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drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-179725-DISC Fracture Compliance Discussions.pdf


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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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addresses these nonlinear-leakoff phenomena, but they do so with
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


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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
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


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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.
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


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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.
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


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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-
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.”


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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
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-


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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.
SPE J. 21(5): 1567–1582. SPE-174902-PA. https://doi.org/10.2118/
174902-PA.


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Interwell-Fracturing Interference: An Eagle Ford Shale-Oil Study.
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


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sure and after-closure solutions without providing a mathematical
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


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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
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


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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-
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


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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
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-


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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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esses, as addressed in the follow-up paper Jung et al. (2016).
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


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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).
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-


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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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drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/SPE-186098-PA Superious Deflection.pdf


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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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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
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-


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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.
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


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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
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


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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-
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


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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.
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-


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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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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).
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


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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
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


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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..............................................................................
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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)
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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tion-time derivative and to investigate certain aspects of real DFIT transients.
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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very small aperture. This is supported by the consistent observation of post-closure linear ﬂow in ﬁeld data sets (linear ﬂow requires an
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


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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
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.


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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
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


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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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derived on the basis of constant-pressure leakoff from a high-storage wellbore/fracture system. The square root of time appears in both
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


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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
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


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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/
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.


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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
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


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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-
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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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-
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


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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.
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


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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
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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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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tion occurs in the superposition-time derivative plot (Eq. 2). It might be argued that the deﬂection in the superposition-time plot adjusts
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


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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.
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


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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-
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-


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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
multiplied by the derivative of ﬂow rate with respect to time. Implicit in the construction of Eq. 23 is the requirement that the constant-.........................................
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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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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-
tion time (gray).REE186098 DOI: 10.2118/186098-PA Date: 26-October-17 Stage: Page: 1050 Total Pages: 11
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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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grals of the constant-rate analytical solution to analyze fracture propagation. This is valid because they restrict their analysis to consider
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


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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
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


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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
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.


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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
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


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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-
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


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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
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


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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
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-


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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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quently curves up to a slope approaching unity, indicating that the pressure is decreasing signiﬁcantly faster than square-root-of-time
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-


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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
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


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(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
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.


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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
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.


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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
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


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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
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


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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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pressure transient. Although superposition-time derivative curves are useful for conventional buildup/falloff tests, they complicate
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


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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
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


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/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
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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Appendix A
Here, I derive equations for the asymptotic behavior of Gða¼1Þfor small Dtand large Dt:Gða¼1Þis deﬁned in Eq. 21. To calculate


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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
The second equality is true because 1.0 becomes inﬁnitesimally small relative toDt
teasDtapproaches inﬁnity. The third equality is


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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 Department of Energy Resources Engineering at Stanford University in 2012, and the Best Paper in Geophysics award from
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
ID:jaganm Time: 13:19 I Path: //chenas03/cenpro/ApplicationFiles/Journals/SA/SPE/REE#/Vol00000/170018/Comp/APPFile/SA-REE#170018November 2017 SPE Reservoir Evaluation & Engineering 1055


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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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Since the introduction of the G-function derivative analysis, prefrac 
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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wf)-log( /H9004t) plots 
and their appropriate diagnostic derivatives.
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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Introduction
Prefrac diagnostic injection test analysis provides critical input data 
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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Relationship of the pre-closure analysis to after-closure analysis 
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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able storage in a transverse fracture system.
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.


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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


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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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Diagnostic Derivative Examples
For each analysis technique various curves are used to help define 
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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For pre-closure analysis, and consistent identification of frac-
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


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G-function (G /H11128p
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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quently been misinterpreted when picking fracture closure, even for the simplest cases. The primary p
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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satisfy the requirement on both the G-function and sqrt( t) plots.
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


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0 5 10 15 20 25
G (Time)750080008500900095001000010500
0200400600800100012001400160018002000
1
P vs. G
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-Nolte After-Closure Analysis (ACA) flow regime identi-fication plot for the normal leakoff example is shown as Fig. 4  
(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


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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 
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


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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 
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


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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
 (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


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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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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 
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


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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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at a single consistent closure event. 
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


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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. 
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


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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.
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


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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


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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
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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In Fig. 18, the pressure derivative departs from the early unit-
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-


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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 
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


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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 
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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In transverse fracture storage a secondary fracture set is opened 
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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At shut-in the secondary fractures will close before the primary 
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


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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 
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


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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
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


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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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p can be as 
low as 0.5 or less. 
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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Conclusions
The use of prefrac injection/falloff diagnostic tests has become 
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


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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.
 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


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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-
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


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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
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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D = dimensionless
 e = end of injection
 f = ﬁ ltrate
 p = pumping
 w = wellbore
 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.


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Castillo, J.L. 1987. Modified Fracture Pressure Decline Analysis Includ-
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.


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Hagoort, J. 1981. Waterflood-induced hydraulic fracturing. PhD thesis, 
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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Talley, G.R., Swindell, T.M., Waters, G.A., and Nolte, K.G. 1999. Field 
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)


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/H9004tt t t
Dp p=−()/.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-l)
The elapsed total time from the start of fracture initiation (not 
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-


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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 
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


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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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R should 
also yield a straight line with intercept equal to pi and slope of mR. 
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,


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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. 
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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The flow rate in Eq. A-9 is assumed to be in barrels per minute 
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


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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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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 
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)


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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
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)


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21 /H9266
ssDCacppi () ( )00−⎛
⎝⎜⎞
⎠⎟
×+⎛
⎝⎜⎞
⎠⎟/H9262
/H9278ckt tte112
/H9004,   . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-11)
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)


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(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
/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)


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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 
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
=⎛
⎝⎜⎞
⎠⎟⌠


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tp taa w f a2∂∂/H9004 / vs. ta where adjusted pseudopressure is defined as
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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terms of adjusted pseudovariables.
Robert D. Barree  is president and principal investigator of Barree 
& 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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Copyright 2007, Society of Petroleum Engineers 
 This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & Gas Technology Symposium held in Denver, Colorado, U.S.A., 16–18 April 2007.  This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as 
presented, have not been reviewed by the Society of Petroleum Engineers and are subject to 
correction by the author(s). The material, as presented, does not necessarily reflect any


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correction by the author(s). The material, as presented, does not necessarily reflect any 
position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than  300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435. 
 
Abstract 
Since the introduction of the G-function derivative analysis, 
pre-frac diagnostic injection tests have become a valuable and 
commonly used technique. Unfo rtunately, the technique is


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pre-frac diagnostic injection tests have become a valuable and 
commonly used technique. Unfo rtunately, 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( ∆p
wf)-log( ∆t) plots and 
their appropriate diagnostic de rivatives. Actual field test 
examples are given for the most common diagnostic curve 
signatures. 
  
Introduction 
Pre-frac diagnostic injection te st analysis provides critical 
input data for fracture design models, and reservoir 
characterization data used to pr edict post-fracture production. 
An accurate post-stimulation prod uction forecast is necessary  
for economic optimization of th e fracture treatment design.


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An accurate post-stimulation prod uction forecast is necessary  
for economic optimization of th e 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 
analysis techniques. In general, a single unique closure event 
will satisfy all diagnostic pl ots or methods. All available 
analysis methods should be used in concert to arrive at a consistent interpretation of fracture closure.  
Relationship of the pre-closur e analysis to after-closure 
analysis results must also be consistent. To correctly perform 
the after-closure analysis the tr ansient 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


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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 generally assumed model for leakoff used in most fracture simulators. Four examples are presented to show the application of 
multiple 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 reservoir 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 fractur e height recession during 
closure, but which can also indicate variable storage in a 
transverse fracture system.


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commonly identified as fractur e height recession during 
closure, but which can also indicate variable storage in a 
transverse fracture system. 
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.
1-4 When appropriate, the after-
closure analysis is presented for each case, as is an empirical 
correlation for permeability from the identified G-function 
closure time.5 A critical part of the analysis is the realization 
that there is a common event indi cating closure that should be 
consistently 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 
identification of after-closure flow regimes, an unambiguous


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The goal of this paper is to provide a method for consistent 
identification of after-closure flow regimes, an unambiguous 
fracture 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 difference curves based on the observed 
bottomhole pressure during falloff minus the “known” 
reservoir pressure.
5,8 While these methods are technically 
correct they can lead to confusin g 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 
present a practical approach to analysis of real, and frequently marginal-quality, pre-fracture field test data. The techniques 
applied are based on some transient test theory. Some of the


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applied are based on some transient test theory. Some of the 
results presented 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 consistent analysis that helps to avoid 
misinterpretation of falloff data, and give the most useful 
information available from diagnostic injection tests. 
Step-rate injection tests and th eir analysis are not included 
in the scope of this paper. Determination of the pressure-
dependent leakoff coefficient is also not described here, as it 
 
SPE 107877 
Holistic Fracture Diagnostics 
R.D. Barree, SPE, and V.L. Barree, Barree & A ssocs. LLC, and D.P. Craig, SPE, Halliburton Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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2  SPE 107877 
has been previously reported.3,4 Only the analysis of pressure 
decline following shut-in of a fracture-rate injection test is 
considered. 
 
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.6  
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 follow ed by a bi-linear flow period 
with the linear flow transient persisting in the fracture while 
reservoir linear flow occurs simu ltaneously.  After the fracture 
transient dissipates the reservoir linear flow period can 
continue for some time, depending on the permeability of the reservoir and the volume of fluid stored in the fracture and


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continue for some time, depending on the permeability of the reservoir and the volume of fluid stored in the fracture and 
subsequently leaked off duri ng closure. After closure the 
pressure transient established around the fracture propagates 
into the reservoir and transitions into elliptical, then 
pseudoradial 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 has pointed out that in some cases 
of moderate reservoir permeability the linear flow regime may not occur, even during fracture extension and early leakoff.
7 
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


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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 closure, leakoff mechan isms, and after-closure flow 
regimes. On each plot the curves  are labeled as the primary (y 
vs. x), the first derivative ( ∂y/∂x), and the semilog derivative 
(∂y/∂(lnx) or x ∂y/∂x). For convenience th e 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


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curves are shown on the same y-axis. 
For pre-closure analysis, and consistent identification of 
fracture closure, three techniques are illustrated for each example: G-function, Square-root of shut-in time, and log-log plot of pressure change with shut-in time. All these analyses 
begin at shut-in. The instantaneous shut-in pressure (ISIP) is taken as the incipient fracture ex tension pressure for all cases. 
When there is significant wellbore afterflow (fluid expansion 
or continued low-rate injection), 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 following sections detail the 
data and analysis for the four  major leakoff type examples.  
Normal Leakoff Behavior


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data and analysis for the four  major leakoff type examples.  
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 fissure overprint in which the flow capacity of the secondary fracture system does not change with pore pressure 
or net stress. After shut-in the fracture is assumed to stop propagating and the fracture su rface 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 derivative).
4 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


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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 ∂pw/∂G) 
from the straight line through the origin. During normal 
leakoff, with constant fractu re surface area and constant 
permeability, the first derivative ( ∂pw/∂G) should also be 
constant.2 The primary pw vs. G curve should follow a straight 
line.1 The example in Figure 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]. 
 
0 5 10 15 20 25
G(Time)750080008500900095001000010500
0200400600800100012001400160018002000
1
P vs. G
GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives
 
Figure 1: Normal leakoff G-function plot 
 
Normal Leakoff Sqrt(t) Analysis 
The sqrt( t) plot has frequently been misinterpreted when


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Derivatives
 
Figure 1: Normal leakoff G-function plot 
 
Normal Leakoff Sqrt(t) Analysis 
The sqrt( t) plot has frequently been misinterpreted when 
picking fracture closure, even  for the simplest cases. The 
primary pw vs. sqrt( t) curve should form a straight line during 
fracture closure, as with th e 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 Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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SPE 107877  3 
amplitude 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 Figure 2, shows that the slope of the 
pressure curve starts low, then increases and reaches a 
maximum rate of decline at the inflection point, then decreases 
again after closure. The first derivative curve in Figure 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


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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 semilog derivative of the P vs. sqrt( t) 
curve. A single closure point mu st satisfy the requirement on 
both the G-function and sqrt( t) plots. 
 
1/24/200704:00 08:00 12:00
1/24/200716:00
Time750080008500900095001000010500
01002003004005006007008009001000
1Pressure
DerivativesP vs. √t
√tdP/d√tv s .  √t
dP/d√tv s .  √tFracture Closure
 
Figure 2: Normal leakoff sqrt(t) plot 
 
Normal Leakoff Log-Log Pressure Derivative 
The log-log plot of pressure change from ISIP versus shut-
in time for the normal leakoff example is shown in Figure 3. 
The heavy curve is the pressu re 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 immediately


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function and sqrt( t) plot. It is common for the pressure 
difference and derivative curves to be parallel immediately 
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 sugges ting possible linear flow coupled 
with changing fracture/wellbor e storage (See Appendix B). 
The separation of the two parallel lines always marks fracture 
closure and is the final confirmation of a consistent closure 
identification. 
 2 3 456789 2 3 456789 2 3 456789 2 3 4567890.1 1 10 100 1000
Time (0 = 8.15)2345678923456789234
101001000
 (m = -1)  (m = 0.632)  BH ISIP = 9998 psi 1Delta-Pressure and Derivative∆P vs. ∆t
∆td∆P/d∆t vs. ∆t
Fracture ClosureRadial Flow
 
Figure 3: Normal leakoff log-log plot 
 
After closure the semilog derivative curve will show a


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∆td∆P/d∆t vs. ∆t
Fracture ClosureRadial Flow
 
Figure 3: Normal leakoff log-log plot 
 
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 pseudoradial 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 identification plot for the normal leakoff example is 
shown as Figure 4.
5  The heavy solid line is the observed 
bottomhole pressure during the falloff minus the initial


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shown as Figure 4.
5  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 sl ope 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 
derivative is not affected by the input reservoir pressure value. 
 
2 3 4 5 6 789 2 3 4 5 6 789 2 3 4 5 6 7890.001 0.01 0.1 1
Square Linear Flow (FL^2)234567892345678923456789
10100100010000
 (m = 1) Delta-Pressure and Derivative∆P vs. FL2
FL2d∆P/dFL2vs. FL2∆P=(pw-pr)
Start of Radial Flow


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10100100010000
 (m = 1) Delta-Pressure and Derivative∆P vs. FL2
FL2d∆P/dFL2vs. FL2∆P=(pw-pr)
Start of Radial Flow
 
Figure 4: Normal leakoff ACA log plot Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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4  SPE 107877 
 
If a pseudoradial flow regime is identified, then the 
Cartesian Radial Flow plot (Figure 5) can be used to 
determine reservoir far- field transmissibility, kh/µ. The 
viscosity used is the far-field mobile fluid viscosity and h is 
the estimated net pay height. For the analysis of the example 
data kh/µ = 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.  
 
 
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*f t
kh = 7.94014 md*ft
Permeability, k = 0.0968 md
Start of Pseudo Radial Time = 2.15 hours 1Pressure
 
Figure 5: Normal leakoff ACA radial flow plot 
 
 
Horner Analysis for Normal Leakoff Example 
If a pseudoradial flow period is identified, then a 
conventional Horner plot can also be used to determine


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If a pseudoradial flow period is identified, then a 
conventional Horner plot can also be used to determine 
reservoir transmissibility. In Figure 6 the Horner slope through 
the radial flow data is 14411 psi. Using an average pump rate 
of 18.4 bpm, kh/µ = 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.  
 
 
2 31
Horner Time72507500775080008250850087509000925095009750
 (m = 14411) 
  (Reservoir = 7476)  1Pressure
 
Figure 6: Normal leakoff Horner Plot 
  
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 available. The correlation 
is described in detail in the Appendix. Figure 7 shows the G-function correlation permeability estimate for the observed 
closure time and other input parameters. The permeability


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closure time and other input parameters. The permeability 
estimate of 0.097 md is consistent with the Horner and ACA results.   
0.0010.010.1110100
0 2 4 6 8 1 01 21 41 61 82 0
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
 
Figure 7: Normal leakoff permeability estimate 
 
Pressure Dependent Leakoff 
Pressure dependent leakoff (PDL) occurs when the fluid 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 flow 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 fissure or


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is a change in the transmissibility of the reservoir fissure or 
fracture system that dominates the fluid loss rate. PDL is only 
apparent when there is substantial stress dependent 
permeability in a composite dual-permeability reservoir. 
 
G-Function for Pressure-Dependent Leakoff 
Figure 9 shows the G-function behavior expected for PDL. 
The primary p
w vs. G curve is concave upward and curved 
while PDL persists. The semilog derivative exhibits the 
characteristic “hump” above the st raight 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.


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departure of the semilog derivative from the straight line 
through the origin.  
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SPE 107877  5 
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
 
Figure 9: PDL G-function plot 
 
 
Sqrt(t) Analysis for PDL 
Interpretation of the sqrt( t) plot in PDL cases has often led 
to incorrect closure picks. Figu re 10 shows an expanded view 
of the sqrt( t) plot for the example with the curves scaled for 
better visibility. Note that the semilog derivative is nearly 
identical in shape and informa tion 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 
derivative is used exclusively to  pick closure. In PDL cases 
the obvious derivative maximum, or most prominent inflection


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derivative 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.  
 
1/24/200700:20 00:40 01:00 01:20 01:40
1/24/200702:00
Time82508500875090009250950097501000010250
0100200300400500
1False ClosurePressure
DerivativesP vs. √t
√tdP/d√t vs. √t
dP/d√t vs. √tFracture Closure
 
Figure 10: PDL Sqrt plot 
 
 Log-Log Pressure Derivative for PDL Example 
Figure 11 shows the log-log plot for the PDL example. The 
normal matrix leakoff period, following the end of PDL,


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Figure 11 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.  
 
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 Derivative∆P vs. ∆t
∆td∆P/d∆tv s .  ∆tFracture closureLinear Flow
Radial Flow
 
Figure 11: PDL log-log plot 
 
After-Closure Analysis for PDL Example 
The ACA log-log plot (Figure 12) shows both the reservoir


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Radial Flow
 
Figure 11: PDL log-log plot 
 
After-Closure Analysis for PDL Example 
The ACA log-log plot (Figure 12) shows both the reservoir 
linear and radial flow periods in their expected locations.  
 
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 Derivative∆P vs. FL2
FL2d∆P/dFL2vs. FL2∆P=(pw-pr)
Start Linear Flow
End Linear Flow
Start Radial Flow
 
Figure 12: PDL ACA log plot 
 
Figures 13 and 14 show the ACA Cartesian plots for the 
linear and radial flow analyses. Both give consistent estimates 
of reservoir pore pressure. Th e pseudoradial flow analysis 
gives a transmissibility of 37.2 md-ft/cp and estimated 
permeability of 0.047 md.  
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6  SPE 107877 
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. 9
End of Pseudo Linear Time = 54.3 9 1 2PressureStart Linear Flow End Linear Flow
 
Figure 13: PDL ACA linear flow 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
 
Figure 14: PDL ACA radial flow plot 
 
Horner Analysis for PDL Example 
For an average pump rate of 6.7 bpm the Horner plot gives 
kh/µ=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


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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  Figure 15 shows the start of 
pseudoradial flow. If a pseudoradial flow period does not 
exist, extrapolation of an apparent straight-line on the Horner 
plot can give extremely inaccura te estimates of pressure and 
flow capacity.  1
Horner Time8000810082008300840085008600870088008900
 (m = 43920) 
  (Reservoir = 8064)  1Pressure
 
Figure 15: PDL Horner plot 
 
 
G-Function Permeability Estimate for PDL Example 
The G-function permeability correlation for the PDL 
example is shown in figure 16. 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


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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.  
 
0.0010.010.1110100
0 2 4 6 8 1 01 21 41 61 82 0
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
 
Figure 16: PDL permeability estimate 
  
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 
fluid to the tip of the fracture, causing continued extension of the fracture length. Much of the pressure decline is therefore


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fluid to the tip of the fracture, causing continued extension of 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.  
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SPE 107877  7 
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 positive slope and the 
slope continues to decrease w ith shut-in time, giving a 
concave-down curvature.3,4 Figure 17 shows a typical case of 
fracture tip extension with minimal leakoff. This is another 
case that is frequently misdiagnosed.  
 
0 5 10 15 20 25 30
G(Time)840086008800900092009400960098001000010200
0255075100125150
1
P vs. G
GdP/dG vs. G
dP/dG vs. GPressure
Derivatives
 
Figure 17: Tip extension G-function plot 
 
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. Figure 18 
shows the sqrt( t) plot for the same data. The first derivative


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often compounded by the use of the sqrt( t) plot. Figure 18 
shows the sqrt( t) plot for the same data. The first derivative 
shows a large maximum very shortly after shut-in. This is 
often mistaken for closure. The semilog derivative 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 ve ry high permeability reservoirs 
and should always be checked using the log-log pressure difference 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
 
Figure 18: Tip extension sqrt(t) plot 
 Log-Log Pressure Derivative Analysis with Tip Extension


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DerivativesP vs. √t xdP/dxvs. √tdP/dxvs. √t
 
Figure 18: Tip extension sqrt(t) 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. 
Figure 19 shows the pressure difference and pressure 
derivative (semilog) for the tip extension example.  
 
2 3 4 5 6789 2 3 4 5 6789 2 3 4 5 6789 21 10 100 1000
Time (0 = 33.7)23456789234567892345
101001000
 (m = 0.25)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative∆P vs. ∆t
∆td∆P/d∆t vs. ∆t
 
Figure 19: Tip extension log-log plot 
 
In Figure 19 the pressure derivative departs from the early 
unit-slope (storage) and establishes a ¼ slope during fracture 
tip extension. 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


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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 driving 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 different mechanisms that can generate a 
similar diagnostic derivative signature during fracture closure. 
Both are caused by an excess st ored volume of fluid in the 
fracture at shut-in relative to the 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 assu mes that leakoff occurs only


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Traditionally this signature has been called “fracture height 
recession”. The usual model assu mes 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 flui d 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 duri ng leakoff, the fracture 
compliance may also decrease, addi ng to the rate of pressure 
loss. 
However, the same signature is observed in many cases 
where fracture height growth out of zone is not observed by 
tracers, inclinometer, or micr o-seismic mapping. Some of


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where fracture height growth out of zone is not observed by 
tracers, inclinometer, or micr o-seismic mapping. Some of 
these cases show treating behavior similar to PDL cases, with Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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8  SPE 107877 
a tendency for rapid screenout and difficulty placing high 
proppant concentration slurries. These observations suggest 
that another mechanism may be responsible for the same 
diagnostic derivative signature. The alternate mechanism is 
called “transverse fracture storage”.  
In transverse fracture storag e a secondary fracture set is 
opened when the fluid pressure exceeds the critical fissure-
opening pressure, just as in PD L. As the secondary fractures 
dilate they create a storage volume for fluid which is taken 
from the primary hydraulic fract ure. While the fracture storage 
volume increases, leakoff can al so 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 modeling studies indicate that the storage mech anism can easily dominate even 
large PDL.


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derivatives show PDL or storage. Numerical modeling studies indicate that the storage mech anism can easily dominate even 
large PDL. 
At shut-in the secondary fractures will close before the 
primary fracture because they ar e 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 d ecreasing the normal rate of 
pressure decline and, in effect , supporting the observed shut-in 
pressure by re-injection of stor ed 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 
permeability dominated leakoff will occur after the end of storage.  
 
G-Function Analysis with Storage 
The characteristic G-function derivative signature is a


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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. Figure 20 shows an example of slight to moderate 
storage. In Figure 20 fracture closure, indicated by the same 
departure of the tangent line from the semilog derivative, 
occurs just after the end of the storage effect.  
  
5 10 15 20
G(Time)750080008500900095001000010500
0200400600800100012001400160018002000
1
P vs. G
GdP/dG vs. G
dP/dG vs. GFracture ClosurePressure
Derivatives
 
Figure 20: Storage G-function plot  
Sqrt(t) Analysis with Storage or Height Recession 
The sqrt( t) plot (Figure 21) shows a clear indication of 
closure based on both the first-derivative inflection point and 
the semilog derivative curve. Pi cking closure in the case of 
storage is not generally a problem. 
 
1/24/200702:00 04:00 06:00 08:00
1/24/200710:00


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the semilog derivative curve. Pi cking closure in the case of 
storage is not generally a problem. 
 
1/24/200702:00 04:00 06:00 08:00
1/24/200710:00
Time750080008500900095001000010500
0100200300400
1Pressure
DerivativesP vs. √t
xdP/dxvs. √t
dP/dxvs. √tFracture Closure
 
Figure 21: Storage sqrt(t) plot 
 
The storage model, whether cau sed by height recession or 
transverse fractures, requires that a larger volume of fluid 
must be leaked-off to reach fr acture closure than is expected 
for a single planar constant-heigh t 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 in 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


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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-tria ngle 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 rp can be as low as 0.5 or less.  
 
Log-Log Pressure Derivative with Storage 
Figure 22 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


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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 previously 
presented cases. All fracture stor age 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.  
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SPE 107877  9 
2 3 4 56789 2 3 4 56789 2 3 4 56789 2 3 4 560.1 1 10 100
Time (0 = 9.416667)234567892345678923
101001000
 (m = 0.5)  (m = -0.5)  BH ISIP = 10000 psi 1Delta-Pressure and Derivative∆P vs. ∆t
∆td∆P/d∆t vs. ∆t
 
Figure 22: Storage log-log plot 
 
 
Conclusions 
The use of pre-frac injection/falloff diagnostic tests has 
become commonplace. Many impor tant decisions regarding 
fracture treatment designs and expectations of post-frac production are based on the results of these tests. In too many cases individual diagnostic plots and analysis techniques are 
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 diagnostic plots.  
3. The conventional analysis of the sqrt( t) plot, using the


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2. A single, unique closure event can be identified on all diagnostic plots.  
3. The conventional analysis of the sqrt( t) plot, using the 
inflection 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. 
5. Flow regimes can be identified using the semilog pressure 
derivative on the log-log plot of ∆p
wf − ∆t 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 ∆pwf − ∆t plot up until 
fracture closure.  
7. Bilinear flow can be identified by parallel ¼ slope lines 
separated by 4x on the log-log ∆pwf − ∆t plot prior to 
fracture closure.


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7. Bilinear flow can be identified by parallel ¼ slope lines 
separated by 4x on the log-log ∆pwf − ∆t plot prior to 
fracture closure. 
8. After closure the pseudolinear reservoir flow period is 
identified by a -1/2 slope of the semilog derivative of the 
pressure difference on the log-log ∆pwf − ∆t 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-
closure 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. 
 
Acknowledgements 
The authors would like to thank Kumar Ramurthy, 
Halliburton, Mike Conway, Stim-Lab, and Stuart Cox,


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Acknowledgements 
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 pre-frac testing has allowed these 
diagnostic analysis procedures to be developed and tested. 
 
Nomenclature 
Af = 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 flow constant, m/Lt5/4, psi·hr3/4 
Cpl = pseudolinear flow constant, m/Lt3/2, psi·hr1/2  
Cpr = pseudoradial flow constant, m/Lt, psi ·hr  
Cfbc = before-closure fracture storage, L4t2/m, bbl/psi 
FL = linear flow time function, dimensionless  
FR = radial flow time function, dimensionless  
g = loss-volume function, dimensionless  
G = G-function, dimensionless  
h = height, L, ft  
k = permeability, L2, md


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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 flow graph, m/Lt2, psia  
mR = slope of data on pseudoradial flow graph, m/Lt2, psia  
p = pressure, m/Lt2, psia  
pwf = fracture pressure measured at wellbore, m/Lt2, psia  
q = flow 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 
 
 
Greek 
β = constant, dimensionless 
∆ = difference, dimensionless 
χ = constant, dimensionless 
µ = viscosity, m/Lt, cp 
φ = porosity, dimensionless 
 
Subscripts 
a = adjusted 
c = closure 
D = dimensionless 
e = end of injection 
f = filtrate 
p = pumping 
0 = end of injection  
w = wellbore 
z = process zone


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c = closure 
D = dimensionless 
e = end of injection 
f = filtrate 
p = pumping 
0 = end of injection  
w = wellbore 
z = process zone 
 
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10  SPE 107877 
References 
 
1. Nolte, K. G.: “Determination of Fracture Parameters from 
Fracturing Pressure Decline, pa per SPE 3841, presented at the 
Annual Technical Conference a nd Exhibition, Las Vegas, NV, 
Sept. 23-26, 1979. 
2. Castillo, J. L.: “Modified Fracture Pressure Decline Analysis 
Including Pressure-Dependent Leakoff,” paper SPE 16417, 
presented at the SPE/DOE Low Permeability Reservoirs Joint 
Symposium, Denver, CO, May 18-19, 1987. 
3. Barree, R. D., and Mukherjee, H. : “Determination of Pressure 
Dependent Leakoff and Its Effect  on Fracture Geometry,” paper 
SPE 36424, presented at the 71st Technical Conference and 
Exhibition, Denver, CO, Oct. 6-9, 1996. 
4. Barree, R.D.: "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 Sympos ium, Denver, Apr. 5-8, 1998.


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presented at the SPE Rocky Mountain Regional/Low-
Permeability Reservoirs Sympos ium, Denver, Apr. 5-8, 1998. 
5. Talley, G. R., Swindell, T. M., Waters, G. A. and Nolte, K. G.: “Field Application of After- Closure Analysis of Fracture 
Calibration Tests,” paper SPE 5 2220, presented at the 1999 SPE 
Mid-Continent Operations Symposium, Oklahoma City, OK, 
March 28–31, 1999. 
6. Cinco-Ley, H., and Samaniego- V., F.: “Transient Pressure 
Analysis for Fractured Wells,” JPT (September 1981) 1749. 
7. Settari, A.: “Coupled Fracture  and Reservoir Modeling,” 
presented at the Workshop on Three Dimensional and Advanced 
Hydraulic Fracture Modeling, held  in conjunction with the Fourth 
North American Rock Mechanics Symposium, July 29, 2000, Seattle, WA. 
8. Craig, D. P. and Bl asingame, T. A.: “Application of a New 
Fracture-Injection/Falloff Mode l Accounting for Propagating, 
Dilated, and Closing Hydrau lic Fractures,” paper SPE 1005778


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Fracture-Injection/Falloff Mode l Accounting for Propagating, 
Dilated, and Closing Hydrau lic Fractures,” paper SPE 1005778 
presented at the SPE Gas Technology Symposium, Calgary, 
Alberta, Canada, May, 15-17, 2006. 
9.  Hagoort, J.: "Waterflood-Induced Hydraulic Facturing," PhD 
Thesis, Delft Technical University, 1981. 
10.  Koning, E.J.L. and Niko, H.: "Fractured Water-Injection Wells: 
A Pressure Falloff Test for Dete rmining Fracture Dimensions," 
paper SPE 14458 presented at the 1985 Annual Technical Conference and Exhibition of the Society of Petroleum 
Engineers, Las Vegas, NV,  September, 22-25, 1985. 
11.  Cinco-Ley, H., Kuchuk, F., Ay oub, J., Samaniego-V, F., and 
Ayestaran, L.: "Analysis of Pr essure Tests Through the Use of 
Instantaneous Source Response Concepts," paper SPE 15476 presented at the 61
st Annual Technical Conference and Exhibition 
of the Society of Petroleum Engineers, New Orleans, LA, 
October, 5-8, 1986.


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st Annual Technical Conference and Exhibition 
of the Society of Petroleum Engineers, New Orleans, LA, 
October, 5-8, 1986. 
  
Appendix A - Definition of diagnostic functions 
 
The G-Function 
The G-function is a representati on 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 fluid efficiency in 
low-permeability formations. Under that assumption the surface area of the fracture is a ssumed to vary linearly with 
time during fracture propagation. The dimensionless pumping 
time used in the G-function is defined as: 
( )/Dp ptt t t∆= − . ..........................................................(A-l) The elapsed total time from the start of fracture initiation 
(not start of pumping) is t and the total pumping time (elapsed


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(not start of pumping) is t and the total pumping time (elapsed 
time from fracture initiation to shut-in) in consistent time units 
is tP. For the assumption of low leakoff the dimensionless time 
(∆tD) is used to compute an intermediate function: 
() ()[ ]5.1 5.1134
D D D t t tg ∆− ∆+ = ∆ . ............................ (A-2) 
The G-function used in the diagnostic plots is derived from 
the intermediate function as follows: 
() () 04
DD Gt gt gπ∆ =⎡∆ − ⎤⎣ ⎦, ....................................... (A-3) 
where g0 is the dimensionless loss-volume function at shut-in 
(t = tp or ∆tD = 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 Identification 
After-closure pressure decline analysis requires the 
identification of fully-developed reservoir pseudolinear and


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After-closure pressure decline analysis requires the 
identification of fully-developed reservoir pseudolinear and 
pseudoradial transient flow regimes. The flow regimes can be identified by characteristic slopes on a log-log plot of 
observed falloff pressure minus reservoir pressure, ( p
w(t) − pi), 
versus the square of the linear-flow time function (F L2) and the 
semilog derivative, (X*dY/dX), of the pressure difference 
curve.5 It is important to note that the guess of reservoir 
pressure, pi, used in construction of the flow 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: 
()cc
c L ttforttttF ≥ =−1sin2,π. ......................... (A-4)  
The linear-flow function al so requires an accurate 
determination of the time requ ired after shut-in to reach


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The linear-flow function al so requires an accurate 
determination of the time requ ired after shut-in to reach 
fracture closure, tc. In the pseudolinear flow period the slope 
of the derivative curve on the log-log plot should be ½. For the 
correct estimate of reservoir 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, pw(t), versus F L should 
yield a straight line with intercept equal to the reservoir pore 
pressure, pi, and with a slope of mL.  
() (),wi L L cptp m F t t−= . ............................................ (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


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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 versus F
R should also yield a straight line with 
intercept equal to pi and slope of mR.  
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SPE 107877  11 
() (),wi R R cptp m F t t−= ............................................. (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 reservoir pressure, and mR is the Cartesian slope of the 
“correct” straight line. The radial-flow function (F R) is given 
by5 
() 6.116, 1ln41,2≅ =⎟⎟
⎠⎞
⎜⎜
⎝⎛
−+ =πχχ
cc
c RtttttF . ......... (A-7)  
In the properly identified ps eudoradial flow period the 
reservoir far-field transmissibility can be determined from the 
slope, fracture closure time, and volume injected during the 
test. 
 
251,000t
RcQ kh
mt µ= . ...................................................... (A-8)  
In Equation A-8 the permeability, k, is in md, net pay 
thickness, h, is in feet, far-field mobile fluid viscosity,  µ, in 
cp, tc in minutes, and the volume injected during the test, Qt, is 
in bbls, and the slope, mR, is in psi.  
 
Horner analysis


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cp, tc in minutes, and the volume injected during the test, Qt, is 
in bbls, and the slope, mR, is in psi.  
 
Horner analysis 
The conventional Horner  analysis uses a Cartesian plot of 
observed pressure versus Horner time, ( tp + ∆t)/∆t, with all 
times in consistent units. The fracture propagation time is tp 
and the elapsed shut-in time is ∆t. As shut-in time approaches 
infinity 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 extrapolation, mH, can be used to 
estimate reservoir transmissibility: 
 
()
Hmq kh 14406.162=µ.....................................................(A-9) 
The flow rate in Equation A-9 is assumed to be in barrels


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()
Hmq kh 14406.162=µ.....................................................(A-9) 
The flow rate in Equation A-9 is assumed to be in barrels 
per minute and is the average rate for the time the fracture was extending. In both Equations A-8 and A-9, the viscosity is the far-field mobile fluid viscosity. The propagation of the 
transient in pseudoradial flow oc curs 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 da ta 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 numerous numerical 
simulations of fracture closure. The correlation is based on the


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permeability has been derived from numerous numerical 
simulations of fracture closure. The correlation is based on the 
observed G-function time at fracture closure: 1.960.0086 0.01
0.038f z
cp
tp
k
GE rcµ
φ=
⎛⎞
⎜⎟⎝⎠. ...................................... (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 incorp orate the effects of reservoir 
temperature and relative permeability in the invaded zone. 
 Note that the correlation gives permeability and not kh.


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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 versus 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 above closure pressure, pc, or 
pz = pISIP − pc. The net extension pressure and Young’s 
Modulus provide a relationship between facture volume 
(width) and surface area during pumping. 
The other parameters in Equation A-10 are defined, except 
the storage ratio, rp. 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 planar  fracture. The storage ratio, rp, 
is the ratio of the area under the G-function semilog derivative 
up until closure divided by the area of the triangle defined by


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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 nor mal matrix leakoff and PDL, rp 
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 (G c) is always delayed by 
the excess fluid volume in th e fracture for either height 
recession or transverse storage.  Likewise the apparent time to 
closure used in the after-closur e analysis will be delayed and 
will cause errors in the reservoir transmissibility estimate. The 
closure time should be corrected by multiplying by rp for all 
permeability and transmissibility estimates.  
 
 
Appendix B - Analytical Solutions and the Log-Log 
Diagnostic Graph 
 
Linear and Bilinear Flow Before-Closure 
Hagoort9 developed a before-closure analytical solution


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Diagnostic Graph 
 
Linear and Bilinear Flow Before-Closure 
Hagoort9 developed a before-closure analytical solution 
accounting for both storage and lin ear formation flow. Koning 
and Niko10 write the solution as 
() ( )22
12DtDf b c DDpt C ee r f c t wDβππ β∆ ⎛⎞=∆− − ⎜⎟ ∆⎝⎠, ................ (B-1) 
where 
( ) () ( )()141.2we we
Dptp t t khptwDqBµ− +∆∆=, ..................................... (B-2) 
20.0002637
D
tfktt
cLφµ∆∆=, ............................................................ (B-3) Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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12  SPE 107877 
2
fbcDCβπ=,  ..................................................................... (B-4) 
and 
225.615 0.8936
2fbc fbc
fbcD
tf tfCC
C
ch L ch Lπφ φ==. ............................................ (B-5) 
Here, Cfbc is the before-closure fracture storage constant, 
which is defined as8 
2
5.615f
fbc
fA
CS=, ................................................................. (B-6) 
where Af  is the fracture area (one wing) and  Sf is the fracture 
stiffness. 
Hagoort's solution predicts that the before-closure falloff is 
a combination of fracture storag e and linear flow. As fracture 
storage becomes small, CfbcD→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 ∆t >> te and long closure times,  
() ( )we weptp t t t−+ ∆ ∝ , ...................................................... (B-7) 
where t = te + ∆t and pw(te) is the instantaneous shut-in


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() ( )we weptp t t t−+ ∆ ∝ , ...................................................... (B-7) 
where t = te + ∆t and pw(te) is the instantaneous shut-in 
pressure. Define a fracture-pressure difference as 
() ( )w f we wep pt pt t ∆= − + ∆ , ................................................... (B-8) 
and a log-log graph of ∆pwf vs. t will exhibit a ½ slope during 
linear flow before closure. An  analytical before-closure 
bilinear flow solution accounti ng for fracture tip extension 
during shut-in does not exist, but field data suggest that during 
before-closure bilinear flow 
14()wfp t ∆∝ ∆ , .................................................................... (B-9) 
and a log-log graph of ∆pwf vs. t will exhibit a ¼ slope during 
bilinear flow before closure. 
 
Pseudoradial, Pseudolinear, and Bilinear Flow After-
Closure 
Craig & Blasingame8 developed an analytical solution for a 
fracture-injection/falloff sequ ence with a propagating and


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Closure 
Craig & Blasingame8 developed an analytical solution for a 
fracture-injection/falloff sequ ence 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 flow is written as  
() 0141.2(24) 1(0) ( ) ()2ti we i
eQp C p p pt t p wsD ackh t tµ+− +∆ − =+∆. ..... (B-10) 
For after-closure pseudolinear flow, 
0141.2(24) 0.0002637 1()(0) ( ) 2t
we ii fQ
pt t ppC p p hL wsD acπ ⎛⎞+∆ − = ⎜⎟+−⎝⎠
121
teckt tµ
φ⎛⎞×⎜⎟+∆ ⎝⎠, ........................................... (B-11) 
and for after-closure bilinear flow, 
1434
0141.2(24)(0.6125)(0.0002637)()()t
we i
i ffQ
pt t ppp C kw acµ⎛⎞+∆ − = ⎜⎟+−⎝⎠  14 3411
teck t tµ⎛⎞ ⎛ ⎞×⎜⎟ ⎜ ⎟+∆ ⎝⎠ ⎝ ⎠. ..................................... (B-12) 
For convenience, define a constant pseudoradial flow term as 
() 0141.2(24)(0) ( )
2ti prQp C p p C wsD ackhµ+− =. ........................... (B-13)


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() 0141.2(24)(0) ( )
2ti prQp C p p C wsD ackhµ+− =. ........................... (B-13) 
For pseudolinear flow, a constant term is defined as 
12
0141.2(24) 0.0002637 1
(0) ( ) 2t
plt i fQ
Cck pC p p hL wsD acµ π
φ⎛⎞ ⎛⎞= ⎜⎟ ⎜⎟+− ⎝⎠ ⎝⎠. (B-14) 
Similarly, for bilinear flow a constant term is written as 
 14 1434
01 141.2(24)(0.6125)(0.0002637)
()t
blt i ffQ
Cck pp C kw acµµ⎛⎞ ⎛⎞= ⎜⎟ ⎜⎟+− ⎝⎠ ⎝⎠. .. (B-15) 
With the new definition, and defining a reservoir-pressure 
difference, ∆pw = pw(t) − pi, the pseudoradial-flow impulse 
solution can be written as 
1wp rpC t−∆= .  ................................................................... (B-16) 
Similarly, the linear-flow impulse solution can be written as 
1/2wp lpC t−∆= , .................................................................. (B-17) 
and the bilinear-flow impulse solution can be written as 
3/4wb lpC t−∆= . ................................................................... (B-18)


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3/4wb lpC t−∆= . ................................................................... (B-18) 
The derivatives with respect to t are written for after-closure 
pseudoradial flow as,  
2
()wprpCtt− ∂∆−=∂, ................................................................ (B-19) 
for after-closure pseudolinear flow as, 
3/2
() 2pl wC ptt− ∂∆−=∂, ............................................................. (B-20) 
and for after-closure bilinear flow as, 
7/4 3
() 4wb lpCtt− ∂∆−=∂. ............................................................ (B-21) 
The semilog derivatives and the impulse derivatives11 are 
calculated by multiplying the after-closure derivatives by t and 
t2, respectively. Table B-1 show s the derivative terms for each 
after-closure flow regime. 
 
Log-Log Diagnostic Graph 
Before-closure linear or bilinear flow are identified by the 
relationship between the fracture-pressure difference, 
∆pwf, and total time, t, but the after-closure flow regimes are


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relationship between the fracture-pressure difference, 
∆pwf, and total time, t, but the after-closure flow regimes are 
identified by the relationship be tween the reservoir-pressure 
difference, ∆pw, and t, which requires knowing the initial 
reservoir pressure, pi. However, by noting that 
wf wp p
tt∂∆ ∂∆=−∂ ∂, ................................................................ (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 
∆p
wf vs. t, wfpt∂∆∂ vs. t, wf tp t∂∆ ∂ vs. t, and Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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SPE 107877  13 
2wf tp t∂∆ ∂ vs. t .  
In gas reservoirs, log-log diagnostic graphs can be prepared by 
plotting ∆pawf vs. ta, awf apt∂∆ ∂ vs. ta, aa w fatp t∂∆ ∂ vs. ta, 
and 2aa w f atp t∂∆ ∂ vs. ta where adjusted pseudopressure is 
defined as 
0pzp d ppap ziµ
µ⌠
⎮
⎮⌡⎛⎞=⎜⎟⎝⎠, ........................................................... (B-23) 
and adjusted pseudotime is defined as ()
0tdttcat ictµµ⌠
⎮
⎮
⌡=. .............................................................. (B-24) 
Since adjusted pseudopressure is a function of initial 
reservoir 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 the estimate using diagnostic and interpretive 
graphs plotted in terms of adjusted pseudovariables. 
 
 
  
 
 
Table B-1. After-closure impulse solutions and derivatives. 
Flow Regime Impulse Solution Derivative Semilog 
Derivative Impulse


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Table B-1. After-closure impulse solutions and derivatives. 
Flow Regime Impulse Solution Derivative Semilog 
Derivative Impulse 
Derivative 
Bilinear 3/4wb lpC t−∆=  7/4 3
4wb lpCtt− ∂∆−=∂ 3/4 3
4wb lpCttt− ∂∆−=∂ 21 / 4 3
4wb lpCttt∂∆−=∂ 
Pseudolinear 1/2wp lpC t−∆=  3/2
2pl wC ptt− ∂∆−=∂ 1/2
2pl wC pttt− ∂∆−=∂ 21 / 2
2pl wC pttt∂∆−=∂ 
Pseudoradial 1wp rpC t−∆=  2 wprpCtt− ∂∆−=∂ 1 wprptC tt− ∂∆−=∂ 2 wprptCt∂∆−=∂ 
 
 
Table B-2. Log-log graph characteristic slopes. 
Before Closure After Closure Log-Log Graph 
Bilinear Linear Bilinear Pseudolinear Pseudoradial 
 vs. 
 vs. wf
awf apt
pt∆
∆ 1/4 1/2 ⎯ ⎯ ⎯ 
 vs. 
 vs. wf
awf a apt t
pt t∂∆ ∂
∂∆ ∂ −3/4 −1/2 −7/4 −3/2 −2 
 vs. 
 vs. wf
aa w fa atp t t
tp t t∂∆ ∂
∂∆ ∂ 1/4 1/2 −3/4 −1/2 −1 
2
2 vs. 
 vs. wf
aa w fa atp t t
tp t t∂∆ ∂
∂∆ ∂ 5/4 3/2 1/4 1/2 0 
  Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/07RMTC/All-07RMTC/SPE-107877-MS/2786245/spe-107877-ms.pdf/1 by University of Houston, Mohamed Gabry on 12 July 2022


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drive/MyDrive/Fracture closure Reviews/Fracture closure Reviews/3-s2.0-B9780323901857000169-main.pdf


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CHAPTER 8
Fracture diagnostic testing
Ali Rezaei1, Mohamed Y. Soliman2and Birol Dindoruk2
1Midstream Production Systems, Houston, TX, United States
2Cullen College of Engineering, Petroleum Engineering, University of Houston, Houston, TX, United States
8.1 Overview
This chapter summarizes several conventional and recent methods to analyze hydraulic
fracture testing in vertical and horizontal wells. These tests are conducted for two pri-
mary purposes: (1) to understand the reservoir properties such as minimum horizontal
stress and permeability; (2) and to optimize the hydraulic fracturing parameters.
Fracture diagnostic techniques are critical preexecution steps toward optimum fracture
design. Using these techniques, one can estimate key parameters such as the target
zone stress regime, fracture confinement height, number of stages, treatment size,
proppant type and amount, and optimum well direction.
8.2 Testing types


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proppant type and amount, and optimum well direction.
8.2 Testing types
In general, three different fracture diagnostics groups exist, namely direct far-field
techniques, direct near-wellbore techniques, and indirect fracture diagnostics ( Cipolla
& Wright 2000 ). The direct far-field techniques are suited to give the big picture
about fracture growth and are conducted in a separate well. The two commonly used
techniques for this type of fracture diagnostics are tiltmeter fracture mapping and
microseismic fracture mapping. On the contrary, the second group of fracture diagnos-
tics techniques is implemented in the same well and gives information about near-
wellbore fracture parameters such as height and width and proppant placement.
Finally, the most commonly used fracture diagnostics group is indirect fracture diag-
nostics. The techniques used in this group can provide estimates about fracture con-
ductivity, dimension, and stress. This group includes fracture modeling, pressure


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ductivity, dimension, and stress. This group includes fracture modeling, pressure
transient testing, and production data analysis ( Cipolla & Wright 2000 ). The pressure
transient testing (sometimes accompanied by rate transient testing) includes step
rate test, pump in-shut in, and pump in/flow back ( Soliman, Craig, Bartko, Rahim, &
Adams, 2005 ). These tests are most widely used because of the abundance in the
recorded data. Pump-in shut-in tests may be referred to as MiniFrac, or diagnostic
fracture injection test (DFIT). There are, however, some differences between the two
tests. Mini-Frac tests were conducted using the fluid to be used in the main fracturing
217Unconventional Shale Gas Development
DOI: https://doi.org/10.1016/B978-0-323-90185-7.00016-9r2022 Elsevier Inc.
All rights reserved.


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treatment, except no proppant was added. The goal of the test was to find the leak-off
coefficient and efficiency of the treatment. The treatment design was consequently
adjusted if necessary. DFIT, on the other ha nd, is conducted to determine reservoir and
fracture properties. Other approaches such as data-based analysis and signal processing
techniques such as wavelet tra nsform have been developed to p erform fracture diagnostics
analysis in recent years. Moreover, other te chniques such as step-rates techniques have
been developed. Those step rate techniques will be briefly discussed. However, the bulk
of the chapter will be devoted to pump-in shut-in test technology.
8.3 Step-rate test
Step-rate test (SRT) is commonly used for estimating the pressure at which the hydrau-
lic fracture starts to propagate (i.e., fracture propagation pressure or FPP). FPP is the
pressure at which a fracture will initiate in an unfractured well, or the existing fracture


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pressure at which a fracture will initiate in an unfractured well, or the existing fracture
in a fractured well starts to propagate. The test is primarily used for identifying the
maximum safe injection pressure without fracturing the reservoir rock ( Singh, Agarwal,
& Krase, 1987 ). Although the SRT has been practiced in the industry for many years,
the first publication on the subject seems to be by Felsenthal (1974) . There are two
types of SRT, step-up and step-down tests. In the following, a review of the two types
is discussed. Also, it has been reported by Savitski and Dudley (2011) that the step rate
test is not efficient in low permeability reservoirs for identifying the closure pressure.
8.3.1 Step-up test
The step-up test is performed on a well with preexisting hydraulic fracture and is used
to estimate the fracture opening pressure. Injection starts at a low rate that is not suffi-
cient to reopen the existing fracture. The injection is increased in a stepwise fashion


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cient to reopen the existing fracture. The injection is increased in a stepwise fashion
until the fracture opens. At each stage, the injection continues until the pressure stabi-
lizes. It is expected that a smaller change in pressure occurs after the fracture reopens,
resulting from the higher injectivity. Plotting the stabilized pressure versus rate yields the
point of change in slope, indicating the reopening pressure as illustrated in Fig. 8.1 .T h e
figure shows that as the fluid loss decreases (lower formation permeability), the difference
between the before closure (BC) and after closure (AC) slopes get smaller, resulting in a
harder fracture opening pressure identification. More discussions and field examples on
the step-up test may be found in Tan, McGowen, Lee, and Soliman (1988) .
8.3.2 Step-down test
In contrary to the step up test, the step-down test is also performed by injection at
a high enough rate to propagate the fracture. As the injection rate is reduced, how-


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a high enough rate to propagate the fracture. As the injection rate is reduced, how-
ever, it stays above the fracture opening pressure. Step down test was proposed by218 Unconventional Shale Gas Development


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Cleary et al. (1993) . The goal of the test is to estimate the friction- and perforation-
dominated pressure during injection. It is hypothesized that the net pressure will be a
function of the rate as expressed below. The exponent B is a strong indication of the
status of the fracture, as given in Tables 8.1, 8.2, 8.3, 8.4 and 8.5 . Depending on the
results of the test, remedial action may be taken.
Δpnet5A3qB
logΔpnet5logA1Blogq
whereΔPnet,q, and Aand Bare net pressure, injection rate, and test constants,
respectively.
Hyden and Stegent (1996) presented field cases using the step-down test for find-
ing the near-wellbore restrictions.
Figure 8.1 A typical pressure-rate cross plot of the step-up test. Courtesy Advantek.
Table 8.1 Indication of potential problems associated with slope Bin the step-down test.
Slope, B Potential problem
0.25 single fracture exists
0.5/C01.0 Presence of a tortuous path
2.0 Insufficient perforation219 Fracture diagnostic testing


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8.4 Pump in/shut-in test
The most commonly used pump-in/shut-in test is the fracture injection test. Fracture
injection fall-off test, also known as DFIT and minifrac test, has been a standardmethod to identify various formation properties such as closure pressure ( BminimumTable 8.2 A summary of the results for field example 1 and its comparison with G-function.
G-function EDP
Closure time (min) 27.47 27.95
Closure pressure (psi) 5822 5790Closure point (#) 93 101Sampling rate 1 second
Total number of data points B1:2310
3
Table 8.3 A summary of the results for field example 2 and its comparison with G-function.
G-function EDP
Closure time (s) 822 761
Closure pressure (psi) 26,756 28,756Closure point (#) 98 37Sampling rate 1 sTotal number of data points B4:6310
3
Table 8.4 A summary of the results for field example 3 and its comparison with G-function.
G-function EDP
Closure time (min) 13.6 13.5


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3
Table 8.4 A summary of the results for field example 3 and its comparison with G-function.
G-function EDP
Closure time (min) 13.6 13.5
Closure pressure (psi) 2656 2702Closure point (#) 101 93Sampling rate 1 sTotal number of data points B10310
3
Table 8.5 A summary of the results for field example 4 and its comparison with G-function.
G-function EDP
Our analysis Closure time (min) 45 46
Closure pressure (psi) 13,539 13,521Closure point (#) 2509 2600
Sampling rate 1 s
Total number of data points 370 310
3220 Unconventional Shale Gas Development


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horizontal stress), formation permeability, and the leak-off type in the last 30 years.
Closure pressure identification is critically important in this test because of two main
reasons. First, it is typically considered as the minimum horizontal stress and affects
the subsequent hydraulic fracturing design as well as analysis of real-time fracture prop-
agation. Second, closure pressure defines two separate classes of analysis, namely BC
analysis and AC analysis ( Soliman et al., 2005; Soliman, Miranda, & Wang, 2010 ).
DFIT involves injecting water without proppant at a sufficient rate and pressure to
create a fracture into a formation, followed by a shut-in period that may last for hours
or even days depending on reservoir permeability. After starting the shut-in period,
the pressure will begin to fall-off due to the fracturing fluid leak-off that causes the
fracture to close gradually, resulting from the far-field compressive stress. The closure


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fracture to close gradually, resulting from the far-field compressive stress. The closure
pressure may be determined by analyzing the decline of the recorded pressure during
the fall-off period.
Fig. 8.2 shows the recorded wellbore pressure during a typical minifrac test. Two
periods, BC (fracture dominated) and AC (reservoir dominated) are separated by a
vertical red dashed line.
8.4.1 Before closure analysis
The methods to analyze the recorded pressure fall-off data before the fracture closure
moment are referred to BC analysis. The main parameters that are estimated using
these techniques are closure pressure, leak-off coefficient, and instantaneous shut-in
pressure. The net pressure at the closure is typically considered as minimum horizontal
stress. The most commonly used method in this category is the DFIT. Various models
have been proposed to analyze DFIT data and determine the fracture closure pressure.


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have been proposed to analyze DFIT data and determine the fracture closure pressure.
Figure 8.2 A Schematic of the DFIT test. After Pokalai et al. (2016) .221 Fracture diagnostic testing


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Nolte (1979) introduced this technology. Further modifications of this technology
were introduced by further authors (e.g., Barree & Mukherjee 1996 ;Craig &
Blasingame 2006 ;Marongiu-Porcu, Ehlig-Economides, & Economides, 2011 ;
Mohamed, Azmy, Sayed, Marongiu-Porcu, & Economides, 2011 ;Soliman, 1986 ).
However, there is a significant disagreement in the calculated closure pressure from
these models in many cases. The primary source of error in the methods used for ana-
lyzing DFIT is their underlying assumptions. Therefore, analyzing DFIT data using a
different methodology independent of premises such as fracture geometry is essential.
8.4.1.1 G-function technique
The first and most widely used method for analyzing BC pressure fall-off was initially
presented in Nolte ’s pioneering work ( Nolte, 1979, 1986 ). The technique, also
known as G-function, expanded by Barree and Mukherjee (1996) , is a tool for deter-


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known as G-function, expanded by Barree and Mukherjee (1996) , is a tool for deter-
mining closure pressure, leak-off coefficient, and fluid efficiency. This technique ’s cen-
tral assumptions are that the leak-off is not pressure-dependent and fracture stiffness is
constant while the fracture is closing. Several modifications to the Nolte G-function
technique have been proposed since then by addressing some of the limitations in the
assumptions ( Castillo, 1987; Barree, Barree, & Craig, 2007; Barree & Mukherjee 1996;
Mukherjee, Larkin, & Kordziel, 1991; Barree et al., 2015 ). The primary purpose of
these modifications was to make the method more applicable to the new cases that
were encountered.
McClure, Blyton, Jung, and Sharma (2014) and McClure, Jung, Cramer, and
Sharma (2016) proposed a technique called the fracture compliance method. In this
technique, the central assumption is that the created fracture does not reach a state of


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technique, the central assumption is that the created fracture does not reach a state of
complete closure. Instead, the fracture retains a finite aperture after asperities come
into contact and result in an infinite fracture conductivity. Although this assumption
may be valid in some cases where the aperture of the created fracture in ultra-tight
formations stays at μdrange, it gives contradictory results to the widely used tangent
methods in cases where reservoir permeability is less than one μD. Also, for such short
fractures that are generated in DFIT, minimum to no asperities mismatch is more
probable, which contradicts the model ’s assumptions. McClure et al. (2019) further
extended their method and recommended steps to analyze DFIT data from low per-
meability formations ranging from tens of nDto tens of μD.Wang and Sharma
(2017) presented a variable compliance method based on averaging the closure point ’s


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(2017) presented a variable compliance method based on averaging the closure point ’s
G-time. Similar to the technique presented by McClure et al. (2014) andWang and
Sharma (2017) concluded that the G-function method underestimates the closure stress.
In another model, Liu and Ehlig-Economides (2015) and Liu, Ehlig-Economides, and
Sun (2016 ) extended the model by Marongiu-Porcu, Retnanto, Economides, and Ehlig-
Economides (2014) to account for wellbore storage, tip extension, and pressure dependant
leak-off, and multiple closures.222 Unconventional Shale Gas Development


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The definition of G-function is given below:
GΔtDðÞ54
πgΔtDðÞ2g0 ðÞ
gΔtDðÞ54
311ΔtD ðÞ1:52Δt1:5
D/C0/C1
;forα51
gΔtDðÞ511ΔtD ðÞ sin2111ΔtD ðÞ1:5/C0/C1
1Δt0:5
D;forα50:5
ΔtD5t2ti
ti
g054
3;forα51
g05π
2;forα50:5
In the above equations, αis a constant that governs the power-law propagation of
the fracture:
A
Ae5t
te/C18/C19α
α51 and 0 :5 in the above equations refer to low and high leak-off regimes,
respectively. A typical G-function plot is shown in Fig. 8.3 . In the plot, a straight tan-
gent line that passes through the origin is fitted on the Gdp
dGversus Gline. The closure
point is picked where the tangent line deviates from the plot. In order to read the clo-
sure pressure, one needs to connect the deviation point to the pressure line (red line
in the figure) using a vertical line (dashed line in the figure).
8.4.1.2 Log /C0log analysis
Another widely used technique for DFIT is known as the log /C0log method
(Marongiu-Porcu et al., 2011, 2014; Mayerhofer & Economides 1997; Mohamed


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Another widely used technique for DFIT is known as the log /C0log method
(Marongiu-Porcu et al., 2011, 2014; Mayerhofer & Economides 1997; Mohamed
et al., 2011; Valkó & Economides, 1995 ). In this method, the closure time is deter-
mined using the log /C0log diagnostic plot, which is used conventionally to analyze tran-
sient well tests and analyze fall-off data. A typical log /C0log analysis of the pressure fall-
off data is shown in Fig. 8.4 . In the figure, the dashed curve (pink) is the semilog
derivative with respect to shut-in time. The slope of the derivative plot before the clo-
sure is an identification of the flow regime before fracture closure (1/2 here indicated223 Fracture diagnostic testing


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Figure 8.3 A normal leakoff G-function plot. ( Barree et al., 2007 )After Barree, R. D., Barree, V. L., &
Craig, D. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium .
Society of Petroleum Engineers.
Figure 8.4 A typical log /C0log analysis of pressure fall-off data. After Barree, R. D., Barree, V. L., &
Craig, D. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium .
Society of Petroleum Engineers.224 Unconventional Shale Gas Development


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linear flow from fracture). In this technique, the separation of the two parallel lines is
an identification of the fracture closure.
8.4.1.3 Square root time method
In this technique, the fracture closure pressure is identified by the first peak in the
pressure plotted versus the square root of time. Fig. 8.5 shows a typical square root
time plot. The primary curve in this plot should form a straight line during the frac-
ture closure (similar to G-Function analysis). Unlike the G-Function analysis, where
the deviation from the straight line was indicating the fracture closure, in this analysis,
the peak of the first derivative of the pressure versus sqrt( t) indicated the fracture clo-
sure point (pink dashed line in the figure).
8.4.1.4 Modified technique to determine abnormalities
Unal et al. (2019c) proposed a signal processing approach for analyzing DFIT based on
the signal processing approach. In the proposed technique, the noise of the recorded


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the signal processing approach. In the proposed technique, the noise of the recorded
pressure using wavelet transform is analyzed. The advantage of this method is that it
does not depend on any assumptions of the currently available models and only relies
on the recorded pressure.
Figure 8.5 An example of a normal Square Root time plot. After Barree, R. D., Barree, V. L., & Craig,
D. (2007). Holistic fracture diagnostics. In Rocky mountain oil & gas technology symposium . Society
of Petroleum Engineers.225 Fracture diagnostic testing


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Other techniques in this area include the one by Mohamed et al. (2011) and
Marongiu-Porcu et al. (2011) , who suggested a global approach for a combined before
and AC analysis. In these approaches, the authors used piece-wise functions. Using
these approach, the BC analysis can be written as:
p-
wD512e2ted1tcD ðÞ s/C0/C1
3q-
D
SK00:5OS/C2/C3
For the AC analysis, an enlarged equivalent wellbore radius was used. Also, the
fluid flow in the reservoir was given as:
p-
wD5K0½rwdOS/C138
S1:5K1ðrwDﬃﬃﬃ
Sp
Þ
The above equation needs to be inverted numerically with the Stehfest (1970)
algorithm. The limitation of this technique is that it can only be used for high perme-
ability formations with short fracture.
8.4.1.5 Transient before closure analysis
Transient flow regimes can occur during fall off test. Cinco-Ley and Samaniego
(1981) graphically presented the major flow regimes around a fractured well. Other
techniques than the conventional methods discussed above have also been developed


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techniques than the conventional methods discussed above have also been developed
in recent years. In a similar way to the traditional G-function analysis, Siddiqui,
Soliman, House, and Ibragimov (2016) developed a new analytical solution that could
be used to determine the closure pressure. In contrast to the conventional G-function
method, which models the fluid leak-off into the formation based on Carter ’s equa-
tion ( Howard & Fast 1957; Hubbert & Willis 1972 ), partial differential equations to
describe the linear leak-off were used in the model. This method requires previous
knowledge of fluid mobility and fracture compliance.
8.4.2 After closure analysis
AC analysis is referred to those analyses that are performed on the pressure fall of data
after the closure point (i.e., when the fracture faces are closed). The conventional well
test analysis can be applied with a high degree of accuracy (Soliman & Dusterhuft,


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test analysis can be applied with a high degree of accuracy (Soliman & Dusterhuft,
2016). This is because that the created fracture during the pumping period of most of
the fracturing tests is small enough that ignoring it does not affect the results tremen-
dously. However, there is one fundamental difference between the developed solu-
tions of the traditional well testing approaches and AC analysis. The difference is that
the solutions of conventional well testing are derived from the constant volume, while
the solution of AC analysis are derived from impulse solution. In cases where the226 Unconventional Shale Gas Development


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pressure fall-off is recorded long enough to reach radial flow, one can obtain reservoir
pressure and permeability using the AC analysis.
Soliman et al. (2004, 2005) presented a technique for AC analysis to distinguish the
dominant flow regime AC. The method could be used to analyze the data in each
region and determine the reservoir pressure and permeability. A summary of the tech-
nique in the case of bilinear flow is given in the following (Soliman & Dusterhuft,
2016):
1.A log /C0log plot of logtinj
Δt/C0/C1
versus log/C0
t@pf0
@t/C1
needs to be plotted. Here tinj,Δt, and
pfoare injection time, time since shut-in, and bottom hole fall off pressure.
2.The data will follow a straight line that indicates the flow regime.
3.The intercept of the log tinj1Δt/C0/C1nversus ( pfo2piÞ, with n being the slope from
the previous step, on a cartesian plot indicates the reservoir pressure.
4.Create a logarithmic plot oftinj
Δt/C0/C1
versus ðpfo2piÞ.


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the previous step, on a cartesian plot indicates the reservoir pressure.
4.Create a logarithmic plot oftinj
Δt/C0/C1
versus ðpfo2piÞ.
5.Assuming the bilinear flow, the intercept will be a function of permeability and
reservoir conductivity.
br5264 :6V
hμðÞ0:75 1
φctk/C18/C190:251
Okfwf !
Also, the end of bilinear flow is a function of both fracture conductivity and
permeability as the following:
tDebf54:55
CfD22:5/C18/C1924
CfD5kfw
kLf
Combining these equations by assuming that the fracture conductivity is low,
the formation permeability can be calculated as:
k5264 :6V
h3μ
br1
2:637tef/C0/C10:25
The fluid properties used in these equations are reservoir fluid properties.
6.If the shut in time gets sufficiently large, the pseudo-radial flow will probably be
observed. For this regime to happen, the dimensionless time from the following
equation should be greater than 1:
t$3:7923103φμCtL2
f
kh
7.A similar approach can be followed for the linear flow.227 Fracture diagnostic testing


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8.5 Pump-in/flow-back test (micro-frac)
The conventional DFIT has several limitations. For example, one needs to wait so
long (days) for the closure to happen. Also, the test gives inaccurate results in the case
of the temperature difference between the test fluid and rocks or when the reservoirs
are naturally fractured. Pump-in flow-back test is designed to address some of the lim-
itations of the conventional pump-in shut-in approaches. During the test, usually, less
than four barrels of fluids are injected into the well (Soliman & Dusterhoft, 2016). A
follow back starts at the end of the test (rather than the shut-in). During the test,
the flow back period is analyzed to find the closure pressure. A typical recorded pres-
sure for the flow back test is shown in Fig. 8.6 . As can be seen, the pressure plot in
the flow back region is concave with two almost constant slopes before and after
points A and B. The slope of the curve above point A represents the stiffness of the


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points A and B. The slope of the curve above point A represents the stiffness of the
system, and the slope below point C represents the stiffness of the fracture alone
(Savitski & Dudley 2011 ). Note that the return rate is very low and should stay con-
stant during the flow back period, as shown in Fig. 8.6B . Different interpretations
have been proposed for picking up the closure pressure at points A, B, or C in this
test (Nolte,1990; Plahn et al., 1997; Soliman & Daneshy, 1991 ).
Savitski and Dudley (2011) showed that in low permeability formations, the con-
tinuous flow back at a constant rate, as shown in Fig. 8.6A , does not result in an
interpretable trend. Hence, an incremental flow back test with pressure stabilization
protocol was suggested. In the suggested approach, instead of a continuous fluid flow,
required by the conventional pump in flow back test, a small incremental volumetric
flow back followed by the pressure stabilization is used. The pressure stabilization


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flow back followed by the pressure stabilization is used. The pressure stabilization
allows for the reaching an equilibrium condition between the wellbore and fracture
pressures. The advantage of this approach is that it removes the uncertainty about the
Figure 8.6 Examples of recorded pressure during the pump-in flow-back test. (A) Schematic of the
pressure (B) a field case reported by Savitski and Dudley (2011) .228 Unconventional Shale Gas Development


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transient flow between the fracture and wellbore and results in a quasiequilibrium situ-
ation. An example of a field case from Savitski and Dudley (2011) using this approach
is shown in Fig. 8.7 .
8.6 New analysis technique
Soliman and Kabir (2012) andSoliman and Gamadi (2012) revisited applications of
some of the mentioned techniques to unconventional and tight reservoirs, respectively.
They concluded that, in general, these methods are not practical for these reservoirs.
To address these limitations, new approaches have emerged in recent years. These
techniques are mainly based on the recorded data from the tests and have limited
assumptions. They include signal processing approaches and data analysis approaches.
In the following, we review the signal processing approach in detail.
8.6.1 Wavelet analysis
Signal processing is a branch of electrical engineering that models and analyzes differ-
ent data types and their representations of physical events. Traditionally, signal proces-


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ent data types and their representations of physical events. Traditionally, signal proces-
sing involves using filters to separate the signal of interest or even remove the
unwanted frequency. The main objective of signal processing is to extract relevant
information from a signal by transforming it. Fourier and wavelet transforms
Figure 8.7 The quasiequilibrium pump in flow back, suggested by Savitski and Dudley (2011) .229 Fracture diagnostic testing


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are conventional techniques for signal analysis. Fourier transform is applied for
stationary signals, signals whose properties do not evolve in time (e.g., sinewaves).
Fourier transform is not recommended for nonstationary signals because any abrupt
change in time would spread out the signal over the whole frequency domain ( Rioul
& Vetterli, 1991 ). Also, the events cannot be located in time using the Fourier
transform.
Wavelet transform is of interest for the analysis of nonstationary signals. Wavelet is
a small wave that grows and decays primarily in a limited time period; on the contrary,
Fourier transform decomposes signal into sin waves, which keep on oscillating up and
down to infinity. The advantage of wavelet transform over Fourier transform in DFIT
is the one can identify the events and their frequencies in time using this technique
(Unal et al., 2019a, 2019b ). Wavelet is a function characterizing by the following
properties ( Percival & Walden, 2000 )


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(Unal et al., 2019a, 2019b ). Wavelet is a function characterizing by the following
properties ( Percival & Walden, 2000 )
The integral of real-valued function ψis equal to zero:
ð1N
2NψuðÞdu50: ð8:1Þ
The square of ψintegrates to unity:
ð1N
2Nψ2uðÞdu51: ð8:2Þ
Based on orthogonality, there are two types of wavelet transforms, continuous and
discrete wavelet transform.
1.Continuous Wavelet Transform (CWT): This transform can be used to obtain a simul-
taneous time-frequency analysis of a signal. CWT is best suited for time-frequency
analysis as these wavelets do not have negative frequency components. Examples
of analytical wavelets are: Morse wavelet, Bump wavelet, and analytical Morlet
wavelet. Continuous wavelet is now widely used in various disciplines, from x-ray
images in the medical field to well-logging in geophysics.
2.Discrete Wavelet Transform (DWT): It is an essential tool needed for studying


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images in the medical field to well-logging in geophysics.
2.Discrete Wavelet Transform (DWT): It is an essential tool needed for studying
time series via wavelet by decomposing the signal into a mutually orthogonal set of
wavelets. The critical usage for discrete wavelet analysis is de-noising and compres-
sion of signals and images. DWT has numerous applications in science, engineer-
ing, mathematics, and computer science. For instance, DWT improved the analysis
of dynamic fracture behavior in unconventional reservoirs. In reservoir engineer-
ing, DWT provided means of inferring inter-well connectivity in water flooding
operations ( Unal et al., 2019c ).230 Unconventional Shale Gas Development


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The base scale in DWT is set to 2. Different scales can be obtained by raising the
base scale to integer values represented as follows
2jðj51;2;3;4...Þ
The translation occurs at integer multiples represented as
2jmðm51;2;3;4...Þ
The DWT analyzes the signal at different frequency bands with different resolutions
by decomposing the signal into coarse approximation and detail information. As shown
inFigure 8.8 , the original signal passes through a high pass filter D1and low pass filter
A1. Half of the samples are eliminated after filtering according to Nyquist ’s rule, which
states that in order to reproduce a signal adequately, it should be periodically sampled
at a rate, that is 2 X, the highest frequency has been recorded ( Cramer & Grenader,
1959). Filtering a signal corresponds to the mathematical operation of convolution of
the signal with an impulse response of the filter. The convolution equation is given by
xn½/C1383hn½/C1385XN
k52Nxk½/C138:h½n2k/C138ð 8:3Þ


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the signal with an impulse response of the filter. The convolution equation is given by
xn½/C1383hn½/C1385XN
k52Nxk½/C138:h½n2k/C138ð 8:3Þ
Figure 8.8 Discrete wavelet multilevel decomposition. The signal is separated to high pass (noise)
and a low pass (approximation) at each level.231 Fracture diagnostic testing


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The low-pass filter halves the resolution but keeps the scale intact. The procedure
can mathematically be expressed as
yn½/C1385XN
k52Nhk½/C138:x½2n2k/C138ð 8:4Þ
The signal is then subsampled by two by discarding every other sample. This pro-
cess continues one level of decomposition and mathematically can be expressed as
yhighk½/C1385X
nxn½/C138:g½2k2n/C138ð 8:5Þ
ylowk½/C1385X
nxn½/C138:g½2k2n/C138ð 8:6Þ
where yhighk½/C138andylowk½/C138are the outputs of the high-pass and low-pass filters, respec-
tively. For the next level of decomposition, the low-pass sub-band A1is iteratively fil-
tered by the same technique into narrower sub-bands AjandDj. With this technique,
the signal of interest can be captured with a few large magnitudes of discrete wavelet
decompositions ( Polikar, 2001 )(Fig. 8.8 ).
In DWT, the total energy of the signal is decomposed into components (i.e., dif-
ferent decomposition levels). Therefore DWT preserves the energy of the signal.


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ferent decomposition levels). Therefore DWT preserves the energy of the signal.
Parseval ’s theorem relates the energy of the total signal to the energy of each compo-
nent and its wavelet coefficients ( Burrus, Gopinath, & Guo, 1998 ). For a discrete sig-
nal with the number of data points, the detail coefficients energy of the signal can be
calculated as follows:
Ed
i5dijj2;i51;2;3;... ;l
where51, 2, 3, ...,lis the wavelet decomposition level up to l.
8.6.1.1 Field example 1
Figure 8.9 shows the analysis of the MiniFrac by G-function of the first field
example case presented in this section ( Eltaleb et al., 2020 ). In this test, an average of
16.8 bpm of treated water was injected into the formation for 15 minutes. Pressure
fall-off was recorded after shut-in for 20 minutes, with a sampling rate of 1 seconds.
Normal leak-off (i.e., fracture area is constant during shut-in) can be recognized from
the derivative shape. The closure can be identified evidently when G3dp/dGdeviates


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the derivative shape. The closure can be identified evidently when G3dp/dGdeviates
from the straight-line, reporting a closure pressure of 5822 psi and closure time of
27.8 minutes. The fracture closure occurred at the 93rd data point in the analysis data
set (Fig. 8.9 ).232 Unconventional Shale Gas Development


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This example is analyzed using our proposed technique. Figure 8.10 shows the
energy distribution plot of the noise at the decomposition level D 5. Two different dis-
tinct regions (A and B) can be identified based on the energy magnitude. It can be
noticed that the energy magnitude is declining, then leveling off, and remains constant
at a minimum level until the end of the job. Note that there are almost six orders of
magnitude reduction in the energy of the noise until the minimum. We hypothesize
that after the closure point, the energy of the noise should be minimum and stay con-
stant. Also, it can be inferred from the energy trend that there is a change in system
properties with respect to time. This change is attributed to the decrease in fracture
volume over time@V
@t/C0/C1
. The closure point is identified from the intersection between
lines connecting the energy levels of regions A and B, as indicated by the dashed line


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lines connecting the energy levels of regions A and B, as indicated by the dashed line
inFigure 8.10 . The intersection point is at the 101st data point, which corresponds to
the closure time of 27.95 minutes and closure pressure of 5790 psi ( Fig. 8.10 ).
A comparison between the results by our proposed technique and G-function anal-
ysis is presented in Table 8.2 . As can be seen, according to our analysis, fracture
closure happens slightly after the closure point by G-function. The reason for this
Closure time = 27.8 min
Closure pressure = 5822 psi
Closure G           = ~ 0.126
Closure point (#)   = 93
Figure 8.9 G-function plot of field example 1.
Data points #Signal Energy, BHPClosure time = 27.95 min & closure pressure of 5790 psi
Closure point (#)   = 101AB
Figure 8.10 Distribution of the noise energy at decomposition level 3 (D 3) for the field example 1
(Eltaleb et al., 2020 ).233 Fracture diagnostic testing


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difference could be that the closure point by our technique is the complete closure of
the fracture, while the closure point by the G-function method is the start of the clos-
ing process.
8.6.1.2 Field example 2
Fig. 8.11 shows the analysis of DFIT by G-function for this case. In this test, an aver-
age of 0.28 bpm of water was injected into the formation for 12 minutes. Pressure
fall-off was recorded after shut-in for B77 minutes, with a sampling rate of 1 second.
As can be seen in the figure, the closure pressure and closure time are estimated to be
26756 psi and 822 seconds, respectively. The closure occurred at the 98th data point
in the data set. This example is relatively straightforward to analyze using the
G-function technique.
The figure below shows the energy distribution plot of decomposition level 3 (D 3).
We can notice that energy magnitude is declining, then leveling off to a minimum


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We can notice that energy magnitude is declining, then leveling off to a minimum
level until the end of the test. The closure point is identified from the intersection of
regions A and B, as indicated by the dashed line in Fig. 8.12 . The intersection point is
at the 37th data point, which corresponds to the closure time of B14 minutes.
The summary of the results of the proposed methodology for field example 2 and
its comparison with G-function analysis is presented in Table 8.3 . As can be seen,
unlike field example 1, our method suggests an earlier fracture closure compared with
the G-function methods. The difference between the closure pressures of the two
methods is about 2000 psi. In this case, our method indicates the G-function underes-
timates the closure pressure. However, as shown in the field Case 1, this is not always
the case as many parameters affect the fracture closure.
Closure time       = 822 
s 
Closure pressure = 26,756 psi 
Closure G           = 0.27


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the case as many parameters affect the fracture closure.
Closure time       = 822 
s 
Closure pressure = 26,756 psi 
Closure G           = 0.27 
Closure point (#)   = 98 
Figure 8.11 G-function plot of the field example 2 ( Eltaleb et al., 2020 ).234 Unconventional Shale Gas Development


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8.6.1.3 Field example 3
Figs. 8.13 and 8.14 show the analysis of DFIT by G-function for the example. In this
DFIT, fracture fluid was injected into the formation for 12 minutes. The Pressure fall-off
was recorded after shut-in for B165 minutes, with a sample rate of 1 second. Also, the
water hammer signature in the early portion of the derivative G3dp/dGis an indication
of a tight formation. The slight increase of the derivative G3dp/dGafter the closure
attributes to fracture growth after pump shut-in. The fracture closure pressure using the
G-function method is estimated to be 2656 psi and closure time of 13.6 minutes.
The analysis of the same data set via the energy distribution plot is shown in
Fig. 8.15 . The closure pressure is explicitly identified when energy magnitude levels
off from region A to region B. The closure point is identified from the intersection
between regions A and B, as indicated by the dashed line in Fig. 8.15 . The
Closure time = 761 s


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between regions A and B, as indicated by the dashed line in Fig. 8.15 . The
Closure time = 761 s
Closure point (#)   = 37B
Data points #Signal Energy, BHPA
Figure 8.12 Distribution of the noise energy at decomposition level 3 (D 3) for the field example 2
(Eltaleb et al., 2020 ).
Figure 8.13 G-function plot of the field example 3 ( Eltaleb et al., 2020 ).235 Fracture diagnostic testing


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intersection point is at the 93rd data point, which corresponds to a closure pressure of
2708 psi and a closure time of 13.5 minutes.
The results of both methods for this case are summarized in Table 8.4 . In this case,
similar to field example 2, the fracture closure happens before the closure identified by
the G-function technique. The closure pressure using our technique, in this case, is
following the methods presented by McClure et al. (2014) andWang and Sharma
(2017) in the sense that the closure happens earlier than the G-function. However,
one needs to compare the methods more closely for a better conclusion. Also, note
that the models presented by Maclure and Wang almost always pick the closure before
the G-function. However, as shown in example 1, this is not always the case with our
method. Moreover, this method is not restricted to any assumption and solely depends
on the recorded data and its quality.
8.6.1.4 Field example 4


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method. Moreover, this method is not restricted to any assumption and solely depends
on the recorded data and its quality.
8.6.1.4 Field example 4
Fig. 8.16 shows an analysis of DFIT of the fourth field example that is analyzed in this
paper. Water is injected with an average rate of 2.8 bbl/min for 3 minutes into a
Closure pressure = 2702 psi & Closure time = 13.5 min
Closure point (#) = 94 BA
Data points #Signal Energy, BHP
Figure 8.15 A summary of the results for field example 3 and its comparison with G-function
(Eltaleb at al., 2020 ).
Closure time = 13.6 min
Closure G            = 0.28
Closure pressure = 2656 psi
Closure point (#) = 101
Figure 8.14 Zoomed G-function plot of the field example 3 ( Eltaleb et al., 2020 ).236 Unconventional Shale Gas Development


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formation that has 140 ft thickness, 9.7% porosity, and 0.58 mD permeability. The
pressure fall-off has been observed for 26 hours, with a sampling rate of 1 second.
The data, beyond 32 hours of shut-in time and corresponding to GB565, is ignored
in the analysis because there is an observed pressure increase due to fluid heating
up (Amini, Soliman, & House, 2015; Soliman, 1986 ). From G3dp/dGsignature
(red line), a natural fracture system can be inferred. However, closure cannot be accu-
rately identified since multiple straight line candidates can be plotted. The possible clo-
sure points corresponding to closure pressures of 13,539, 13,068, and 12,815 psi. We
were given analysis for this case by an oil company, indicating the closure pressure as
13,544 psi (first tangent line).
The analysis of this DFIT using the proposed technique yields more affirmative
results about the interpretation of the pressure data. The decomposition level (D 3) has


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results about the interpretation of the pressure data. The decomposition level (D 3) has
been used in this example. As shown in Fig. 8.17 , two regions of energy levels are rec-
ognized as A and B as follows:
Region A : The energy decline describes the closing of the hydraulic fracture. The
intersection between regions A and B would be the minimum stress for hydraulic
fractures (closure pressure 513,521 psi, closure time 546 minutes, and closure
point (#) 52600).
Region B : The energy levels off as the pressure continue to decline but no further
reduction in the noise, indicating no physical changes are happening in the system,
and the hydraulic fracture is completely closed.
A summary of results from our proposed methodology and G-function is presented
inTable 8.5 . As can be seen, in this case, although multiple closures were possible on
theG3dp/dGplot, the wavelet technique showed an obvious minimum energy level
T closure =45 min
P closure   = 13,539  psi 
T closure =294 min


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theG3dp/dGplot, the wavelet technique showed an obvious minimum energy level
T closure =45 min
P closure   = 13,539  psi 
T closure =294 min
P closure   = 13,068  psi 
T closure   = 713 min
P closure   = 12,815  psi 
Figure 8.16 G-function plot of the field example 4 ( Eltaleb et al., 2020 ).237 Fracture diagnostic testing


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that matched with the G-function analysis. This is an advantage of our proposed tech-
nique as it does not limit it to a specific rock type and job parameter.
8.6.2 Effect of natural fracture and temperature
Current techniques for estimating the closure stress often contradict each other, reach-
ing up to 2000 psi difference in the identified closure stress. This discrepancy is due to
two main reasons, (1) the underlying assumptions of these techniques, (2) ultra-tight
permeability, natural fractures, and heat exchange.
Eltaleb, Rezaei, Soliman, and Dindoruk (2021) proposed a signal processing tech-
nique for analyzing DFIT in geothermal reservoirs, where it showed several advantages
over traditional methods in evaluating in-situ stresses. Because of the changing temper-
ature during the test, conventional techniques may yield inaccurate results. To show
the temperature effect on DFIT in geothermal reservoirs, two FORGE field cases are


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the temperature effect on DFIT in geothermal reservoirs, two FORGE field cases are
analyzed using the conventional and modified G-function analyses and the proposed
signal processing technique.
8.6.3 Utah FORGE example (effect of heat exchange)
A series of injection tests were performed in 2019 to estimate in situ stresses and for-
mation parameters. Nine different test cycles have been performed, including pump-
in/flow back and pump-in/shut-in. For this section, we focused on the pump in
shut-in only.
In this test, water was injected at the rate of 15 BPM into FORGE formation
through a 3.5-inch diameter tubing in Well 58 /C032 in April 2019. Series of fracture
injection/shut-in tests and Injection/flow-back tests were performed, as shown in
Fig. 8.18 . In this study, the results from DFIT 1 and DFIT 2 are analyzed.
Both temperature and pressure data have been analyzed for the first and second
DFITs using the developed methodology before considering the temperature and


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DFITs using the developed methodology before considering the temperature and
compressibility effect. Also, the pressure was analyzed after applying the corrections
to pressure. The energies of the noise in pressure and temperature were plotted
Data points #Signal Energy, BHPFracture closure pressure = 13,521                       psi & Closure time = 46
minute
Closure point (#) = 2600AB
Figure 8.17 Distribution of the noise energy at decomposition level 3 (D 3) for the field example 4
(Eltaleb et al., 2020 ).238 Unconventional Shale Gas Development


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against time. Our methodology for interpretation of pressure energy for unconven-
tional reservoirs shows a negative slope in the energy distribution plot that is indicative
of energy change in the system attributed to fracture closure. However, in geothermal
reservoirs, the expected change of injected fluid temperature has a significant impact
on the observed pressure signal. It, therefore, may lead to an ambiguous estimation of
the rock in-situ stresses. Consequently, it is essential to include the effect of the frac-
turing fluid ’s temperature change in DFIT. As shown in Fig. 8.19 , during water injec-
tion into the hot formation, heat convection dominated as the temperature decreases.
The temperature energy shows two distinctive regions, indicated by the yellow and
black lines in Fig. 8.19 . Contrary to the pressure energy, the recorded temperature
noise energy AC decays in a constant slope until the end of the test, which attributes


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noise energy AC decays in a constant slope until the end of the test, which attributes
to a decrease in the heat exchange rate as the system reaches equilibrium.
Detailed analyses of the two DFITs in Fig. 8.19 are given in the following section.
8.6.4 DFIT 1
Fig. 8.20 shows the G-function of the pressure analysis in DFIT 1. As can be seen, the
closure, in this case, is equal to B4200 psi, which is 200 psi lower than what was
identified from the wavelet analysis in Fig. 8.19 . The closure pressure using our tech-
nique, in this case, is in accordance with the methods presented by McClure et al.
(2014) andWang and Sharma (2017) in the sense that the closure happens earlier than
the G-function (i.e., G-function underestimates closure).
Figure 8.18 The diagnostic test on Utah FORGE geothermal reservoir (granite) ( Eltaleb et al., 2021 ).239 Fracture diagnostic testing


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Figure 8.19 Energy distribution of the noise in the recorded pressure and temperature in the semi-
log plot ( Eltaleb et al., 2021 ).
Figure 8.20 G-function analysis of DFIT 1 before applying the corrections ( Eltaleb et al., 2021 ).240 Unconventional Shale Gas Development


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The energy distribution of the noise in the pressure and temperature data are plot-
ted in Fig. 8.21 . As can be seen, there are two drops in the level of noise in the pres-
sure data. These drops correspond to 4400 psi and 3800 psi. After the second drop, the
level of noise in the pressure data remains constant. However, the noise in the temper-
ature shows a different trend. The temperature curve shows only one sharp drop in
the noise and a continuous drop in the noise after that.
We think that the first drop in the pressure noise is the closure pressure, and the
second drop might be due to the shearing of the natural fracture planes, as discussed
byNadimi et al. (2020) . The temperature plot confirms the closure because, BC, the
majority of the noise will be from the conduction and convection, with the former
become dominant. However, after the fracture closure, the noise in the temperature
data will be from conduction and vanish as the system reaches equilibrium.


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data will be from conduction and vanish as the system reaches equilibrium.
The method that was developed by Soliman (1986) accounts for the fracturing fluid
compressibility and temperature. The technique works up until the closure moment.
According to the method, the fracturing temperature and compressibility cause the
pressure fall-off to be higher than what it should be in reality. In this study, the thermal
conductivity and fluid compressibility were assumed to be 300 310261/psi and
50310261//C14F, respectively. Fig. 8.22 shows the corrected pressure for DFIT 1.
Figure 8.21 Energy distribution of the noise in the pressure and temperature for DFIT 1 ( Eltaleb
et al., 2021 ).241 Fracture diagnostic testing


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Based on the corrected pressure, one may apply the conventional G-function
again. After applying the correction for temperature, the closure pressure would be
picked at 3900 psi, which is 400 psi lower than the closure identified from the wavelet
in DFIT 1. This shows that the conventional G-Function is not suited for ultra-tight
and hot reservoirs ( Fig. 8.23 ).
The energy distribution of the noise in the corrected pressure is not plotted, but it
can be shown that the correction of the pressure will not affect the wavelet results.
This is an advantage of our technique over conventional analytical methods.
8.6.5 DFIT 2
In this injection test cycle, 183 bbl was pumped into the formation. The pumping
schedule started with pumping 8 at 3800 psi; then, the rate was increased to 15 bpm at
5000 psi. The pump stopped and the pressure was allowed to fall off for B15 hours.
The conventional G-function analysis of DFIT 2 without considering the effect of


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The conventional G-function analysis of DFIT 2 without considering the effect of
the temperature and fracturing fluid ’s compressibility is shown in Fig. 8.24 . In this
case, the closure using G-function analysis is identified as 4300 psi.
The distribution of the noise energy in the recorded pressure and temperature
is shown in Fig. 8.25 . As can be seen in this example, three levels of energy are
Figure 8.22 Corrected pressure for compressibility and temperature for DFIT 1 ( Eltaleb et al., 2021 ).242 Unconventional Shale Gas Development


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distinguished. The vertical line on the pressure plot shows where the closure pressure
should be picked from our proposed methodology. The closure, in this case, is equal to
4350 psi, which is in accordance with the previous analysis on the FORGE. Also, we
observed a constant level of noise energy before the closure in the temperature noise.
Figure 8.23 G-Function analysis of the corrected pressure of DFIT 1 ( Eltaleb et al., 2021 ).
Figure 8.24 G-function analysis of DFIT 2 before applying the corrections ( Eltaleb et al., 2021 ).243 Fracture diagnostic testing


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Figure 8.25 Energy distribution of the noise in the pressure and temperature for DFIT 2 ( Eltaleb
et al., 2021 ).
Figure 8.26 Corrected pressure for compressibility and temperature for DFIT 2 ( Eltaleb et al., 2021 ).244 Unconventional Shale Gas Development


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This constant noise level is due to convection. After the closure point, the temperature
noise reduces gradually until the temperature stabilizes to the in-situ reservoir
temperature.
The correction of the pressure, accounting for the temperature and fluid compress-
ibility, is shown in Fig. 8.26 . As can be seen, the pressure difference can reach up to
1000 psi that can result in erroneous conclusions.
The G-Function analysis of the corrected pressure is shown in Fig. 8.27 . As can
be seen, the G-function method again underestimates the closure. After applying the
temperature effect, its underestimation is more apparent.
8.7 Summary
In this chapter, conventional and newly developed approaches for fracture diagnostics
are reviewed. Fracture tests are typical pre- and posttreatment tools to estimate several
reservoirs and design variables. Typically, in all of the tests, a small amount of fluid is
pumped into the reservoir, and a short fracture is created. The diagnostics can be cate-


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pumped into the reservoir, and a short fracture is created. The diagnostics can be cate-
gorized into three main groups: direct far-field techniques, direct near-wellbore tech-
niques, and indirect fracture diagnostics. In this chapter, we reviewed the indirect
fracture diagnostics methods that are conducted on recorded response pressure and
rate from the wellbore. Sometimes, a combination of several methods is good for hav-
ing a better understanding of the interesting parameters.
Figure 8.27 G-Function analysis of the corrected pressure of DFIT 2 ( Eltaleb et al., 2021 ).245 Fracture diagnostic testing


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SPE 16953906
Diagnostic Fracture Injection Tests: Common Mistakes, Misfires, and 
Misdiagnoses 
R.D. Barree, J.L. Miskimins, and J.V. Gilbert, Barree & Associates 
Copyright 2014, Society of Petroleum Engineers 
 
This paper was prepared for presentation at the SPE Western North American and Rocky Mountain Joint Regional Meeting held in Denver, Colorado, USA , 16–18 April 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 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 consent of the Society of Petroleum Engineers is prohi bited. Permission to


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reproduce in print is restricted to a n abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. 
 
 
Abstract 
Over the last twenty years, Diagnostic Fracture Injection Tests, or DFIT’s, have evolved into commonly  used techniques that 
can provide valuable information about the reservoir, as well as hydraulic fracture treatment parameters. Thousands are 
pumped every year in both conventional and unconventional reservoirs. Unfortunately, many tests that are pumped provide 
poor or no results due to either problematic data acquisition or incorrect analysis of the acquired data. 
This paper discusses common issues and mistakes made while acquiring DFIT data. Guidelines on how to avoid these 
errors and secure the best possible data are provided including data resolution, pump rates, test duration, and fluid selection.


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errors and secure the best possible data are provided including data resolution, pump rates, test duration, and fluid selection. 
Rules of thumb are provided to estimate the time required to reach fracture closure and establish stable reservoir transients for 
analysis. 
The last part of the paper addresses potential (and commonly observed) problems in the analysis of the DFIT. The se 
issues can be magnified in tight gas and shale reservoirs due the long data acquisition times and the subtle pressure transients 
that can occur. Specific issues that are discussed include poor ISIP data from perforation restriction, loss of hydrostatic head, 
gas entry and the resulting phase segregation, the use of gelled fluids, and errors in after closure analysis. 
 
 
Introduction 
Diagnostic fracture inject ion tests or DFIT’s are small pump-in treatments performed to gather data to help design follow-up


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Introduction 
Diagnostic fracture inject ion tests or DFIT’s are small pump-in treatments performed to gather data to help design follow-up 
hydraulic fracturing treatments, as well as to characterize the subject reservoir. DFIT’s have their basis in co nventional mini-
frac treatments; however, they are subtly different and are intended to acquire significantly more data for fracture design and 
execution and reservoir description. Conventional mini-frac treatments have historically been focused on acquiring very 
specific treatment desig n parameters, such as fluid efficiencies and leakoff values, however, DFIT’s are intended to expand 
that role and acquire additional data such as reservoir pore pressure, detailed closure and fracture gradients, process zone 
stresses (PZS), transmissibility values which can be converted into reservoir permeability values, and leak-off mechanisms.


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stresses (PZS), transmissibility values which can be converted into reservoir permeability values, and leak-off mechanisms. 
The advent of unconventional reservoirs have added even more value to DFIT testing, as most of the information gained is 
comparable to traditional pressure transient tests, which are impractical to run in tight sand and shale systems since the time 
to analyzable pseudo-radial flow can be months, if not years. 
A generic DFIT test is shown in Figure 1. The wellbore is first filled at low to moderate rate, until a positive surface 
pressure response is observed. With low to moderate wellbore compressibility, the pressure should rise quickly until initial 
breakdown occurs. Breakdown will either be indicated by a sharp drop in pressure as a new fracture initiates (as at point 1 in 
Figure 1), or a plateau in pressure as existing fractures are opened and extended. Once a breakdown event is observed, the


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Figure 1), or a plateau in pressure as existing fractures are opened and extended. Once a breakdown event is observed, the 
injection rate should be increased to the maximum rate allowed by available horsepower, or to 75% of the planned treatment 
rate of the main frac, whichever is achievable. A constant rate is then held for 3-5 minutes (step 2 in Figure 1) , after which a 
rapid step-down can be conducted. The step-down test is separate from the actual DFIT but is commonly run at the end of the 
pump-in to determine perforation and ne ar-wellbore frictional pressure losses. The rate is then immediately reduced to zero, 
to obtain the instantaneous shut-in pressure (ISIP, shown as step 4) and the falloff pressure is monitored for as long as 
possible or as long as necessary to acquire the desired data. Even when high friction is not expected, the step-down is 
recommended to make identification of ISIP easier. More specific details of how to pump DFIT tests and recommended


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recommended to make identification of ISIP easier. More specific details of how to pump DFIT tests and recommended 
acquisition procedures are discussed later in this paper. 
 
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2  SPE 169539  
 
 
Figure 1:  Generic DFIT procedure. Initial breakdown is achieved at point 1; a constant  rate is held for 3 -5 minutes indicated by  point 
2; point 3 shows a rapid step -down; point 4 indicates ISIP; and point 5 shows falloff.  
 
 
This paper not intend ed to provide specific  instructions on the basics of how to  analyze  DFIT tests, as that information is 
described in detail in other papers ( Barree, et al, 2009; Barree, 1998 ). Rather, as the title implies, it aims to aid in the 
acquisition  of useful and analyzable data , as well as  provide hints to common problems that occur either during the pumping 
of the test or during the analysis of the results . It is not uncommon to hear statements that disparage the test , however, most 
of the  problems  can be attributed to failed te sts that do not provide any useful data  because they we re pumped  or analyzed


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of the  problems  can be attributed to failed te sts that do not provide any useful data  because they we re pumped  or analyzed  
incorrectly . This is a valid concern – these tests cost additional time and money, and if they fail to provide usable  information 
value is lost or d estroyed . However, more often than not, it is the quality of the acquired data  and how the acquisition does 
(or does not) take place  that i s the root issue. I f appropriate steps are taken and quality data acquired, the tests are quite valid 
and correct int erpretation can shed light on what the reservoir is trying to  say. 
This paper tries to address these acquisition issues, provide guidance on acquiring quality data, and discuss some common 
mistakes that happen during analysis procedures. If these steps are  addressed, the success rate of the treatments should 
increase significantly.  Most of t he examples in this paper focus  on unconventional, tight hydrocarbon systems, however, all


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increase significantly.  Most of t he examples in this paper focus  on unconventional, tight hydrocarbon systems, however, all 
points are equally  applicable to conventional  reservoirs . 
 
 
Recommended Acquis ition Procedure  
Every time fluid is pumped into a formation, there is a risk of formation damage.  Therefore, fluid injections of any kind 
should be minimized  and be conducted under controlled conditions. However, at the same time, enough fluid must be 
injected under fracturing conditions  to adequately contact the reservoir and provide conditions under which the desi red 
parameters can be measured.  
A DFIT is a fracture test, and must be conducted at a stable fracturing rate. I t does not take extremely h igh rates to initiate 
and propagate a fracture  in low to moderate permeability formations . In milli -Darcy range permeabilities, rates of 1 -2 BP M


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and propagate a fracture  in low to moderate permeability formations . In milli -Darcy range permeabilities, rates of 1 -2 BP M 
are frequently high enough to exceed  fracturing pressure. Approximate  rates that cause fracture extension  in a giv en situation 
can be calculated using radial Darcy’s Law , as shown in Equation 1. To extend a fracture the injection rate must be high 
enough to generate a pressure differential  (psafe) greater than the net fracture extension pressure, FG*D -p. 
For micro - and nano -Darcy systems, the necessary rates are even lower. Figure 2  shows a graph relating the generated 
pressure for a variety of reservoir permeabilities. In an effort to pump enough fluid to adequately contact the reservoir, do so 
in a relatively short amount of time, and provide a safety factor, rates of 5 -6 BPM, or even slightly higher, 7-8 BPM, have 
proven to  be very successful in providing good results. This maximum rate is recommended to be held for 3 -5 minutes, as


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proven to  be very successful in providing good results. This maximum rate is recommended to be held for 3 -5 minutes, as 
shown in Figure 1, which results i n ~600-1800 gallons of fluid being injected.  
 
) (ln] )* [( 10 917.46
max,
srrp p D FGkh Xq
wesafe
i


    (1) 
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SPE 169539   3 
 
Where,  qi,max  =  injection rate in bbl/min  
k  =  permeability of undamaged formation, md  
h  =  net thickness, ft  
FG  =  fracture gradient, psi/ft  
D  =  depth, ft  
Δpsafe  =  safety margin, psi  
p  =  reservoir pressure, psi  
μ  =  viscosity of injected fluid, cp  
β  =  formation volume factor  
re  =  drainage radius, ft  
rw  =  wellbore radius, ft  
s  =  skin 
 
 
 
 
Figure 2 : Rate necessary to initiate a fracture at various reservoir permeabilities. Assumes a reservoir depth of 7000 ft, fracture 
gradient of 0.8 psi/ft,  normal hydrostatic pore gradient,  formation thickness  of 20 ft, and a water fluid system.  
 
Data resolution fo r the entire injection and falloff period is a critical co mponent. Common pumping gauge resolutions of 
5-10 psi are not adequate for DFIT acquisition , since the intent of the treatment is look for subtle variations in the derivative


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5-10 psi are not adequate for DFIT acquisition , since the intent of the treatment is look for subtle variations in the derivative 
of the pressure versus time. Low resolution gauges can miss th ese data and render the test unusable . It is recommended to use 
gauges that have a resolution of at least 0.01 – 0.10 psi, with a sampling rate of one measurement per second during the 
pumping period  and through fract ure closure . This sampling rate can then be expanded during the falloff period after closure . 
During the post-closure falloff period, a sampling rate of up to one measurement every 30 seconds is adequate  and achievable 
with appropriate gauge memory systems . 
Two k ey parameter s in the analysis of a DFIT  is the time and volume of injection  (Barree, et al, 2009; Barree, 1998) , 
therefore, the injection schedule  must be recording precisely. Ideally, it is measured digitally in conjunction with the pressure


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therefore, the injection schedule  must be recording precisely. Ideally, it is measured digitally in conjunction with the pressure 
during the injection period, however, if this is not possible or is not done, extremely detailed notes must be kept that detail 
the pumping start/stop times and any rate changes  and times of such made during the procedure . If the data are recorded in 
detail, a rate schedule can then be recreated during analysis.  
Fluid selection for the treatment is a common question – what is the best option? In general, a Newtonian, non -wall 
building fluid should be used. This is one area that differs significantly from traditional mini -fracs, which commonly inject 
gelled or other non -Newtonian fluid systems , specifically to measure fluid efficiency . For DFIT purposes, water, diesel, or 
some other type of non -wall building fluid should be u sed. The reason for t his is demonstrated by Figures 3 -5. Figure 3


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some other type of non -wall building fluid should be u sed. The reason for t his is demonstrated by Figures 3 -5. Figure 3  
demonstrates t he conventional fluid loss model as a one -dimensional solution for linear transient flow with constant pressure 
boundary conditions. The f luid pressure at the fracture face is assumed constant with time and the far -field pore pressure is 
assumed to be constant. Initially the pressure gradient, and the associated leakoff rate, is very high. With time, the transient 
moves further into the reservoir and the g radient (and rate) decr ease. This solution shows  rate decreasing linearly with the 
square -root of time.  
0.0000010.000010.00010.0010.010.1110100
0.0001 0.001 0.01 0.1 1 10 100 1000
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4  SPE 169539  
Figure 4  then shows leakoff  modeled as a combination of series flows. The figure roughly describes a high permeability 
far-field reservoir zone, a near -fracture invaded zone,  and a thin wall filter -cake zone. In series flow , the total pressure drop 
through the system is the sum of the pressure drops through each zone. Using Darcy’s Law  in linear fashion , each pressure 
drop can be determined from the length and permeability of each zone. When even a thin film of very high flow resistance is 
present, such as the filter -cake, the flow capacity of the least conductive region dominates the system. When this filter -cake is 
deposited on a  fracture wall, as demonstrated in Figure 5 , most of the pressure drop is taken across the filter -cake during 
leakoff. The far -field pressure grad ient is much less than expected  when compu ted based on the leakoff rate, and t he after -


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leakoff. The far -field pressure grad ient is much less than expected  when compu ted based on the leakoff rate, and t he after -
closure analysis yields an estimate of reservoir flow capacity , i.e. k h, that is much too high and is inconsistent with the 
observed closure time.  
 
 
 
Figure 3 : Conventional fluid loss model as a one -dimensional solution for linear transient flow with constant pressure boundary 
conditions . 
 
 
 
 
Figure 4 : Leakoff  modeled as a combination of series flows  for permeabilities of 100 md, 1 md, and 0.001 md.  
 
 
 
Figure 5 : When a filter -cake is deposited on a fracture wall most of the pressure drop is taken across the filter -cake during leakoff  
unlike that shown in Figure 3 which de monstrates the pressure drop in an undamaged system.  
 
As a last acquisition component , isolation of the wellhead after pumping ends is required in order to eliminate any


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As a last acquisition component , isolation of the wellhead after pumping ends is required in order to eliminate any  
inadvertent pressure transients that might be caused by mechanical or logistical issues. Since DFIT analysis uses derivatives, 
any large changes or “bumps” in press ure, can inadvertently alter, invalidate the test , or render it impossible to analyze . Any 
pressure bleedoff that might occur during rig -down  can render the test useless. Likewise, if pressure measurements are being 
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SPE 169539   5 
acquired using long-term gauges that will remain in place after the pumping equipment is removed, care must be taken during 
other r outine field operations that these  gauges are not disturbed in any way.  
 
 
Comm on Acquisition Issues  
As noted above, a DFIT consists of i nject ing water or other clean fluid above fra cture extension pressure, creating  a stable 
fracture geometry, shut ting down injection and watch ing the pressure decay to evaluate leakoff mechanism and find fracture 
closure, and then evaluating  the after-closure pressure decline using appropriate reservoir transient solution s. If any of these 
steps are compromised during acquisition of the data, the data c an be unusable. This section discusses common acquisition 
issue s. 
As noted in the previous section and shown in Figures 3 -5, it should be stressed that a Newtonian, non -wall building fluid


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issue s. 
As noted in the previous section and shown in Figures 3 -5, it should be stressed that a Newtonian, non -wall building fluid 
must be used  when pumping the treatment , if any reservoir propertie s are to be obtained from the analysis . Using gelled or 
other non -Newtonian fluids can disrupt the after -closure pressure gradient and ma sk the reservoir flow capacity. Another  
common issue that is also a  holdover from old style mini -frac treatments is the use of multiple pump -ins. DFIT’s should be 
considered  as “one and done”. If the treatment is not pumped correctly the first time, a second attempt is almost pointless, as 
the reservoir conditions have now been altered by the first injection and , in low pe rmeability systems , the created pressure 
transients will not have dissipated by the time the second attempt is usually made.  The assumptions required for valid


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transients will not have dissipated by the time the second attempt is usually made.  The assumptions required for valid 
analysis of a pulse test do not allow for superposition of multiple injections.  A second attempt can possibly be made if the 
first attempt was not too invasive, however, remember that the pressure transient may take days if not weeks to fully 
dissipate.  
Not acquiring falloff data for a long enough period is also a com mon acquisition is sue. Specifically h ow long to plan on 
letting the falloff period be recorded for different reservoir types is disc ussed in following sections of this paper, however, it 
suffices to say here that the time period will be longer than the pumping equipment wil l physically stay on location. 
Therefore, secondary gauges that remain after the pumping equipment is removed are generally required. A common issue


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Therefore, secondary gauges that remain after the pumping equipment is removed are generally required. A common issue 
with these secondary, long -term gauges is that they are not opened to the wellhead when the treatment is pu mped. Since these 
secondary gauge s are commonly calibrated to a slightly different level than the pump gauges, if the two data sets are 
overlapped there will often be a  difference in the readings . Since the analysis of DFIT’s calls for pressure derivatives , any 
difference in readings will show as a slope change and will provide false indications of reservoir conditions on the analysis . 
The secondary long -term gauges must be opened to the wellhead during the pumping period also to eliminate this issue.  In 
any case, accurate calibration of all pressure gauges should be tested and assured.  
Additional field operational issues that can cause acquisition problems are frozen  or temperature -affected  wellheads and


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Additional field operational issues that can cause acquisition problems are frozen  or temperature -affected  wellheads and 
offset reservoir interruptions. Pre cautions should be  taken for wells in winter conditions to prevent freezing of the surface 
gauges. Even if the gauges don’t fully freeze, temperature fluctuations can impact the pressure readings, therefore, insulati ng 
the wellheads during the falloff periods is often helpf ul to prevent these pressure movements. Thermal fluid expansion or 
contraction in the wellbore will affect bo th surface and down -hole gauges.  Some people assume that a down -hole gauge will 
be in a constant temperature environment, and will therefore not be  affected. In a low compressibility fluid -filled wellbore the 
coefficient of thermal expansion, coupled with density driven convection currents, will still cause a pressure change 
throughout the wellbore volume .


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throughout the wellbore volume . 
Again, the behavior of pressure transients i s critical to that analysis of the treatment – if the pressure changes are caused 
by anything but the reservoir  and fracture response , the treatment can provide false readings. This is also the case if the 
reservoir itself is upset by any pressure changes such as pumping offset hydraulic fracturing treatments. A quiet environment 
in both the surface and downhole environments is needed for optimal data acquisition.  
One final acquisition precaution relates to the step -down portion of the test. It is strongly recommended to use a step -
down test versus a step -up test for t he reasons displayed in Figure 6. Figure 6  shows the treating pressure for increasing rates 
(overlapped by the pink line) and then the treating pressure for decreasing rates. The hysteresis in the curve is obvious and


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(overlapped by the pink line) and then the treating pressure for decreasing rates. The hysteresis in the curve is obvious and 
indicates that during the initial rate increase period, the wellbore is still experiencing breakdown periods and changes in t he 
overall net pressure and fracture extension period.  In many cases, the number of perforations open to i njection changes with 
each increase in pump rate. The step -down period does a better job of measuring the frictional change information that is 
desired rather than the complexity of friction and fracture extension components combined.  
 
 
Common Analysis Mis takes  
In addition to the acquisition issues previously mentioned , there are also several common analysis mistakes after the DFIT 
data is acquired. The following sections discuss some of these more common errors and ways that they can be avoided (if 
possibl e). 
 
Incorrect ISIP Determination


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possibl e). 
 
Incorrect ISIP Determination  
Analysis of a pressure falloff begins with determination of the instantaneous shut -in pressure (ISIP). This value is taken to 
represent the fracture extension pressure, when corrected to bottomhole conditions, and the frac ture extension gradient. The 
difference between the ISIP and determined closure pressure is taken as the net pressure in the fracture, or as a measure of the Downloaded from http://onepetro.org/SPEWRM/proceedings-pdf/14WRRM/All-14WRRM/SPE-169539-MS/1542530/spe-169539-ms.pdf/1 by University of Houston, Mohamed Gabry on 18 July 2022


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6  SPE 169539  
resistance of the rock to facture extension, or process -zone stress  (PZS) . Since net pressure dete rmines fracture width, and to 
a large degree height (due to the expectation that confining stress contrast controls containment) the accurate determination  of 
ISIP is critical to any later analysis and fracture geometry modeling.  
 
 
Figur e 6: A step -rate analysis chart showing increasing rate (step -up) measurements versus decreasing rate (step -down) 
measurements. Note the hysteresis exhibited between the two curves.  
 
 
In many cases the ISIP is not obvious, and does not occur immediately after shut-in. When a substantial near -well 
pressure drop exists, either through poor perforation efficiency or, more commonly, high near -wellbore tortuosity, injection 
at the fracture entrance may not stop for some time after the surface injection stops. The am ount of continued injection after


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at the fracture entrance may not stop for some time after the surface injection stops. The am ount of continued injection after 
shut-in, the amount of excess frictional pressure that must dissipate, and the time required for decompression of the wellbore 
fluids depends on the severity of the near -well restriction to flow and the compressibility of the wellbore volume.  
Figure 7 shows a fairly extreme case with an apparent ISIP of about 3640 psi at surface. The pressure falls nearly 1800 psi 
in the first few minutes after shut -in, with an injection rate of only 4 bpm. This DFIT was conducted through perforations in 
the toe of a cemented horizontal well. Inside diameter of the pipe was 4.0 inches and total depth of the well was 
approximately 15,000 feet. The wellbore was filled with fresh water for the test. Before any meaningful analysis can be done,  
an ISIP must be selected somewhere between 3640 psi and 1750 psi. If a value significantly below the pressure at shut -in


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an ISIP must be selected somewhere between 3640 psi and 1750 psi. If a value significantly below the pressure at shut -in 
(3640 psi) is used, the excess pressure drop must be explained.  
 
 
 
 
 
Figure 7 : Rate and pressure chart for a test with extremely high entry friction . 
20002500300035004000450050005500
0 5 10 15 20 25 30 35 40
Pump rate, bpmTreating Pressure, psi
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SPE 169539   7 
 
Total compressibility of the wellbore described can be estimated from the water compressibility (3.0e -6/psi) and wellbore 
volume (233 bbl). Expansion of the pipe should also be considered. Using the thi n-walled cylinder equation (Eq. 2 ), the 
change in volume of the wellbore can be estimated, where d is the pipe diameter, t is the wall thickness, p is the internal 
pressure change, and E is the modulus of the steel (30.0e6 psi).  
 
d
tEpd
22
     (2) 
 
Where,  p  =  internal pressure  change, psi  
d  =  pipe diameter, in  
t  =  wall thickness, in  
E = modulus of steel, psi  
 
Combining the water and pipe compressibility, the pressure of the fluid in the wellbore will change by approximately 
1170 psi for every 1 bbl of volume change. This re lationship can be used with an estimate of pressure drop due to tortuosity, 
assuming a square -root of rate linear relation, to estimate the frictional pressure drop driven by fluid expansion, and the


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assuming a square -root of rate linear relation, to estimate the frictional pressure drop driven by fluid expansion, and the 
expansion rate after shut -down. With an injection rate o f 4 bpm at shut -in, a frictional pressure drop of 1940 psi result s from 
a tortuosity factor of 970 psi/ (bpm). In the first second after shut -in, this rate decreases the wellbore volume by 0.0667 bbls, 
causing the pressure to drop from 3640 to 3562 psi. As  the fluid continues to expand, the injection rate drops, and the inlet 
frictional pressure decays. Figure 8  shows a plot of the pressure decay computed from this simple relation, compared to the 
observed pressure decline in the test.  
 
 
 
Figure 8 : Actual  (thin blue line)  and computed  (thick blue line)  pressure decay after shut -in, accounting for near -well tortuosity and 
wellbore decompression . 
 
The thin blue line is the  measured surface pressure for the test and the heavy line is the computed wellbore pressure with


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The thin blue line is the  measured surface pressure for the test and the heavy line is the computed wellbore pressure with 
the frictional pressure drop calculated from a constant tortuosity factor. The injection rate declines from 4 bpm at the inst ant 
of shut -in to 1.0 bpm after 2 5 seconds, and finally to less than 0.05 bpm after 200 seconds. Perforation pressure drop and 
other  factors are ignored in this analysis. The main point is that wellbore storage and decompression can drive injection after 
shut-in for a long time, in excess  of several minutes, and can account for large pressure drops. With large wellbore volumes, 
typical of extended reach horizontal wells and larger pipe sizes, this effect becomes more pronounced. Any gas or more 
compressible fluid in the wellbore will exten d the blow -down period even longer.  
Now that the source and duration of the early pressure  decay is accounted for, what is the actual representative ISIP? One


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Now that the source and duration of the early pressure  decay is accounted for, what is the actual representative ISIP? One 
method that can help is to extrapolate the pressure versus log of time curve, as shown in Figure 9, back to the point of shut -in. 
This construction suggests that some effect of wellbore fluid expansion may be felt for as long as 18 minutes after shut -in, 
and gives an estimated effective ISIP of 1766 psi. This may be a lower -bound of ISIP, but is a goo d starting estimate.  
 
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8  SPE 169539  
 
 
Figure 9 : Extrapolation of the pressure versus log time to approximate ISIP . 
 
The validity of the ISIP estimate can often be verified by examination of the log -log plot of pressure change from the 
assumed ISIP after shut -in and th e semi -log derivative of the pressure difference curve versus shut -in time. This plot, shown 
in Figure 10 , should produce parallel lines of the pressure difference (blue Delta Pressure) and derivative (red) during the pre -
closure period. The magnitude of t he derivative should be the pressure difference multiplied by the slope of the derivative 
curve. In the example, when the pressure difference curve is 300 psi at closure and the derivative is roughly 0.28 (the slope ) 
times that value. Note that any change to the estimated ISIP value will move the pressure difference curve but will have no 
effect on the derivative curve slope or position on the plot. When a stable linear derivative is apparent through the pre -closure


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effect on the derivative curve slope or position on the plot. When a stable linear derivative is apparent through the pre -closure 
data, this method provides an effective f inal check, but is not always infallible.  
 
 
 
Figure 10 : Log -log plot of pressure difference and derivative versus time . 
 
Basic Selection of Closure  
Basic DFIT analysis should be covered in sufficient detail in SPE 107877 (2009), but some mistakes in the most basic 
interpretation of the G -function diagnostic semi -log derivative continue to be made by many analysts. Figure 11 shows a 
DFIT with variable fracture compliance, or storage during closure, where the closure time and pressure have been mistakenly 
diagnosed.  In the original  analysis the clo sure time and pressure were identified by the positions of the red lines, at a closure


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diagnosed.  In the original  analysis the clo sure time and pressure were identified by the positions of the red lines, at a closure 
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SPE 169539   9 
time of G=14 and P=3960 psi. The correct closure time is G=32 and P=3640 psi. This discrepancy wi ll obviously affect any 
interpr etation of closure stress for calibration of a geomechanical model or stress profile, and will impact estimates of 
formation flow capacity and post -frac production. The red arrows in Figure 11 indicate other incorrect closure events that are 
often selected  by mistake. The one correct closure event is shown by the blue arrow, after the end of the variable compliance 
and linear closure periods. The error in closure stress also leads to an error in the interpretation of fracture net extensio n 
pressure. This af fects the predicted width of the fracture, and therefore directly impacts the computed fracture length, through 
conservation of injected volume. Predicted fracture height may also be affected, depending on the strength and stress of the


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conservation of injected volume. Predicted fracture height may also be affected, depending on the strength and stress of the 
confining layers. I n short, if the wrong analysis results are applied to the calibration of any fracture design model, the results 
will be incorrect.  
 
 
Figure 11: G -Function  derivative plot for variable compliance during closure, with incorrect closure picks.  
 
 
Effect of Injection Time versus Pump Rate  
Assuming a high enough injection rate is used so that the rock is dilated to its ultimate capacity, the width of the fracture  
will be determined by the net pressure generated, modulus of the rock, and some characteristic dim ension of the fracture. The 
theoretical basis of the G -function analysis assumes that the fracture adheres to Perkins -Kern -Nordgren geometry 
assumptions (Nolte, 1979). While this assumption is not necessarily valid, it implies that the generated fracture h eight will be


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assumptions (Nolte, 1979). While this assumption is not necessarily valid, it implies that the generated fracture h eight will be 
the controlling dimension. This leads to a somewhat circular argument as net pressure, which is  proportional  to fracture 
width, and fracture compliance (H/E) are directly linked in this model. A confined height, for a constant modulus, means the 
net pressure must be higher for a given frac width. A higher net pressure implies the ability to break through more barriers 
and possibly generate more fracture height, thus increasing compliance. The system should stabilize at an equilibrium 
fracture height, width, and net pressure for a given injection rate and fluid system.   
For a DFIT to be valid , an injection rate must be selected that will generate a net pressure and fracture geometry that is 
representative of the main frac. In practical terms, that means that the highest possible injection rate for the test will le ad to


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representative of the main frac. In practical terms, that means that the highest possible injection rate for the test will le ad to 
the most representative r esults. Once a stable net pressure and fracture geometry is achieved, a higher injection rate will cause 
the fracture to grow faster, but should not alter the width or height significantly. This assumption deviates from the classi cal 
PKN assumptions wherei n all net pressure is generated solely through frictional pressure gradient of the moving fluid. More 
realistic analysis of DFIT results requires that there be some resistance to fracture propagation inherent in the rock. This has 
been referred to as “proc ess zone stress,” “fracture toughness,” “rock strength,” and by other names, although each carries 
certain different connotations regarding the mechanism of resistance to fracture growth. A very low injection rate will


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certain different connotations regarding the mechanism of resistance to fracture growth. A very low injection rate will 
develop less net pressure, therefore less fracture height and width, and will not strain the rock to its ultimate limit. This can 
result in missing the evolution of secondary shear fractures  that can affect compliance, leakoff rate, and system permeability. 
The original theoretical models ass umed the rock to be linearly elastic and homogeneous, with no consideration that shear 
failure could exist.  
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10  SPE 169539  
Assuming a stable net pressure and fracture geometry are achieved, the fracture volume to surface area, or fracture width, 
will remain constant for  any duration of injection. That implies that a fracture with height of 50 feet and length of 300 feet 
will close in the same time as a fracture 50 feet tall and 600 feet in length , as long as leakoff flux per unit area is the same in 
both cases . For this reason the time to closure is not a function of injected volume, and is insensitive to rate as long as a high 
enough rate is achieved to reach a stable failure and fracture propagation condition. In fact, time to closure is a function of 
pumpi ng time and n ot volume injected.  
The theoretical calculation for time to closure depends on the integration of the leakoff rate through  each ele ment of the 
fracture face as it s being created, and during the closure process. For single -phase flow, with no damage to the face of the


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fracture face as it s being created, and during the closure process. For single -phase flow, with no damage to the face of the 
fracture (such as a gel filtercake), the leakoff rate through each element of fracture face should decrease linearly with the  
square -root of time since exposure of the face element.  For a slow injection rate each element of the face of the fra cture is 
exposed for a longer time, so the leakoff rate from the total fracture face will be much slower at shut -in than for the high rate 
injection case.  
 
 
Figure 1 2: Pressure transients established by low rate (A) and high rate (B) injection.  
 
Figure 1 2 shows conceptually the radius of investigation of the leakoff pressure transients at three times for a low rate (A) 
and high rate (B) injection scenario. The center point  represents the wellbore or point of fracture initiation. The inner set of


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and high rate (B) injection scenario. The center point  represents the wellbore or point of fracture initiation. The inner set of 
points re present the fracture half-length  for some point near the middle of the injection period. In case A , the pressure 
transient has penetrated a much larger distance into the formation , due to the longer injection time at low rate,  and the leakoff 
rate is much lower than in case B. In the case B example, the same fracture length is reached at a much shorter injection time 
so the pressure transient cannot travel as far from the fracture face, and leakoff rate is faster than in case A. The second set of 
points rep resents the fracture position at the end of injection. In case A , the pressure transients along the length of the fracture 
have travelled a greater distance into the formation, leakoff rate through each element of fracture surface is much lower


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have travelled a greater distance into the formation, leakoff rate through each element of fracture surface is much lower 
(following the square -root of time function), and therefore time to reach closure is significantly longer than in case B. In the 
high rate case, the entire fracture length is established with a short time of exposure o f all elements of the fracture surface and 
leakof f rate through each element remains high.  
For the same injected fluid volume the fracture of case B may be slightly longer than the case A fracture, but the 
difference is not linear with rate. In case B , the leakoff rate is higher throughout the injection  period but the fluid efficiency is 
also higher. All this assumes a similar net pressure and fracture compliance in both cases A and B .  
A second result of the impact of rate is apparent from Figure 12. In the low rate (case A) condition, the pressure tran sient


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A second result of the impact of rate is apparent from Figure 12. In the low rate (case A) condition, the pressure tran sient 
at closure, represented by the outer ellipse, has reached a nearly pseudo -radial flow case very quickly after closure. In case B 
the transient remains in pseudo -linear flow for a very long time after closure. For this reason many people, generally th ose 
concerned with application of classical pressure transient theory, recommend very low injection rates. This recommendation 
is based solely on the desire to establish a pseudo -radial flow period within a practical test duration. Unfortunately, in the re al 
world where rocks are not isotropic, homogeneous, and linearly  elastic, with no  inherent resistance to fracture growth, the use 
of a very low injection rate makes the test essentially useless for fracture diagnostics. These tests seek to approach a matr ix


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of a very low injection rate makes the test essentially useless for fracture diagnostics. These tests seek to approach a matr ix 
injection/falloff test and should not be considered  as valid diagnostic  fracture injection tests as the information critical to 
fracture design will all be compromised. Another failing of the low rate test is that the height of the zone contacted will r arely 
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SPE 169539   11 
(in the real world that means effectively never) be the perforated p ay height. Therefore any permeability or even KH estimate 
will probably not match post -frac production.  
 
Size of Generated Fractures  
Most people consider a short -duration fracture injection test to be too small to effectively evaluate enough reservoir 
volume to be applicable for post -frac production . Figure 1 3 shows the estimated fracture  half that would be generated for a 30 
foot fracture height, with a fracture width of 0.1 inches, given 100% fluid efficiency. These may seem to be arbitrary input 
assumpt ions, but are reasonable for water injection in “typical” rocks of 3 -4 million psi Young’s Modulus with a net pressure 
of 1000 psi and very low system permeability. The estimated fracture length can be directly multiplied by fluid efficiency to  
correct for  leakoff in higher permeability systems, and corrected by the ratio of created to assumed fracture height, if it is 
known.


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correct for  leakoff in higher permeability systems, and corrected by the ratio of created to assumed fracture height, if it is 
known.  
 
 
 
Figure 1 3: Fracture half -length created as a function of pumping time and rate . 
 
Note that for the recommended rate of 10 bpm, a pumping time of 5 minutes generates a fracture half -length of more than 
600 feet, when leakoff during the 5 -minute injection period is negligible. This is very likely the case in a shale or ultra -tight 
sand or carbonate reservoir s. Extending the pumping ti me to 10 minutes increases the created length to almost 1200 feet and 
increases the time to closure by increasing the average exposure time of all fracture face elements. Clearly, in tight rocks,  
these are not “small” fractures and the formation surface ar ea exposed is statistically significant. For the 5 -minute, 10 bpm 
injection case , the surface are a of one of the four fracture wing faces exposed is 18,000 square feet.


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injection case , the surface are a of one of the four fracture wing faces exposed is 18,000 square feet.  
 
Design of Test Duration  
Longer pumping times allow more time for penetration of the i nduced pressure transient. Given a stable fracture 
geometry, the time required to close the fracture by leaking off a fixed volume to surface -area ratio of fluid in the fracture can 
be related to the system permeability. A good approximation for time t o closure is given in Equation 3 : 
 
kttp
c3
     (3) 
 
Where,  tc =  time to fracture closure, min  
tp =  pumping time, min  
k =  permeability of leakoff system, md  
 
Where t c is the time to reach fracture closure (minutes), t p is the pumping time (minutes), and k is the permeability of the 
leakoff system connected to the fracture face, including any secondary permeability enhancement due to induced shear fa ilure


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leakoff system connected to the fracture face, including any secondary permeability enhancement due to induced shear fa ilure 
of the host rock. Figure 1 4 is a plot of the estimated closure tim e for pumping times of 2 to 20 minutes ove r a range of system 
permeabilities  from 100 nano -darcies (0.0001 md) to 1 md. For convenience the time scale has been converted to hours of 
shut-in.  
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12  SPE 169539  
 
 
 
Figure 1 4: Approximate time to reach fracture closure for va rious pumping times . 
 
Note that for a 5 -minute injection period into a 0.1 md formation, fracture closure occurs in about 0.2 hours (12 minutes). 
In these moderately high permeability formations, where this technique was first applied, extending the pumpin g time may 
not be a serious concern. For this 0.1 md rock, a pump time of 20 minutes will still reach fracture closure in about one hour  of 
shut-in time. Note that for higher permeability formations closure can happen too quickly to be easily identified wh en other 
factors such as wellbore friction and compressibility are considered.  
At 0.01 md, the 5 -minute pumping time requires 2 -3 hours to reach closure. For a 0.001 md formation, which is not 
atypical for what are considered commercial tight silt or shal e reservoirs, closure can take up to 24 hours after a 5 -minute


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atypical for what are considered commercial tight silt or shal e reservoirs, closure can take up to 24 hours after a 5 -minute 
injection. If the system permeability is actually in the 100 -nano -darcy range, closure can take up to ten days following a 5 -
minute injection. When system permeability is this low , the created fracture length will be large, and the pressure transient 
established at closure will remain in a pseudo -linear flow regime for a very long time.  
Once the fracture has closed, some time is required for a stable reservoir transient flow regime to be establ ished. On a log -
log plot of pressure change versus shut -in time, a safe and realistic lag time is ½ of a log cycle in time after closure. That 
roughly means that no data should be analyzed for any reservoir flow regime, including pseudo -linear flow, until three times 
the closure time has elapsed. So, using the 0.01 md, 5 -minute injection case, allow 2 -3 hours for closure and 6 -9 hours after


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the closure time has elapsed. So, using the 0.01 md, 5 -minute injection case, allow 2 -3 hours for closure and 6 -9 hours after 
shut-in to the start of a valid reservoir flow regime. For the 0.001 md case, that is 24 hours to closure and 3 days o f shut -in 
for any useful reservoir information.  
A goo d example of the time lag required to reach a stable reservo ir transient  is shown in Figure 1 5. After some non -linear 
leakoff behavior (including some mechanical noise in the data), there is an obvious break in the semi -log derivative curve  at 
the closure time of 52 minutes after shut -in.  The derivative continues to curve until the first indication of a valid reservoir 
pseudo -linear flow transient, at about 185 minutes after shut -in. The derivative cont inues to steepen, and approaches a 
negative unit slope that some may be tempted to analyze as a pseudo -radial reservoir flow regime. For the purposes of this


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negative unit slope that some may be tempted to analyze as a pseudo -radial reservoir flow regime. For the purposes of this 
discussion, the important consideration is that any reservoir transient must take time to develop.  Expecting a sharp break in 
the derivative curve immediately after closure to represent a reservoir flow transient is simply not reasonable.  
 
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SPE 169539   13 
 
 
Figure 1 5: Log -log pressure change versus shut -in time plot showing closure and start of pseudo -linear flow.  
 
If we follow the assumptions for the PKN fracture geometry, and assume that linear transient flow governs fracture 
leakoff flow up until closure, then the time required for the induced pressure transient to establish a pseudo -radial flow 
regime can be comp uted from the definition of t Dxf. For pseudo -radial flow , a value of t Dxf = 1.0  (at least)  must be reached i n 
the falloff period. Equation 4  gives the time, in hours, to reach pseudo -radial flow (t pr) as a function of reservoir storativity, 
permeability, and fracture length.  
 
ktxCtDxfft
pr0002637.02
     (4) 
 
Where,  tpr =  time to reach pseudo -radial flow, hours  
φ =  porosity, fraction  
μ = fluid viscosity, cp  
Ct = total reservoir compressibility, 1/psi  
xf = fracture half -length, ft  
k =  permeability of leakoff system, md


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Ct = total reservoir compressibility, 1/psi  
xf = fracture half -length, ft  
k =  permeability of leakoff system, md  
 
 
Equation 4  has been solved for t Dxf=1 for a gas reservoir case with viscosity of 0.02 cp, porosity of 8%, and total reservoir 
compressibility of 3.0e -5/psi. The time, in days, require d to reach the fully developed pseudo -radial flow period, according to 
these a ssumptions, is shown in Figure 1 6. Daltaban and Wall (1998) suggest that pseudo -radial flow may not be identified 
until t Dxf=2 for an infinite conductivity fracture, or t Dxf=3 fo r a uniform flux fracture.  The assumptions used in Figure 1 6 are, 
therefore, somewhat optimistic.  
Consider a moderate permeability DFIT in a 0.1 md systems with a pump time of 5 minutes at 10 bpm. The generated 
fracture will close in about 15 minutes and have a fluid efficiency of about 50%. According to Figure 13, the generated


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fracture will close in about 15 minutes and have a fluid efficiency of about 50%. According to Figure 13, the generated 
fracture half-length  would be approximately 300 feet, for a fracture height of 30 feet (the default assumed height in the plot). 
Using the rule of thumb for start of after closure analysis, a pseudo -linear transient may develop in about 45 minutes after 
shut-in. If the created f racture is planar, constant height, and follows the assumptions for a homogeneous system permeability 
under li near flow during closure, Figure 16 shows that a pseudo -radial flow regime should theoretically not be established 
until 2 -3 days after shut -in.  
 
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14  SPE 169539  
 
 
Figure 16: Time, in days, to theoretically reach pseudo-radial reservoir transient flow conditions for a gas reservoir for various 
system permeabilities. 
 
Conducting the same analysis with a 0.001 md system permeability leads to a predicted closure time of 27 hours, start of 
pseudo-linear flow after 83 hours, or 3.5 days, and start of pseudo-radial flow after about 1000 days of shut-in for a created 
fracture half-length of about 500 feet (83% efficiency). This implies that expecting pseudo-radial flow behavior in a DFIT in 
unconventional reservoirs is unreasonable. No combination of pump rate and pumping time will establish a stable fracture 
geometry and result in stabilized pseudo-radial flow in a practical test duration with system permeability less than about 0.01 
md. Attempting to induce early onset of pseudo-radial flow by conducting the test at very low rate, or very short duration,


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md. Attempting to induce early onset of pseudo-radial flow by conducting the test at very low rate, or very short duration, 
compromises the geometry of the fracture and the validity of any data associated with fracture extension criteria. If a test is to 
be conducted as an injection/falloff test for reservoir properties, then it should not be confused as a diagnostic of fracture 
extension or closure conditions. Low rate, and small volume tests are then compromised by the inability to achieve 
breakdown of many perforations or uniform injection over a predictable formation height. 
Why do some DFITs appear to enter a pseudo-radial flow period, based on the negative unit slope of the semi-log 
derivative on the log-log pressure-time plot, or based on the after-closure flow regime analysis plot? The most plausible 
answer is that fractures do not adhere to the simple planar geometry of a PKN fracture that is assumed in the derivation of the


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answer is that fractures do not adhere to the simple planar geometry of a PKN fracture that is assumed in the derivation of the 
G-function equations and the conventional after-closure solution of the diffusivity equation. A more likely, or at least 
plausible, image of the fracture system created in an unconventional reservoir is shown in map view in Figure 1 .  
 
 
 
Figure 1: Complex, non-planar, fracture system with coalescing transient pressure fields. 
 
If a complex, non-planar fracture system develops during injection, leakoff will occur through each face of all fracture 
strands. Even with high volumetric fluid efficiency, the primary fracture will be much shorter than predicted for a planar 
fracture, as shown in Figure 12, and normally assumed in theory. Time to closure, based on the G-function, will be


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fracture, as shown in Figure 12, and normally assumed in theory. Time to closure, based on the G-function, will be 
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SPE 169539   15 
proportional to the overall surface area to fracture volume ratio at shut -in, whether the fracture is planar  or not. For this 
reason, the permeability estimated from the G -function closure time, and given in Barree et al, 2009,  is a reliable estimate of 
the permeability of the system contacted by the induced fracture(s). The leakoff rate will be affected slightl y by the 
overlapping of the transient pressure envelopes around each fracture, but time required to leak off the fluid volume in the 
fracture system should be related to net pressure and system flow capacity or permeability. Some crossflow between fracture  
stands opening against different normal stresses should be expected. This can lead to the variable storage signature seen  in G-
function derivative analysis. Low compliance in the fracture systems coupled with a large exposed surface area can also lead


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function derivative analysis. Low compliance in the fracture systems coupled with a large exposed surface area can also lead 
to the pressure -dependent permeability signature. Since both of these phenomena are commonly observed, it is reasonable to 
assume that complex fracture systems in unconventional reservoirs are common.  
As shut -in time progresses the overlapping transient pres sure envelopes with coalesce into what appears to be a more 
radial flow geometry, illustrated by the outer ellipse in Figure 16. As this pressure transient dissipates in to the far -field it will 
appear as a radial flow signature on the diagnostic derivativ e plots. However, this radial flow pattern does not conform to the 
assumptions inherent in the after -closure pseudo -radial flow solution. This solution is based on the assumption of a single 
planar fracture with a transient evolving from a linear flow patt ern along the face of a single created fracture, which


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planar fracture with a transient evolving from a linear flow patt ern along the face of a single created fracture, which 
transitions to a radial flow pattern over a dimensionless time proportional to the assumptions in Equation 4 and Figure 12. 
The net result is that an apparent pseudo -radial flow regime develops much fa ster than would be the case for a single planar 
fracture of equivalent fluid efficiency. Any transmissibility derived from the apparent pseudo -radial flow time function may 
be orders of magnitude too high to represent the actual flow capacity of the reserv oir system. The problem and errors in 
interpretation are compounded when the radial flow or Horner analysis techniques are applied when a true pseudo -radial flow 
pattern does not exist at all.  
 
Phase Segregation and Gas Entry  
During the falloff period , while the well is sitting undisturbed, gas entry from the formation can occur.  For the ideal case


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During the falloff period , while the well is sitting undisturbed, gas entry from the formation can occur.  For the ideal case 
of a sealed wellbore under isothermal conditions, the volume of the gas bubble remains constant as it rises. With no mass 
transfer from the gas to the wellbore fluid, the moles of gas in the bubble remains constant, therefore its pressure remains 
constant as it rises. If a single gas bubble floats from the perfs to the surface under these conditions , the surface pressure wil l 
rise to the original bottomhole press ure and the pressure at the perf oration s will double. In reality, leakoff from the well is not 
identically zero, the increased pressure generated by the rising bubble causes an increase in leakoff rate, the gas temperatu re 
decreases somewhat during transit , and some gas may dissol ve in the wellbore fluid, therefore,  a very small gas bubble


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decreases somewhat during transit , and some gas may dissol ve in the wellbore fluid, therefore,  a very small gas bubble 
entering at the perforations can cause a large pressure upset  as demonstrated in Figure 1 8. Figure 1 9 shows the effects of gas 
entry on an actual treatment.  
 
 
 
Figure 1 8: Schematic showing a gas bubble that enters from the perforations can generate increased pressures at it rises to the 
surface.  
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16  SPE 169539  
 
Figure 1 9: The complete data set for a long -term injection test in an over -pressured formation is shown. The minimum pressure 
recorded at surface was 743 psi after 2.5 hours of falloff. The pressure increases after that time until about 9 hours of fal loff.  For 
purposes of analysis, the end of the valid data occurs before the minimum pressure point at 2.5 hours. The later pressur e rise is 
caused by entry of small gas bubbles at the perforations which rise in the wellbore fluid column.  
 
 
Loss of Hydrostatic Head  
The loss of the hydrostatic head effectively terminates a DFIT test. Bottomhole gauges can and are installed to acquire data in 
certain circumstances, however, these will not aid in extending the test if the hydrostatic column starts to fall. In these cases, 
the bottomhole gauge will record  the weight of the fluid standing above the gauge. The fluid column is supported by a  partial


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the bottomhole gauge will record  the weight of the fluid standing above the gauge. The fluid column is supported by a  partial 
vacuum between the fluid level in the well and the sealed wellhead. The rate of fluid level fall is then controlled more by t he 
rate of vaporization of the wellbore fluid than the reser voir transmissibility. Figure 20 shows a typical pressure resp onse for a 
bottomhole gauge in a well with falling fluid level.  
The data for the first few hours of falloff accurately represent the ISIP and initial leakoff rate and closure of the fractur e. 
The remaining data, covering about a week of shut -in, shows almost no change in the pressure on the gauge because the fluid 
head  above the gauge is nearly constant. Analysis of this part of the pressure falloff will give no  useful reservoir or  fracture 
information. Figure 2 1 shows the log -log plot for this falloff, and provides a useful clue for identifying the falling fluid level


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information. Figure 2 1 shows the log -log plot for this falloff, and provides a useful clue for identifying the falling fluid level 
case. As the fluid level begins to drop, the pressure deriv ative (magenta line in Figure 2 1) exhibits a large negative slope. 
Once the fluid level in the well stabilizes, and begins to drop at a rate controlled by vaporization of the fluid at the surf ace, 
the derivative assumes a positive, nearly straight -line, slope of about ½. During this period , the fluid at the surface of the 
standing column is effectively boiling (fluid pressure equals vapor pressure), and the heat of phase transformation causes th e 
fluid temperature to drop. This increases fluid density, driving a convection cell which replaces the colder surface fluid with  
warmer fluid from below. The important point here, is that analysis of the pressure -time data in this case cannot provide 
useful r eservoir or fracture data.


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useful r eservoir or fracture data.  
 
 
Figure 20: Bottomhole gauge response with falling fluid level, or surface pressure on vacuum.  
3/22/200322:00
3/23/200300:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00
3/23/200318:00
Time0100020003000400050006000A
0.00.51.01.52.02.53.03.5B Wellhead Pressure (psi) Slurry Rate (bpm)A B
Customer: Job Date: Ticket #:
Well Desc: UWI:GohWin v1.6.3
18-Dec-06 15:35
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SPE 169539   17 
 
 
 
Figure 2 1: Log -log derivative an d pressure difference for a bottomhole gauge  with falling fluid level.  
 
There are three practic al ways to avoid this problem: f irst, use a fluid of low enough density so that hydrostatic wellbore 
head is less than reservoir pressure. That can be achieved in some cases by using oil (a surface gauge can then be used) if t he 
reservoir is not badly unde r-pressured. Second, use a bottomhole gauge and bottomhole shut-in or isolation tool (that work s). 
The example in Figures 20 and 2 1 was run with a bottomhole shut -in tool and memory gauge, but the shut -in tool leaked. The 
third methods is to vent the wellh ead when the surface pressure drops to zero. This will allow the fluid level to fall to a 
pressure in equilibrium with the reservoir. Fracture closure will often occur before the wellhead pressure drops to zero, and


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pressure in equilibrium with the reservoir. Fracture closure will often occur before the wellhead pressure drops to zero, and  
the closure time and pressure will be wi thin acceptable accuracy. After closure there will be an extended transient period as 
the fluid level seeks equilibrium. After that, the pressure decay to reservoir conditions  will be analyzable, but all reservoir 
transients will be delayed. Corrections ca n be made for the added injection volume in the after -closure calculations.  
 
Common Mistakes in After -Closure Analysis  
The following example is one of many and shows some extremely common mistakes in DFIT after -closure analysis. The 
example is not intende d to disparage any particular software or analyst and identifying information about the specific test has 
been hidden or removed whenever possible. The test data were acquired with a botto mhole gauge and the test was pumped


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been hidden or removed whenever possible. The test data were acquired with a botto mhole gauge and the test was pumped 
through the toe perforations in a horizontal well. Figure 2 2 (a and b) show the G -function diagnostic derivatives for the test , 
computed in two different software packages. The result s are essentially the same, with closu re pressures of 22.463 and 
22.419 MPa , respectively. The semi -log derivative shows variable storage and a long after -closure transient period. This part 
of the analysis is consistent across almost all software. The slight  difference in the value of G at cl osure may be due to slight 
differences in the computed pumping time.  
 
 
  
(a)                                                                                                  (b)  
 
Figure 2 2: G-function diagnostic derivative plots from different software anal ysis systems .


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Figure 2 2: G-function diagnostic derivative plots from different software anal ysis systems . 
 
Once ISIP and closure are identified, the net extension pressure can be computed from the difference. In this ca se, the 
ISIP is slightly different due to wellbore expansion effects of short duration. The next step in the analysis is the log -log plot 
of pressure change from IS IP versus shut -in time. Figure 2 3 (a and b) show these plots from the two analysis packages.  
 
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18  SPE 169539  
 
(a)                                                                                             (b) 
 
Figure 2 3: Log -log pressure  change versus time after shut -in, with semi -log derivative . 
 
After the end of the variable storage period, and for a very brief time prior to closure, the plot in Figure 2 3(a) shows a 
positive ½ slope derivative corresponding to a linear fracture leakoff mechanism. After closure the derivative falls on a 
negative slope that is slightly shallower than -1/2. Through the entire test , the flow regime does not develop into an ideal 
pseudo -linear flow regime, and certainly shows no evidence whatsoever of a pseud o-radial f low regime. The plot in Figure 
23(b) identifies the late time slope at -1/2 (shown in the green text box), which is an approximation.  
The next step in the analysis should be to extrapolate the after -closure pressure decline to estimate reservoir  pore pressure.


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The next step in the analysis should be to extrapolate the after -closure pressure decline to estimate reservoir  pore pressure. 
Since the pseudo -linear flow regime is not fully developed, the extrapolation of the Cartesian linear flow plot will probably 
give a pressure estimate that is slightly too high. Use of the Cartesian radial flow plot i s not warranted at all.  Figure 2 4(a) 
shows the construction and extrapolation of the Cartesian linear flow plot, with an extrapolation to a pore pressure of 17.5 5 
MPa. Figure 2 4(b) shows the construction and extrapolation of the Cartesian radial flow plot that was presented in t he other 
analysis. This extrapolation gives a reservoir pore pressure of 18.03 MPa.  
 
 
 
(a)                                                                                                  (b) 
 
Figure 2 4: Pressure versus linear and radial time functions for linear extrapolation to reservoir pore pressure .


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Figure 2 4: Pressure versus linear and radial time functions for linear extrapolation to reservoir pore pressure . 
 
In this case , the difference in the estimated pressure is probably insignificant (about 3%), although in many cases the 
same error in flow regime identification can lead to pressure errors of up to 30%. A m ore serious  error develops if the wrong 
flow regime, and wrong pore pressure are used in the Horner or radial -flow solution for reservoir transmissibility. The best 
way to assure that the after -closure flow regime is correctly identified is to use the Tall ey et al (1999) log -log flow regime 
plot. This plot displays the difference between calculated bottomhole pressure and static reservoir pressure (p* or p i) as a 
function of the square of the linear -flow time function. Entering the wrong pore pressure chang es the shape and position of 
the pressure difference curve, but does not alter the shape and position of the derivative curve.


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the pressure difference curve, but does not alter the shape and position of the derivative curve.  
Figure 2 5 (a and b) show the two after -closure flow regime plots for the same test data. Figure 2 5(a) shows the plot using 
the po re pressure estimated from the linear flow extrapolation. The straight line connecting the beginning and end of the 
assumed linear flow period has the required slope of ½. Clearly the derivative falls on a significantly lower sl ope, which is 
consistent with the slope of the  semi -log derivative in Figure 2 4(a). The pressure difference curve is forced to fall on the ½ 
slope and be separated from the derivative by approximately 2x because of the selection of the pore pressure value derived 
from the extrapol ation of the Cart esian linear flow plot (Figure 2 4(a)). 
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SPE 169539   19 
 
 
 
(a)                                                                                          (b) 
 
Figure 2 5: Log -log of well pressure minus static reservoir pressure versus square of the linear flow time function . 
 
Figure 25 (b) shows the plot with the pore pressure picked from the Cartesian radial flow extrapolation. This value forces 
the pressure difference curve to approach a -1 slope, and to fall on top of the derivative curve. Clearly, the deriva tive is not on 
a -1 slope, and is still, in fact, of a slope less than -1/2. There appears to be a consistent trend in the industry to force a 
pseudo -radial solution out of a DFIT analysis, even when the data clearly show that no such flow regime exists. T he final 
results of this error in flow regime identification, and the forcing of an imposed incorrect analysis is most obvious in the


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results of this error in flow regime identification, and the forcing of an imposed incorrect analysis is most obvious in the 
computed reservoir transmissibility. Figure 25(b) indicates an estimate permeability of more than 0.1 md, which is consist ent 
with a transmissibility of more than 0.8 md -m/cp. These results can be reproduced in the software used for the case (a) results 
when the same mistakes are made. The closure time of G=20, from Figure 21, is more consistent with a system permeability 
of less than 0.002 md and transmissibility of less than 0.014 md -m/cp.  
Applying the other relations described in this paper to this example gives a clearer interpretation of a consistent result. 
The average pump rate for the test was 0.6 m3/m (3.7 bpm) with a pump time of 9.5 minutes. Fluid efficiency estimated from 
closure time is 90% with a transverse storage factor of 91%. Based on Figure 13, this should generate a fracture half-length


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closure time is 90% with a transverse storage factor of 91%. Based on Figure 13, this should generate a fracture half-length  
of about 100m (320 ft). The rule of thumb for closure from Figure 14, assuming a permeability of about 0.002 md, is 10 
hours. Actual closure time was 688 minutes (11.4 hours). A valid linear flow reservoir transient may be expected after 30 
hours, but a fully developed pseudo -linear flow regime never developed, even up to the  end of the test at 330 hours (13.8 
days). According to Figure 16, a pseudo -radial flow regime consistent with a single planar fracture should not be expected 
before 300 days. Clearly any results obtained from a forced radial flow analysis are not valid.  
A final comment about the use of the Horner plot in after -closure analysis is warranted. The Horner plot is only valid, and 
should only be relied upon, when a fully developed pseudo -radial flow regime is identified  in the data. The conditions


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should only be relied upon, when a fully developed pseudo -radial flow regime is identified  in the data. The conditions 
described abo ve for the use of the Cartesian radial flow analysis must also be met for the Horner extrapolation to be valid. In 
practical terms, that means that extrapolation of an apparent straight line at the end of a Horner plot will almost always be  
wrong. Both the  Cartesian radial flow analysis and Horner plots can be extrapolated to the pore pressure derived from a 
pseudo -linear flow analysis (if the pseudo -linear flow regime exists) , to obtain a maximum upper bound of reservoir 
transmissibility. Be aware that thi s estimate may be at least an order of magnitude too high.  
 
Conclusions  
DFIT tests can provide a wide variety of data to aid in designing hydraulic fracturing treatments and characterizing the 
subject reservoir. However, there are many factors associated with data collection and analysis that can result in poor or


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subject reservoir. However, there are many factors associated with data collection and analysis that can result in poor or 
incorrect results. This paper attempts to describe  some of the common problems  and to help prevent some common errors  
often observed in DFIT execution and analysis. These issues (and their possib le solutions) include:  
 
 DFIT’s should be pumped with a New tonian, non -wall building fluid;  
 Multiple pump -ins must be avoided;  
 The tests must be planned to allow for closure and  enough falloff time to enter linear flow at a minimum. This 
time period may be several days, if not weeks, in low permeability systems;   
 Use step -down  not step -up tests  to determine near wellbore and perforation friction;  
 Correct ISIP determination  is cri tical to further analysis of the test. Wellbore compressibility and other factors 
must be taken into consideration;


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must be taken into consideration;  
 The tests must b e of adequate duration and rate  (shortest practical duration at highest practical rate) ; 
 The possibility of p hase segregatio n and gas entry  must be considered;  
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20  SPE 169539  
 Loss of hydrostatic head  will likely render the test useless , even with bottomhole gauges,  unless certain 
additional steps are taken; and,  
 Not al l tests result in pseudo -radial flow regimes , and, i n fact, reaching pseud o-radial conditions is more the 
exception to the ru le. There are numerous reasons for f alse indications of pseudo -radial. To avoid these, full 
integration of the test results must be considered and the physics of the solution must be respected.  
 
Considerat ion of these points and the other s listed in this paper will lead to i mproved data acquisition and analysis , which 
in turn,  will lead to better tests, better results, and ultimately better predictions of production and reserve recovery.  
 
References  
Barree,  R.D. 1998. Applications of Pre -Frac Injection/Falloff Tests in Fissured Reservoirs – Field Examples. Paper SPE 39932 presented  at


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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 and Exhibition, Denver, CO, 5 -8 April.  
Barree, R.D., Ba rree, V.L, and Craig, D.P. 2009. Holistic Fracture Diagnostics: Consistent Interpretation of Prefrac Injection Tests Using 
Multiple Analysis Methods SPE Prod & Oper 24(3): 396 -406. SPE 107877 -PA. DOI 10.2118/107877 -PA. 
Daltaban, T. S. and Wall, C. G. 1998. Fundamenta l and Applied Pressure Analysis, Imperial College Press,  p438 . 
Talley, G. R., Swindell, T. M., Waters, G. A. and Nolte, K. G. 1999. Field Application of After -Closure Analysis of Fracture Calibration 
Tests. Paper SPE 52220 presented a t the SPE Mid -Continent Operations Symposium, Oklahoma City, OK, 28 -31 March.  
Nolte, K. G. 1979. Determination of Fracture Parameters from Fracturing Pressure Decline. Paper SPE 3841 presented at the Ann ual


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Nolte, K. G. 1979. Determination of Fracture Parameters from Fracturing Pressure Decline. Paper SPE 3841 presented at the Ann ual 
Technical Conference and Exhibition, Las Vegas, NV, 23 -26 Sept. 23 -26. 
 
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Transient Pressure Analysis 
for Fractured Wells 
Heber Cinco-Ley, * SPE, Stanford U. 
Fernando Samaniego-V., * SPE, Inst. Mexicano del Petr6leo 
Summary 
A new technique is presented for analyzing pressure 
transient data for wells intercepted by a finite­
conductivity vertical fracture. This method is based 
on the bilinear flow theory, which considers transient 
linear flow in both fracture and formation. It is 
demonstrated that a graph of Pwf vs. tv. produces a 
straight line whose slope is inversely proportional to 
hf(kfbf} Yz. New type curves are presented that 
overcome the uniqueness problem exhibited by other 
type curves. 
Introduction 
A large amount of information concerning well test 
analysis has appeared in the literature over the last 
three decades. As a result of developments in this 
area, three monographs 1 ,2,3 and one book4 have 
been published covering different aspects of pressure 
transient analysis. Ramey 5 also has presented a


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been published covering different aspects of pressure 
transient analysis. Ramey 5 also has presented a 
review on the state of the art. 
The analysis of pressure data for fractured wells 
has deserved special attention because of the number 
of wells that have been stimulated by hydraulic 
fracturing techniques. A summary of the work done 
on flow toward fractured wells' was presented by 
Raghavan 6 in 1977. 
It was recognized early that intercepting fractures 
can strongly affect the transient flow behavior of a 
well7-9 and that, consequently, the application of 
classical methods 10-12 to the analysis of transient 
pressure data in this situation may produce erroneous 
results. Several methods 13-24 were proposed to solve 
this problem. 
These analysis techniques consider a well in­
tersected by either an infinite-conductivity vertical 
fracture or a uniform-flux vertical fracture. Cinco­
Ley et al.25 demonstrated that the assumption of


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fracture or a uniform-flux vertical fracture. Cinco­
Ley et al.25 demonstrated that the assumption of 
infinite fracture conductivity is valid whenever 
*Now with Petr61eos Mexicanos and U. Nacional de Mexico. 
0149·2136/8110009·7490$00.25 
Copyright 1981 Society of Petroleum Engineers of AIME 
SEPTEMBER 1981 the dimensionless fracture conductivity 
(kfbflkxf) ~300; all other cases; such as those 
represented by long or poorly conductive fractures, 
must be analyzed by considering a finite-conductivity 
fracture model. 
Exploitation of low-permeability gas reservoirs has 
required stimulation of wells by massive hydraulic 
fracturing (MHF) techniques. Vertical fractures of 
large horizontal extension are created as a result of 
this operation; consequently, pressure drop along the 
fracture cannot be ne~lected. 
Several papers25-3 have been published on the 
behavior of finite-conductivity vertical fractures. 
Type-curve matching has been proposed as an


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Several papers25-3 have been published on the 
behavior of finite-conductivity vertical fractures. 
Type-curve matching has been proposed as an 
analysis method under these conditions; however, 
some regions of the curves present a uniqueness 
problem in the analysis. Barker and Ramey31 in­
dicated that the use of published type curves becomes 
practical when a large span of pressure data is 
available. 
The purpose of this work is to present a new in­
terpretation technique for early-time pressure data 
for a well intercepted by a finite-conductivity vertical 
fracture, including the criteria to determine the end 
of wellbore storage effects. In addition, new type 
curves are discussed to overcome the uniqueness 
problem exhibited by previous curves at intermediate 
and large time values. 
Transient Pressure Behavior 
for Fractured Wells 
Consider a vertically fractured well producing at a 
constant flow rate, q, in an infinite, isotropic,


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for Fractured Wells 
Consider a vertically fractured well producing at a 
constant flow rate, q, in an infinite, isotropic, 
homogeneous, horizontal reservoir that contains a 
slightly compressible fluid of constant com­
pressibility c, and viscosity p.. The porous medium 
has a permeability k, porosity 1>, thickness h, and 
initial pressure Pi' 
Let us assume that the well is intercepted by an 
undeformable, fully penetrating vertical fracture of 
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TABLE 1 -SI PREFERRED UNITS, CUSTOMARY UNITS, 
AND UNIT CONVERSION CONSTANTS 
Parameter 
or Variable 
k 
h 
qo 
qg 
p. 
B 
<P 
Ct 
p 
m(p) 
t 
IMPERMEABLE 
BOUNDARIES USED IN THESE SYSTEMS 
SI Preferred Units 
p'm2 
m 
m3/d 
m3/d 
Pa·s 
m3/m3 
fraction 
Pa-1 
kPa 
kPa2/Pa·s 
hours 
1,842 
1,293 
3.6x 10-9 
34.97 
24.57 
0.3918 
0.275 
K 
m3/Pa Customary Units 
md 
ft 
STBID 
Mscf/D 
cp 
RB/STB 
fraction 
psi-1 
psi 
psi2/cp 
hours 
141.2 
1,424 
2.637 x 10-4 
44.1 
444.75 
8.128 
81.97 
oR 
cu ft/psi 
WELLBORE 
FRACTURE 
h 
Fig. 1 -Finite·conductivity vertical fracture in an infinite· 
slab reservoir. 
, CASE 
f:>wD 
A 
F B D E 
"', I< 
tDXf 
Fig. 2 -Low·log graph of typical cases for fractured wells. 
!750 half-length xf' width bf, permeability kf' porosity 
r/>f' and total compressibility Cft (Fig. 1). The 
properties of both the reservoir and the fracture are 
independent of pressure and the flow in the entire 
system obeys Darcy's law. In addition, it is con­


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independent of pressure and the flow in the entire 
system obeys Darcy's law. In addition, it is con­
venient to assume that pressure gradients are small, 
gravity effects are negligible, and flow entering the 
well bore comes only through the fracture. 
With these assumptions, the unsteady-state flow in 
the system can be described by the diffusivity 
equation.! For details on the boundary conditions 
and a semianalytical solution for this flow problem, 
see Ref. 25. 
The general solution for the well bore flowing 
pressure P wf for oil is given by25 
kh(Pi-Pwf) 
----!....-B~- =PWD(tDxf,'Y/fD,CfDf) ...... (1) aoq J), 
and, for gas, 
where 
{3kt 
tDxf = r/>J),ctx}' ......................• (3) 
'Y/fD=~' ..•...................... (4) 
kr/>jCft 
and 
CfDf= ~r;~ ....................... (5) 
P wD represents the dimensionless pressure drop; it is 
a function of dimensionless time, t Dx ' dimen­
sionless fracture hydraulic diffusivity, 'Y/ib, and di­


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a function of dimensionless time, t Dx ' dimen­
sionless fracture hydraulic diffusivity, 'Y/ib, and di­
mensionless fracture storage capacity, CfDJ. ao' ag, 
and {3 are unit conversion constants (see Table 1). 
Cinco-Ley et at.25 showed that for practical values 
of dimensionless time the pressure behavior depends 
on two parameters only: the dimensionless time t Dx.r' 
and the dimensionless fracture conductivity 
(kfb f) D· The former was defined in Eq. 3 and the 
latter may be obtained from Eqs. 4 and 5. By 
definition, the dimensionless fracture conductivity is 
kb (kfbf) D = ~k , ..................... (6) xf 
and it appears to be related to CfDf and 'Y/fD as 
follows. 
(kfbf)D=7rCfDf'Y/fD. . ................. (7) 
Fig. 2 shows the general behavior of a well with a 
finite-conductivity vertical fracture. There, we show 
a log-log graph of dimensionless wellbore pressure, 
PwD' vs. dimensionless time, tDX[" For the sake of 
simplicity only two cases are presented. Case 1


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PwD' vs. dimensionless time, tDX[" For the sake of 
simplicity only two cases are presented. Case 1 
represents the behavior of a low-conductivity 
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~L--=~c5well ) \ 
Fracture 
(0) 
FRACTURE LINEAR FLOW 
Fracture -----_/ 
(c ) 
FORMATION LINEAR FL()\~ (b) 
B I LIN EAR FLO W 
\ / 
Fracture , . 
I 
Well 
\ ~ 
PSEUDO-RADIAL FLOW 
Fig. 3 -Flow periods for a vertically fractured well. 
fracture (kfbf) D =0.1; Case 2 considers a highly 
conductive fracture, (kfbf) D = 500. These cases 
were selected because they exhibit all the features of 
the transient pressure behavior for a fractured well. 
An analysis of Fig. 2 shows that the transient 
behavior of a well with a finite-conductivity vertical 
fracture includes several flow periods. Initially, there 
is a/racture linear flow characterized by a half-slope 
straight iine; after a transition flow period, the 
system mayor may not exhibit a bilinear flow period, 
indicated by a one-fourth-slope straight line. As time 
increases, a formation linear flow period might 
develop. Eventually, in all cases, the system reaches a 
pseudoradial flow period (see Fig. 3).


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increases, a formation linear flow period might 
develop. Eventually, in all cases, the system reaches a 
pseudoradial flow period (see Fig. 3). 
Points A, F, and L in Fig. 2 represent the end of 
the fracture linear flow period (half-slope straight 
line). The bilinear flow period (one-quarter-slope 
straight line) is defined by Segments B-C and G-H; 
this behavior is not present when the fracture has a 
high storage capacity and a high conductivity (lower 
curve in Case 2). 
The formation linear flow period is shown by the 
half-slope straight line between Points I and J, and it 
is only exhibited by fractures of high conductivity, 
(k fb f) D ~ 300. Points D and K show the beginning 
of the pseudoradial flow period. 
We present a detailed description of both the 
fractured linear flow and the bilinear flow. The 
formation linear flow and the pseudoradial flow have 
been discussed in the literature. 14,15,17,19,22,25 
SEPTEMBER 1981 Fracture Linear Flow Period


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been discussed in the literature. 14,15,17,19,22,25 
SEPTEMBER 1981 Fracture Linear Flow Period 
This behavior occurs at very small values of dimen­
sionless time, and it is exhibited by all cases. During 
this flow period, most of the fluid entering the well­
bore comes from the expansion of the system within 
the fracture and the flow is essentially linear, as 
shown in Fig. 3a. 
The dimensionless pressure response at the 
wellbore is given by· 
P D -2 ..j 1'('YIfDtD ............ (8) w -(kfbf) D -, 'Xf 
Hence, for oil, 
_ _ 0ffoqB I p,t 
Pwf-Pi b h ..,J k cJ> c ' ... ---. ---... (9) 
f f1ft 
and for gas, 
_ 0ffgqT I t m(pwf) -m(Pi) -b h ..,J k cJ> , ••••• (10) 
j f jp,Ct 
where 0ffo and 0ffg are unit conversion constants. 
These equations indicate that a log-log graph of 
pressure vs. time yields a straight line whose slope is 
equal to one half. A graph of pressure or pseudo­
pressure vs. the square root of time also gives a 
straight line whose slope depends on the fracture


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pressure vs. the square root of time also gives a 
straight line whose slope depends on the fracture 
characteristics excluding the fracture half-length xf-
The fracture linear flow ends when 
·See Appendix A for derivation. 
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2 
P wD ----End of Straight 
Line O.21T 
Fig. 4 -PwD vs. tb~l for a well with a finite·conductivity 
vertical fracture. 
_ 0.01 (kfbf) b tDxf- 2 ...........•..... (11) 
'YIfD 
Unfortunately, this flow period occurs at a time too 
early to be of practical use. 
Bilinear Flow Period 
To the best of our knowledge, this is a new type of 
flow behavior that has not been considered in the 
literature. It is called bilinear flow because two linear 
flows occur simultaneously. One flow is a linear 
incompressiblejlow withinthefracture and the other 
is a linear compressible flow in the formation, as 
shown in Fig. 3b. A bilinear flow exists, as shown in 
Appendix A, whenever most of the fluid entering the 
well bore comes from the formation and fracture tip 
effects have not yet affected the well behavior. 
Let us now examine this behavior in a log-log 
graph of PwD vs. tDx (Fig. 2). In Case 1 
[ (kjb f) D = 0.1], the bili6ear flow exists between 
Pomts Band C after a transition flow period


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graph of PwD vs. tDx (Fig. 2). In Case 1 
[ (kjb f) D = 0.1], the bili6ear flow exists between 
Pomts Band C after a transition flow period 
represented by Segment A-B. The pressure behavior 
for the bilinear jlow exhibits a straight line whose 
slope is equal to one fourth. The duration of this 
period depends on both (kfbf) D and CfPf. Case 2 
[(kfbf) D = 500] mayor may not exhibit the bilinear 
flow period, as shown by the upper and lower curves. 
The upper curve for Case 2 corresponds to a low 
value for CfDf and does exhibit bilinear flow over a 
short perioo of time (Segment G-H); however, the 
1752 -4 10 
Fig. 5 -Dimensionless time for the end of the bilinear flow 
period vs. dimensionless fracture conductivity. 
lower curve corresponding to a high value of CfDf does not exhibit the bilinear flow behavior because 
fracture tip effects are felt before this flow regime is 
established (Segment L-I). 
The dimensionless wellbore pressure for the 
bilinear flow period is given by*


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established (Segment L-I). 
The dimensionless wellbore pressure for the 
bilinear flow period is given by* 
7r \1.1 
PwD = r(1.25)..)2(k
fb
f) D t DXf' ......... (12) 
or 
2.45 II. 
PwD = ..) (kfb
f) D t DXj" ................ (13) 
This equation indicates that a graph of PwD vs. 
{j t Dx produces a straight line whose slope is 
2.451..) (kfb f) D, intercepting the origin. Fig. 4 
presents that type of graph for different values of 
(kfbf)D· . 
An important feature of this graph is that after the 
bilinear flow period (straight-line portion), the curves 
for (kfbf) D :51.6 are concave downward and the 
curves for (kfbf) D > 1.6 are concave upward. 
The end 0 the straight-line portion of the curves 
depends on the fracture conductivity and may be 
expressed by 
0.1 
·See Appendix A for derivation. 
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0. 
<J 
o 
Fig. 6 -Graph for analysis of pressure data of bilinear 
flow. 
. [ ] -1.53 tDebf=O.0205 (kfbf}D-1.5 
for 1.6$ (kfbf) D $3, ........... (l4b) 
and 
t _ [ 4.55 _ 2.5] -4 
Debf--J (kfbf) D 
for (kiJf) D $1.6 ............... (l4c) 
Fig. 5 shows a graphical representation of these 
equations. 
Bilinear Fiow Analysis 
Bilinear flow is exhibited by finite-conductivity 
fractures with a small dimensionless storage capacity; 
CfDf. Any attempt to analyze pressure data observed 
over this flow period using conventional methods (p 
vs. Yt or p vs. log t) will produce erroneous results. 
We present the appropriate analysis method based on 
the bilinear flow theory. 
Basic Equations and Graphs 
From Eq. 13, the pressure drop for oil may be ex­
pressed as 
SEPTEMBER 1981 Q 
<l 
Ol o -1 
Slope = 1/4 
Log t 
Fig. 7 -Log·log graph of pressure data for bilinear flow. 
and, for gas, 
where t:.p is the pressure (pseudopressure) change for 
a given test. Both 0bfo and 0bfg are unit conversion


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and, for gas, 
where t:.p is the pressure (pseudopressure) change for 
a given test. Both 0bfo and 0bfg are unit conversion 
constants and are given in Table 1. 
Eqs. 15 and 16 indicate that the pressure change is 
both inversely proportional to the square root of the 
fracture conductivity and directly proportional to the 
fourth root of time. 
According to Eqs. 15 and 16, a graph of t:.p vs. Vi 
produces a straight line passing through the origin, 
whose slope, m bf' for oil, is given by 
m -°bfoqBp, 
bf-h(kfbf} Y2(~p,Ctk} y. , ............ (17) 
and, for gas, 
m - °bfgqT 
bf-h(kfb
f} 'I2(~p,ctk} y., .•.......... (18) 
Hence, the product h(kfbf} '12 can be estimated by 
using the following equations. For oil, 
h (k b ) Y2 = °bfoqBp, (19) f f mbf(~p,Ctk} y., •........... 
and, for gas, 
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[lP 
o 
Fig. 8 -Bilinear·flow graph for a fracture with a flow 
restriction near the well bore. 
These equations indicate that values of reservoir 
properties must be available to estimate the group 
h (kfbf) Y2 • Note that this analysis technique tends to 
decrease the effect of the error introduced when poor 
information on reservoir properties (i.e., k, cp, and 
et) is used. 
All comments on the concavity of the curve in Fig. 
4 are valid for the curve in Fig. 6. 
From Eqs. 13 and 14, if (kfbf) D ??3, the 
dimensionless pressure drop at the end of the bilinear 
flow period is given by 
1.38 
(PwD}ebf=- (kfbf}D ................. (21) 
Hence, the dimensionless fracture conductivity can 
be estimated using the following equation. 
1.38 (kfbf) D=- ................. (22) 
(PwD}ebf 
(p wD ) ebf can be calculated using Eq. 1 or Eq. 2 and 
(1¥J) ebf or &n (p) ebf obtained from the bilinear 
flow graph. 
From Eqs. 15 and 16, a graph of log 1¥J vs. log t


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(1¥J) ebf or &n (p) ebf obtained from the bilinear 
flow graph. 
From Eqs. 15 and 16, a graph of log 1¥J vs. log t 
yields a quarter-slope straight line (Fig. 7) that can be 
. used as a diagnostic tool for bilinear flow detection. 
Extensions and Limitations 
The region disturbed during bilinear flow includes 
only the fracture and its vicinities because it occurs at 
early time, even in partially penetrating fracture 
systems. Thus, the equations and graphs discussed in 
the previous section for bilinear flow can be extended 
to cases where the fracture does not penetrate the 
1754 llP 
,. /" \\'.~ 
<#\<1' \.-
s'-
~\.-O'\ 
\{c'r<!- ~ 
",\\.-\ ~ 
fvC,'\ ~ 
I<!-"'/""''''-:~ 
",.. 
",.. PRESSURE BEHAVIOR WITH WELLBORE 
/" /" ?---- STORAGE EFFECTS 
4 « 
Fig. 9 -Wellbore storage effect on the bilinear flow graph. 
entire thickness of the formation. This is possible 
simply by using the fracture height, hf' instead ofthe 
formation thickness, h.


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entire thickness of the formation. This is possible 
simply by using the fracture height, hf' instead ofthe 
formation thickness, h. 
In cases where a flow restriction (low conductivity, 
turbulent flow) exists within the fracture in the 
vicinity of the well bore, bilinear flow still occurs and 
the pressure data analysis discussed cim be applied 
(see Fig. 8). An extra pressure drop, 1¥Js' is created in 
this case and the straight-line portion does not in­
tercept the origin. These situations distort the 
straight-line portion in the log-log graph. 
If well bore storage affects the system, the bilinear 
flow pressure behavior may be masked, as shown in 
Fig. 9, and data analysis becomes difficult (if not 
impossible) with current interpretation methods. 
Flow Regime Identification 
and New Type Curves 
The pressure behavior of a fractured well may exhibit 
several flow periods for practical values of time: 
bilinear flow, formation linear flow, and 
pseudoradial flow.


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several flow periods for practical values of time: 
bilinear flow, formation linear flow, and 
pseudoradial flow. 
Pressure data for each flow period should be 
analyzed using a specific interpretation method (i.e., 
Pw vs. Vi, Pw vs. Vi, and Pw vs. log t for bilinear, 
linear, and pseudoradial flows, respectively). 
The log-log graph has been used commonly as a 
diagnostic tool to detect different flow regimes in a 
transient pressure test. The use of type curves in the 
analysis of pressure data for fractured wells 
represents a major step in that area. The first type 
curves for fractured wells were presented by 
Gringarten et. af.19 Cinco-Ley et af.25 showed that 
the infinite-conductivity vertical fracture solution of 
Gringarten and Ramey17 behaves like the solution 
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for a finite-conductivity vertical fracture of 
(k fb f ) D ~ 300; furthermore, they showed that the 
umform-flux vertical fracture behaves as a variable­
conductivity vertical fracture. The type-curve 
analysis offers an advantage over the specific analysis 
methods already mentioned because it can be applied 
to interpret at once pressure test data that correspond 
to different flow periods. In addition, the type-curve 
analysis could indicate when the different graphical 
techniques apply. 
Experience has shown that, in some cases, ap­
plication of the type curves available for finite­
conductivity vertical fractures does not yield unique 
results. This is because the shape of pressure 
behavior curves is similar for different values of 
dimensionless fracture conductivity over some 
regions of the type curves. A close inspection of Fig. 
2 indicates that the uniqueness problem can exist if 
pressure data to be analyzed occur at either the


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2 indicates that the uniqueness problem can exist if 
pressure data to be analyzed occur at either the 
bilinear flow period (one-fourth slope) or the 
pseudoradial flow period. 
A convenient presentation of the type curve 
published by Cinco-Ley et al., 25 Ramey et al., 31 and 
Agarwal et al.29 is given in Fig. 10. We show a graph 
of log [PwD(kfbf}vl vs. log [tDx (kfbf}~l.* The 
main feature of this graph is that for all values of 
(k fb J ) D the behavior of both bilinear flow (quarter 
slope) and the formation linear flow (half slope) is 
given by a single curve. Note that there is a transition 
period between the bilinear and linear flows. The 
dashed line in this figure indicates the approximate 
start of the pseudoradial flow period (semilog 
straight line). Also shown are the end of the bilinear 
flow and the start of the formation linear flow; the 
time for the end of the bilinear flow from Fig. lO 
agrees with the results presented in Fig. 5. The groups


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time for the end of the bilinear flow from Fig. lO 
agrees with the results presented in Fig. 5. The groups 
of variables used in Fig. lO were derived in Appendix 
B, where it is shown that, for some values of fracture 
conductivity, bilinear flow ends when fracture tip 
effects are felt at the wellbore. 
Although many curves are presented for 
(kfbf) D > 207r, the shape of these lines is essentially 
the same. The only difference is the duration of the 
formation linear flow (half slope); that is, the higher 
the fracture conductivity, the longer the linear flow 
period. 
The beginning of the formation linear flow occurs 
at tDxf (kfbf) b::::: 102; that is, 
100 
t Dblf == 2 • • ••••••••.•••••.•.••• (23) 
(kfbf) D 
The end of this flow period is given by19 
tDelf==0.016 .......................... (24) 
Hence, the fracture conductivity may be estimated as 
follows: 
lO (kfbf}D== ~, ................... (25) 
'ItDblf 
• A large-scale copy of this graph may be obtained from the authors.


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follows: 
lO (kfbf}D== ~, ................... (25) 
'ItDblf 
• A large-scale copy of this graph may be obtained from the authors. 
SEPTEMBER 1981 slope=-} 
'~~~~-'~6~2~--~'070~--,O~2~--~,~~~-,~~~~~,o8 
2 t DX
f (kf bflD 
Fig. 10 -Type curve for vertically fractured wells. 
or 
(kfbf}D==1.25XlO-2j telf ........... (26) 
tblf 
These equations apply when (k fbf) D ~ 100. 
Fig. lO can be used as a type curve to analyze 
pressure data for a fractured well. Pressure data on a 
graph of log tlp vs. log t is matched on a type curve to 
determine (tlp}M' [PwD(kfbf}DlM, (t}M' [tDx 
. (kfbf}blM' [(k/b/}DlM, (tebf}M' (tblj}M' anl~ 
(tbssl J M' From thiS mformation, we can estimate the 
following. 
Dimensionless Fracture Conductivity. 
Formation Permeability. For oil, 
ko = _Q!o,,-q_B_J1._ [PwD' (kfbf) DlM 
h(tlp}M [(kfbf}vlM ....... (27) 
and, for gas, 
k -Q!gqT [PwD' (kfbf) DlM 
g-h[~(p}lM [(kfbf}vlM ..... (28) 
Fracture Half-Length. 
... (29) 
Fracture Conductivity. 
End of Bilinear Flow.


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k -Q!gqT [PwD' (kfbf) DlM 
g-h[~(p}lM [(kfbf}vlM ..... (28) 
Fracture Half-Length. 
... (29) 
Fracture Conductivity. 
End of Bilinear Flow. 
Beginning of Formation Linear Flow . 
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End of Wellbore 
Storage Effects 
Fig. 11 -Type curve for well bore storage under bilinear 
flow conditions. 
-:: 102 
.D 
.x-
a 
o.~ 0. 
<l 
01 
0 
---' 
'fA Log t ----Approximate start 
of Semi log Straight 
Line 
10-2 L----'--------'-_-'----'--_1-----'--------'-_-'-----L_L----'--------'-
10-4 10-2 102 104 106 108 
2 
t DX
f ( kf bflD 
Fig. 12 -Type-curve matching for data in bilinear and 
transitional flow. 
Q 
<l 
OJ o 
---' 
Log t 
Fig. 13 -Pressure data for a half·slope straight line in a 
10g·log graph. 
1756 Beginning of Semilog Straight Line. 
These results can be obtained if a large span of 
pressure data is available. It should be kept in mind 
that specific analysis graphs must be used for dif­
ferent flow regimes to obtain a better estimate of 
both fracture and reservoir parameters. 
Now we discuss cases where all pressure data fall 
on a very small portion of the type curve and a 
complete set of information may not be obtained.


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Now we discuss cases where all pressure data fall 
on a very small portion of the type curve and a 
complete set of information may not be obtained. 
Case 1. Pressure data exhibit one-fourth slope on a 
log-log graph (Fig. 7). According to the discussion 
in the previous section, these data correspond to the 
bilinear flow period and a unique match with Fig. 10 
cannot be attained. The bilinear flow type of analysis 
is the only method available for this case to obtain 
information regarding the fracture characteristics 
(kfbf)· 
A minimum value for fracture half-length, xf' can 
be estimated from Eqs. 14a through 14c for tiie end 
of the bilinear flow; that is, for (kfbf) D ~ 3, we 
have 
> 411O(3(k fbf)2tebf xf-"<I c/>/l-ctk .. ............... (31) 
Generally, wellbore storage affects a test at early 
time; thus, it is expected to have pressure data 
distorted by this effect, causing deviation from the 
one-fourth slope characteristic of this flow period.


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distorted by this effect, causing deviation from the 
one-fourth slope characteristic of this flow period. 
Fig. 11 may be applied to analyze pressure data for 
this case even if the duration of the test is not enough 
to reach the one-fourth slope portion. It is important 
to note that pressure behavior in Fig. 11 for both 
wellbore-storage-dominated and bilinear-flow 
portions is given by a single curve that completely 
eliminates the uniqueness matching problem. The 
correlating parameters FI (PwD) and F2 (tDxl) used 
in this figure are derived in Appendix C and aefined 
in Fig. 11. 
The end of well bore stora~e effects in Fig. 11 
occurs when F 2 (t Dx f) "" 2 x 10 ; that yields 
17.25 3/ c4 tews :::::--"<1 2 4 ••••••••••• (32) 
(3 (kfbf) h c/>ctk 
From observation of the results presented in Fig. 11, 
we can see that the end of well bore storage effects 
occurs three log cycles after the end of the unit slope. 
This criterion is useful to determine whether the


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occurs three log cycles after the end of the unit slope. 
This criterion is useful to determine whether the 
proper straight-line portion for bilinear flow analysis 
is chosen (see Fig. 9). If Fig. 11 is used as a type 
curve, the following information may be obtained: 
[FI (PwD»)M' [F2(tDXf»)M' (i¥J)M' and (t)M' 
Hence, we can estimate the following. 
WeI/bore Storage Constant. For oil, 
C= 27rDto(3qB/l-(t)M [FI (PwD»)M , ...... (33) 
(i¥J)M [F2(tDXf»)M 
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1m.,....,.,""""""""" 
~ +I+++t~ftHlttiH 
7 
1++l+l~+l+~f!HlH-
4 
Fig. 14 -Effective wellbore radius vs. dimensionless fracture conductivity for a vertical fracture. 
and, for gas, 
C= 2'1rcxg{3qT(t)M [FI (PwD)lM , ....... (34) 
[.::lm(p)lM [F2(tDXf)lM 
Fracture Conductivity. For oil, 
_ 0.4 I ~ \ cxoqBp,[FI (PwD) lM '13 
kfbf- h2 ...j ¢ctk l (flp) M j' 
........................ (35) 
and, for gas, 
k b = 0.4 J ~ \ cxgqI1FI (PwD)lM'I 3. 
f f h2 ¢ctk l [.::lm(p)lM j 
........................ (36) 
Case 2. Pressure data partially match the curve for 
the transition period between bilinear and linear 
flows (Fig. 12). In this case, the remaining portion 
of the data may correspond to bilinear or linear flow 
and the type-curve match is unique because the 
transition period has a characteristic shape. This 
comment is valid for dimensionless fracture con­
ductivities, (kfbf) D ~57f. 
From the type-curve match of pressure data for 
this case in Fig. 10, we obtain IPwD (kfbf) DlM'


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ductivities, (kfbf) D ~57f. 
From the type-curve match of pressure data for 
this case in Fig. 10, we obtain IPwD (kfbf) DlM' 
[tDxf (kfbf) blM, (M) M' and (flp) M' 
Hence, for oil, 
SEPTEMBER 1981 and, for gas, 
(kfbf)_ cxgqT rpWD(kfbf)D]M' .(38) xf -h[.::lm(p)lM t 
Fracture half-length and fracture conductivity. For 
oil or gas, 
.... (39) 
and 
Since the formation permeability generally is 
known from prefracture tests, the dimensionless 
fracture conductivity can be estimated by using 
results from Eq. 37 or Eq. 38. Some of these results 
also can be obtained from a specific analysis method 
corresponding to the flow period exhibited by data 
other than the transition flow region (i.e., bilinear 
flow or linear formation flow, as discussed in the 
bilinear flow analysis section and in Ref. 14). 
If all pressure data fall on the transition period of 
the curve, type-curve matching (Fig. 10) is the only 
analysis method available.


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If all pressure data fall on the transition period of 
the curve, type-curve matching (Fig. 10) is the only 
analysis method available. 
Case 3. Pressure data exhibit a half-slope line on a 
log-log graph. (Fig. 13). For this case there is no 
unique match with Fig. 10; however, the linear flow 
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P wD 
Fig. 15 -Type curve for a finite-conductivity vertical 
fracture. 
analysis presented by Clark 14 can be applied to 
obtain fracture half-length if formation permeability 
is known. In addition, a minimum value for the 
dimensionless fracture conductivity, (kfb J) D.' can 
be estimated using Eq. 26. Note that tb/f In Fig. 13 
represents a maximum value for the time of the 
beginning of the linear flo~ period and t elf represents 
a minimum value for the tIme of the end of the half 
slope. 
If well bore storage effects are present at early 
times in a test for this case, the analysis can be made 
using the type curve presented by Ramey and 
Gringarten.21 
Case 4. Pressure data partially match the curve jor 
the pseudoradial flow period. If a large span of 
pressure data is not available, a unique match would 
not be obtained by using Fig. 10 because curves for different values of fracture conductivity are similarly 
shaped for the pseudoradial flow period. However,


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shaped for the pseudoradial flow period. However, 
the transient pressure behavior shown in Fig. 10 can 
be correlated to analyze these cases better. 
For the pseudoradial flow period, a fractured well 
behaves like an unfractured well with an effective 
wellbore radius being a function of dimensionless 
fracture conductivity, (kfbf) D. Fig. 14 presents a 
graph of dimensionless effective wellbore radius, 
r~/xf' vs. dimensionless fracture conductivity, 
(kfbf) D. Notice that for large values of 
(k b f) D ( > 300), the dimensionless effective 
wefibore radius is 0.5, as mentioned by Prats et al. 8 
If the dimensionless time is defined by using r~ 
instead of xf, a graph of Pwq vs. tDr~ t;>rovides a 
single curve for the pseudoradlal flow penod for all 
values of dimensionless fracture conductivity (see 
Fig. 15). This curve provides an excellent tool for 
type-curve analysis of pressure data partially falling 
in the pseudoradial flow period because the


