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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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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 [
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.


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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 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
on well XX22. Due to the limit volume of sampler container the cycling time for influx is short: 17 sec


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- 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).
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


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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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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,
measuring pressure using well pressure buildup curve interpretation method for producer after hydraulicfracturing leads to significant oil production losses.


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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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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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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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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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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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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 with reservoir pressure


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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
wij – measurement accuracy at hydrodynamic logging, unit fraction


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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,
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.


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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
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 fracture tip extension. In


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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 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


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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
G*dP/dG may appear concave up or down ( Soliman and Kabir, 2012 ;Cramer and Nguyen, 2013 ). In some


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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 (
McClure and Horne, 2013 ). The simulations


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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
al., 2007 ;Soliman and Kabir, 2012 ;Padmakar, 2013 ;Cramer and Nguyen, 2013 ;Wallace et al., 2014 ).


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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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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)
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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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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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
following shut-in, pressure dependent leakoff, and fluid storage from transverse or dilated naturalfractures.


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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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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
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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(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).
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


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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
and/H9268n,Eref is the 90% closure stress for void aperture. /H9268n=is the effective normal stress, equal to the


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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.
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
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.


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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
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


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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
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.


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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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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
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


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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.,
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).


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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 (
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.


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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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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
be constant at 20 microns. When fracture elements opened, their hydraulic aperture could grow higherthan e


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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) .
Wellbore storage


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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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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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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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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.
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


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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
/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


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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)
where his the fracture height, p*is the slope of


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(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
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.


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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.
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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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
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


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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
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


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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
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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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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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
closure, and then an abrupt drop in pressure afterclosure. Wellbore storage has an effect when thepressure in the wellbore is changing (


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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
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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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”
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 (


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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
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


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residual aperture percentage had been higher (as in Simulation HRAP ), the transition would have been less
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
between storage and leakoff. In Simulation CMT , the much higher permeability caused the fracture that


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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
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


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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
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


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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
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 (
Figure 13 ). Because the walls begin to come in contact while


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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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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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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 ).
Figure 16—Comparison of two methods for identifying closure stress.


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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.
Figure 12 ,Figure 13 , and Figure


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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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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
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 (


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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” fracture, which may explain


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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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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:
1. In low permeability formations, a concave up deviation of the G*dP/dG curve from a linear trend


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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
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-


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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
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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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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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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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
Background. DFITs are used for a variety of purposes, including
estimating the magnitude of the minimum principal stress in a for-


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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
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.


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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
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-


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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
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).


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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
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


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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-
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


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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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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-
sion. These interpretations are based on numerical simulations


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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)
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


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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
(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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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
integrating the pressure data after shut-in to ﬁnd the effectivepressure response, I. This can be written as
IðDPÞ¼ð
teþDt
0½PðtÞ/C0Pinit/C138dt; ð4Þ


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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.
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


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(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
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


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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
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-


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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
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


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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
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


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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
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-


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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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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
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


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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
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-


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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-
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


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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-
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.


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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
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


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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-
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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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
derivative when closure occurred. Closure clearly and unambigu-


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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)
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


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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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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
Marongiu-Porcu et al. (2011), the pressure trend before closure
can be used to estimate the leakoff coefﬁcient and the fracture size


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(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-
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-


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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
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.


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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-
lation value of 1.01 mm. With the actual value of average aperture


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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
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.


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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
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.


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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.
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.


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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-
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.


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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
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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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
Distance (m)Distance (m)


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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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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
similar to that of the tangent method (green arrow). The Bourdet


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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
Fig. 8—Distribution of aperture during Simulation G3 at 4.4788 hours after shut-in (27.8749 G-time). The upper ﬁgure shows aper-


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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,
before dropping down to a fracture-propagation pressure approxi-


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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-
lations (not shown in this paper) show that with multiple
propagating-fracture strands, the transient looks similar, but the


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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
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


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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
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


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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
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-


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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
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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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.
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


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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-
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


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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.
•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


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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-
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


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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.
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),
when it begins to increase sharply. Thus, according to fracture-


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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
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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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
13.25 L/s (5 bbl/min), and injection 6B, which involved 63.6 m3


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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
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


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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-
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


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(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
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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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
injection ﬂuid has a different viscosity and compressibility than


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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
estimated from core or geophysical logs.
Regardless of these uncertainties, the estimates for closure
pressure that are based on the simulations matches are always


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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
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,


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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.
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-


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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
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-


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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
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


key: chroma:document
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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)
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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key: chroma:document
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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
Vfr¼volume of ﬂuid in fracture, m3


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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
n¼remote-loading normal stress, MPa
rn,ref¼reference normal stress, MPa
/init¼initial porosity, unitless
/¼porosity, unitless


key: chroma:document
string_value:
/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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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-
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-


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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
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


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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
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


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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-
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


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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
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


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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
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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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-
sis indicates that the dominant effect in the total-storage coefﬁ-
cient is likely to be the fracture stiffness, the wellbore storage has


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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
(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


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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
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


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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
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-


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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
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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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
petroleum engineering, and a PhD degree in energy-resources


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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
Rocky Mountain Region from 2004 to 2007.Mukul M. Sharma is a professor and holds the “Tex” Moncrief


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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
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-


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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
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The Spurious Deflection on Log-Log
Superposition-Time Derivative Plots
of Diagnostic Fracture-Injection Tests
Mark McClure, McClure Geomechanics
Summary
Log-log superposition-time derivative plots are used to identify ﬂow regimes in well tests with variable rate. The use of superposition
time adjusts for the effect of the prior rate history, and (under some conditions) shows what the transient would have looked like if the
test had been performed at a constant rate. In this report, I show that if these plots are used to interpret shut-in transients from diagnostic
fracture-injection tests (DFITs), the superposition-time derivative has an upward deﬂection that does not represent actual reservoir or
transient behavior. I review mathematical properties of the superposition-time derivative. I derive equations for the pressure transient ina simpliﬁed model DFIT in which closure does not occur. I show that the onset of late-time impulse ﬂow is controlled by injection vol-


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ume and formation, wellbore, and fracture properties, not the duration of injection (as implied by the deﬁnition of superposition time).
Log-log superposition-time derivative plots of DFITs exhibit a slope of 3/2 at intermediate time. However, pressure change never scales
with a 3/2 power of time. One form of the G-function superﬁcially resembles a superposition-time function constructed by summing
constant-rate solutions with 3/2 power scaling. However, this is not a mathematically or physically valid interpretation. The 3/2 powerarises from a spatial integration of the Carter leakoff solution. There is not a mathematical, physical, or practical justiﬁcation for plot-
ting DFIT pressure-time data in a way that creates a 3/2 slope. I conclude by providing a ﬁeld example and practical recommendations
for DFIT interpretation.
Introduction
The log-log derivative plot is widely used in well-test analysis. The slope of the derivative curve, d p/d[ln( t)], is used to diagnose


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ﬂow regimes–it is more precise to call d p/d[ln( t)] the semilog derivative, but it is common convention in well-test analysis to call
it simply the “derivative curve.” In tests that involve varying rate, such as buildup tests and falloff tests, the derivative is oftencalculated with respect to an effective time function, sometimes called superposition time. In inﬁnite-acting radial ﬂow, the superposi-
tion-time derivative curve shows what the transient would have looked like if the test had been performed at a constant rate (Ehlig-
Economides 1988).
In recent years, log-log derivative plots have been used for the interpretation of DFITs. In a DFIT, 1–100 bbl of water is injected for
several minutes, creating a small hydraulic fracture. The term “DFIT” speciﬁcally refers to a fracturing test performed in a low-perme-
ability formation in a completed well. After shut-in, the transient is monitored to estimate the fracture-closure pressure, formation per-
meability, formation pressure, and other properties.


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meability, formation pressure, and other properties.
The calculation of formation properties requires the identiﬁcation of ﬂow regimes, which is assisted by log-log derivative plots.
Practitioners often interpret DFITs with log-log plots with derivative taken with respect to superposition time (Mohamed et al. 2011;
Marongiu-Porcu et al. 2011; Soliman and Kabir 2012; Bachman et al. 2012; Hawkes et al. 2013; Wallace et al. 2016; Ibrahim and
Pieprzica 2016; Liu et al. 2016). However, unlike in a conventional well test, the superposition-time derivative curve does not show
what the transient would have looked like if the test had been performed at a constant rate. The early-time transient is controlled by the
interplay of fracture compliance and leakoff in a way that has no analogy to a conventional well test. The late-time transient is similarto an impulse (or slug) test, in which a volume of ﬂuid is instantaneously placed into the well, causing an immediate pressure response


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and then a decay back to the formation pressure (Craig and Blasingame 2006). The onset of the impulse ﬂow regime causes a downward
deﬂection in the derivative curve.
As shown in the Properties of the Superposition-Time Derivative section, taking the derivative with respect to superposition time,
instead of shut-in time, causes an upward deﬂection on the log-log plot at the point when shut-in time is equal to injection time. This isa mathematical property of how the derivative is calculated. In a conventional well test; the deﬂection adjusts for the effect of the prior
rate history because it occurs at approximately the same time as the transition to impulse ﬂow. In a DFIT transient, there is no relation-
ship between the timing of the superposition-time derivative deﬂection and the onset of impulse ﬂow. In the Simpliﬁed Model Problem
section, I show that the timing of the transition to late-time impulse behavior is controlled by injection volume and the formation, well-


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bore, and fracture properties, and is unrelated to injection time. To be precise, the timing of the onset of impulse ﬂow actually has aweak (but nonzero) relationship with injection time, but this is a minor effect that does not need to be examined in this report. van den
Hoek (2016) also discusses issues with interpreting DFITs with superposition-time derivative plots.
As discussed in the Properties of Nolte’s G-Function section, the superposition-time derivative deﬂection occurs at the same time as
a transition from a one slope to a 1/2 slope in an ideal DFIT shut-in transient. However, the superposition-time derivative deﬂection
adds unity to the slope, whereas the ideal transient behavior experiences a decrease in slope of 1/2, not unity. In any case, it would not
assist the interpretation to calculate the derivative in a way that masks the “ideal” transition from unit slope to 1/2 slope. In actual ﬁeld
data, the ideal early-time slope of unity is not typically observed.


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In the Properties of Nolte’s G-Function section, I review the properties of Nolte’s G-function, which describes the scaling of pres-
sure with time in an idealized DFIT shut-in transient. One form of the G-function contains terms with 3/2 powers of time and superﬁ-
cially resembles conventional well-test analysis solutions constructed by applying time superposition. However, I show that the 3/2
power terms arise from a spatial integration of Carter’s leakoff equation, not from using time superposition to sum constant-rate
solutions.
Copyright VC2017 Society of Petroleum Engineers
Original SPE manuscript received for review 5 November 2016. Revised manuscript received for review 1 March 2017. Paper (SPE 186098) peer approved 15 March 2017.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1045 Total Pages: 11
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In the Practical Implications section, I provide practical recommendations for interpreting DFIT transients. In the Field Example
section, I provide a ﬁeld example. I conclude that there is no physical, mathematical, or practical reason to plot DFIT transients in a
way that implies 3/2 power scaling of pressure with time.
Properties of the Superposition-Time Derivative
Superposition time can be deﬁned in a variety of ways. For interpretation of DFITs, it is common to deﬁne superposition time as
(Mohamed et al. 2011; Marongiu-Porcu et al. 2011; Liu et al. 2016)
s¼teþDt
Dt; ð1Þ
where teis duration of injection and Dtis the time elapsed since the end of injection. This deﬁnition of superposition time is identical to
the deﬁnition of Horner time. A related parameter, Agarwal time, is deﬁned as (Agarwal 1980)
ta¼teDt
teþDt¼1
1
teþ1
Dt¼te
s: ð2Þ
The ﬁnal equality in Eq. 2 shows that there is a simple relationship between Agarwal time and superposition time (or Horner time).


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Agarwal time is equal to one-half the harmonic average of teandDt. Agarwal time has the property that it is approximately equal to Dt
whenDt/C28teand approximately equal to tewhenDt/C29te.I fDtis much larger than te,taking the derivative with respect to Agarwal
time is equivalent to adding unity to the slope of the d P/d[ln(Dt)] curve on the log-log plot. This is true because
tadp
dta¼tadp
dDtdDt
dta¼tadp
dDt1
t2
a1
ta/C01
te/C18/C192¼tadp
dDtDt2
t2
a¼Dt2
tadP
dDt/C25Dt2
tedp
dDt; ð3Þ
where the ﬁnal step is taken because taasymptotically approaches teforDt>>te. Therefore, for Dt>>te,
d log tadp
dta/C18/C19/C20/C21
d½logðDtÞ/C138/C25d logDt2
tedp
dDt/C18/C19/C20/C21
d½logðDtÞ/C138¼d log Dtdp
dDt/C18/C19/C20/C21
d½logðDtÞ/C138þ1: ð4Þ
On a log-log plot, the derivative curve with derivative taken with respect to Agarwal time is the same as the derivative curve with
derivative taken with respect to superposition time because teis a constant and (from Eq. 2)
dp
d½logðtaÞ/C138¼/C0dp
d½logðsÞ/C138: ð5Þ


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dp
d½logðtaÞ/C138¼/C0dp
d½logðsÞ/C138: ð5Þ
The change in sign does not affect the log-log plot because the derivative is commonly plotted as positive on the log-log plot.
Fig. 1 shows a log-log plot of the function DP¼ﬃﬃtp,along with derivatives taken with respect to time and superposition time.
Superposition time is calculated assuming teis equal to 0.0833 hours (5 minutes), which is a typical injection duration for a DFIT.
When time is approximately equal to te,the superposition semilog derivative curves upward to a slope of 3/2; the pressure plot and the
derivative with respect to time continue with a slope of 1/2..............................................................................
..................................................................
...................................
..........................................
.....................................................................
1×10–3
1×10–41×10–31×10–21×10–1
HoursΔP (blue), d ΔP/d(ln( t)) (orange), d ΔP/d(ln( τ)) (grey)


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HoursΔP (blue), d ΔP/d(ln( t)) (orange), d ΔP/d(ln( τ)) (grey)
1×1001×1011×1021×10–21×10–11×1001×1011×1021×1031×104
Fig. 1—Log-log diagnostic plot showing the function DP5ﬃﬃ ﬃ
tp
, along with derivative taken with respect to time and superposition
time. The superposition-time derivative curves upward when time teis approximately equal to 0.0833 hours.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1046 Total Pages: 11
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Simplified Model Problem
Problem Definition. I analyze an idealized DFIT model problem with the following assumptions:
1. A single fracture forms.
2. The fracture forms effectively instantaneously before the start of the shut-in transient.
3. The volume of ﬂuid stored in the fracture is linearly proportional to the difference between formation pressure and fracture
pressure.
4. The fracture is effectively inﬁnite-conductivity so that pressure gradient caused by ﬂow along the fracture can be neglected.5. Thermal effects can be neglected.
6. Leakoff is linear away from the fracture, and can be modeled assuming single-phase, single-component ﬂow.
7. The fracture-surface area is constant during shut-in.
Even though these assumptions are simpliﬁed relative to actual DFITs, they are sufﬁcient to analyze the properties of the superposi-
tion-time derivative and to investigate certain aspects of real DFIT transients.


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Assumption 3 neglects the effect of fracture closure. In reality, when the fracture mechanically closes (asperities in contact), the
storage coefﬁcient changes because fracture compliance becomes controlled by a nonlinear joint-closure process related to the rough-
ness and stiffness of fracture asperities (Barton et al. 1985; McClure et al. 2016).
Assumptions 4 and 7 are idealizations, but they are reasonable approximations of real DFIT transients in very-low-permeability for-
mations, suitable for the purposes of this analysis. Fractures are inﬁnite-conductivity as long as they have dimensionless fracture con-
ductivity greater than 300. In nanodarcy permeability, fractures retain dimensionless conductivity greater than 300 even if they have
very small aperture. This is supported by the consistent observation of post-closure linear ﬂow in ﬁeld data sets (linear ﬂow requires an


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inﬁnite-conductivity fracture). The fracture may cease to be inﬁnite-conductivity after closure in formations that are not very low per-meability, or late in the transient when the effective normal stress on the fracture becomes high because of pressure decline. Also, there
may be fracture-tip extension immediately after shut-in.
Analytical Solution of the Model Problem. From Assumption 2, we assume the fracture forms instantaneously. At the start of the
transient, the fracture contains a volume of ﬂuid equal to C
tðPf;init/C0PinitÞ, where Pinitis the initial formation pressure, Pf,initis the initial
pressure in the fracture, and Ctis a total storage coefﬁcient equal to
Ct¼CwþA
Sf; ð6Þ
where Cwis the wellbore-storage coefﬁcient and Sfis the fracture stiffness. The compressibility of the ﬂuid in the fracture has a very
small contribution that is neglected in Eq. 6 for simplicity.
Before the fracture pressure has changed signiﬁcantly, the rate of ﬂuid ﬂow out of the fracture into the matrix can be described


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approximately as 1D ﬂow from a constant-pressure boundary condition (equivalent to Carter leakoff). The cumulative ﬂow of ﬂuid
from the fracture into the matrix is therefore (Howard and Fast 1957)
VLs¼4Aﬃﬃﬃﬃﬃ
Dtp
ðpf;init/C0pinitÞﬃﬃﬃﬃﬃﬃﬃﬃﬃ
/ctk
pls
; ð7Þ
where VLsis cumulative volume of ﬂuid leaked off during shut-in, Dtis shut-in time, pf,initis the initial pressure in the fracture, pinitis
the initial formation pressure, /is the formation porosity, ctis the formation total compressibility, kis the formation permeability, and
lis the ﬂuid viscosity. Note that Eq. 7 is valid only because the fracture is assumed to form instantaneously.
Eqs. 6 and 7 can be combined to write the overall material balance on the system:
Ctðp/C0pinitÞ¼Ctðpf;init/C0pinitÞ/C04Aﬃﬃﬃﬃﬃ
Dtp
ðpf;init/C0pinitÞﬃﬃﬃﬃﬃﬃﬃﬃﬃ
kct/
pls
; ð8Þ
where pis the ﬂuid pressure in the fracture at time Dt:
If we deﬁne dimensionless time as
tD¼16A
Ct/C18/C192kct/
plDt; ð9Þ
and we deﬁne dimensionless pressure as
pD¼p/C0pinit
pf;init/C0pinit; ð10Þ


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pD¼p/C0pinit
pf;init/C0pinit; ð10Þ
then we can write the material-balance equation as a simple expression giving the evolution of pressure with respect to time,
pDE¼1/C0ﬃﬃﬃﬃ ﬃtDp: ð11Þ
Eq. 8 can also be written as,
Ctðp/C0pf;initÞ¼/C0 VLs: ð12Þ
Eq. 12 shows that if Ctis constant, then the change in pressure during shut-in is proportional to the cumulative leakoff volume dur-
ing shut-in. If the fracture forms instantaneously, then VLsscales withﬃﬃﬃﬃﬃ
Dtp
, as shown in Eq. 11. As discussed in the Properties of Nolte’s
G-Function section, Nolte’s G-function is based on a generalization of Eq. 11 that accounts for the leakoff that occurs as the fracture
propagates during injection. If fracture-propagation time is considered, then VLsscales with the G-function during shut-in. This is why
plots of pressure vs. Gare (ideally) linear before closure and can be used to infer leakoff coefﬁcient if Ctis assumed to be known (Nolte...........................................................................


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1979; Nolte and Smith 1981). When closure occurs, this affects both the leakoff rate and Ct, which causes pressure to deviate from line-
arity. Whether pressure begins to increase more rapidly or less rapidly after closure depends on the relative effect of closure on leakoff
rate and the Ct(discussed in the Appendix of McClure et al. 2016). A variety of other “nonideal” processes may affect the pressure
trend during a real DFIT transient.
The square-root-of-time behavior in Eq. 11 is not the same as the well-test solution for linear ﬂow (Gringarten et al. 1974). The
Gringarten et al. (1974) linear-ﬂow solution is derived for constant-rate injection and neglects wellbore/fracture storage. Eq. 11 is
derived on the basis of constant-pressure leakoff from a high-storage wellbore/fracture system. The square root of time appears in both


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solutions because diffusivity has units of length squared per time, and so square-root-of time scaling is frequently present in mathemati-cal relationships involving the diffusivity equation (such as the expression for radius of investigation).
After a period of time, the pressure in the fracture drops sufﬁciently so that leakoff is no longer well-approximated by Carter leakoff,
which assumes constant pressure in the fracture (Howard and Fast 1957). The late-time pressure transient can be found by taking the
dimensionless time derivative of the nondimensionalized solution for constant-rate injection (Houze et al. 1988; Ayoub et al. 1988; Gu
et al. 1993). The solution for constant-rate injection into an inﬁnite-conductivity fracture, neglecting the ﬂuid storage of the wellbore, is(Gringarten et al. 1974)
p¼p
initþq
Aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃl
pkct/rﬃﬃﬃﬃﬃ
Dtp
; ð13Þ
where qis the volumetric injection rate. Taking the derivative of the nondimensionalized form of this equation, we can ﬁnd the solution


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for the injection of a volume of ﬂuid Vsluginto the system:
p¼pinitþVslug
2Aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃl
pkct/r1ﬃﬃﬃﬃﬃ
Dtp : ð14Þ
For our problem,
Vslug¼Ctðpf;init/C0pinitÞ: ð15Þ
Therefore,
p¼pinitþCt
2Aðpf;init/C0pinitÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
pkct/s
1ﬃﬃﬃﬃﬃ
Dtp : ð16Þ
With the same dimensionless variables as before, Eq. 16 can be written as
pDL¼2
p1ﬃﬃﬃﬃ ﬃtDp : ð17Þ
Eq. 11 describes the early-time behavior of the transient, and Eq. 17 describes the late-time behavior of the transient. There is not a
simple expression to describe the evolution of pressure during the transition between these two limiting cases. However, we can ﬁnd asimple estimate for the time at which the transition occurs.
Enforcing equality of the derivatives at the transition time,
dpDE
dtd¼dpDL
dtdfortD¼tD;trans; ð18Þ
results in the expression
tD;trans¼2
p; ð19Þ
which in dimensional form is
Dttrans¼1
8Ct
A/C18/C192l
kct/: ð20Þ
The transition to impulse ﬂow marks the peak in the d p/d[ln(Dt)] curve, which occurs at the same time as the peak in the d P/


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d[ln( G)] curve and the transition from 3/2 slope to 1/2 slope on the log-log plot of d p/d[ln( s)]. This provides a physical interpretation
for the peak in the d p/d[ln( G)] curve in G-function plots: the onset of impulse ﬂow. This occurs as the leakoff rate deviates from Carter
leakoff (which is derived from the assumption that pressure in the fracture is constant). In real transients, Ctchanges strongly when the
fracture closes, and so Eq. 20 cannot be applied directly. However, Eq. 20 demonstrates that the onset of the peak in d p/d[ln( G)] is con-
trolled by the interplay of formation and fracture properties in causing the onset of impulse ﬂow.
Numerical Solution to the Model Problem. To conﬁrm the results in the Analytical Solution of the Model Problem section, I solve


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the model problem with a numerical simulator. The simulator calculates leakoff rate from the fracture by solving the diffusivity equa-tion with a 1D ﬁnite-difference grid with logarithmically spaced elements. On one side of the model, a special “fracture” element is
placed with ﬂuid-storage coefﬁcient equal to C
t. The fracture element is initialized at an elevated pressure pf,init, and the matrix elements
are initialized with pressure pinit. Away from the fracture element, the grid is sufﬁciently large that it is effectively inﬁnite for the dura-
tion of the simulation.
Fig. 2 shows a log-log diagnostic plot of pressure and the derivative taken with respect to shut-in time. The numerical solution is
shown with solid lines, and the two approximate analytical solutions (Eqs. 11 and 17) are shown with dashed lines. The comparison
shows that the analytical results are very accurate at early and late time. The transition between the two solutions occurs at approxi-


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mately when the derivatives of the two analytical solutions are equal, consistent with Eq. 19...................................................................
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Fig. 3 shows the numerical solution to the model problem with derivative taken with respect to shut-in time and superposition time.
As discussed previously, the model problem neglects the effect of fracture closure and assumes that the fracture formed instantaneously
during injection. To calculate superposition time, it is necessary to deﬁne a value for te. In Fig. 1, superposition time is calculated
assuming teis equal to 0.0833 hours, or 5 minutes, which is a typical injection duration in a DFIT. The superposition-time plot curves
upward when Dtis equal to te. The derivative with respect to time continues with a slope of 1/2 (indicating square-root-of-time scaling)
until the onset of impulse ﬂow at approximately 15 hours. The onset of impulse ﬂow causes a transition from a 1/2 slope to a –1/2 slope
on the actual-time derivative curve and a 3/2 slope to a 1/2 slope on the superposition-time derivative curve. The superposition-time de-


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rivative plot does not quite reach a full slope of 3/2 because the pressure trend begins deviating downward from pure square-root-of-
time scaling as it approaches the onset of impulse ﬂow.
Properties of Nolte’s G-Function
In this section, I review properties of Nolte’s G-function (Nolte 1979; Nolte and Smith 1981; Castillo 1987; Gulrajani and Nolte 2000)
to show that there is not a physical or mathematical justiﬁcation for plotting it in a way that implies 3/2 power scaling with respect
to time.
The G-function is deﬁned so that it is proportional to the cumulative leakoff volume during shut-in, given certain simplifying
assumptions: (a) the leakoff rate at a given location along the fracture is described by Carter leakoff (leakoff rate is proportional to the
inverse of the square root of leakoff duration); (b) during injection, the fracture length is equal to the duration of injection to the powera; and (c) the fracture immediately stops propagating after shut-in.


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If the fracture is assumed to grow linearly with time during injection ( a¼1), then Nolte (1979) showed that the G-function
scales as
Gða¼1Þ/ 1þDt
te/C18/C191:5
/C0Dt
te/C18/C191:5
/C01: ð21Þ .........................................................
ΔP (blue), d ΔP/d(ln( t)) (orange),
dΔP/d(ln( τ)) (grey)
1×10–21×10–11×1001×1011×1021×1031×1041×105
1×10–41×10–31×10–21×10–1
Hours1×1001×1011×1021×1031×104
Fig. 3—Log-log diagnostic plot showing the numerical solution to the model problem described in the Simpliﬁed Model Problem
section. The model problem is based on the simplifying assumption that the fracture does not mechanically close. The plot showschange in pressure (blue) and the derivative curves with derivative taken with respect to shut-in time (orange) and superpositiontime (gray).
ΔP (blue), d ΔP/d(ln( t)) (orange),
dΔP/d(ln( τ)) (grey)
1×10–21×10–11×1001×1011×1021×103
1×10–41×10–31×10–21×10–1
Hours1×1001×1011×1021×1031×104


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1×10–41×10–31×10–21×10–1
Hours1×1001×1011×1021×1031×104
Fig. 2—Log-log diagnostic plot with derivative taken with respect to time showing the numerical solution to the model problem
(blue lines) and the early-time and late-time analytical solutions to the model problem (green lines). The solid lines show change in
pressure, and the dashed lines show the derivative curves.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1049 Total Pages: 11
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Even though Eq. 21 contains terms with a 3/2 power, it never scales according to a 3/2 power. At early time, Eq. 21 scales linearly
withDt;and at late time, it scales with the square root of Dt:Derivations of these properties are provided in Appendix A, and they are
shown graphically in Fig. 4.
Nolte (1979) also deﬁned a function gthat is proportional to the total volume of ﬂuid leakoff from the start of injection (rather than
total leakoff volume after shut-in). As shown by Eq. 12, pressure after shut-in scales with leakoff volume during shut-in, not total leak-off volume (Gulrajani and Nolte 2000). Nolte (1979) derived the G-function by subtracting gat shut-in from gat a point in time after
shut-in.
In Eq. 21, the transition between linear and square-root scaling occurs at Dt¼t
e:This is the same time at which the spurious deﬂec-
tion occurs in the superposition-time derivative plot (Eq. 2). It might be argued that the deﬂection in the superposition-time plot adjusts


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for this downward bend in the G-function that occurs near the same time (though it is unclear why this would assist in an analysis).
However, taking the derivative with respect to superposition time adds unity to the slope of the curve, whereas the transition in Eq. 21
causes the slope of the line to reduce by 1/2, not unity. In real DFIT transients, various complex processes cause deviation from ideality
in the minutes after shut-in. As a result, the early-time unit slope predicted by Eq. 21 is not typically observed.
Fig. 4 shows Gða¼1Þ(Eq. 21) and the derivative curves taken with respect to time and superposition time (for teis equal to 0.0833
hours). All three curves show a slope of unity at early time, indicating linear scaling. At approximately 0.0833 hours, the G-function
curve bends toward a slope of 1/2. The derivative curve with respect to time also bends to a slope of 1/2. The superposition-time deriva-
tive curve bends upward to a slope of 3/2.


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tive curve bends upward to a slope of 3/2.
If the fracture is assumed to grow according to the square root of time ( a¼0:5), then the G-function scales (Nolte 1979) as
Gða¼0:5Þ/ 1þDt
te/C18/C19
arcsin 1 þDt
te/C18/C19/C00:5"#
þDt
te/C18/C190:5
/C0p
2: ð22Þ
In this form of the G-function, there are not terms with a 3/2 power. Gða¼0:5Þhas the same asymptotic scaling as Gða¼1:0Þ: lin-
ear at early-time and square root of Dtat late time.
Eqs. 21 and 22 scale linearly with time at early time because shut-in does not cause a discontinuous change in leakoff rate. By deﬁni-
tion, continuous functions have a constant derivative when evaluated over a sufﬁciently short interval. This implies that leakoff rate is
approximately constant at early time after shut-in. Eq. 12 shows that if leakoff rate is constant, then pressure change will be linear with
shut-in time.
Eqs. 21 and 22 scale with the square root of time at late time because the difference in leakoff duration at each point along the frac-


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ture becomes negligible. As a result, the leakoff rate asymptotically becomes the same everywhere along the fracture. This is analogousto St. Venant’s principle of mechanical loading, which states that complex applied loads are asymptotically equivalent to point loading
at sufﬁcient distance.
Eq. 21 appears superﬁcially similar to superposition-time solutions constructed by addition of constant-rate solutions. However, it is
not mathematically or physically valid to interpret Eq. 21 in this way. For example, the linear-ﬂow buildup-test transient can be con-
structed from superposition by use of the Gringarten et al. (1974) constant-rate solution and scales as
DP/1þDt
te/C18/C190:5
/C0Dt
te/C18/C190:5
: ð23Þ
Eq. 23 is a speciﬁc case of the more general method of performing a time-convolution integral of a constant-rate analytical solution


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multiplied by the derivative of ﬂow rate with respect to time. Implicit in the construction of Eq. 23 is the requirement that the constant-.........................................
................................................................
ΔP (blue), d ΔP/d(ln( t)) (orange), d ΔP/d(ln( τ)) (grey)
1×10–2
1×10–31×10–11×1001×1011×1021×1031×1041×105
1×10–41×10–31×10–21×10–1
Hours1×1001×1011×102
Fig. 4—Log-log diagnostic plot showing G(a51) (blue) and the derivative curve taken with respect to time (orange) and superposi-
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rate solution is not itself changing over time. For example, the line source in the Gringarten et al. (1974) solution is ﬁxed in size, shape,
and location. All conventional well-test solutions (inﬁnite-acting radial ﬂow, bilinear, and others) assume that the source does not
change geometry over time. In contrast, in the hydraulic-fracturing problem, the fracture grows during the injection period prior.
van den Hoek (2002) and Sections 3.2.3 and 4.2.2 from Koning (1987) analyze the propagation of a hydraulic fracture with very
long injection time and very high leakoff. Under these conditions, the fracture-propagation rate is slow, and the injection rate is approx-
imately equal to the leakoff rate. Under these conditions, they ﬁnd that it is a reasonable approximation to use time-convolution inte-
grals of the constant-rate analytical solution to analyze fracture propagation. This is valid because they restrict their analysis to consider


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only a very-high-leakoff, slow-propagation scenario. DFITs are the opposite: very low leakoff, rapid propagation, and short injectiontime. In a DFIT, until Dtis much greater than t
e,the changing fracture geometry during injection cannot be neglected.
According to the assumptions underlying Gða¼1:0Þ, the leakoff duration at each point along the fracture (the duration of time since
the fracture formed at that location) is equal to Dtþx
v, where vis the fracture-propagation velocity (assumed constant during injection
because a¼1) and xis position along the fracture (where the tip of the fracture is located at xequal to 0 and the center is at xequal to
a). Using Carter leakoff, the cumulative leakoff volume (including ﬂuid leaked off during injection) is equal to
VLðDtÞ¼8hðpf;init/C0pinitÞﬃﬃﬃﬃﬃﬃﬃﬃﬃ
/ctk
plsða
0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Dtþx
vr
dx/ða
0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Dtþx
vr
dx/1þDt
te/C18/C191:5
/C0Dt
te/C18/C191:5
; ð24Þ
where his the fracture height and teis equal toa


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; ð24Þ
where his the fracture height and teis equal toa
v. Eq. 24 is proportional to the g-function for total leakoff volume (for a¼1:0). The cu-
mulative leakoff volume during the shut-in period is equal to VLminus the cumulative leakoff volume at shut-in ( Dt¼0Þ, which recov-
ersGða¼1Þ:
VLsðDtÞ¼VLðDtÞ/C0VLð0Þ/ 1þDt
te/C18/C191:5
/C0Dt
te/C18/C191:5
/C01: ð25Þ
The origin of the 3/2 power in Gða¼1Þ(Eq. 21) is a spatial integration of the Carter leakoff solution over the length of the fracture.
It is not derived in a way that is analogous to Eq. 23, and Eq. 21 cannot be interpreted as being constructed from the sum of constant-rate analytical solutions.
Practical Implications
Log-log superposition-time derivative plots can be used to diagnose ﬂow regimes in DFIT transients, but the interpreter must be aware
that the use of superposition time causes the plot to curve upward when Dtis equal to t
e. The G-function experiences a downward bend


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e. The G-function experiences a downward bend
at the same time, but the downward bend is a decrease in slope of 1/2, and the superposition-time bend causes an increase in slope of
unity. The interpreter must understand the complex relationship between the slope of the curve and the scaling of pressure with time,and must be aware that there is no physical signiﬁcance to the 3/2 slope on the plot.
To avoid confusion, I recommend interpreting DFIT transients from the basic pressure-transient interpretation plot—the log-log plot
with derivative taken with respect to actual-time. To assist the identiﬁcation of closure and other preclosure trends, plots of pressure
andG/C2dG/dPvs.Gcan be used.
When the derivative is taken with respect to shut-in time in a log-log plot, there is a simple mathematical interpretation of the slope
of the derivative curve. If the pressure change scales with respect to t
b, then the slope of the derivative curve on a log-log plot is equal


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tob. If the pressure change scales with respect to logarithmic time, then the slope is zero. In contrast, if the derivative is taken with
respect to superposition time, then after the spurious deﬂection, the superposition-time derivative curve has a slope of bþ1. Before the
onset of impulse ﬂow, the slope of bþ1 does not have a physical interpretation. In conventional (nonfracturing) buildup/falloff transi-
ents, the onset of impulse ﬂow occurs at approximately the same time as the deﬂection, which is why the superposition derivativemethod can be used to show what the transient “would have been” if it was a constant-rate test. But in DFIT transients, the deﬂection
occurs much earlier than the onset of impulse ﬂow, and the superposition-time derivative curve does not have the same interpretation.
Interpreters are tempted to identify a change in ﬂow regime at the spurious deﬂection. If the deﬂection occurs in a part of the tran-


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sient where the derivative is curving because of an actual pressure-transient response, the real response is distorted by the overprint of
the spurious deﬂection and will be more difﬁcult to interpret.
Late-time impulse ﬂow regimes can be identiﬁed easily on derivative plots, regardless of whether the derivative is taken with respect
to shut-in time or superposition time. The late-time impulse solution is the derivative of the constant-rate solution (Houze et al. 1988).
If pressure change in a constant-rate solution scales with respect to t
bor the logarithm of time, then, for the impulse solution, the slope
of the derivative curve (taken with respect to shut-in time) is the slope of the constant-rate solution minus unity. For example, impulse
radial ﬂow has a slope of /C01, and impulse linear has a slope of /C01/2.
Field Example
Fig. 5 shows a log-log diagnostic plot for the ﬁeld data set described by McClure et al. (2016). The data are from a DFIT in a vertical


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well in a formation with strong stress contrast between layers (causing fracture-height conﬁnement), high stress anisotropy (precluding
the possibility of “opening transverse fractures”), drilling-induced axial tensile fractures visible in an image log perpendicular to the min-
imum principal stress (indicating that fractures will not be forced to rotate into the far-ﬁeld orientation as they propagate away from the
wellbore), and very low permeability (approximately 100 nd). The injection volume was 11.5 bbl. These conditions are as close as practi-cally possible to being an “ideal” DFIT in very tight rock. Nevertheless, there is strong deviation from the ideal G-function response.
Typically, the transient would be interpreted in conjunction with a plot of G/C2dP/dGvs.G. However, for simplicity, I provide only the
log-log diagnostic plot. Fig. 2 from McClure et al. (2016) provides a G-function plot of these data. For calculating superposition time, the
value of t


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value of t
eis determined by dividing the total injection volume by the ﬁnal injection rate and is equal to 0.0383 hours (138 seconds).
The ﬁrst 0.01 hours (36 seconds) is dominated by water hammer in the wellbore, causing sharp pressure oscillations. The superposi-
tion-time derivative curve deﬂects upward at approximately te(0.0383) hours of shut-in. As discussed in the Properties of the Superpo-
sition-Time Derivative section, the upward curve of the superposition-time derivative curve is a consequence of how the derivative is
calculated and should not be interpreted as representing a reservoir process. The actual-time derivative curve is nearly ﬂat from 0.01 to.......................
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0.1 hours. The early-time slope of unity predicted by the G-function is not apparent. A variety of processes, with conﬂicting effects,
may be occurring during the early-time transient: reheating of the ﬂuid in the wellbore caused by conduction from the surrounding for-
mation, fracture-tip extension (continued propagation after shut-in), and many other possible effects. Because of the variety of complexprocesses that may be occurring in the early minutes of shut-in, it is difﬁcult to draw conclusions from this part of the transient. The
nearly ﬂat actual-time derivative from 0.01 to 0.1 hours should not be interpreted as representing radial ﬂow.
The actual-time derivative curve gently curves upward to a slope of 1/2, centered at approximately 1.0 hours. However, it subse-
quently curves up to a slope approaching unity, indicating that the pressure is decreasing signiﬁcantly faster than square-root-of-time


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scaling. The superposition-time plot curves upward in tandem, approaching a slope of two. In the superposition-time plot, it would be
difﬁcult to identify that the slope has deviated from 3/2 slope without the help of the dashed yellow guide line. It is visually easier to
distinguish between a slope of 1/2 and unity than a slope of 3/2 and two.
A curving-upward pressure derivative, indicating pressure scaling more rapid than square root of time, has been the subject of dis-
cussion in the literature. On a G-function plot, this phenomenon manifests as curving upward G/C2dP/dG. Barree and Miskimins (2016)
call this behavior “delayed leakoff” and hypothesize that it could be caused by wellbore effects, near-wellbore fracture complexity, clo-
sure of transverse fractures, or poroelastic effects. McClure et al. (2016) provide a different explanation that is based on analytical and


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numerical calculations that provide a match to the ﬁeld data shown in Fig. 5. Their explanation is that fracture compliance decreases(stiffness increases) as the fracture walls come into contact, accelerating the rate of pressure decline. Eventually, the leakoff rate devi-
ates sufﬁciently from Carter leakoff that the rate of pressure decline decelerates, causing the onset of impulse ﬂow.
Conventionally, fracture closure is picked from G-function plots by drawing a tangent to the G/C2dP/dGcurve from the origin
(Barree and Miskimins 2016). This has the effect of picking closure very close to the peak in G/C2dP/dG. The peak in G/C2dP/dGoccurs
at exactly the same time as the peak in the derivative plot and at the same time as the transition from 3/2 to 1/2 slope on the superposi-tion-time derivative plot. Consequently, picking closure at the start of the 1/2 slope on a superposition-time derivative plot is equivalent


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to the tangent method of picking closure from a G/C2dP/dGcurve. The Simpliﬁed Model Problem section demonstrates that this transi-
tion occurs at the onset of impulse ﬂow. Thus, the conventional closure pick is equivalent to picking closure at the onset of impulseﬂow. In contrast, McClure et al. (2016) suggest picking closure earlier, at the deviation from square-root-of-time scaling. In Fig. 5, the
two different methods provide substantially different closure picks, either at approximately 0.5–1.5 hours or at approximately 10 hours.
If the transient does not curve upward past square-root-of-time scaling, the methods described by McClure et al. (2016) and Barree
and Miskimins (2016) yield similar closure picks. McClure et al. (2016) explain that if permeability is not very low, closure can reduce


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dimensionless fracture conductivity sufﬁciently that closure causes a decrease in leakoff rate (as ﬂuid continues to ﬂow into the fracturebecause of wellbore storage) and weakening of the hydraulic connection between the fracture and the wellbore. The decrease in leakoff
rate can overwhelm the effect of closure on compliance, resulting in a decrease in the rate of leakoff when closure occurs, causing
Dt/C2dP/dDtandG/C2dP/dGto curve downward and peak, rather than curve upward.
In real DFITs, there is often a period of rapid pressure drop at the start of the transient (e.g., Fig. 3 from Liu et al. 2016). Fracture-tip
extension could explain this phenomenon, but sometimes the pressure drop occurs over thousands of psi. It is unlikely that tip-extensioncould explain such large pressure changes. An alternative explanation is that fractures start at the wellbore with orientation different


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from their far-ﬁeld orientation and then are forced to rotate as they propagate away from the well. After shut-in, pressure drops as ﬂuid
ﬂows out of the misoriented fractures near the wellbore (which require high pressure to remain open during injection) into the far-ﬁeld
fracture (which remains open at a lower pressure). Figs. 13 and 17 from Jung et al. (2016) show numerical simulations of this process.
Conclusions
The following conclusions are drawn regarding the use of superposition-time derivative plots for DFIT shut-in transients:
•Superposition-time derivative plots of DFIT data commonly exhibit a 3/2 slope. This slope does not reﬂect the actual scaling of pres-
sure with time and does not have a physical or theoretical interpretation. For the particular case of constant propagation velocity, the
G-function contains terms with a 3/2 power, but these terms arise from an integration of the Carter leakoff solution along the length


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of the fracture, and are not analogous to buildup/falloff solutions constructed from superposition of constant-rate solutions.
1×10–3 1×10–2 1×10–1
Hours1×100 1×101 1×102 1×103ΔP (blue), d ΔP/d(ln( t)) (orange),
dΔP/d(ln( τ)) (grey)
1×1011×1021×1031×1041×1051×1061×107
Fig. 5—Log-log diagnostic plot showing the ﬁeld data set described by McClure et al. (2016). The plot shows change in pressure
(blue), and the derivative curve taken with respect to time (orange) and superposition time (gray). The dashed yellow lines are pro-vided for reference to show slopes of 1/2 and 3/2.REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1052 Total Pages: 11
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•The superposition-time derivative plot deﬂects upward, adding unity to the slope, when shut-in time is equal to injection time. The ideal
G-function response transitions from a slope of unity to a slope of 1/2 (decrease of 1/2) at approximately the same time as the deﬂection.
•The transition from 3/2 to 1/2 slope on a superposition-time plot (equivalent to the transition from 1/2 to –1/2 slope if the derivative
is taken with respect to time) indicates the onset of impulse ﬂow. This occurs even in an idealized model problem in which closure
does not occur. Closure and the onset of impulse ﬂow are related, but they should not be regarded as the same thing, and they do not
necessarily occur at the same time.
•The slope of log-log plots with the derivative taken with respect to actual shut-in time has a simple and intuitive relationship with the
pressure transient. Although superposition-time derivative curves are useful for conventional buildup/falloff tests, they complicate


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the interpretation of DFIT transients and do not aid in the interpretation.
•I recommend interpreting DFIT transients with log-log diagnostic plots with the derivative taken with respect to actual shut-in time
and plots of pressure and G/C2dG/dPvs.G.
Nomenclature
a¼fracture half-length, m
A¼fracture area, m2
ct¼formation total compressibility, MPa–1
Ct¼total-storage coefﬁcient, m3/MPa
Cw¼wellbore-storage coefﬁcient, m3/MPa
g¼g-time function (dimensionless ﬂuid-loss volume function), dimensionless
G¼G-time function (dimensionless ﬂuid-loss time function), dimensionless
h¼fracture height, m
k¼formation permeability, m2
p¼pressure, MPa
pD¼dimensionless pressure, dimensionless
pDE¼solution for early-time dimensionless pressure, dimensionless
pDL¼solution for late-time dimensionless pressure, dimensionless
pf,init¼ﬁnal fracture pressure, MPa
pinit¼initial pressure, MPa
q¼volumetric injection rate, m3/s
Sf¼fracture stiffness, MPa/m
t¼time, seconds
ta¼Agarwal time, dimensionless
tD¼dimensionless time, dimensionless


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tD¼dimensionless time, dimensionless
tD,trans¼time at the transition between early-time and late-time transient, seconds
te¼duration of injection, seconds
v¼fracture-propagation velocity, m/s
Vf¼volume of ﬂuid in the fracture, m3
VL¼total cumulative leakoff volume from the start of injection, m3
VLs¼total cumulative leakoff volume from shut-in, m3
Vslug¼total injection volume, m3
x¼distance from the location of the ﬁnal fracture tip, m
a¼fracture power-law growth exponent, dimensionless
b¼exponent, dimensionless
Dp¼change in pressure since the start of shut-in, seconds
Dt¼time elapsed since the start of shut-in, seconds
/¼formation porosity, dimensionless
l¼ﬂuid viscosity, MPa /C1s
s¼superposition time (identical to Horner time), dimensionless
Acknowledgments
Gratitude is expressed to Roland Horne for informally reviewing this manuscript, and the journal editor and two anonymous reviewers


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deserve thanks for their thoughtful comments. Dave Cramer and ConocoPhillips deserve appreciation for providing permission to repro-
duce the ﬁeld data shown in Fig. 5.
References
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Barree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-Linear Derivatives in Diagnostic Fracture Injection Test Analysis. Presented a t
the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 9–11 February. SPE-179134-MS. https://doi.org/10.2118/
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Appendix A
Here, I derive equations for the asymptotic behavior of Gða¼1Þfor small Dtand large Dt:Gða¼1Þis deﬁned in Eq. 21. To calculate
the behavior at small Dt;take the Taylor-series expansion of the 1 þDt
te/C18/C191:5
term centered at Dt¼0:
Gða¼1;Dt!0Þ//C0Dt
te/C18/C191:5
/C01þ1þ0
te/C18/C191:5
þ1:5
te1þ0
te/C18/C190:5
Dtþ0:75
t2
e1þ0
te/C18/C19/C00:5
Dt2þ… ðA-1Þ
The two constant terms cancel out. The lowest-order term in Eq. A-1 (the term that decreases most slowly as Dtapproaches zero) is
linear with Dt:Therefore,
limDt!0Gða¼1Þ/1:5Dt
te: ðA-2Þ
To calculate the behavior at large Dt;note that
limDt!1Dt
teþ1/C18/C191:5
/C0Dt
te/C18/C191:5
¼limDt!1Dt
teþ1/C18/C191:5
/C0Dt
te/C18/C191:5
1¼limD!0Dt
teþD/C18/C191:5
/C0Dt
te/C18/C191:5
D¼1:5ﬃﬃﬃﬃﬃ
Dt
tes
: ðA-3Þ


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teþD/C18/C191:5
/C0Dt
te/C18/C191:5
D¼1:5ﬃﬃﬃﬃﬃ
Dt
tes
: ðA-3Þ
The second equality is true because 1.0 becomes inﬁnitesimally small relative toDt
teasDtapproaches inﬁnity. The third equality is
true from the deﬁnition of a derivative and the power rule of differentiation. Finally, as Dtapproaches inﬁnity, the “–1” term in Eq. 21
becomes negligibly small; therefore,................
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limDt!1Gða¼1Þ/1:5ﬃﬃﬃﬃﬃ
Dt
tes
: ðA-4Þ
Mark McClure is the founder and chief executive officer of McClure Geomechanics and is a consulting professor at Stanford Uni-
versity. He was an assistant professor in the Department of Petroleum and Geosystems Engineering at the University of Texas atAustin from 2013 to 2015. McClure’s research focuses on modeling and characterization of hydraulic fracturing, induced seismic-
ity, and enhanced geothermal systems. He was honored with the SPE Regional Completions Optimization and Technology
Award from the Southwest North American Region in 2015 and an Outstanding Service as a Technical Editor award for SPE Res-
ervoir Evaluation & Engineering in 2014, the Hank Ramey Award for Outstanding Research and Service to the Department from
the Department of Energy Resources Engineering at Stanford University in 2012, and the Best Paper in Geophysics award from


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the Society of Exploration Geophysicists in 2011. McClure holds a BS degree in chemical engineering, an MS degree in petroleum
engineering, and a PhD degree in energy-resources engineering, all from Stanford University................................................................REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1055 Total Pages: 11
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Discussion of The
Fracture-Compliance Method for
Picking Closure Pressure From
Diagnostic Fracture-Injection Tests
Matteo Marongiu Porcu, Schlumberger; Christine Ehlig-Economides, University of Houston;
and Robert D. Barree, Barree & Associates
We would like to open this discussion on SPE-179725-PA
(McClure et al. 2016) directly from its title. The “fracture-compli-
ance method” does not allow determination of the closure pres-
sure associated with minimum principal in-situ stress. What
McClure et al. refer to as “closure pressure” is in fact the manifes-
tation of known physical phenomena that are generally addressed
as nonlinear leakoff, departing from Carter’s ideal leakoff
(Howard and Fast 1957), which all occur at higher to much higher
pressures than the minimum principal in-situ stress.
McClure et al. do cite the essential published work that
addresses these nonlinear-leakoff phenomena, but they do so with
a very dismissive tone. Barree and Miskimins (2016) deﬁned


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a very dismissive tone. Barree and Miskimins (2016) deﬁned
these as “accelerated leakoff” and “delayed leakoff,” and pro-
vided a very clear description of all physical causes of these pres-sure-falloff behaviors.
In particular, Barree and Miskimins (2016) dedicated a consid-
erable effort to describing features of diagnostic fracture-injection
tests (DFITs) in multistage/multicluster hydraulically fractured
horizontal wells in naturally fractured/laminated shale formationsthat are not expected in vertical-well DFITs. In such settings, it is
highly unrealistic to expect ideal planar biwing propagation of
induced hydraulic fractures. Instead, it is possible to have highlynonideal conﬁgurations, including sublongitudinal coalescence of
discrete cracks along the wellbore, strong near-wellbore tortuosity,
and hydraulic-fracture complexity (Weng et al. 2011) because of
the severe interaction with near-wellbore stresses possibly altered


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by drilling, pre-existing natural fractures, joints, and genericplanes of weakness. These nonideal fracture features (near-well-
bore and far-ﬁeld) are arbitrarily oriented, and, at the moment of
injection shut-in, are held open against a higher stress than the
minimal principal stress. Consequently, they will be the ﬁrst ones
to gradually close during the pressure falloff, and, during theirown closure process, they will keep transmitting their higher pres-
sure to the primary fracture (orthogonal to the minimum principal
in-situ stress), causing the slower or delayed leakoff behavior
symptomatic of the upward concavity in a Cartesian ( G-function)
plot of the GdP/dGderivative function vs. the G-time function.
Thus, application of the “fracture-compliance method” might
lead to identiﬁcation of a closure event ascribable to the nonideal
conﬁgurations described above, but not to a ﬁnal primary-fractureclosure that would reﬂect the minimum principal in-situ stress.


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The authors confuse this point in their Conclusions 2 through
4 by claiming that the picks by other cited analysis methods
underestimate minimum stress, while stating in their Conclusion 7
that the closure pressure (picked from their method) can be asmuch as 1 to 2 MPa (145 to 290 psi) greater than the minimum-
principal-stress value. The other analysis methods cited in Con-
clusions 2 through 4 will all pick the same closure event, correctly
associated to the minimum principal in-situ stress.
The numerical model used by the authors to present the variable
compliance simulations exposes signiﬁcant limitations, the ﬁrst ofwhich is that the model is incapable of handling such features as
normal leakoff behavior; faster or accelerated leakoff scenariossuch as pressure-dependent leakoff that produce downward con-
cavities in the GdP/dGderivative function; and multiple closure
scenarios consistent with multiple straight-line portions that appearin DFIT data illustrated in Mohamed et al. (2011) and others.


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A second model limitation, which is really a serious model
ﬂaw, is shown by obvious discontinuities occurring immediately
after what the authors pick as the closure pressure. The authors
acknowledge that such discontinuities are not seen in DFIT
responses. They attempt to avoid the model discontinuity by mod-eling a fracture with a residual aperture exceeding 0.5 mm at the
closure pressure claimed to correspond to the minimum principal
stress. Few if any analysts would envision such a fracture width
as closed. Even so, to completely remove the model discontinuity,
they add wellbore-storage and tip-extension model features. Onlywith all this (including residual fracture aperture at minimum
principal stress exceeding 0.5 mm) do they produce simulations
remotely similar to typical DFIT data.
From our experience from falloff-test analyses performed on
Eagle Ford, Wolfcamp, Haynesville, Fayetteville, Marcellus, andHorn River shale wells, the real closure events indicating the min-


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imum principal in-situ stresses systematically occur at equivalent
stress gradients 0.05 to 0.1 psi/ft lower than the gradients associ-ated with nonideal-conﬁgurations events (described above) that
one would pick using the “fracture-compliance method.”
For instance, the extra 0.05 psi/ft (and often higher) in the mini-
mum principal in-situ-stress gradient that we systematically obtain
when we attempt to pick closure events using the “fracture-com-pliance method” in a vast number of Eagle Ford and Wolfcamp
calibration tests is impossible to reconcile within state-of-the-art
mechanical Earth models [refer to, among others, the example of
mechanical Earth models built and used in Berard et al. (2015) and
Marongiu Porcu et al. (2016)] that were built using anisotropicacoustic borehole measurements calibrated with actual core-
derived mechanical properties and pore pressures either deter-
mined from after-closure analysis or measured with MDT tools.
References


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References
Barree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-
Linear Derivatives in Diagnostic Fracture Injection Test Analysis. Pa-
per presented at the SPE Hydraulic Fracturing Technology Confer-
ence, The Woodlands, Texas, 9–11 February. SPE-179134-MS.
https://doi.org/10.2118/179134-MS.
Berard, T., Desroches, J., Yang, Y. et al. 2015. High-Resolution 3D Struc-
tural Geomechanics Modeling for Hydraulic Fracturing. Paper pre-
sented at the SPE Hydraulic Fracturing Technology Conference, The
Woodlands, Texas, 3–5 February. SPE-173362-MS. https://doi.org/10.2118/173362-MS.
Howard, G. C. and Fast, C. R. 1957. Optimum Fluid Characteristics for
Fracture Extension. Drilling and Production Practices, New York, 1
January, API-57-261, 261-270 (Appendix by Carter, E. D.). American
Petroleum Institute.
Marongiu Porcu, M., Lee, D., Shan, D. et al. 2016. Advanced Modeling of
Interwell-Fracturing Interference: An Eagle Ford Shale-Oil Study.


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SPE J. 21(5): 1567–1582. SPE-174902-PA. https://doi.org/10.2118/
174902-PA.
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McClure, M. W., Jung, H., Cramer, D. D. et al. 2016. The Fracture-Com-
pliance Method for Picking Closure Pressure From Diagnostic Frac-ture-Injection Tests. SPE J. 21(4): 1321–1339. SPE-179725-PA.
https://doi.org/10.2118/179725-PA.
Mohamed, I. M., Nasralla, R. A., Sayed, M. A. et al. 2011. Evaluation of
After-Closure Analysis Techniques for Tight and Shale Gas Forma-
tions. Paper presented at the SPE Hydraulic Fracturing TechnologyConference, The Woodlands, Texas, 24–26 January. SPE-140136-MS.
https://doi.org/10.2118/140136-MS.
Weng, X., Kresse, O., Cohen, C. -E. et al. 2011. Modeling of Hydraulic-
Fracture-Network Propagation in a Naturally Fractured Formation.
SPE Prod & Oper 26(4): 368–380. SPE-140253-PA. https://doi.org/
10.2118/140253-PA.J179725 DOI: 10.2118/179725-PA Date: 11-May-17 Stage: Page: 2 Total Pages: 3
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Reply to the Discussion of The
Fracture-Compliance Method for
Picking Closure Pressure From
Diagnostic Fracture-Injection Tests
Mark McClure, McClure Geomechanics; Hojung Jung and Mukul Sharma, University of Texas at Austin;
and Dave Cramer, ConocoPhillips
McClure et al. (2016) provides a mathematically consistent
description of an ideal diagnostic fracture-injection test (DFIT)
pressure transient. We set out the governing and constitutive
equations of the coupled ﬂuid-ﬂow and fracture-mechanics prob-lem, use realistic input parameters, and solve the equations. The
result is a close match to a real DFIT pressure transient. Our work
is the ﬁrst to provide a single continuous calculation that realisti-
cally describes an entire DFIT pressure transient—before, during,
and after closure. Previous investigators have focused on onlyparts of the transient and/or have interpolated between before-clo-
sure and after-closure solutions without providing a mathematical
description of the closure process itself.


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description of the closure process itself.
The appendix in McClure et al. (2016) derives equations
demonstrating that the effect of closure on the pressure derivative
depends on the relative effect on system compliance (which
decreases as the asperities in the fracture come into contact) and
on the leakoff rate from wellbore/fracture storage (which is
related to the dimensionless fracture conductivity as closure
occurs). If the compliance effect dominates, then the pressure
decreases more rapidly in response to closure. If the fracture con-
ductivity effect dominates, the pressure decreases less rapidly in
response to closure. These principles should form the basis
for DFIT interpretation, whether or not more-complicated (and
difﬁcult to validate) “nonideal” processes are believed to be tak-
ing place.
A key aspect of our model (and difference from most prior
work) is that we allow the fracture to have residual aperture after
the aperities come into contact. Is this reasonable? The strongest


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proof is that linear ﬂow regimes are routinely observed after clo-
sure in DFITs. Linear ﬂow regimes require an inﬁnite conductiv-
ity fracture. In McClure et al. (2016), we provide a brief
calculation showing that residual aperture of only tens of microns
is sufﬁcient for a fracture to maintain inﬁnite conductivity in a
formation with very low permeability. There is a large scientiﬁc
literature demonstrating the ability of mechanically closed frac-tures to store and conduct ﬂuid. We used a fracture closure law
and input values based on this literature.
In their discussion, Marongiu Porcu et al. state that our paper
does not address the wide range of “nonideal” processes that are
possible during DFITs. In a follow-up paper (Jung et al. 2016),we performed simulations of a variety of nonideal processes,
including near-wellbore tortuosity and opening of transverse frac-
tures. These results add signiﬁcant new insight, but none of them
contradict the basic ﬁndings of McClure et al. (2016). The simula-


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tions demonstrate that near-wellbore tortuosity causes a rapiddrop in pressure at the start of the shut-in period, before the tran-
sient settles into a “normal” preclosure trend (Jung et al. 2016,
Fig. 17). This causes an early peak in G/C2dP/dGbefore the main
closure [as noted in McClure et al. (2016) and by Barree and
Miskimins (2016)]. Jung et al. (2016) showed that mechanical
opening/closing of transverse fractures away from the well is
unlikely to occur unless there is very low stress anisotropy. If it
does occur, the transverse fractures will close early in the tran-sient. This ﬁnding contradicts the conventional DFIT interpreta-
tion that “closure of transverse fractures” causes G/C2dP/dG to
curve upward later in the transient (Barree and Miskimins 2016).
However, review of the literature indicates that this conventional
interpretation is based on heuristic arguments, not a rigorous
mathematical solution to the equations governing the problem.
Marongiu P


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Marongiu P
orcu et al. note in their discussion that we did not
consider nonideal leakoff processes and cite discussion from
Barree and Miskimins (2016). Barree and Miskimins (2016) state
that they will not “dwell on derivative calculation methods and
theoretical elastic model derivations” and instead “present some
real physical mechanisms.” They provide a list of nonideal proc-
esses that could hypothetically occur during a DFIT. Merely pro-
viding a list of alternative hypotheses does not refute our analysis;
they do not provide mathematical analysis or validating data.
Marongiu P et al. also state in their discussion that our
numerical model is incapable of handling pressure-dependent
leakoff or multiple closures. The model can handle all these proc-
esses, as addressed in the follow-up paper Jung et al. (2016).
Marongiu Porcu et al. note that some of our simulations had a


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Marongiu Porcu et al. note that some of our simulations had a
discontinuity in the pressure derivative at closure. As stated in thepaper, those simulations intentionally used unrealistic input pa-rameters and were included for sensitivity analysis and to demon-strate the effect of compliance on the pressure trend; they were notintended to be realistic. We also provided simulations that appearvery realistic and do not have discontinuity in the pressure deriva-tive, including one simulation that closely matches ﬁeld data.
Finally, Marongiu Porcu et al. state that our closure picks can-
not be reconciled with “state-of-the-art mechanical Earth models”and cite two papers. We reviewed those papers and did not identify
any discussion that refuted our results.
A convincing theory of DFIT transients should lay out the gov-
erning and constitutive equations of ﬂuid ﬂow and fracture
mechanics, solve them, and provide a match to ﬁeld data with a
single, continuous calculation (before, during, and after closure).


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Our work is the ﬁrst to have demonstrated this. We stand by our
conclusions.
References
Barree, R. D. and Miskimins, J. L. 2016. Physical Explanation of Non-
Linear Derivatives in Diagnostic Fracture Injection Test Analysis.
Paper presented at the SPE Hydraulic Fracturing Technology Con-
ference, The Woodlands, Texas, 9–11 February. SPE-179134-MS.
https://doi.org/10.2118/179134-MS.
Jung, H., Sharma, M. M., Cramer, D. D. et al. 2016. Re-examining Inter-
pretations of Non-Ideal Behavior During Diagnostic Fracture InjectionTests. Journal of Petroleum Science and Engineering 145: 114–136.
https://doi.org/10.1016/j.petrol.2016.03.016.
McClure, M. W., Jung, H., Cramer, D. D. et al. 2016. The Fracture-Com-
pliance Method for Picking Closure Pressure From Diagnostic Frac-
ture-Injection Tests. SPE J. 21(4): 1321–1339. SPE-179725-PA.
https://doi.org/10.2118/179725-PA.J179725 DOI: 10.2118/179725-PA Date: 11-May-17 Stage: Page: 3 Total Pages: 3


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ID:jaganm Time: 19:06 I Path: S:/J###/Vol00000/170027/Comp/APPFile/SA-J###1700272017 SPE Journal 3orcu


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396 August 2009 SPE Production & OperationsHolistic Fracture Diagnostics: Consistent 
Interpretation of Prefrac Injection Tests 
Using Multiple Analysis Methods
R.D. Barree,  SPE, and V .L. Barree,  Barree & Associates LLC; and D.P . Craig,  SPE, Halliburton
Copyright © 2009 Society of Petroleum Engineers
This paper (SPE 107877) was accepted for presentation at the 2007 SPE Rocky Mountain 
Oil & Gas Technology Symposium, Denver, 16 –18 April, and revised for publication. Original manuscript received for review 12 February 2007. Revised manuscript received for review 2 April 2008. Paper peer approved 4 April 2008.Summary
Since the introduction of the G-function derivative analysis, prefrac


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diagnostic injection tests have become a valuable and commonly used technique. Unfortunately, the technique is frequently misapplied or misinterpreted, leading to confusion and misdiagnosis of fracturing parameters. This paper presents a consistent method of analysis of the G-function, its derivatives, and its relationship to other diagnostic techniques including square-root(time) and log( /H9004p
wf)-log( /H9004t) plots 
and their appropriate diagnostic derivatives.
Four field test examples are given for the most common diag-


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Four field test examples are given for the most common diag-
nostic curve signatures. These show how multiple analysis methods can be applied to consistently interpret closure pressure and time, as well as pre- and post-closure flow regimes and reservoir proper-ties from the test data. The cases include normal constant-area and constant permeability leakoff, pressure dependent fissure leakoff, fracture tip extension, and variable fracture storage. In some cases conventionally accepted analysis methods, such as the Sqrt(time) plot, can lead to misleading interpretations. A single consistent approach to analysis is described for each case. The example cases can be used to build a foundation for consistent and less ambiguous analysis of any complex fracture injection/falloff test.
Introduction
Prefrac diagnostic injection test analysis provides critical input data


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for fracture design models, and reservoir characterization data used to predict post-fracture production. An accurate post-stimulation production forecast is necessary for economic optimization of the fracture treatment design. Reliable results require an accurate and consistent interpretation of the test data. In many cases closure is mistakenly identified through misapplication of one or more analy-sis techniques. In general, a single unique closure event will satisfy all diagnostic plots or methods. All available analysis methods should be used in concert to arrive at a consistent interpretation of fracture closure.
Relationship of the pre-closure analysis to after-closure analysis


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results must also be consistent. To correctly perform the after-closure analysis the transient flow regime must be correctly identified. Flow regime identification has been a consistent problem in many analyses. There remains no consensus regarding methods to identify reservoir transient flow regimes after fracture closure. The method presented here is not universally accepted but appears to fit the gen-erally assumed model for leakoff used in most fracture simulators.
Four examples are presented to show the application of mul-


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Four examples are presented to show the application of mul-
tiple diagnostic analysis methods. The first illustrates the expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after shut-in. The second example shows pressure dependent leakoff (PDL) in a reser-voir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary fractures or fissures. The third example shows fracture tip extension after shut-in. These cases generally show definable fracture closure. The fourth example shows what has been commonly identified as fracture height recession during closure, but which can also indicate vari-
able storage in a transverse fracture system.
For each example the analysis will be demonstrated using the


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For each example the analysis will be demonstrated using the 
G-function and its diagnostic derivatives, the sqrt(time) and its derivatives, and the log-log plot of pressure change after shut-in and its derivatives (Nolte 1979; Castillo 1987; Barree and Mukherjee 1996; Barree 1998). When appropriate, the after-closure analy-sis is presented for each case, as is an empirical correlation for permeability from the identified G-function closure time (Talley et al. 1999). A critical part of the analysis is the realization that there is a common event indicating closure that should be consis-tently identified by all diagnostic methods. To reach a conclusion all analyses must give consistent results. 
The goal of this paper is to provide a method for consistent


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The goal of this paper is to provide a method for consistent 
identification of after-closure flow regimes, an unambiguous frac-ture closure time and stress, and a reasonable engineering estimate of reservoir flow capacity from the pressure falloff data, without requiring assumptions such as a known reservoir pressure. Other methods, based on sound transient test theory, require pressure dif-ference curves based on the observed bottomhole pressure during falloff minus the “known” reservoir pressure (Talley et al. 1999; Craig and Blasingame 2006). While these methods are technically correct they can lead to confusing results at times, especially in low permeability reservoirs when pore pressure is difficult to determine accurately prior to stimulation.
This is not a transient test analysis paper but is intended to


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This is not a transient test analysis paper but is intended to 
present a practical approach to analysis of real, and frequently marginal-quality, prefracture field test data. The techniques applied are based on some transient test theory. Some of the results pre-sented here are still under debate and development. The methods shown have been tested and, we believe, proven in the analysis of hundreds of tests. Application of these methods provides consis-tent analysis that helps to avoid misinterpretation of falloff data and gives the most useful information available from diagnostic injection tests.
Step-rate injection tests and their analysis are not included in the 
scope of this paper. Determination of the pressure-dependent leakoff coefficient is also not described here, because it has been previ-ously reported (Barree and Mukherjee 1996; Barree 1998). Only the analysis of pressure decline following shut-in of a fracture-rate injection test is considered.
Transient Flow Regimes During and After


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Transient Flow Regimes During and After 
Fracture Closure
Several transient flow regimes may occur during a falloff test after 
injection at fracture rate. The major flow regimes are graphically illustrated in the classic paper by Cinco-Ley and Samaniego (1981). 
Immediately after shut-in the pressure gradient along the length 
of the fracture dissipates in a short-duration linear flow period. In a long fracture in low permeability rock the initial fracture linear flow can be followed by a bi-linear flow period with the linear flow transient persisting in the fracture while reservoir linear flow occurs simultaneously. After the fracture transient dissipates, the reservoir linear flow period can continue for some time, depend-ing on the permeability of the reservoir and the volume of fluid stored in the fracture and subsequently leaked off during closure. After closure the pressure transient established around the fracture propagates into the reservoir and transitions into elliptical, then


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August 2009 SPE Production & Operations 397pseudoradial flow. Each of these flow regimes has a characteristic 
appearance on various diagnostic plots.
Fluid leakoff from a propagating fracture is normally modeled 
assuming one-dimensional linear flow perpendicular to the fracture face. Settari (2000) has pointed out that in some cases of moder-ate reservoir permeability the linear flow regime may not occur, even during fracture extension and early leakoff. During fracture extension and shut-in, the transient may already be in transition to elliptical or pseudoradial flow. In this case analyses based on an assumed pseudolinear flow regime will give incorrect results. In all cases an understanding of the flow regime and its relation to the fracture geometry is critical to arriving at a consistent interpretation of the fracture falloff test.
Diagnostic Derivative Examples
For each analysis technique various curves are used to help define


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closure, leakoff mechanisms, and after-closure flow regimes. On each plot the curves are labeled as the primary (y vs. x), the first derivative ( /H11128y//H11128x), and the semilog derivative ( /H11128y//H11128(lnx) or 
x/H11128y//H11128x). For convenience the primary curve is plotted on the left 
y-axis and all derivatives are plotted on the right y-axis for all Cartesian plots. For the log-log plot all curves are shown on the same y-axis.
For pre-closure analysis, and consistent identification of frac-


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ture closure, three techniques are illustrated for each example: G-function, Square-root of shut-in time, and log-log plot of pres-sure change with shut-in time. All these analyses begin at shut-in. The instantaneous shut-in pressure (ISIP) is taken as the incipient fracture extension pressure for all cases. When there is significant wellbore afterflow (fluid expansion or continued low-rate injec-tion), or severe near-well pressure drop, the ISIP can be difficult to interpret accurately and may be too high to represent actual fracture extension pressure. In all the examples in the paper the pressures have been offset to an approximate ISIP of 10,000 psi to remove any relation to the original field test data. The follow-ing sections detail the data and analysis for the four major leakoff type examples.
Normal Leakoff Behavior.  Normal leakoff is observed when the


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Normal Leakoff Behavior.  Normal leakoff is observed when the 
composite reservoir system permeability is constant. The reservoir may exhibit only matrix permeability or have a secondary natural fracture or ﬁ  ssure overprint in which the ﬂ  ow capacity of the 
secondary fracture system does not change with pore pressure or net stress. After shut-in the fracture is assumed to stop propagat-ing and the fracture surface area open to leakoff remains constant during closure.
Normal Leakoff G-Function. As noted in previous papers, 
the expected signature of the G-function semilog derivative is a straight-line through the origin (zero G-function and zero deriva-tive) (Barree 1998). In all cases the correct straight line tangent to the semilog derivative of the pressure vs. G-function curve must pass through the origin. Fracture closure is identified by the departure of the semi-log derivative of pressure with respect to 
G-function (G /H11128p
w//H11128G) from the straight line through the origin.


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w//H11128G) from the straight line through the origin. 
During normal leakoff, with constant fracture surface area and constant permeability, the first derivative ( /H11128p
w//H11128G) should also be 
constant (Castillo 1987). The primary pw vs. G curve should follow 
a straight line (Nolte 1979). The example in Fig. 1  shows some 
slight deviation from the perfect constant leakoff but is a good example of the expected curve shapes with a clear indication of closure at G
c = 2.31. The closure event is marked by the dashed 
vertical line [1].
Normal Leakoff Sqrt(t) Analysis . The sqrt( t) plot has fre-
quently been misinterpreted when picking fracture closure, even for the simplest cases. The primary p
w vs. sqrt( t) curve should form


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w vs. sqrt( t) curve should form 
a straight line during fracture closure, as with the G-function plot. Some users suggest that the closure is identified by the departure of the data from the straight line trend, similar to the way the G-function closure is picked. This is incorrect and leads to a later closure and lower apparent closure pressure. The correct indication of closure is the inflection point on the p
w vs. sqrt( t) plot. 
The best way to find the inflection point is to plot the first 
derivative of pw vs. sqrt( t) and find the point of maximum ampli-
tude of the derivative. Many fracture-pressure analysis software packages plot the inverse of the actual first derivative and show the inflection point as the minimum of the derivative. The plot in Fig. 2,  shows that the slope of the pressure curve starts low, then


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increases and reaches a maximum rate of decline at the inflection point, then decreases again after closure. The first derivative curve in Fig. 2 is plotted with the proper sign. The dashed vertical line [1] is the G-function closure pick that is synchronized in time and pressure with the sqrt( t) plot. Clearly the consistent closure lies 
at the inflection point and not at the point of departure from the straight line tangent to the pressure curve.
The semilog derivative of the pressure curve is also shown on 
the sqrt( t) plot. This curve is equivalent to the semilog derivative of 
the G-function for most low-perm cases. The closure pick falls at the departure from the straight line through the origin on the semi-log derivative of the P vs. sqrt( t) curve. A single closure point must 
satisfy the requirement on both the G-function and sqrt( t) plots.
Normal Leakoff Log-Log Pressure Derivative. The log-log


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Normal Leakoff Log-Log Pressure Derivative. The log-log 
plot of pressure change from ISIP vs. shut-in time for the normal leakoff example is shown in Fig. 3.  The heavy curve is the pres-
sure difference and the dashed curve is its semilog derivative with respect to shut-in time. The vertical dashed line is the unique closure pick from the G-function and sqrt( t) plot. It is common for 
the pressure difference and derivative curves to be parallel immedi-ately before closure. The slope of these parallel lines is diagnostic of the flow regime established during leakoff before closure. In many cases a near-perfect ½ slope is observed, strongly suggesting linear flow from the fracture. In this example the slope is greater than ½ suggesting possible linear flow coupled with changing fracture/wellbore storage (See Appendix B). The separation of the 
0 5 10 15 20 25
G (Time)750080008500900095001000010500
0200400600800100012001400160018002000
1
P vs. G
GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives


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GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives
Fig. 1—Normal leakoff G-function plot.1/24/200704:00 08:00 12:00
1/24/200716:00
Time750080008500900095001000010500
01002003004005006007008009001000
1Pressure
DerivativesP vs. √t
√tdP/d√t vs. √t
dP/d√t vs. √tFracture Closure
Fig. 2—Normal leakoff sqrt( t) plot.


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398 August 2009 SPE Production & Operationstwo parallel lines always marks fracture closure and is the final 
confirmation of a consistent closure identification.
After closure the semilog derivative curve will show a slope 
of −1/2 in a fully developed reservoir pseudolinear flow regime and a slope of −1 in fully developed pseudoradial flow. In the example the derivative slope is −1 indicating that reservoir pseu-doradial flow was observed. The late-time data shows a drop in the derivative probably caused by wellbore effects such as gas entry and phase segregation. The use of the semilog derivative of the log-log plot for after-closure flow regime identification, as well as closure confirmation, is a powerful new addition to fracture pressure decline diagnostics.
After-Closure Analysis for Normal Leakoff Example. The 
Talley-Nolte After-Closure Analysis (ACA) flow regime identi-fication plot for the normal leakoff example is shown as Fig. 4


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(Talley et al. 1999). The heavy solid line is the observed bottomhole pressure during the falloff minus the initial reservoir pressure. The slope of the semi-log derivative of the pressure difference function (dashed line) is 1.0 during the identified pseudoradial flow period. If a linear-flow period existed in this data set a derivative slope of ½ would exist. It is critical to remember that the slope of the pressure difference curve on this plot is determined solely by the guess of reservoir pressure used to construct the plot. The slope of the deriva-tive is not affected by the input reservoir pressure value.
If a pseudoradial flow regime is identified, then the Cartesian 
Radial Flow plot (Fig. 5)  can be used to determine reservoir 
far-field transmissibility, kh/
/H9262. The viscosity used is the far-field 
mobile fluid viscosity and h is the estimated net pay height. For the


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analysis of the example data kh//H9262 = 299 md-ft/cp. For gas viscosity at reservoir temperature, kh = 7.9 md-ft. For the assumed net pay, 
the effective reservoir permeability is 0.097 md. 
Horner Analysis for Normal Leakoff Example. If a pseudo-
radial flow period is identified, then a conventional Horner plot can also be used to determine reservoir transmissibility. In Fig. 6 
the Horner slope through the radial flow data is 14411 psi. Using an average pump rate of 18.4 bpm, kh/
/H9262 = 298 md-ft/cp. For the 
assumed gas viscosity kh = 7.9 md-ft. Using the same assumed 
net gives k = 0.097 md. This result is consistent with the ACA 
results. 
G-Function Permeability Estimate. An empirical correlation 
has also been developed to estimate formation permeability from the G-function closure time when after-closure data is not avail-able. The correlation is described in detail in the Appendix. Fig. 7


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shows the G-function correlation permeability estimate for the observed closure time and other input parameters. The perme-ability estimate of 0.097 md is consistent with the Horner and ACA results. 
Pressure-Dependent Leakoff.  Pressure dependent leakoff (PDL) 
occurs when the ﬂ  uid loss rate changes with pore pressure or net 
effective stress in the rock surrounding the fracture. PDL is not caused by the normal change in transient pressure gradient during leakoff. This is part of the normal leakoff mode and is handled by the one-dimensional linear ﬂ  ow solution of the diffusivity 
equation used to model fracture leakoff in a constant permeability system. The pressure dependence referred to here is a change in the transmissibility of the reservoir ﬁ  ssure or fracture system that 
dominates the ﬂ  uid loss rate. PDL is only apparent when there 2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 67890.001 0.01 0.1 1
Square Linear Flow (FL^2)234567892345678923456789
10100100010000


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10100100010000
 (m = 1) Delta-Pressure and DerivativeΔP vs. FL2
FL2dΔP/dFL2vs. FL2ΔP= (pw–pr)
Start of Radial Flow
Fig. 4—Normal leakoff ACA log plot.
2 3 1
Horner Time72507500775080008250850087509000925095009750
 (m = 14411) 
  (Reservoir = 7476)  1Pressure
Fig. 6—Normal leakoff Horner Plot. 
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Radial Flow Time Function72007400760078008000820084008600880090009200
 (m = 4814.2) Results
Reservoir Pressure  = 7475.68 psi
Transmissibility, kh/μ = 298.94991 md*ft
kh = 7.94014 md*ft
Permeability, k = 0.0968 md
Start of Pseudo Radial Time = 2.15 hours 1Pressure
Fig. 5—Normal leakoff ACA radial flow plot.2 3 456789 2 3 456789 2 3 456789 2 3 456789
0.1 1 10 100 1000
Time (0 = 8.15)2345678923456789234
101001000
 (m = –1)  (m = 0.632)  BH ISIP = 9998 psi 1Delta-Pressure and DerivativeΔP vs. Δt
ΔtdΔP/dΔtv s .  Δt
Fracture ClosureRadial Flow
Fig. 3—Normal leakoff log-log plot.


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August 2009 SPE Production & Operations 399is substantial stress dependent permeability in a composite dual-
permeability reservoir.
G-Function for Pressure-Dependent Leakoff.  Fig. 8 shows 
the G-function behavior expected for PDL. The primary pw vs. 
G curve is concave upward and curved while PDL persists. The semilog derivative exhibits the characteristic “hump” above the straight line extrapolated to the derivative origin. The end of PDL and the critical fissure opening pressure corresponds to the end of the “hump” and the beginning of the straight line representing matrix dominated leakoff. Fracture closure is still shown by the departure of the semilog derivative from the straight line through the origin. 
Sqrt(t) Analysis for PDL. Interpretation of the sqrt(t) plot in 
PDL cases has often led to incorrect closure picks. Fig. 9 shows an 
expanded view of the sqrt( t) plot for the example with the curves


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scaled for better visibility. Note that the semilog derivative is nearly identical in shape and information content to the G-function semilog derivative. It clearly shows the PDL “hump” and closure, which has been synchronized to the G-function result. 
Incorrect closure picks on the sqrt( t) plot will not occur if the 
semilog derivative is used. Problems arise when the first deriva-tive is used exclusively to pick closure. In PDL cases the obvious derivative maximum, or most prominent inflection point, is caused by the changing leakoff associated with PDL and does not indicate fracture closure. The false closure indication is shown on the plot. Many fracture diagnostic tests have been badly misdiagnosed because the early and incorrect closure was picked because of dependence on only the sqrt( t) plot. This example clearly illustrates why all available diagnostic plots must be used in concert to arrive 
at a single consistent closure event. 
Log-Log Pressure Derivative for PDL Example. Fig. 10 shows


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Log-Log Pressure Derivative for PDL Example. Fig. 10 shows 
the log-log plot for the PDL example. The normal matrix leakoff period, following the end of PDL, appears as a perfect ½ slope of the semilog derivative with a parallel pressure difference curve exactly 2-times the magnitude of the derivative. The parallel trend ends at the identified closure time and pressure difference. In this example a well-defined −½ slope, or reservoir pseudolinear flow period, is 
shown shortly after closure. The later data approach a slope of −1, 
which indicates pseudoradial flow has been established. 
After-Closure Analysis for PDL Example. The ACA log-log 
plot (Fig. 11)  shows both the reservoir linear and radial flow peri-
ods in their expected locations. 
Figs. 12 and 13  show the ACA Cartesian plots for the linear and 
radial flow analyses. Both give consistent estimates of reservoir pore pressure. The pseudoradial flow analysis gives a transmissibil-ity of 37.2 md-ft/cp and estimated permeability of 0.047 md.


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Horner Analysis for PDL Example. For an average pump rate 
of 6.7 bpm the Horner plot gives kh/
/H9262 = 35.72 md-ft/cp. The Horner 
estimated permeability is 0.046 md compared to 0.047 md from the ACA Radial Flow analysis. Pore pressure estimated from the Horner plot is also consistent with both the linear and radial analyses because a well-developed pseudoradial flow period does exist in this case. The vertical dotted line in Fig. 14  shows the start of pseudora-
dial flow. If a pseudoradial flow period does not exist, extrapolation of an apparent straight-line on the Horner plot can give extremely 
inaccurate estimates of pressure and flow capacity. 0.0010.010.1110100
0              2              4               6              8             10            12            14             16            18            20
GcData Input
rp 1
φ 0.09 V/V
ct 7.50E-05 psi–1
E3 . 5 M p s i
μ 1c p
Gc 2.44
Pz 966.0 psiEstimated Permeability = 0.0974 mdPermeability, md
Fig. 7—Normal leakoff permeability estimate.


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Fig. 7—Normal leakoff permeability estimate.
2 3 456789 2 3 456789 2 3 456789 2 3 4567890.1 1 10 100 1000
Time (0 = 9.133333)23456789234567892
101001000
 (m = 0.5) 
(m = –1) (m = –0.5)  BH ISIP = 10000 psi 1Pressure Difference and DerivativeΔP vs. Δt
ΔtdΔP/dΔtv s .  ΔtFracture ClosureLinear Flow
Radial Flow
Fig. 10—PDL log-log plot. 
0 2 4 6 8 10 12 14 16 18 20
G (Time)8250850087509000925095009750100001025010500
01002003004005006007008009001000
21
P vs. G
GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives
Fig. 8—PDL G-function plot.
1/24/200700:20 00:40 01:00 01:20 01:40
1/24/200702:00
Time825085008750900092509500975010000102 50
0100200300400500
1False ClosurePressure
DerivativesP vs. √t
√tdP/d√t vs. √t
dP/d√t vs. √tFracture Closure
Fig. 9—PDL Sqrt plot.


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400 August 2009 SPE Production & OperationsG-Function Permeability Estimate for PDL Example. The 
G-function permeability correlation for the PDL example is shown in Fig. 15. It also gives a consistent permeability of 0.045 md. The 
impact of the accelerated leakoff during PDL gives an estimate of the composite reservoir effective permeability. Note that the injected fluid viscosity is used for the permeability estimate based on closure time. Fracture Tip Extension.  In very low permeability reservoirs 
the decline in wellbore pressure observed after shut-in may be caused by the dissipation of the pressure transient established in the fracture during pumping. The near-well pressure decreases as the fracture closes, which results in a decrease of fracture width at the well. The closing of the fracture volumetrically displaces ﬂ uid to the tip of the fracture, causing continued extension of


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the fracture length. Much of the pressure decline is therefore not related to leakoff but to the dissipation of the linear transient along the fracture length. 
G-Function Analysis for Tip Extension. During fracture tip 
extension the G-function derivatives fail to develop any straight-line trends. The primary P vs. G curve is concave upward, as is the first derivative. The semilog derivative starts with a large posi-tive slope, and the slope continues to decrease with shut-in time, giving a concave-down curvature (Barree and Mukherjee 1996; Barree 1998). Fig. 16  shows a typical case of fracture tip exten-
sion with minimal leakoff. This is another case that is frequently misdiagnosed. 
Sqrt(t) Analysis With Fracture Tip Extension. Many times the 
first break in the semilog derivative curve has been misinterpreted as a closure event. The mistake is often compounded by the use of the sqrt( t) plot.  Fig. 17  shows the sqrt( t) plot for the same data.


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The first derivative shows a large maximum very shortly after shut-in. This is often mistaken for closure. The semilog deriva-tive on the sqrt( t) plot helps to avoid this mistaken closure pick, 
and shows the same continuously increasing trend as seen on the G-function semilog derivative plot. In low permeability systems it is generally safe to assume that as long as the semilog derivative is still rising, the fracture has not yet closed. This is not true in 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Linear Flow Time Function8000820084008600880090009200
 (m = 1438.5) 
Results
Reservoir Pressure = 8056.66 psi
Start of Pseudo Linear Time = 15.9End of Pseudo Linear Time = 54.391 2PressureStart Linear Flow End Linear Flow
Fig. 12—PDL ACA linear flow plot.2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 67890.001 0.01 0.1 1
Square Linear Flow (FL^2)234567892345678923456789
10100100010000
 (m = 1) 
 (m = 0.5) 
1 2 3Delta-Pressure and DerivativeΔP vs. FL2
FL2dΔP/dFL2vs. FL2ΔP=(pw–pr)
Start Linear Flow
End Linear Flow


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FL2dΔP/dFL2vs. FL2ΔP=(pw–pr)
Start Linear Flow
End Linear Flow
Start Radial Flow
Fig. 11—PDL ACA log plot.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Radial Flow Time Function8000820084008600880090009200
 (m = 11373) Results
Reservoir Pressure = 8068.81 psi
Transmissibility, kh/μ = 37.21984 m
kh = 0.93764 md*ft
Permeability, k = 0.0469 md
Start of Pseudo Radial Time = 11.2 6 1Pressure
Fig. 13—PDL ACA radial flow plot.1Horner Time8000810082008300840085008600870088008900
 (m = 43920) 
  (Reservoir = 8064)  1Pressure
Fig. 14—PDL Horner plot.
0.0010.010.1110100
0              2              4              6               8             10            12            14             16            18            20
GcData Input
rp 1
φ 0.08 V/V
ct 6.00E-05 psi–1
E5 M p s i
μ 1c p
Gc 2.9
Pz 841.0 psiEstimated Permeability = 0.0453 mdPermeability, md
Fig. 15—PDL permeability estimate.


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August 2009 SPE Production & Operations 401very high permeability reservoirs and should always be checked 
using the log-log pressure difference plot. 
Log-Log Pressure Derivative Analysis With Tip Extension. 
The log-log plot of pressure change after shut-in is particularly useful for diagnosing fracture tip extension. Fig. 18  shows the 
pressure difference and pressure derivative (semilog) for the tip extension example. 
In Fig. 18, the pressure derivative departs from the early unit-


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slope (storage) and establishes a ¼ slope during fracture tip exten-sion. The pressure difference curve falls on a parallel ¼ slope line separated by 4-times the magnitude of the derivative. The ¼ slope signature is diagnostic of bilinear flow representing a continued dissipation of the linear pressure transient along the fracture length (extension and concomitant fluid flow) and some linear flow driv-ing minimal leakoff. For tip extension to occur the leakoff rate to the formation must be low. As long as the parallel ¼ slope trend continues, the fracture has not closed and is still in the process of extending. Closure cannot be determined and no after-closure analysis can be conducted.
Height Recession or Transverse Storage.  There are two differ-
ent mechanisms that can generate a similar diagnostic derivative signature during fracture closure. Both are caused by an excess stored volume of ﬂ  uid in the fracture at shut-in relative to the


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expected surface area of the fracture for a planar, constant-height geometry model. 
Traditionally this signature has been called “fracture height 
recession”. The usual model assumes that leakoff occurs only through a thin permeable bed and that the fracture extends in height to cover impermeable strata with no leakoff. At shut-in there is a large volume of fluid stored in the fracture and the leakoff rate relative to the stored volume is small, hence the rate of pressure decline is likewise small. As the fracture empties, the rate of leakoff 
relative to the remaining stored fluid accelerates and the pressure declines more rapidly. If the fracture height changes during leakoff, the fracture compliance may also decrease, adding to the rate of pressure loss.
However, the same signature is observed in many cases where


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However, the same signature is observed in many cases where 
fracture height growth out of zone is not observed by tracers, inclinometer, or micro-seismic mapping. Some of these cases show treating behavior similar to PDL cases, with a tendency for rapid screenout and difficulty placing high proppant concentration slur-ries. These observations suggest that another mechanism may be responsible for the same diagnostic derivative signature. The alter-nate mechanism is called “transverse fracture storage”. 
In transverse fracture storage a secondary fracture set is opened


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when the fluid pressure exceeds the critical fissure-opening pres-sure, just as in PDL. As the secondary fractures dilate they create a storage volume for fluid which is taken from the primary hydraulic fracture. While the fracture storage volume increases, leakoff can also be accelerated so PDL and storage are aspects of the same coupled mechanism of fissure dilation. The relative magnitude of the enhanced leakoff and storage mechanisms determines whether the G-function derivatives show PDL or storage. Numerical model-ing studies indicate that the storage mechanism can easily domi-nate even large PDL.
At shut-in the secondary fractures will close before the primary


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fracture because they are held open against a stress higher than the minimum in-situ horizontal stress. As they close fluid will be expelled from the transverse storage volume back into the main fracture decreasing the normal rate of pressure decline and, in effect, supporting the observed shut-in pressure by re-injection of stored fluid. Accelerated leakoff can still occur at the same time but if the storage and expulsion mechanism exceeds the enhanced leakoff rate then the only signature observed during falloff will be storage. In many cases a period of linear, constant area, constant matrix perme-ability dominated leakoff will occur after the end of storage. 
G-Function Analysis With Storage. The characteristic G-function 
derivative signature is a “belly” below the straight line through the origin and tangent to the semilog derivative of p
w vs. G at the 
point of fracture closure. Fig. 19  shows an example of slight to


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moderate storage. In Fig. 19, fracture closure, indicated by the same departure of the tangent line from the semilog derivative, occurs just after the end of the storage effect. 
Sqrt(t) Analysis With Storage or Height Recession. The sqrt( t) 
plot (Fig. 20)  shows a clear indication of closure based on both the 
first-derivative inflection point and the semilog derivative curve. Picking closure in the case of storage is not generally a problem.
The storage model, whether caused by height recession or 
transverse fractures, requires that a larger volume of fluid must be leaked-off to reach fracture closure than is expected for a single planar constant-height fracture. In either case the time to reach fracture closure is delayed by the excess fluid volume that must be lost. Any estimation of reservoir permeability will give an incorrect result if the uncorrected closure time (either G
c or time-to closure 0 5 10 15 20 25 30
G (Time)840086008800900092009400960098001000010200
0255075100125150
1
P vs. G


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0255075100125150
1
P vs. G
GdP/dG vs. G
dP/dG vs. GPressure
Derivatives
Fig. 16—Tip extension G-function plot.1/25/200704:00 08:00 12:00 16:00
1/25/200720:00
Time840086008800900092009400960098001000010200
0102030405060708090100
1Incorrect ClosurePressure
DerivativesP vs. √t xdP/dxvs. √tdP/dxvs. √t
Fig. 17—Tip extension sqrt( t) plot.
2 3 4 56789 2 3 4 56789 2 3 4 56789 21 10 100 1000
Time (0 = 33.7)23456789234567892345
101001000
 (m = 0.25)  BH ISIP = 10000 psi 1Delta-Pressure and DerivativeΔP vs. Δt
ΔtdΔP/dΔtv s .  Δt
Fig. 18—Tip extension log-log plot.


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402 August 2009 SPE Production & Operationsin minutes, tc) is used. The observed closure time must be corrected 
by multiplying by the storage ratio, rp. The magnitude of rp can 
be determined by taking the ratio of the area under the G-function semilog derivative up to the closure time, divided by the area of the right-triangle formed by the tangent line through the origin at closure. For normal leakoff and PDL the value of r
p is set to 1.0 
even though the ratio of the areas will be greater than 1 for the PDL case. It is possible that the closure time for PDL leakoff is proportional to the composite system permeability including both the matrix and fractures. For severe cases of storage r
p can be as 
low as 0.5 or less. 
Log-Log Pressure Derivative With Storage. Fig. 21  shows


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Log-Log Pressure Derivative With Storage. Fig. 21  shows 
the log-log plot of pressure difference and semilog derivative for the storage case. Prior to closure, and while transverse storage is dominant, the semilog derivative approaches a unit slope, with the pressure difference curve nearly parallel. In some cases the two curves lie together on a single unit-slope line. In this case the curves are separated slightly and the slope is not exactly 1.0. After closure the reservoir transient signature is defined as in the previ-ously presented cases. All fracture storage effects are eliminated and the reservoir pseudolinear flow period is shown by a −1/2 slope with a pseudoradial flow period indicated by a −1 slope of the semi-log derivative. 
Conclusions
The use of prefrac injection/falloff diagnostic tests has become


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commonplace. Many important decisions regarding fracture treat-ment designs and expectations of post-frac production are based on the results of these tests. In too many cases individual diagnostic plots and analysis techniques are misapplied, leading to incorrect 
interpretations. The analyses presented here lead to the following conclusions: 1.  With consistent application of all available pressure decline 
diagnostics, a single unambiguous determination of fracture closure time and pressure can be made. 
 2.  A single, unique closure event can be identified on all diag-
nostic plots. 
 3.  The conventional analysis of the sqrt( t) plot, using the inflec-
tion point identified by the first derivative, gives incorrect indications of closure for cases of PDL and tip extension and should not be relied upon.
 4.  A modified sqrt( t) analysis, using the semilog derivative, is 
equivalent to the G-function analysis and helps avoid incorrect closure picks in cases of PDL and tip extension.


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5.  Flow regimes can be identified using the semilog pressure 
derivative on the log-log plot of /H9004p
wf − /H9004t during the shut-in 
period following the fracture injection test.
 6.  As in conventional transient test analysis, a pseudolinear flow 
period is identified by parallel ½ slope lines, separated by 2x, on the log-log /H9004p
wf − /H9004t plot up until fracture closure. 
 7.  Bilinear flow can be identified by parallel ¼ slope lines 
separated by 4x on the log-log /H9004pwf − /H9004t plot prior to fracture 
closure.
 8.  After closure the pseudolinear reservoir flow period is identi-
fied by a −1/2 slope of the semilog derivative of the pressure difference on the log-log /H9004p
wf − /H9004t plot, and a −3/2 slope of 
the first derivative of the pressure difference with shut-in time on the same plot.
 9.  Pseudoradial flow is identified by a −1 slope of the semilog 
derivative on the log-log plot.
10.  When a stable pseudolinear flow period exists, the after-clo-


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sure Cartesian plot of the linear flow function can be used to estimate reservoir pressure.
11.  When a pseudoradial flow period exists, both the conventional 
Horner analysis and after-closure radial-flow analysis can be used to determine reservoir transmissibility and pore pressure.
Nomenclature
 A
f = fracture area, L2, ft2
 B = formation volume factor, L3/L3, RB/STB
 ct = total compressibility, Lt2/m, psi−1
 Cac = after-closure storage, L4t2/m, bbl/psi
 Cbl = bilinear ﬂ  ow constant, m/Lt5/4, psi·hr3/4
 Cfbc = before-closure fracture storage, L4t2/m, bbl/psi
 Cpl = pseudolinear ﬂ  ow constant, m/Lt3/2, psi·hr1/2 
 Cpr = pseudoradial ﬂ  ow constant, m/Lt, psi·hr 
 FL = linear ﬂ  ow time function, dimensionless 
 FR = radial ﬂ  ow time function, dimensionless 5 10 15 20
G (Time)750080008500900095001000010500
0200400600800100012001400160018002000
1
P vs. G
GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives
Fig. 19—Storage G-function plot.1/24/200702:00 04:00 06:00 08:00
1/24/200710:00


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1/24/200710:00
Time750080008500900095001000010500
0100200300400
1Pressure
DerivativesP vs. √t
xdP/dxvs. √t
dP/dxvs. √tFracture Closure
Fig. 20—Storage sqrt( t) plot.
2 3 456789 2 3 456789 2 3 456789 2 3 4560.1 1 10 100
Time (0 = 9.416667)234567892345678923
101001000
 (m = 0.5)  (m = –0.5)  BH ISIP = 10000 psi 1Delta-Pressure and DerivativeΔP vs. Δt
ΔtdΔP/dΔtv s .  Δt
Fig. 21—Storage log-log plot.


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August 2009 SPE Production & Operations 403 g = loss-volume function, dimensionless 
 G = G-function, dimensionless 
 h = height, L, ft 
 k = permeability, L2, md 
 Lf = fracture half-length, L, ft 
 mH = slope of data on Horner plot, m/Lt2, psia
 mL = slope of data on pseudolinear ﬂ  ow graph, m/Lt2, psia
 mR = slope of data on pseudoradial ﬂ  ow graph, m/Lt2, psia
 p = pressure, m/Lt2, psia
 pwf = fracture pressure measured at wellbore, m/Lt2, psia
 q = ﬂ ow rate, L3/t, bbl/D
 Qt = total injection volume, L3, bbl
 rp = storage ratio, dimensionless
 Sf = fracture stiffness, m/L2t2, psi/ft
 t = time, hr
 ta = adjusted pseudotime, hr
 /H9252 = constant, dimensionless
 /H9004 = difference, dimensionless
 /H9262 = viscosity, m/Lt, cp
 /H9278 = porosity, dimensionless
 /H9273 = constant, dimensionless
Subscripts
 a = adjusted
 c = closure
 D = dimensionless
 e = end of injection
 f = ﬁ ltrate
 p = pumping
 w = wellbore
 z = process zone
 0 = end of injection
Acknowledgments


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z = process zone
 0 = end of injection
Acknowledgments
The authors would like to thank Kumar Ramurthy, Halliburton, 
Mike Conway, Stim-Lab, and Stuart Cox, Marathon, for their discussion and contribution to the procedures described. Sincere thanks are also due to the many operators whose diligence in prefrac testing has allowed these diagnostic analysis procedures to be developed and tested.
References
Barree, R.D. 1998. Applications of Pre-Frac Injection/Falloff Tests in 
Fissured Reservoirs—Field Examples. Paper SPE 39932 presented at the SPE Rocky Mountain Regional/Low-Permeability Reservoirs Symposium, Denver, 5–8 April. DOI: 10.2118/39932-MS.
Barree, R.D. and Mukherjee, H. 1996. Determination of Pressure Depen-
dent Leakoff and its Effect on Fracture Geometry. Paper SPE 36424 presented at the SPE Annual Technical Conference and Exhibition, Denver, 6–9 October. DOI: 10.2118/36424-MS.
Castillo, J.L. 1987. Modified Fracture Pressure Decline Analysis Includ-


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ing Pressure-Dependent Leakoff. Paper SPE 16417 presented at the Low-Permeability Reservoirs Symposium, Denver, 18–19 May. DOI: 10.2118/16417-MS.
Cinco-Ley, H. and Samaniego-V ., F. 1981. Transient Pressure Analysis for 
Fractured Wells. J. Pet Tech  33 (9): 1749–1766. SPE-7490-PA. DOI: 
10.2118/7490-PA.
Cinco-Ley, H., Kuchuk, F., Ayoub, J., Samaniego-V ., F., and Ayestaran, 
L. 1986. Analysis of Pressure Tests Through the Use of Instantaneous Source Response Concepts. Paper SPE 15476 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 Octo-ber. DOI: 10.2118/15476-MS.
Craig, D.P. and Blasingame, T.A. 2006. Application of a New Fracture-
Injection/Falloff Model Accounting for Propagating, Dilated, and Closing Hydraulic Fractures. Paper SPE 100578 presented at the SPE Gas Technology Symposium, Calgary, 15–17 May. DOI: 10.2118/ 100578-MS.
Hagoort, J. 1981. Waterflood-induced hydraulic fracturing. PhD thesis,


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Delft Technical University, Delft, The Netherlands.Koning, E.J.L. and Niko, H. 1985. Fractured Water-Injection Wells: A 
Pressure Falloff Test for Determining Fracture Dimensions. Paper SPE 14458 presented at the SPE Annual Technical Conference and Exhibi-tion, Las Vegas, Nevada, USA, 22–25 September. DOI: 10.2118/14458-MS.
Nolte, K.G. 1979. Determination of Fracture Parameters from Fractur-
ing Pressure Decline. Paper SPE 8341 presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, USA, 23–26 September. DOI: 10.2118/8341-MS.
Settari, A. 2000. Coupled Fracture and Reservoir Modeling. Presented at 
the Workshop on Three Dimensional and Advanced Hydraulic Frac-ture Modeling, Fourth North American Rock Mechanics Symposium, Seattle, Washington, USA, 29 July.
Talley, G.R., Swindell, T.M., Waters, G.A., and Nolte, K.G. 1999. Field


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Application of After-Closure Analysis of Fracture Calibration Tests. Paper SPE 52220 presented at the SPE Mid-Continent Operations Symposium, Oklahoma City, Oklahoma, USA, 28–31 March. DOI: 10.2118/52220-MS.
Appendix A—Definition of Diagnostic 
Functions
The G-Function.  The G-function is a representation of the elapsed 
time after shut-in normalized to the duration of fracture extension. Corrections are made for the superposition of variable leakoff times while the fracture is growing. The form of the G-function used in this paper assumes high ﬂ  uid efﬁ  ciency in low-permeability 
formations. Under that assumption the surface area of the fracture is assumed to vary linearly with time during fracture propaga-tion. The dimensionless pumping time used in the G-function is deﬁ ned as:
/H9004tt t t
Dp p=−()/.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-l)
The elapsed total time from the start of fracture initiation (not


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start of pumping) is t and the total pumping time (elapsed time from fracture initiation to shut-in) in consistent time units is t
P. For 
the assumption of low leakoff the dimensionless time ( /H9004tD) is used 
to compute an intermediate function:
gt t tDD D/H9004/H9004 /H9004() =+() − ⎡⎣⎤⎦4
3115 15 . ..   . . . . . . . . . . . . . . . . . .(A-2)
The G-function used in the diagnostic plots is derived from the 
intermediate function as follows:
Gt gt gDD/H9004/H9004() = () − ⎡⎣⎤⎦4
0/H9266,  . . . . . . . . . . . . . . . . . . . . . . .(A-3)
where g0 is the dimensionless loss-volume function at shut-in ( t = tp 
or /H9004tD = 0). All derivatives are calculated using a central difference 
function of pressure and G-function (normalized shut-in time).
After-Closure Analysis and Flow Regime Identiﬁ  cation.  After-
closure pressure decline analysis requires the identiﬁ  cation of fully-
developed reservoir pseudolinear and pseudoradial transient ﬂ  ow


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regimes. The ﬂ  ow regimes can be identiﬁ  ed by characteristic slopes 
on a log-log plot of observed falloff pressure minus reservoir pressure, (p
w(t) − pi), vs. the square of the linear-ﬂ  ow time function ( FL2) and 
the semilog derivative, (X*dY/dX), of the pressure difference curve Talley et al.). It is important to note that the guess of reservoir pressure, p
i, used in construction of the ﬂ  ow regime plot severely impacts the 
slope and magnitude of the pressure difference curve. The pressure derivative, because of the difference function used to generate it, is not affected by the initial guess of reservoir pressure. 
The linear-flow time function is defined by:
Ft tt
tfor t tLcc
c ,s i n() =≥− 21
/H9266.  .  . . . . . . . . . . . . . . . . .(A-4)
The linear-flow function also requires an accurate determina-
tion of the time required after shut-in to reach fracture closure, tc.


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404 August 2009 SPE Production & OperationsIn the pseudolinear flow period the slope of the derivative curve 
on the log-log plot should be ½. For the correct estimate of reser-voir pore pressure, the pressure difference curve should also have a slope of ½ and should be exactly twice the magnitude of the derivative. If a stable pseudolinear flow period is identified then a Cartesian plot of observed pressure during the falloff, p
w(t), vs. 
FL should yield a straight line with intercept equal to the reservoir 
pore pressure, pi, and with a slope of mL. 
pt p m Ft twi L L c()−= (),.   . . . . . . . . . . . . . . . . . . . . . . . . .(A-5)
If a pseudoradial flow period exists, the slope of the derivative 
and correct pressure difference curves on the log-log flow regime plot should both be 1.0 and the two curves should coincide. In the pseudoradial flow period, a Cartesian plot of pressure vs. F
R should 
also yield a straight line with intercept equal to pi and slope of mR.


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pt p m Ft twi R R c()−= (),.   . . . . . . . . . . . . . . . . . . . . . . . . .(A-6)
In these equations, tc is the time to fracture closure with time 
zero set as the beginning of fracture extension, pi is the initial reser-
voir pressure, and mR is the Cartesian slope of the “correct” straight 
line. The radial-flow function ( FR) is given by (Talley et al.)
Ft tt
ttRcc
c,l n , .() =+−⎛
⎝⎜⎞
⎠⎟=≅1
4116162/H9273/H9273/H9266.  . . . . . . . . . . .(A-7)
In the properly identified pseudoradial flow period the reservoir 
far-field transmissibility can be determined from the slope, fracture closure time, and volume injected during the test.
kh Q
mtt
Rc /H9262=251 000, .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-8)
In Eq. A-8, the permeability, k, is in md, net pay thickness, h, 
is in feet, far-field mobile fluid viscosity,  /H9262, in cp, tc in minutes, 
and the volume injected during the test, Qt, is in bbls, and the 
slope, mR, is in psi.


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slope, mR, is in psi. 
Horner Analysis.  The conventional Horner analysis uses a Cartesian 
plot of observed pressure vs. Horner time, ( tp + /H9004t)//H9004t, with all times 
in consistent units. The fracture propagation time is tp and the elapsed 
shut-in time is /H9004t. As shut-in time approaches inﬁ  nity the Horner 
time function approaches 1. A straight-line extrapolation of the Horner plot to the intercept at a Horner time of 1.0 gives an estimate of reservoir pressure. The slope of the correct straight-line extrapola-tion, m
H, can be used to estimate reservoir transmissibility:
kh q
mH /H9262=() 162 6 1440..   . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-9)
The flow rate in Eq. A-9 is assumed to be in barrels per minute


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and is the average rate for the time the fracture was extending. In both Eqs. A-8 and A-9, the viscosity is the far-field mobile fluid viscosity. The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity.
The major problem with the Horner analysis is that the results 
are only valid if the data used to extrapolate the apparent straight line are actually in fully developed pseudoradial flow. There is no way to determine the validity of the Horner analysis or to determine the flow regime within the Horner plot itself.
G-Function Permeability Estimate.  An empirical function to 
approximate formation permeability has been derived from numer-ous numerical simulations of fracture closure. The correlation is based on the observed G-function time at fracture closure:
kp
cGE rfz
tcp=
⎛
⎝⎜⎞
⎠⎟0 0086 0 01
0 03819 6..
../H9262


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kp
cGE rfz
tcp=
⎛
⎝⎜⎞
⎠⎟0 0086 0 01
0 03819 6..
../H9262
/H9278.   . . . . . . . . . . . . . . . . . . . . . .(A-10)The rate of fluid loss from the fracture before closure is dominated 
by the mobility of the injected fluid instead of the far-field viscosity. The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected fluid at leakoff temperature and the relative permeability to the leakoff fluid. The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid mobility. As a general rule, an assumed injected fluid viscosity of 1.0 cp is used to incorporate the effects of reservoir temperature and relative permeability in the invaded zone.
Note that the correlation gives permeability and not kh. The 
time to closure is related to fracture volume vs. created area so both the fracture height and length do not appear in the equation. The process zone stress, P
z, is the net fracture extension pressure


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z, is the net fracture extension pressure 
above closure pressure, pc, or pz = pISIP − pc. The net extension pres-
sure and Young’s Modulus provide a relationship between facture volume (width) and surface area during pumping.
The other parameters in Eq. A-10 are defined, except the stor-
age ratio, r
p. This parameter represents the amount of excess fluid 
that must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height pla-nar fracture. The storage ratio, r
p, is the ratio of the area under the 
G-function semilog derivative up until closure divided by the area of the triangle defined by the straight-line (normal leakoff) tangent to the semilog derivative at closure. For normal matrix leakoff and PDL, r
p is therefore 1.0. For the transverse storage and height recession 
signature, rp is some value less than 1.0. The observed G-function 
closure time ( Gc) is always delayed by the excess fluid volume in the


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fracture for either height recession or transverse storage. Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility estimate. The closure time should be corrected by multiplying by r
p 
for all permeability and transmissibility estimates. 
Appendix B—Analytical Solutions and the 
Log-Log Diagnostic Graph
Linear and Bilinear Flow Before Closure.  Hagoort (1981) developed 
a before-closure analytical solution accounting for both storage and linear formation ﬂ  ow. Koning and Niko (1985) write the solution as
pwDtC ee r f c tDf b c Dt
DD =−⎛
⎝⎜⎞
⎠⎟ − () () /H9266/H9266/H9252/H9252/H9004 /H9004/H9004
212
2, 
 . . . . . . . . . . . . . . . . . . . . . . (B-1)
where
pwDtkhpt pt t
qBDwe we()() ( )
./H9004/H9004=−+ ()
141 2 /H9262,   . . . . . . . . . . . . . . (B-2)
/H9004/H9004tkt
cLD
tf=0 0002637
2.
/H9278/H9262,  . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)
/H9252/H9266=2


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/H9252/H9266=2
CfbcD,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)
and
CC
ch LC
ch LfbcDfbc
tffbc
tf==5 615
20 8936
22..
/H9266/H9278 /H9278.  . . . . . . . . . . . . . . . . . . . (B-5)
Here, Cfbc is the before-closure fracture storage constant, which is 
defined as (Craig and Blasingame 2006)
CA
Sfbcf
f=2
5 615.,   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-6)
where Af is the fracture area (one wing), and  Sf is the fracture 
stiffness.


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August 2009 SPE Production & Operations 405Hagoort’s solution predicts that the before-closure falloff is a 
combination of fracture storage and linear flow. As fracture storage becomes small, C
fbcD→0, linear flow will dominate the before-
closure pressure falloff. In most "hard rock" environments, CfbcD 
is small, and for a fracture-injection/falloff sequence with /H9004t >> te 
and long closure times, 
pt pt t twe we() ( )−+ ∝ /H9004 ,  . . . . . . . . . . . . . . . . . . . . . . . . . (B-7)
where t = te + /H9004t, and pw(te) is the instantaneous shut-in pressure. 
Define a fracture-pressure difference as
/H9004/H9004pp t p ttw f we we=− + () ( ) ,  . . . . . . . . . . . . . . . . . . . . . . . (B-8)
and a log-log graph of /H9004pwf vs. t will exhibit a ½ slope during 
linear flow before closure. An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist, but field data suggest that during before-closure bilinear flow


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/H9004/H9004ptwf∝()14,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-9)
and a log-log graph of /H9004pwf vs. t will exhibit a ¼ slope during 
bilinear flow before closure.
Pseudoradial, Pseudolinear, and Bilinear Flow After Closure.  
Craig and Blasingame (2006) developed an analytical solution for a fracture-injection/falloff sequence with a propagating and closing hydraulic fracture, and they derived the “complete” after-closure impulse solutions accounting for fracture storage. The after-closure impulse solution for pseudoradial ﬂ  ow is written as 
pt t pkhQpwsDCacpwe it ().( )() ( +− = +− /H9004141 2 24
200/H9262pp
tti
e) . ( )+1
/H9004
 . . . . . . . . . . . . . . . . . . . . . (B-10)
For after-closure pseudolinear flow,
pt t p
hLQ
pwwe i
ft()
.( ) .+−
=+/H9004
141 2 24 0 0002637
21 /H9266
ssDCacppi () ( )00−⎛
⎝⎜⎞
⎠⎟
×+⎛
⎝⎜⎞
⎠⎟/H9262
/H9278ckt tte112
/H9004,   . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-11)
and for after-closure bilinear flow,
pt t pwe i()


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and for after-closure bilinear flow,
pt t pwe i()
.( ) (. ) (. )+−
=/H9004
141 2 24 0 6125 0 000263714
kkwQ
pp Cac fft
i +−⎛
⎝⎜⎞
⎠⎟()034/H9262
×⎛
⎝⎜⎞
⎠⎟+⎛
⎝⎜⎞
⎠⎟1114 34
/H9262ck t tte /H9004.   . . . . . . . . . . . . . . . . . . . . . . . (B-12)For convenience, define a constant pseudoradial flow term as
CkhQpwsDCacppprti = +−()141 2 24
200.( )() ( )/H9262.   . . . . . (B-13)
For pseudolinear flow, a constant term is defined as
C
hLQ
pwsDCacpl
ft=+141 2 24 0 0002637
21
0.( ) .
() (/H9266
ppp ck i t 012
−⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜⎞
⎠⎟ )./H9262
/H9278
 . . . . . . . . . . . . . . . . . . . . . (B-14)
Similarly, for bilinear flow a constant term is written as
C
kwQ
bl
fft=+141 2 24 0 6125 0 000263714.( ) (. ) (. )
(ppp Cacck i t 034141
−⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜⎞
⎠⎟ ). /H9262/H9262
 . . . . . . . . . . . . . . . . . . . . . (B-15)
With the new definition, and defining a reservoir-pressure differ-
ence, /H9004pw = pw(t) − pi, the pseudoradial-flow impulse solution can 
be written as


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be written as
/H9004pC twp r=−1.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-16)
Similarly, the linear-flow impulse solution can be written as
/H9004pC twp l=−12/,   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-17)
and the bilinear-flow impulse solution can be written as
/H9004pC twb l=−34/.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-18)
The derivatives with respect to t are written for after-closure 
pseudoradial flow as, 
−∂
∂=− /H9004p
tCtw
pr()2,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-19)
for after-closure pseudolinear flow as,
−∂
∂=− /H9004p
tCtw pl
()/
232,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-20)
and for after-closure bilinear flow as,
−∂
∂=− /H9004p
tCtwb l
()/ 3
474.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-21)
The semilog derivatives and the impulse derivatives (Cinco-Ley


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The semilog derivatives and the impulse derivatives (Cinco-Ley 
et al. 1986) are calculated by multiplying the after-closure deriva-tives by t and t
2, respectively. Table B-1  shows the derivative terms 
for each after-closure flow regime.
Log-Log Diagnostic Graph.  Before-closure linear or bilinear ﬂ  ow 
are identiﬁ  ed by the relationship between the fracture-pressure 
TABLE B-1—AFTER-CLOSURE IMPULSE SOLUTIONS AND DERIVATIVES  
Flow Regime  Impulse Solution  Derivative  Semilog Derivative  Impulse Derivative  
Bilinear     
Pseudolinear     
Pseudoradial


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406 August 2009 SPE Production & Operationsdifference, /H9004pwf, and total time, t, but the after-closure ﬂ  ow regimes 
are identiﬁ  ed by the relationship between the reservoir-pressure 
difference, /H9004pw, and t, which requires knowing the initial reservoir 
pressure, pi. However, by noting that
∂
∂=−∂
∂/H9004 /H9004 p
tp
twf w,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-22)
a log-log diagnostic graph can be prepared that is independent of 
initial reservoir pressure but can be used to identify before- and 
after-closure flow regimes. Table B-2  shows the characteristic 
slopes of each flow regime for log-log graphs of /H9004pwf vs. t, ∂∂/H9004ptwf/ 
vs. t, tp twf∂∂/H9004 / vs. t, and tptwf2∂∂/H9004 / vs. t. 
In gas reservoirs, log-log diagnostic graphs can be prepared 
by plotting /H9004pawf vs. ta, ∂∂/H9004ptawf a/ vs. ta, tp taa w f a∂∂/H9004 / vs. ta, and 
tp taa w f a2∂∂/H9004 / vs. ta where adjusted pseudopressure is defined as
paz
pipdp
zp
=⎛
⎝⎜⎞
⎠⎟⌠
⌡⎮
⎮/H9262


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paz
pipdp
zp
=⎛
⎝⎜⎞
⎠⎟⌠
⌡⎮
⎮/H9262
/H92620,  . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-23)
and adjusted pseudotime is defined as
tactidt
ctt
=⌠
⌡⎮
⎮⎮()/H9262/H92620.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-24)
Because adjusted pseudopressure is a function of initial reser-
voir pressure, and if initial reservoir pressure is unknown, it’s often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining TABLE B-2—LOG-LOG GRAPH CHARACTERISTIC SLOPES 
Before Closure  After Closure  
Log-Log Graph  Bilinear  Linear  Bilinear  Pseudolinear  Pseudoradial  
 1/4 1/2 – – – 
 –3/4 –1/2 –7/4 –3/2 –2 
 1/4 1/2 –3/4 –1/2 –1 
 5/4 3/2 1/4 1/2 0 
the estimate using diagnostic and interpretive graphs plotted in 
terms of adjusted pseudovariables.
Robert D. Barree  is president and principal investigator of Barree


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& Associates, a consulting firm specializing in stimulation and well performance optimization. He has been involved in the development of hydraulic fracture technology since 1980 and is the primary author of the fully 3D hydraulic fracture simulator GOHFER. Barree is also the author of more than 50 technical publications and has served as SPE Distinguished Lecturer on the topic of new philosophies in hydraulic fracturing. He holds a BS degree in petroleum engineering from the Pennsylvania State University and PhD degree from Colorado School of Mines. Valencia Barree  is a partner at Barree & Associates LLC,


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primarily working in well-test analysis and software development for the Stim Lab Proppant Consortium (including Predict-K) and GOHFER. Previously, she has focused on software development for core test analysis and instrumentation under the company name Custom Scientific Tools. Developed software includes tools for the acquisition and analysis of relative permeability and capillary pressure data, as well as many other customized testing procedures. She holds a BS degree in mineral engineering mathematics from the Colorado School of Mines. David P . Craig  currently writes a technical blog at www.


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refrac.net and is researching complex-fracture propagation models for Halliburton. In 2000, Craig was awarded the Henry Matson Technical Service Award from the Denver Section of SPE in “recognition of significant contributions in engineering design and diagnostics of hydraulic fractures,” and in 2006 he received the Award of Excellence for Outstanding Research from the Harold Vance Department of Petroleum Engineering at Texas A&M University. He holds a BS degree from Texas Tech University and MS and PhD degrees from Texas A&M University, all in petroleum engineering.


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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
compliance) is assumed to be constant during fracture closure. Fracture closure is a gradual process that


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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
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


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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
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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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.
2016), stimulation design (Ramurthy et al. 2011), development of reservoir models (Mirani et al. 2016;


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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
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


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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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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
dimensionless time Δt D.tp is the total pump time.r p is the productive fracture ratio, which is the ratio of


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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
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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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
constant during fracture closure. This is why G-function based models are only used for before closure


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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
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


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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)
At the boundary of the fracture surface, both Darcy's flow and material balance have to be honored


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(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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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
into contact at asperities, and the other is hydraulic closure, where fracture pressure in the closed section


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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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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
pressure drop can be observed during early-time of shut-in in low permeability reservoirs, but it is the result


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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
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


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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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8 SPE-187348-MS
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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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
width and the net closure stress for fractured rocks, based on the work of Barton et al. (1985):
(22)


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(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).
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


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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.
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


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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
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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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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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
tiltmeter demonstrates that soon after shut-in, the measured displacement declines linearly with pressure.


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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
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,


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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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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
"normal leak-off behavior" to account for different fracture boundary conditions, while other assumptions


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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 
 , 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
deflection. This occurs if some mechanisms that accelerate pressure drop (e.g., the gradual increase of


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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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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
indicate that as the fracture width decreases, the rough fracture walls will come into contact sooner if the


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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.
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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= 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
Next, we examine how contact reference stress affects the pressure decline response. Fig. 12 and Fig. 13


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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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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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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


