-
ing at the University of Texas at Austin, where he has been for thepast 30 years. He served as chair of the Department from 2001 to2005. Sharma’s current research interests include improved oil re-
covery, injection-water management, hydraulic fracturing, for-
mation damage, and petrophysics. He has published more than350 journal articles and conference proceedings and holds 15patents. Sharma has a bachelor of technology degree in chemi-
cal engineering from the Indian Institute of Technology and MS
and PhD degrees in chemical and petroleum engineering,respectively, from the University of Southern California. Amonghis many awards, he is the recipient of the 2009 Anthony Lucas
Gold Medal, the 2004 SPE Faculty Distinguished Achievement


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Gold Medal, the 2004 SPE Faculty Distinguished Achievement
Award, the 2002 SPE Lester C. Uren Award, and the 1998 SPE For-mation Evaluation Award. Sharma served as an SPE DistinguishedLecturer in 2002, has served on the editorial boards of many jour-
nals, and has taught and consulted for industry worldwide.J179725 DOI: 10.2118/179725-PA Date: 2-August-16 Stage: Page: 1339 Total Pages: 19
ID:jaganm Time: 18:40 I Path: S:/J###/Vol00000/160007/Comp/APPFile/SA-J###160007August 2016 SPE Journal 1339


