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


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18 SPE-187348-MS
Figure 19—Pressure decline response for different wellbore volume with a PKN geometry
Fig. 20 shows the fracturing pressure and its derivatives for different initial reservoir pressure on G-
function and square root of time plots. It can be observed when the initial reservoir pressure is low; the
pressure declines more rapidly. This is also reflected by Eq. (18), a lower reservoir pressure leads to a higher
leak-off rate. However, when the reservoir pressure is over pressurized with high initial pore pressure, the
pressure decline trend resembles a "normal-leak-off behavior". In this case (P 0=28MPa), one can still notice
that there exist a subtle "bump" in the pressure derivatives that indicates an increase in fracture stiffness as
reflected in Fig. 10. For small values of σ ref, this increase can be too gradual to be noticeable on G-function
and square root of time plots. This example deomonstrates that long closure time not only can be attributed


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


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


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


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in-situ stress implicitly coupled with pressure, it is not possible to analytically estimate the minimum in-situ
stress from a single diagnostic plot. Based on the analysis of our synthetic cases, it is clear that the "tangent
line method" underestimates minimum in-situ stress and the "compliance method" overestimates closure in
all cases if the closure pressure is interpreted as the minimum in-situ stress.
Our analysis shows that the answer lies between these two limiting cases, regardless of parameters
used in simulation. To keep the analysis pratical and not resorting to a complex inverse problem, we
propose a "variable compliance method," where the minimum in-situ stress can be obtained by averaging
the dimensionless G-time or the square root of time of these two methods and extrapolating back to the
pressure curve that corresponds to the averaged G-time or the square root of time. An exception is when


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pressure derivative plot resembles the so-called "normal leak-off behavior," the changes in fracture fracture/
compliance may not be detectable on diagnostic plots and the "tangent line method" gives an estimation of
minimum in-situ stress within an acceptable error.
The estimated minimum in-situ stress using "variable compliance method" is also included in Table 3.
When we compare the estimated minimum in-situ stress obtained from the different methods with the input
minimum in-situ stress, Table 3 shows that the "variable compliance method" gives significantly lower
errors. The maximum error does not exceed a few 100 KPa, while the "tangent line method" yield errors
that are an order of magnitude larger.
Even though we have focused on the PKN fracture geometry in this section, these conclusions are also
applicable to a KGD fracture geometry. We can obtain the results for a KGD geometry if we replace fracture


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height with fracture length in calculating the evolution of fracture stiffness, as discussed in the Appendix B.


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


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


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


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is only limited to a few cases where fracture height is large or contact width is small. For a typical DFIT,
the injeciton volume is normally only a few cubic meters and the resulting fracture height or radius is often
limited. In addition, depending on rock mineralogy and heterogeneity, the height of asperities can range
from hundreds to thousands of micrometers (Bhide et al. 2014; Fredd et al. 2000; Sakaguchi et al. 2006;
Wells and Davatzes 2015; Warpinski et al. 1993; Zou et al. 2015), and, as shown by the field tiltimeter
survey, the residual fracture width can be as large as 20-30% of the maximum fracture width during pumping
(Warpinski et al. 2002). Remember is determined by the tallest asperities, so it is challenging to calibrate
the minimum in-situ stress using the "compliance method". In addtion, if the fracture has penetetrated
into barrier layers with higher minimum pricinpal stress (e.g., Multilayer Case1 in Table 4), the fracture


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


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


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plots. A larger fracture radius leads to lower fracture stiffness and this, in turn, results in a lower pressure
decline rate.
Figure 26—Pressure decline response for different fracture radiiwith a radial geometry.


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


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


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


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


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


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


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wellhead can be much higher than expected during fracture propagationduring theearly time of shut-in
period, because significant pressure drop can happen in the near-wellbore region and along the horizontal
wellbore, due to friction and tortuous/complex fracture path that initiated from perforation clusters. So
pressure data from this period cannot be used to infer fracturing parameters and reservoir properties. If a
rate step down test can be performed before complete shut-in, the relationship between early-time excessive
pressure drop and flow rate can be quantified.
Log-log plots of pressure derivatives are often used to determine flow regimes for after closure analysis
and assist the interpretation of closure stress (Barree et al. 2019; Hawkes et al. 2012; Soliman et al. 2015).
Even though Well-A was shut-in for nearly four weeks, pseudo-radial flow is still absent, as shown in
Fig. 31. This is a clear evidence of extremely low formation permeability. Without pseudo-radial flow,


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formation pore pressure and permeability cannot be determined independently with enough confidence
using just after closure analysis alone. One has to analyze the whole spectrum of DFIT data to reach a
consistent interpretation. Using late time data from the linear flow regime, and plotting Pf vs 
  on
Cartesian coordinates, the intercept on the y-axis when Δt → ∞ is the estimated initial pore pressure, which
is 29.1 MPa. This indicates it is indeed an over-pressurized reservoir. From geological and petrophysical


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


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on G-function and the square root of time plots. The results indicate that our simulated pressure matches
extremely well with the field data for the entire duration of the test, excluding the first hour of shut-in. Our
matched reservoir permeability is 220 nd, which is within the range of other petrophysical measurements.
The matched contact width and contact reference stress are 0.7 mm and 3 MPa, respectively, and Fig. 33
shows the corresponding fracture and fracture-wellbore system stiffness, based on the matched fracture
geometry, wellbore volume and contact parameters. Because the DFIT was conducted at the toe of a
horizontal well, it is no surprise that the wellbore storage effect is significant, considering such a large
contrast between the wellbore and fracture volume. We can also observe that the fracture stiffness starts
to increase noticeably when pressure drops to 34 MPa, over 2 MPa above the minimum in-situ stress. It


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is apparent that the so-called "normal leak-off behavior" is just a coincidence when the effect of pressure
dependent leak-off at the fracture surface and variable fracture compliance counterbalance each other during
the fracture closure process.
Figure 32—Matched pressure decline response for Well-A on G-function and square root of time plots


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


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shut-in). It should be observed that in the early-time pressure transient response, the pressure derivative
starts to deviate upwards at around 55 MPa. Using "variable compliance method" through averaging the G-
function or the square root of time that corresponding to a pressure of 50.4 MPa and 55 MPa, respectively,
we can obtain the estimated minimum in-situ stress of 52.8 MPa by picking the pressure that corresponds
to the averaged G-function or square root of time.
Figure 34—Pressure decline response from Well-B on G-function and square root of time plots
From Fig. 35, we can observe that only linear flow emerged after closure. Using late time data from the
linear flow regime, the estimated initial pore pressure is 33.7MPa. Geological and petrophysical studies
indicate that the thickness of the target formation is 5 m with an average porosity of 0.03 and in-situ fluid
viscosity of 0.28 cp. The Young's modulus is 39.5 GPa, the Poisson's ratio is 0.25 and formation total


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compressibility is 1.9e-3 MPa−1. Hydraulic fracture modeling shows that the fracture is well contained within
the target formation with roughly 34 m fracture half length. Based on this information and assuming PKN
geometry, the pressure decline response can be matched globally using our global DFIT model by adjusting
reservoir permeability and contact parameters.


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


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


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compensate leak-off and pressure drop. In this field case, with fracture height contained in 5 m, the fracture
stiffness should be at least 5.36 MPa/mm (From Table 1) even without considering the contact of fracture
surfaces. If the fracture geometry and reservoir properties can be constrained with enough confidence, the
estimated contact parameters and fracture stiffness evolution can be used to reconstruct the dynamic fracture


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


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demonstrated that the square root of time plot gives the same quantitative information obtained from the
G-function plot.
A new global DFIT model is proposed, which for the first time, has the capability to predict pressure
decline response across the entire duration of a DFIT test using a single, coherent mathematic framework in
low permeability formations. We show how each factor impacts the pressure decline response and the effects
of previous overlooked coupled mechanisms are examined and discussed, along with synthetic and field
case studies. The content of this article promotes our conceptual understanding of pressure transient and
coupled behaviors during DIFT and enables us to revisit DFIT analysis from a new perspective. Conclusions
reached from the analysis presented in this paper include the following.
1.Carter's leak-off, (leak-off rate is proportional to 
 ), is applicable during stable fracture
propagation, when net pressure is roughly constant and injection time is limited. However, during the


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fracture closure peroid, the fluid pressure in the fracture declines significantly, and the leak-off rate
becomes fracture pressure dependent and ceases to be proportional to 
 .
2. The so-called "normal leak-off," with a straight line of pressure and its derivatives on G-function and
the square root of time plots, is not "normal" at all. This is sometimes observed in data sets when
mechanisms that accelerate pressure decline (e.g., increase in fracture stiffness during closure) offset
those that decelerate pressure decline (e.g., pressure dependent leak-off at the fracture surface and
wellbore storage effects). So any "before closure" analysis based on this conventional "normal leak-
off" assumption to estimate fracture geometry and formation permeability should be re-examined.
3. Hydraulic fractures do not close as two smooth parallel plates, but rather close progressively from
edges to the center on asperities and roughness surfaces, as confirmed by our modeling results and


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


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


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effective stress. This adds tremendous value to DFIT analysis for completion design and estimates
of well productivity and EUR.
Nomenclature
a = Half fracture height (PKN model) or half fracture length (KGD model), L, m
Af =Fracture surface area (one face of one wing),L2, m2
ct =Formation total compressibility, Lt2/m, 1/Pa
cW =Water compressibility, Lt 2/m, 1/Pa
CL =Carter's leak-off coefficient, 
 , 
E = Young's modulus, m/Lt 2, Pa
E′ = Plane strain Young's modulus, m/Lt 2, Pa
g(Δt D) =Dimensionless g-function of time
G(Δt D) =Dimensionless G-function of time
hf =Fracture height, L, m
ISIP = Instant shut-in pressure, m/Lt2, Pa
k = Formation permeability, L2, m2
P = Pressure, m/Lt2, Pa
Pf =Fracturing pressure, m/Lt2, Pa
Pnet =Fracturing net pressure, m/Lt 2, Pa
P0 =Initial reservoir pressure, m/Lt2, Pa
qf =Leak-off rate (one wing), L3/t, m3/s
rD =Normalized radius
rp =Productive surface ratio
rf =Local fracture radius, L, m
Rf =Fracture radius, L, m


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


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


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process to generate the special BCA plot and predict fall-off pressure is complicated and cumbersome,
prone to mistakes that can lead to dimension inconsistency ( Soliman et al. 2013). In addition, their approach
used a constant-rate solution of an infinite-conductivity well introduced by Gringarten et al. (1974). The
results obtained by use of the constant-rate solution are questionable to account for the fracture pressure
dependent leak-off behavior. In this appendix, we derive the solutions of pressure decline under fracture
pressure dependent leak-off using superposition based on a constant-pressure solution.
The basic principle of superposition is that the response of a system to a number of perturbations is equal
to the sum of the responses to each of the perturbation as if they were present by themselves if the responses
are linearly related to the perturbations. From Eq. (18), we can calculate the leak-off rate under a constant
fracture pressure P f:
(A.1)


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fracture pressure P f:
(A.1)
Divide the shut-in time Δt into n time steps, and the leak-off rate at the nth time step can be determined
based on superposition:
(A.2)
The pressure difference P f,j - P f,j-1 is the pressure in the fracture at time step j minus the pressure in the
formation at the fracture-formation interface, which equals to the pressure in the fracture at the previous
time step. Combine Eq. (A.2) with Eq. (10) via material balance, we can get:
(A.3)
let P f,0 = P 0, Pf,1 = ISIP, Δt 1 = 0 then for n ≥ 2, integrate Eq. (A.3) we can obtain:
(A.4)
Eq. (A.4) can be solved explicitly with marching time interval, for example, when n = 2
(A.5)


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SPE-187348-MS 35
On closer observation of Eq. (A.5), we can realize that it is exactly Carter's leak-off solution to Eq. (19),
because during the first time step, no superposition is needed. The fracture pressure dependent leak-off
coefficient can be recovered as:
(A.6)
It should be mentioned that Eq. (A.4) is only valid when Sf is constant, where the linear response system
is the prerequisite of using superposition principal.


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


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


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


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


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Similar to Fig. B1, a radial fracture can be discretized into n segments, as shown in Fig. B3. In each
segment, there exists a uniform net pressure/stress that acts on the opposite fracture surface. If n is large
enough, the net pressure/stress distribution can be approximated to any distribution of P Net(r).


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


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


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In this Picard iteration scheme, α r is a relaxation factor that is less than or equal to 1.0. It controls how
much the net pressure/stress should be corrected within iterations. Smaller α r can avoid over-correction
and improve convergence stability, but more iterations may be required with small value of α r. When the
changes in net pressure/stress between iterations is less than the tolerance value, the estimation of contact
stress across the entire fracture faces is converged and the final net pressure/stress distributionwith the
corresponding fracture width profile can be obtained. So given initial fracture geometry, rock properties
and contact law parameters, the fracture width profile can be uniquely determined under certain fracturing
pressure. By discretizing fracturing pressure within a certain range, the relationship between fracture volume
vs. pressure and fracture stiffness vs. pressure can also be obtained.


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SPE-185649-MS
Reinterpretation of Fracture Closure Dynamics During Diagnostic Fracture
Injection Tests
B. Zanganeh and C. R. Clarkson, University of Calgary; R. V. Hawkes, Trican Well Service Ltd.
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April 2017.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written


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consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
A fit-for-purpose, fully coupled stress-pore pressure simulation model (Abaqus®) is used to simulate
diagnostic fracture injection tests (DFITs) and generate before closure pressure responses. The simulated
responses are used to explain field observations, and to propose a new concept: progressive fracture closure.
The cohesive zone model (CZM) is used to model fracture propagation and closure associated with
DFITs. The customized model in Abaqus® is capable of modeling all the physical processes involved
in a typical DFIT including: porous media deformation? fluid flow inside the reservoir? hydraulic
fracture initiation, propagation and closure? compliance change before and after closure; residual fracture


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


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


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


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


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slope on the Bourdet-derivative (with
respect to superposition time), and picked closure as the deviation from the 
  slope. Bachman et al. (2012)
presented a systematic PTA approach, and a combination of diagnostic plots, to identify various flow
regimes before and after fracture closure. Van Den Hoek (2016) and McClure et al. (2016) demonstrated
limitations of superposition time and Agarwal's time (1980) for analyzing DFITs where the pumping time
is small. They stated that the Bourdet-derivative (with respect to superposition time and Agarwal's time)
starts to exceed one at shut-in times equal to roughly one-tenth of the pump time and that the 
  slope is
not related to fracture closure.
Van Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by conducting experiments
on different rock types. They observed that fracture closure happens at the tip first, and then towards the
wellbore. They reported a break on a plot of pressure versus G-function due to the decrease of fracture


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compliance at closure, and argued that, instead of a deviation from a straight line on the pressure derivative
(with regard to G-function), the local minimum should be selected as closure pressure. Further, they stated
that plastic deformation can cause a fracture to remain open in the vicinity of the wellbore. McClure et
al. (2014) and McClure et al. (2016) investigated the effect of fracture compliance on pressure behavior
using a numerical simulator. Their modeling of after-closure compliance behavior was based on the Barton-
Bandis (1985) model, which is an empirical relationship. They demonstrated that the previous "fracture
height recession" signature presented by Barree et al. (2009) is the natural behavior of closure caused by
a change in fracture compliance. Based the "fracture compliance method," closure is picked as the start of
upward deviation on the 
  curve. McClure et al. (2016) argued that previously-presented methods tend
to underestimate the closure pressure.


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to underestimate the closure pressure.
Van Den Hoek (2016) presented a PTA approach for modeling pressure behavior in DFITs and
waterflood-induced fractures based on simplified numerical and analytical solutions. In his work, fracture
growth rate was a predefined input into the simulator, the Carter model was used as the leakoff model for
the DFIT, and closure was modeled as a gradual decline of fracture compliance representing a combination
of "hinge" (constant length) and "zipper" (length recession) closure.


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


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In this study, the CZM implemented in the Abaqus® finite element program is used to model fracture
initiation, propagation and closure for DFITs. A pressure-dependent leakoff model is coupled with the
program through user subroutines. The model is used to generate synthetic pressure responses, which in
turn are compared to field observations and used to study fracture closure dynamics.
Theory and Methods
In the following, key components of the simulation model used in this study are described, as is the model
setup used for generating synthetic DFITs.
Cohesive Zone Model (CZM)
With the CZM, the fracture is modeled as a gradual separation between two material (rock) surfaces. This
separation is modeled as a progressive degradation of cohesive strength along the cohesive layer, which is
a pre-defined surface embedded in the rock and follows a traction-separation law ( Abaqus Analysis User's
Guide 2016; Chen 2012). With traction-separation behavior (Fig. 1(a)), prior to damage initiation, cohesive


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elements exhibit a reversible linear elastic response (characterized by the cohesive layer stiffness, Kn) until
the traction reaches the maximum tensile strength (T n0) with a separation of δ0. After damage initiation,
the cohesive traction evolves from the maximum strength (T n0) down to zero where the element is fully
damaged with the separation of δf. The damage evolution is governed by the critical energy release rate ( Gc)
that is equal to the area under traction-separation curve. Fig. 1(b) is a schematic of fracture propagation
in a cohesive layer, demonstrating the fully damaged cohesive zone (filled with fracturing fluid) and the
fracture process zone (damage initiation and evolution).
Figure 1—(a) Typical traction-separation behavior; (b) Damage evolution in cohesive elements.
Different guidelines have been presented for selecting the cohesive element mesh sizes and stiffness.
Turon et al. (2007) suggested the following equation for the cohesive layer stiffness:


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


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flow inside the fracture is:
(2)
where w is the fracture opening, and q inj is the injection rate into the fracture.
Figure 2—Plan view schematic of one wing of the fracture.
Tangential Flow
The tangential flow is formulated based on Poiseuille's law:
(3)
where w is the fracture opening, μ is the fluid viscosity and Pf is the fluid pressure along the fracture length.
Leakoff Model
The Carter leak-off model (Howard and Fast 1957) has been used extensively for modeling of hydraulic
fracture propagation and in the analytical solutions for DFIT analysis. It is a one dimensional pressure-
independent leak-off model, applicable to high-viscosity fracturing fluids causing the formation of a low
permeability filter cake on the fracture walls and is described by:
(4)
where qleak(x) is leakoff rate at point x, Ccarter is the Carter leakoff coefficient, t is the time elapsed since the
start of injection and τ(x) is the time when the fracture is created at point x (exposure time). The Carter leakoff


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coefficient depends on filter cake created on fracture walls, and it is not related to formation properties such
as reservoir permeability and pressure.


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SPE-185649-MS 5
In this study, a rigorous leakoff model based on the solution of the 1-D diffusion equation in a half-space
with an imposed pressure history at the boundary is used (Sarvaramin and Garagash 2015):
(5)
where qleak(x) is leakoff rate at point x, S is the storage coefficient and αl is the hydraulic diffusivity. Unlike the
Carter model, this model (Sarvaramin and Garagash 2015) is pressure-dependent, and it is directly related
to formation properties such as permeability, porosity and total compressibility. The final approximation
of this model is similar to Howard and Fast (1957)'s solution for leakoff of low viscosity fracturing fluids,
except that the pressure difference term is not constant, and it is a function of time and location.
Model Setup and Description
In this study, a 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure
in a DFIT. Fig. 3 provides a schematic of the simulation model. Minimum and maximum horizontal stresses


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are acting in the Y and X directions, respectively. In the Base Case model, wellbore storage is neglected,
and the fluid is injected directly at the injection point (perforation). In a more complex scenario, a block
representing the wellbore storage is considered. The fluid is injected into the wellbore block, which is
connected to the model using pipe elements. The pipe elements also model possible pressure drops during
the test.
Figure 3—Plan view schematic of the simulation model setup.
In order to consider the appropriate far-field boundary conditions and model propagation of the fracture in
an infinite medium, the model is surrounded by infinite elements. Infinite elements are used in cases where
the modeled region is small in size compared to the surrounding medium (formation), and they provide
quiet boundaries in the simulation (Abaqus Analysis User's Guide 2016). Chen (2012) and Haddad and
Sepehrnoori (2015) have discussed the advantages of using infinite elements in improving the accuracy


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and runtime of the simulation. For well-testing applications, we have observed improvements in the
representation of pressure transient behaviors, particularly in after-closure flow regimes and removal of
unrealistic boundary effects, using infinite elements.
The cohesive elements are embedded in the formation rock, and act as the potential pathway for hydraulic
fracture growth. During the injection of high pressure fluid, once the required criteria (as discussed in the
previous section) are reached, the cohesive elements can break and act as a hydraulic fracture. During shut-
in, the fracture aperture reduces to closure aperture (w closure), which is a predefined input into the simulator.
After closure, the aperture can further decrease below the wclosure, but it will always be a positive value,


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6 SPE-185649-MS
meaning the fracture retains a finite aperture and conductivity even after closure. In low permeability
formations, even a few microns of residual aperture results in an infinite conductive fracture relative to
matrix permeability.
Table 1 lists properties of the Base Case simulation model. In order to compare simulation results with
field observations, four other models with the same reservoir and rock mechanical properties, but different
settings, are considered (Table 2). In Model 2, the cohesive layer stiffness is reduced to observe its effect on
fracture compliance and pressure behavior. In Model 3, the first cohesive element connected to the injection
point remains open during shut-in and closure. In Model 4, wellbore storage is included that is comparable
with real field data.
Table 1—Base Case model simulation properties
Input parameter Value Input parameter Value
Permeability, md 0.025 Initial porosity, % 10
Young's modulus, GPa 20.7 Fracture fluid viscosity, cp 1


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


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Base Case
Fig. 4(a) illustrates the pressure behavior during injection for the Base Case simulation model. Because no
wellbore is included in the Base Case model, pressure increases quickly during injection, until it reaches the
breakdown point pressure. At this point, damage initiates (fracture initiation), and then propagates along
the cohesive element (fracture propagation). The smooth pressure behavior during propagation indicates
proper mesh size of the model and cohesive elements. Fig. 4(b) illustrates the fracture aperture profile during
propagation and early shut-in time. Fracture tip extension occurs for approximately 1 meter after shut-in.


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SPE-185649-MS 7
Figure 4—Base Case model: (a) pressure profile during injection and fracture propagation; (b)
aperture profile during injection and illustration of tip extension during the early shut-in period.
Fig. 5(a) illustrates fracture opening at a typical element along the length of the fracture (e.g. x=1)
during pressure falloff. As the fluid inside the fracture leaks off to the surrounding formation and pressure
inside the fracture decreases, the fracture opening gradually reduces to the closure aperture. The fracture
aperture further reduces after the closure, but will always be a positive value, meaning the fracture retains
a finite aperture and conductivity even after closure. As demonstrated in Fig. 5(a), the rate of aperture
change with time before closure, 
 , is significantly different from after closure behavior,
. The simulated response is similar to the experimental results of Van Dam et al. (2002).
Based on the continuity equation of fluid flow in the fracture ( Eq. 2), the change of


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at the time of closure
causes an abrupt change in pressure. The same behavior is observed for change of the aperture with pressure
(Fig. 5 (b)) in our simulated case. This is in agreement with the concept of compliance contrast at closure
presented previously by Van Dam et al. (2002) and McClure et al. (2014).
Figure 5—Base Case model: fracture aperture behavior at x=1 during falloff with (a) time and (b) pressure.
Progressive Fracture Closure.   Fig. 6 illustrates the accumulated leakoff volume from the fracture face
along the fracture length for a typical time during falloff (e.g. 15 minutes). The accumulated leakoff volume
is higher at the perforation compared to at the fracture tip. Due to the low permeability of the formation, this
results in a non-uniform pressure profile and effective stress (on fracture walls) along the fracture length
(Fig. 6). The effective stress is maximum at the fracture tip (x=11) and gradually decreases along the fracture


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


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model used herein (Eq. 5), the 
  curve slightly deviates from the straight line with a concave downward
trend. The start of the upward deviation (derivative spike) on the 
  plot is correlated with tip closure, and
the end of derivate spike indicates a fully closed fracture (closure at the wellbore).


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


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


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


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


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


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


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


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


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


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SPE-185649-MS 13
signatures are very similar to Model 4 except that the progressive closure period is longer. The early unit
slope (Fig. 14) and fluctuations indicate fracture volume increase and tip extension due to afterflow. The
transition from a unit slope represents the start of hinge closure. Similar to Model 4, tip extension occurs
during this period as hinge closure pushes the fluid to the tip of the fracture. Progressive fracture closure
starts with PPD violation (or upward deviation on derivative plots) at 22.7 MPa and ends at 19.5 MPa. The
progressive closure period lasts around 5 hours and the difference between closure pressure estimates is
significant.
Figure 14—Field Example 1: before-closure analysis using PTA plot.
Figure 15—Field Example 1: before-closure analysis using the G-function plot.
Field Example 2
This well has been analyzed previously by Hawkes et al. (2013; Well 058L). Again, the overall trend is very
similar to Model 4 (Fig. 16). The upward deviation at the end of the


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slope was of particular interest in the
previous analysis, and no explanation was provided at the time for this feature.


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14 SPE-185649-MS
Figure 16—Field Example 2: before-closure analysis using PTA plot (modified from Hawkes et al., 2013; Well 058L).
The early time unit slope indicates fracture expansion and tip extension due to wellbore storage. At the
end of the unit slope, hinge closure starts and is accompanied by tip extension. After this period (considering
that falloff time in this case is large enough), the 
  slope appears on the Bourdet-derivative. Consistent
with our modeling results, the upward deviation at the end of the 
  slope, that is consistent with the start
of the PPD violation, corresponds to the progressive fracture closure period. The start of the PPD violation
indicates tip closure (52.8 MPa) and the end of PPD violation represents a fully closed fracture (50.9 MPa).
The progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure pressure.
Field Example 3
In this test, pumping time was relatively long (19 minutes). Combined with the low permeability of the


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formation, this probably resulted in a long fracture requiring more time to close. The pressure values
are recorded using surface gauges and significant ambient temperature variations between day and night
occurred during the falloff period. As shown in Fig. 17(a), there are severe fluctuations in the G-function
derivative plot. These functions can result in misinterpretation of the falloff period (e.g. near wellbore
complexities, tip extension, pressure dependent leakoff, etc.); however, when ambient temperature is
plotted, it is observed that pressure fluctuations are a direct effect of ambient temperature variation. To
remove the temperature effect from the surface pressure data, the following equation is used to account for
fluid volume change with pressure and temperature:
(6)
where Preading is the recorded pressure at surface, T is the current ambient temperature, T0 is the ambient
temperature at time of shut-in, β is the thermal expansion coefficient of wellbore fluid and Cf is the


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compressibility of wellbore fluid. In this equation, temperature dependency of β and Cf is neglected which
can further increase its accuracy.


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


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


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


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Other PTA derivative plots are useful in the identification of flow regimes before the start of progressive
closure and explaining other PPD violations. For instance, PPD violations happening before closure during
the unit slope on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are
caused by fracture expansion and tip extension due to wellbore storage.
In previous studies, the magnitude and duration of the tip extension period has been underestimated.
The use of the cohesive zone model in this study enabled us to capture this process accurately and the
simulations are consistent with field observations. The model results illustrate that tip extension can occur
during early fracture expansion period due to wellbore storage and during the hinge closure (and can last
for several hours).
The afterflow caused by wellbore storage can mask the 
  change at closure, and its transition from


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change at closure, and its transition from
the tip of the fracture to near the wellbore. Further, it may result in excessive tip extension and a larger
fracture which delays after closure radial flow. Therefore, removing wellbore storage using downhole shut-
in not only accelerates closure and radial flow, but also provides a clear signature of compliance change
and progressive closure.
A simple method is presented for removing the effect of ambient temperature change on pressure values
without applying large derivative windows for smoothing. The temperature effect may be interpreted
incorrectly as tip extension or pressure dependent leakoff. Further, using large derivative windows results
in loss of accuracy and removal of signatures such as tip extension and PPD violation.
Acknowledgements
The authors would like to thank Dassault Systemes Simulia Corporation for providing Abaqus® program
and Seven Generations Energy Corporation for providing field data. The authors would also like to


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acknowledge Dr. Jack R. Jones for his valuable advice and comments. The sponsors of Tight Oil Consortium
are acknowledged for their support. Partial funding for this project was obtained through an NSERC
Collaborative Research and Development grant (CRDPJ 452752-13) led by Clarkson.
Nomenclature
Ccarter Carte leakoff coefficient, m3/sec0.5
CfWellbore fluid compressibility, Pa-1
EYoung's modulus, Pa
GG-function time, dimensionless
GcCritical energy release rate, Pa.m
KnStiffness of the cohesive layer, Pa/m
mSlope of straight line, dimensionless
PPressure, Pa
PfFluid pressure in the cohesive element, Pa


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SPE-185649-MS 17
qfracGap flow rate within the cohesive element, m3/sec
qinjInjection rate into the cohesive element, m3/sec
qleakLeakoff rate from cohesive element, m3/sec
SStorage coefficient, Pa-1
ShmaxMaximum horizontal stress, Pa
ShminMinimum horizontal stress, Pa
taAgarwal's time, dimensionless
tadjacent Thickness of the adjacent sub-laminate, m
Tn0Maximum tensile strength, MPa
tpPumping time, sec
wFracture opening (aperture), m
wclosure Closure aperture, m
Greek Variables
αCoefficient for calculation of cohesive layer stiffness, dimensionless
αlHydraulic diffusivity, m2/sec
βThermal expansion coefficient, °C-1
δ0Separation at damage initiation, m
δfSeparation at failure, m
ΔtShut-in time, sec
τ(x) Exposure time, sec
μFracturing fluid viscosity, cp
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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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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 
frequently misapplied or misinterpreted leading to confusion


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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. 
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


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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 
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


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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. 
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


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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 
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


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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 
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


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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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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 
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.


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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 
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


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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 
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


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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 
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


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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
 
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


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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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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 
before closure. The slope of these parallel lines is diagnostic 
of the flow regime established during leakoff before closure.


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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 
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


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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 
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


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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
 
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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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 
estimate of 0.097 md is consistent with the Horner and ACA results.   
0.0010.010.1110100


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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 
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.


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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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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 
point, is caused by the changing leakoff associated with PDL


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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, 
appears as a perfect ½ slope of the semilog derivative with a 
parallel pressure difference curve exactly 2-times the


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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 
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)


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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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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 
viscosity is used for the permeability estimate based on 
closure time.  
 
0.0010.010.1110100


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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 
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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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 
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


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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 
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


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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 
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


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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 
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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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 
“belly” below the straight line through the origin and tangent 
to the semilog derivative of p
w vs. G at the point of fracture


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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
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


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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 
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


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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 
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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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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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. 
8. After closure the pseudolinear reservoir flow period is


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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, 
Marathon, for their discussion and contribution to the procedures described. Sincere thanks are also due to the many


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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  
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


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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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10  SPE 107877 
References 
 
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Fracturing Pressure Decline, pa per SPE 3841, presented at the 
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3. Barree, R. D., and Mukherjee, H. : “Determination of Pressure 
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SPE 36424, presented at the 71st Technical Conference and 
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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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Permeability Reservoirs Sympos ium, Denver, Apr. 5-8, 1998. 
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Calibration Tests,” paper SPE 5 2220, presented at the 1999 SPE 
Mid-Continent Operations Symposium, Oklahoma City, OK, 
March 28–31, 1999. 
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North American Rock Mechanics Symposium, July 29, 2000, Seattle, WA. 
8. Craig, D. P. and Bl asingame, T. A.: “Application of a New 
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presented at the SPE Gas Technology Symposium, Calgary, 
Alberta, Canada, May, 15-17, 2006.


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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. 
  
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.


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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 
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


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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 
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


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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 
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


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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 
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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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 
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


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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 
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


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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. 
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


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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 
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


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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 
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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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 
closing hydraulic fracture, and they derived the "complete" 
after-closure impulse solutions accounting for fracture storage.


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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) 
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


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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) 
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


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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 
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


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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 
Derivative


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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 
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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
In general, three different fracture diagnostics groups exist, namely direct far-field


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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
transient testing, and production data analysis ( Cipolla & Wright 2000 ). The pressure
transient testing (sometimes accompanied by rate transient testing) includes step


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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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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
until the fracture opens. At each stage, the injection continues until the pressure stabi-


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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-
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
Closure pressure (psi) 2656 2702Closure point (#) 101 93Sampling rate 1 sTotal number of data points B10310
3


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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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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.
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-
mining closure pressure, leak-off coefficient, and fluid efficiency. This technique ’s cen-


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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
complete closure. Instead, the fracture retains a finite aperture after asperities come
into contact and result in an infinite fracture conductivity. Although this assumption


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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
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-


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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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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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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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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,
2016). This is because that the created fracture during the pumping period of most of


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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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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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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
allows for the reaching an equilibrium condition between the wellbore and fracture


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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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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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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
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-


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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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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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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
from the straight-line, reporting a closure pressure of 5822 psi and closure time of


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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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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
level until the end of the test. The closure point is identified from the intersection of


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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 
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
Closure point (#)   = 37B


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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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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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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
P closure   = 13,068  psi 
T closure   = 713 min
P closure   = 12,815  psi


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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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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
compressibility effect. Also, the pressure was analyzed after applying the corrections


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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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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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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 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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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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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 
hydraulic fracturing treatments, as well as to characterize the subject reservoir. DFIT’s have their basis in co nventional mini-


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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. 
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.


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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 
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


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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 
acquisition procedures are discussed later in this paper. 
 
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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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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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 
points are equally  applicable to conventional  reservoirs . 
 
 
Recommended Acquis ition Procedure


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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 
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


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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 
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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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 
during the injection period, however, if this is not possible or is not done, extremely detailed notes must be kept that detail


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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  
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


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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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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  
inadvertent pressure transients that might be caused by mechanical or logistical issues. Since DFIT analysis uses derivatives,


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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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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 
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.


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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 
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 .


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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 
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


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coefficient of thermal expansion, coupled with density driven convection currents, will still cause a pressure change 
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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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  
Analysis of a pressure falloff begins with determination of the instantaneous shut -in pressure (ISIP). This value is taken to


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represent the fracture extension pressure, when corrected to bottomhole conditions, and the frac ture extension gradient. The 
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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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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 
(3640 psi) is used, the excess pressure drop must be explained.


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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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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 
the frictional pressure drop calculated from a constant tortuosity factor. The injection rate declines from 4 bpm at the inst ant


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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 
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,


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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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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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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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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 
the controlling dimension. This leads to a somewhat circular argument as net pressure, which is  proportional  to fracture


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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 
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


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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 
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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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 (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 
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


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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 
(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


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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 
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


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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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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.  
 
Design of Test Duration  
Longer pumping times allow more time for penetration of the i nduced pressure transient. Given a stable fracture


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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 
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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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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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 
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


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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 
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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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 
fracture half-length  would be approximately 300 feet, for a fracture height of 30 feet (the default assumed height in the plot).


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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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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 
G-function equations and the conventional after-closure solution of the diffusivity equation. A more likely, or at least


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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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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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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 
transitions to a radial flow pattern over a dimensionless time proportional to the assumptions in Equation 4 and Figure 12.


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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 
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


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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 
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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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 
case. As the fluid level begins to drop, the pressure deriv ative (magenta line in Figure 2 1) exhibits a large negative slope.


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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.  
 
 
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 #:


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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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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 
through the toe perforations in a horizontal well. Figure 2 2 (a and b) show the G -function diagnostic derivatives for the test ,


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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 . 
 
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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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 . 
 
In this case , the difference in the estimated pressure is probably insignificant (about 3%), although in many cases the


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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.  
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


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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 
computed reservoir transmissibility. Figure 25(b) indicates an estimate permeability of more than 0.1 md, which is consist ent


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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  
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


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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 
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


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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 
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;


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 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;  
 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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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 
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 
review on the state of the art.


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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 
infinite fracture conductivity is valid whenever 
*Now with Petr61eos Mexicanos and U. Nacional de Mexico. 
0149·2136/8110009·7490$00.25


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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 
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.


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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, 
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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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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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
mensionless fracture storage capacity, CfDJ. ao' ag, 
and {3 are unit conversion constants (see Table 1).


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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 
represents the behavior of a low-conductivity


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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). 
Points A, F, and L in Fig. 2 represent the end of


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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 
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


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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 
characteristics excluding the fracture half-length xf-
The fracture linear flow ends when 
·See Appendix A for derivation.


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·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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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* 
7r \1.1 
PwD = r(1.25)..)2(k
fb
f) D t DXf' ......... (12) 
or 
2.45 II. 
PwD = ..) (kfb


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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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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 
yields a quarter-slope straight line (Fig. 7) that can be


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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. 
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


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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. 
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


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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 
bilinear flow period (one-fourth slope) or the


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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 
of variables used in Fig. lO were derived in Appendix 
B, where it is shown that, for some values of fracture


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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. 
SEPTEMBER 1981 slope=-} 
'~~~~-'~6~2~--~'070~--,O~2~--~,~~~-,~~~~~,o8 
2 t DX
f (kf bflD 
Fig. 10 -Type curve for vertically fractured wells. 
or


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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. 
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. 
Case 1. Pressure data exhibit one-fourth slope on a


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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. 
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


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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 
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.


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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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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. 
Case 3. Pressure data exhibit a half-slope line on a 
log-log graph. (Fig. 13). For this case there is no


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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 
remaining data must follow one of the curves for


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remaining data must follow one of the curves for 
different fracture conductivities. Fig. 14 must be 
used as an auxiliary curve to determine (kfbf) D 
when using Fig. 15. 
Application of Fig. 15 to match pressure data 
provides (PwD)M' (tDr )M' (tlp)M' (t)M' and 
[(kfbf) D]M. Hence, th~ following equations are 
given. 
Reservoir Permeability. For oil, 
DloqBp. k= (PwD) M' ................ (41) h(tlp) M 
102r----------,----------~----------1I----------r_------~ 
.-
til a. 
a. 
E 
<J o 
.D MdLCh Point 
[ot]M c 1 hour 
= 0.35 hour 
t '2 _ 5 hour ebf 
10~----------~----------~----------_t-------
I 
I 
I 
I 
,-- ___ ~_O. 
I 
I o 
I 0 I - -,­
I 
I 
Beginning of Senilog Straight Line 10. 
lo10-1~----~I--~~-----L---L._----i---~------~~~----~~ 3 2 1 101 107 '-_ .!.O-_ I 10-. I 10- _ 1 __ 
- - - - - - - - - - 2 1 102 /). t [hoursl 
10-2 10-1 t DXf (kf bflD 10 
Fig. 16 -Type-curve matching for Well A.


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Fig. 16 -Type-curve matching for Well A. 
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and, for gas, 
(XgqT k= (PwD) M. . ............ (42) 
h[~(p}lM 
Effective WeI/bore Radius. For oil or gas, 
r' =J (3k(t) M w cPJlCt (tDr ~) M· ................. (43) 
By using [(kfbj) DlM in Fig. 14, we obtain 
(r~/xj) Fig. 14; hence, 
r' Xj= [' j ..................... (44) 
r wlXj Fig. 14 
The pressure data falling in the pseudoradial flow 
period also must be analyzed using the semilog 
methods to estimate k and r ~ . 
The discussion in this section clearly indicates (as 
mentioned by Agarwal et af.29) that caution and 
diligence should be exercised when applying the type­
curve matching technique. To perform the type of 
analysis presented in the four cases, it is necessary to 
have prefracture information. 
It should be kept in mind that application of the 
type-curve analysis method to pressure buildup tests 
is appropriate when the producing time is large. 
Examples of Application 
Three examples illustrate the application of several of 
the methods and theory previously discussed. 
Well A


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the methods and theory previously discussed. 
Well A 
A buildup test was conducted in this fractured well 
producing in a low-permeability reservoir. Table 2 
presents the information and results of the analysis 
for this test. Fig. 16 shows a log-log graph of pressure 
data. matching the type curve given in Fig. 10. Notice 
that the first data points are influenced by well bore 
storage and the rest of the data fall in both the 
bilinear and the transition flow periods. The match­
point results also are presented in Fig. 16. A 
minimum value for (k jb j ) D can be estimated from 
the position of the last data point with respect to the 
type curves; for this case, (kjbj}D min =511". 
The end of well bore storage occurs at ap­
proximately 0.35 hour and the end of bilinear flow is 
at 2.5 hours. We also see that the formation linear 
flow period was not reached in this test. 
Since the test was not long enough to match a 
curve for a specific value of (kjbj) D' this example


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curve for a specific value of (kjbj) D' this example 
corresponds to Case 2 in the type-curve-analysis 
section. 
Using the pressure data match and Eq. 38, we 
obtain 
kjbj _ 1,424x7,350x690x6.5 
Xj - 118x109 
=0.3978 md-ftlft. 
SEPTEMBER 1981 TABLE 2 -TEST INFORMATION AND ANALYSIS RESULTS 
FORWELLA 
Reservoir Data 
Production rate q, MscflD 
Producing time tp, hours 
Formation thickness h, ft 
Porosity cJ>, fraction 
Permeability k, md 
Formation temperature T, oR 
Average gas viscosity 1-', cp 
Total compressibility ct, psi-1 
Flowing well bore pressure Pwl, psia 
Analysis Results 
k,b
" md-ft 
x" ft 
(k:~'). md-ftlft 
(klbl)D 
,;.., ft Type-Curve 
Analysis 
148 
373 
0.3978 
15.9 
171.6 7,350 
2,640 
118 
0.1 
0.025 
690 
0.0252 
0.129x 10-3 
1,320 
Bilinear 
Flow Analysis 
154 
368 
0.4185 
-16.71 
-169.3 
From the time-match information and Eq. 39, we can 
calculate 
J 2.637 x 10 4 x 1 
. 0.1 x 0.025 x 1.29 x 10-4 x 0.025 x 3.69 x 10-2 
=373 ft. 
Now, application of Eq. 40 yields 
We also can estimate


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Now, application of Eq. 40 yields 
We also can estimate 
(kjbj}D= kjbf ..!.. 
Xj k 
0.3978 
= 0.025 = 15.9. 
From Fig. 14, r~/xj=0.46. 
:. r~=373x0.46=171.6ft. 
Fi~. 17 shows the bilinear flow graph [~(p) vs. 
llt ~ 1 for this example. Based on the information 
provided by Fig. 16, the correct straight line is 
drawn. The slope of this line is 1.62 X 108 psi2/cp­
hr ~, and at the end of bilinear flow 
llm(p}ebj=2.05x 108 psi2/cp-hr~. Notice that the 
pressure curve after the end of the bilinear flow 
period is concave upward, indicating that 
(kjbj) D > 1.6. 
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'" 0. 10 ~ 
E • 
<J 
o Cucve concave upwards ... (kfbf) D > 1.6 
1~ V. IllI) (hour I 
Fig. 17 -Bilinear flow graph for Well A. 
TABLE 3 -TEST INFORMATION AND ANALYSIS RESULTS 
FOR WEll B ' 
Reservoir Data 
Production rate q, MscflD 
Producing time tp, hours 
Formation thickness h, ft 
Porosity C/>, fraction 
Permeability k, md 
Formation temperature T, OR 
Average gas viscosity p., cp 
Total compressibility ct, psi-1 
Flowing wellbore pressure Pwt, psia 1,675 
1,800 
85 
0.11 
0.0045 
675 
0.025 
0.152x 10-3 
1,250 
Analysis Results Type·Curve Bilinear 
Analysis Flow Analysis 
ktbt, md·ft 
Xt, ft 
(k:~t), md.ftlft 
(ktbt)D 
r';", ft 
From Eq. 20, 95.3 
;;::692 
::50.1377 
::530.6 
::5325 
(kjbj) Yz = (444.75 X 7,350 X 690)/11.62 X 108 X 118 
·(0.1 x 0.0252 x 1.29 x 10-4 xO.025)1~, 
:. (kjbj) = 154md-ft. 
Using Eqs. 22 and 2, 
(k b ) D "'" 1.38 
j j (0.025X1l8X2.05XI08) 
1,424 X 7,350 x 690 
"'" 16.71. 
1760 109 
0: u 
N~ 
'" 0. 
Ci 108 
E 
<J Slope = 1/4 
II t Ihours)


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0: u 
N~ 
'" 0. 
Ci 108 
E 
<J Slope = 1/4 
II t Ihours) 
Fig. 18 -Log·log graph of pressure data for Well B. 
Hence, 
154 ---- "'" 368 ft. 0.025 x 16.71 
From Fig. 14, r:V1xj = 0.46. 
:.r:V =368 x 0.46 = 169.3 ft. 
We see that the results obtained by using both the 
bilinear flow and the type-curve methods are ap­
proximately the same; this fact increases confidence 
in the analysis performed. 
WellB 
A buildup test was run after fracturing this gas well. 
Information about the test and results of analysis are 
presented in Table 3. A log-log graph of pressure 
data (Fig. 18) indicates that the test was completely 
dominated by bilinear flow (quarter slope), 
corresponding this example to Case 1 in the type­
curve-analysis section. 
The bilinear flow graph (Fig. 19) yields a straight 
line whose slope, m bj' can be used to calculate 
(kjbj) usingEq. 20: 
or (kjbj) Yz = (444.75 x 1,675 x 675)/[0.92 x 108 
x 85 x (0.11 x 0.025 x 1.52x 10-4 
x 0.0045) ~], 
If we assume that the last data point corresponds to


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If we assume that the last data point corresponds to 
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0. u 
"''::' U) 
0. 
co 9 
Q 
E 
<J 2 
(.H) last = 18 hours 
mbf = 0.92 x 108 ~, 
cp hrV" 
2 
1;4 1/4 (lit) (hour ) 
Fig. 19 -Bilinear flow graph for Well B. 3 
the end of the bilinear flow period and (k fb f) D ~ 3, 
then from Eq. 31, 
~ lOx2.637xlO-4X(95.3)2 X18 
xf ~ 0.11 x 0.025 x 1.52 x lO -4 x 0.0045 
=692 ft, 
and 
k b 95.3 ~ !S --=0.1377 md-ft/ft. xf 692 
FromEq.6, 
( kfbf) 95.3 
(k fb f) D = kx f !S -0-.0-04-5-x-6-9-2 ' 
From Fig. 14, r~/xf!S0.47; hence, r~ !S0.47 X 692, 
r~ !S325 ft. 
Welle 
After a flowing time of 1,890 hours, a buildup test 
was run on this fractured oil well. Information for 
the test and analysis results are presented in Table 4. 
Fig. 20 shows a log-log graph of the pressure data; 
from this graph we can see that neither a one-fourth 
slope nor a half slope is exhibited by the data. This 
figure also shows that pressure data match the curve 
for (kfbf) D = 27r given in Fig. 15 and the last six 
points fall on the semilog straight line.


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points fall on the semilog straight line. 
From the pressure match obtained in Fig. 20 and 
Eq. 41, we estimate 
k= 141.2 x 220 x 1.2 x 0.8 x 0.34 
49x 100 
=2.07md. 
SEPTEMBER 1981 101 r-----r---,---,---,---,-----, 
Beg~nning of Semllog 
Straight Llnc 
1031-- - -1-- -1--1-- - -
I I I !>latch POl.nt 
I (tlp)1'-! = 100 psi 
(PWO)M '" 0.34 ___ 1 ___ J._ 
I {6t)M = 1 hr 
1 (tDr~)M = 0.19 
1 I ' 
10 '----- -L ___ 1 ____ 1 ___ ..J 
10-1 I 10 102 103 
LIt (hours) 
10-2 L,-----'-:----'--,-----'--~---'7-----'-;;-----.J 
10-3 10-2 10-1 10 1 102 103 
Fig. 20 -Type·curve matching for Well C. 
TABLE 4 -TEST INFORMATION AND ANALYSIS RESULTS 
FORWELLC 
Reservoir Data 
Production rate q, STBID 
Producing time tp, hours 
Formation thickness h, ft 
Porosity </>, fraction 
Viscosity /1-, cp 
Total compressibility ct, psi-1 
Formation volume factor 80, bbllSTB 
Well bore radius r w, ft 
Flowing well bore pressure Pw', pSia 
Analysis Results 
k,md 
(k,btlo 
x" ft 
k,b" md-ft 
r;', ft 
S Type-Curve 
Analysis 
2.07 
211" 
88.7


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k,b" md-ft 
r;', ft 
S Type-Curve 
Analysis 
2.07 
211" 
88.7 
1,156 
36.89 
-4.99 220 
1,890 
49 
0.15 
0.8 
17.6x 10-6 
1.2 
0.25 
1,704 
Semi log 
Analysis 
2.28 
30.37 
-4.8 
Using information from the time match in Eq. 43, 
r' _~ 2.637x 10 4 x2.07x 1 
w-0.15xO.8xI7.6xlO 6xO.19 
= 36.9 ft. 
From Fig. 14, r~/xf=0.415; hence, 
36.9 
xf= 0.415 = 88.9 ft. 
The skin factor is estimated by 
r 0.25 s=ln ~ =In-- = -4 99 
r~ 36.9 .. 
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7.--------,-------,--------,---r----, 
6 
5 
2. 4 
'" ~ m = 307 psi/cycle 
-3 
Q. 
<l 
2 
o 
o 
10-1 00 
o 00 
{l>p} : -47 psi // 1hr 
101 
lit Ihours/ 
Fig. 21 -Semi log graph for Well C. 
The fracture conductivity is 
kjbj= {kjbj)nkXj 
=27rx 2.07 x 88.9= 1,156.2 md-ft. 
Fig. 21 is a semilog graph for this example. The 
correct semilog straight line has a slope m = 307 
psi/cycle and (ilp) Ihr = -47 psi. The formation 
permeability can be calculated as 
k = _16_2_._6_qB_fJ. 
mh 
162.6 x 220 x 1.2 x 0.8 = =2.28md. 307x49 
The skin factor is 
S=1.151[{Llp)lhr -log( k 2)+3.2275J 
m 1>fJ.ctrw 
[-47 = 1.151 -307 
2.28 
-log 0.15 xO.8 x 17.6x 10-6 x (0.25)2 
+ 3.2275 J = -4.8. 
Finally, the effective wellbore radius is 
r~ =r we-s =0.25 e4.8 
= 30.37 ft. 
The results provided by both the type-curve 
analysis and semilog analysis methods are in good 
agreement. From these examples it is demonstrated 
that type-curve analysis, when applied properly, 
provides an excellent diagnostic tool and a technique


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provides an excellent diagnostic tool and a technique 
to estimate both reservoir and fracture parameters. 
1762 Conclusions 
Based on the material presented in this work, the 
following remarks are pertinent. 
1. The transient flow behavior of a vertically 
fractured well may exhibit four flow periods: (a) 
fracture linear flow, (b) bilinear flow, (c) formation 
linear flow, and (d) pseudoradial flow. Bilinear flow 
is a new type of flow that has not been considered 
before. 
2. A new technique is presented to analyze data in 
the bilinear flow period. It is shown that, during this 
flow period, a graph of pwj[or m (Pwj)] vs. Vi yields 
a straight line whose slope is inversely proportional 
to h j{kjbj) Yz. 
3. New type curves are now available for pressure 
analysis of fractured wells. The uniqueness problem 
in the analysis is reduced considerably with the use of 
these type curves. 
4. Prefracture information about the reservoir is 
necessary to estimate fracture parameters.


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necessary to estimate fracture parameters. 
5. The type-curve analysis method must be used 
simultaneously with the specific analysis methods 
(Pwfvs. Vt,Pwjvs.1t, andpwjvs.log t) to produce 
reliable results. 
Acknowledgments 
Part of this work was develo,ped at Stanford U. and 
the Inst. Mexicano del Petroleo. We are grateful to 
many people, especially H.J. Ramey Jr. for his 
encouragement and helpful comments. 
Nomenclature 
b j = fracture width 
B = formation volume factor 
c = compressibility 
C = well bore storage coefficient 
FI ,F2 = correlating parameters for well bore 
storage 
h = formation thickness 
h j = fracture height 
k = permeability 
kjbj = fracture conductivity 
(k jb j) n = dimensionless fracture conductivity 
m = slope of semilog straight line, gas 
pseudopressure 
mbj = slope of straight line for bilinear flow 
P = pressure 
q = well flow rate 
r w = wellbore radius 
r~ = effective well bore radius 
s = skin factor or Laplace space variable 
t = time 
ilt = shut-in time


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t = time 
ilt = shut-in time 
T = reservoir temperature 
X,Y = space coordinates 
Xj = fracture half-length 
(X,{3 = unit conversion constants 
r = gamma function 
'Y/ = hydraulic diffusivity 
1> = porosity 
fJ. = fluid viscosity 
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Subscripts 
b= beginning 
bf = bilinear flow 
g gas 
D= dimensionless 
e = end 
f= fracture, flowing 
i = initial 
/f= linear flow 
0 oil 
t total 
xf based on xf 
w wellbore 
References 
1. Matthews, C.S. and Russell, D.G.: Pressure Buildup and 
Flow Tests in Wells, Monograph Series, SPE, DaJlas (1967) 1. 
2. Ramey, H.J. Jr., Kumar, A., and Gulati, M.S.: Gas Well Test 
Analysis Under Water-Drive Conditions, AGA, Arlington, 
VA (1973). 
3. Earlougher, R.C. Jr.: Advances in Well Test Analysis, 
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4. Theory and Practice of Testing of Gas Wells, third edition, 
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5. Ramey, H.J. Jr.: "Practical Use of Modern Well Test 
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California Regional Meeting, Long Beach, April 8-9, 1976. 
6. Raghavan, R.: "Pressure Behavior of Wells Intercepting 
Fractures," Proc., Invitational Well-Testing Symposium, 
Berkeley, CA, Oct. 19-21, 1977.


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Berkeley, CA, Oct. 19-21, 1977. 
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AIME,225. 
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12. Matthews, C.S., Brons, F., and Hazebroek, P.: "A Method 
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Reservoir," Trans., AIME (1954) 201,182-191.


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Reservoir," Trans., AIME (1954) 201,182-191. 
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14. Clark, K.K.: "Transient Pressure Testing of Fractured Water 
Injection WeJls," J. Pet. Tech. (June 1968) 639-643; Trans., 
AIME,243. 
15. Millheim, K.K. and Cichowicz, L.: "Testing and Analyzing 
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1968) 193-198; Trans., AIME, 243. 
16. Raghavan, R., Cady, G.V., and Ramey, H.J. Jr.: "Well-Test 
Analysis for VerticaJly Fractured Wells," J. Pet. Tech. (Aug. 
1972) 1014-1020; Trans., AIME, 253. 
17. Gringarten, A.C. and Ramey, H.J. Jr.: "Unsteady-State 
Pressure Distributions Created by a WeJl With a Single In­
finite-Conductivity Vertical Fracture, Soc. Pet. Eng. J. (Aug. 
1974) 347-360; Trans., AIME, 257. 
18. Gringarten, A.C. and Ramey, H.J. Jr.: "Unsteady-State 
Pressure Distributions Created by a Well With a Single


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Pressure Distributions Created by a Well With a Single 
Horizontal Fracture, Partial Penetration or Restricted En­
try," Soc. Pet. Eng. J. (Aug. 1974) 413-426; Trans., AIME, 
257. 
19. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: 
"Applied Pressure Analysis for Fractured WeJls," J. Pet. 
Tech. (July 1975) 887-892; Trans., AIME, 259. 
SEPTEMBER 1981 20. Wattenbarger, R.A. and Ramey, H.J. Jr.: "Well Test In­
terpretations of Vertically Fractured Gas WeJls," J. Pet. 
Tech. (May 1969) 625-632; Trans., AIME, 246. 
21. Ramey, H.J. Jr. and Gringarten, A.C.: "Effect of High­
Volume Vertical Fractures on Geothermal Steam WeJl 
Behavior," Proc., Second United Nations Symposium on the 
Use and Development of Geothermal Energy, San Francisco, 
May 20-29, 1975. 
22. Cinco, H., Ramey, H.J. Jr., and Miller, F.: "Unsteady-State 
Pressure Distribution Created by a Well With an Inclined 
Fracture," paper SPE 5591 presented at the SPE 50th Annual 
Technical Conference and Exhibition, Dallas, Sept. 30-0ct. 3, 
1975.


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1975. 
23. Raghavan, R. and Hadinoto, N.: "Analysis of Pressure Data 
for Fractured Wells: The Constant-Pressure Outer Boun­
dary," Soc. Pet. Eng. J. (April 1978) 139-150; Trans., AIME, 
265. 
24. Raghavan, R., Uraiet, A., and Thomas, G.W.: "Vertical 
Fracture Height: Effect on Transient Flow Behavior," Soc. 
Pet. Eng. J. (Aug. 1978) 265-277. 
25. Cinco, H., Samaniego, F., and Domlnguez, N.: "Transient 
Pressure Behavior for a Well with a Finite-Conductivity 
Vertical Fracture," Soc. Pet. Eng. J. (Aug. 1978) 253-264. 
26. Holditch, S.A. and Morse, R.A.: "The Effects of Non-Darcy 
Flow on the Behavior of HydraulicaJly Fractured Wells," J. 
Pet. Tech. (Oct. 1976) 1169-1178. 
27. Ramey, H.J. Jr., Barker, B., Arihara, N., Mao, M.L., and 
Marques, J.K.: "Pressure Transient Testing of Hydraulically 
Fractured Wells," paper presented at American Society 
Topical Meeting, Golden, CO, April 12-14, 1977. 
28. Cinco, H. and Samaniego, F.: "Effect of Wellbore Storage


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28. Cinco, H. and Samaniego, F.: "Effect of Wellbore Storage 
and Damage on the Transient Pressure Behavior of Vertically 
Fractured WeJls," paper SPE 6752 presented at the SPE 52nd 
Annual Technical Conference and Exhibition, Denver, Oct. 9-
12,1977. 
29. Agarwal, R.G., Carter, R.D., and Pollock, C.B.: 
"Evaluation and Prediction of Performance of Low­
Permeability Gas WeJls Stimulated by Massive Hydraulic 
Fracturing," J. Pet. Tech. (March 1979) 362-372; Trans., 
AIME,267. 
30. Barker, B.: "Transient Flow to Finite-Conductivity Vertical 
Fractures," PhD dissertation, Stanford U., Palo Alto, CA 
1977. 
31. Barker, B.J. and Ramey, H.J. Jr.: "Transient Flow to Finite­
Conductivity Vertical Fractures," paper SPE 7489 presented 
at the SPE 53rd Annual Technical Conference and Exhibition, 
Houston, Oct. 1-3, 1978. 
32. Scott, J.O.: "A New Method for Determining Flow 
Characteristics of Fractured WeJls -Application to Gas WeJls 
in Tight Formations," paper 78-T-2, AGA 1978 Transmission


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in Tight Formations," paper 78-T-2, AGA 1978 Transmission 
Conference, Montreal, T-279-186. 
APPENDIX A 
Short-Time Transient Pressure 
Behavior for a Well With 
a Finite-Conductivity Fracture 
Derivation of Solution 
Let us consider the system described in the text of this 
paper. A fractured well produces at a constant flow 
rate from an infinite reservoir. At small time values, 
the pressure behavior of the system is not affected by 
the tips of the fracture. In addition, the flow in the 
formation is essentially linear and perpendicular to 
the fracture plane. The reservoir and the fracture 
may be treated as two different homogeneous 
regions. 25 
If the flow within the fracture is assumed to be 
linear, the pressure behavior can be described by 
a2pfD 
a 2 + xD 2 apD I 
(kfbf) D aYD YD=O 
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= _1_ ap[D .................. (A-I) 
7JjD atDXj 
for O<xD < 00, tDxj >0. 
Initial Condition. 
PfD =0, tDXj=O, O:s;xD <00. . .......... (A-2) 
Boundary Conditions. 
apfD I = - 7r , tDx >0, 
aXD XD=O (kfbf)D f 
. . . . . . . . . . . . . . (A-3) 
and 
lim PfD =0, tDxj >0, ............... (A-4) 
XD-OO 
where, in oilfield units, 
_ kh[Pi-Pf] 
PjD -141.2 qBJ.I. 
and kh [Pi -p] 
P D = 141.2 qBJ.I. 
x xD=-' xf 
Y YD=-· xf 
The variables not included above are defined in the 
text. P and P f represent the formation pressure and 
the fracture pressure, respectively, x is the flow 
direction in the fracture, and Y is the formation flow 
direction perpendicular to the fracture plane. 
The transient flow in the formation may be 
described by 
a2p ap D_~·O<y <00· ~-!:I ' D ' aYD utDxj 
tDxj>O. . ................... (A-5) 
Initial Condition. 
PD =0; O<YD <00; tDXj=O ............. (A-6) 
Boundary Conditions. 
PD IYD=O =PfD; tDXj>O, .............. (A-7) 
and 
lim PD =0; tDxj>O ................. (A-8) 
YD-OO


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and 
lim PD =0; tDxj>O ................. (A-8) 
YD-OO 
1764 The two partial differential equations are already 
coupled by the boundary conditions. Application of 
the Laplace transformation with respect to. time to 
Eqs. A-I through A-8 and simplification yields 
a2pfD 2 apD I + --
aXD2 (kfbf) D aYD Yd=O 
s _ 
= -PfD; O<xD < 00 ........... (A-9) 
7JjD 
Boundary Conditions . 
apfD I = - 7r , ••••••••••• (A-lO) 
aXD Xd=O (kfbf) D 
lim PjD =0, ..................... (A-ll) 
XD-OO 
and 
a2pD 
aYD2 =SPD; O<YD<oo ............ (A-12) 
Boundary Conditions. 
PD IYD=O =PfD' .................... (A-l3) 
lim PD=O' ..................... . (A-14) 
YD-OO 
where 
PjD (xD' s) =£tDXj [PjD (XD,IDXj)]' 
and 
PD (YD' s) = £tDXj [PD (YD' tDXj) ]. 
Now let us define 
PD (r,s) = £YD [PD (YD,s)]. 
Application of the Laplace transformation, with 
respect tOYD, to Eq. A-12 yields 
2 ~ I ap D I ~ (A 15) r PD-rPD YD=O--a-- =sPD···-
'YD YD=O 
By solving for PD and considering Eq. A-l3, we 
obtain 
_ oPD I rPfD+ -a-


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obtain 
_ oPD I rPfD+ -a-
~ 'YD YD=O (A 16) PD= 2 • ...•.•.•.. -r -s 
The inversion of this equation with respect to Y D 
yields 
a-
P D = P jD cosh (...;s Y D) + : D I 
'YD YD=O 
................. (A-17) 
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From Eqs. A-14 and A-17, we can write 
apD I - -- = -PfDYs. . ............. (A-18) 
aYD YD=O 
Next, substitution of Eq. A-18 into Eq. A-9 allows 
us to write a partial differential equation with only 
one dependent variable: 
a2pfD _ S 2Ys_ 
2 - -+ PfD· ........ (A-19) 
aXD 'YlfD (kfbf) D 
The solution of Eq. A-19 with boundary conditions 
given by Eqs. A-tO and A-II can be expressed as 
[ [ S 2Ys] Yz 7rexp -xD -+ 
-_ 'YlfD (kfbf) D ] prD-
p S 2Ys Yz 
(kfbf}DS[-+ b ] 
'YlfD (kf f) D 
....................... (A-20) 
The pressure at the wellbore PwD is calculated at 
xD =0; thus, 
7r 
PwD = S 2Ys Y2·· (A-21) 
(kfbf}D s[-+ ] 
'YlfD (kfbf}D 
Finally, the Laplace inversion of Eq. A-21 with 
respect to 1 DXf produces 
PwD= ~ rDXf 
(kfbf) D 0 
....................... (A-22) 
Unfortunately, Eq. A-22 is too complex to analyze 
both the short-and long-time behavior of the 
solution. Eq. A-21 may be used for this purpose. 
Short-Time Behavior 
The short-time approximation of the solution can be


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The short-time approximation of the solution can be 
obtained from Eq. A-21 by taking the limit as the 
variable S approaches infinity. Thus, 
--~ PwD-(k
fb
f}DS312 ................. (A-23) 
Inversion of this formula produces Eq. 8. 
Long-Time Behavior 
We can obtain the solution for large values of time by 
taking the limit of Eq. A-21 as S approaches zero; 
hence, 
7r 
PwD =::. ..J2(k
fb
f) D s5/4· ............. (A-24) 
Inverting Eq. A-24 produces Eq. 12. This solution 
also may be obtained by considering incompressible 
flow within the fracture. 
SEPTEMBER 1981 Total Formation Flow 
Eqs. A-18 and A-20 can be used to calculate the 
fraction of the well flow rate that is being produced 
from the formation. The formula for this case ap­
pears to be 
q formation = 1 _ exp( 41 DXr 2 ) 
q 7r2CfDf 
( 2..JIDxr ) . erfc . . ........ (A-25) 
7rCfDf 
For small values of 1 Dx' the amount of fluid 
coming from the formation {s negligible; however, at 
large values of 1 Dx' most of the produced fluid


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large values of 1 Dx' most of the produced fluid 
comes from the formation. This illustrates the 
physical behavior of the system. 
APPENDIXB 
Derivation of Correlation Parameters 
for Bilinear and Linear Flows 
If we consider a fracture of length xf' Eq. A-4 
becomes 
apfD I =0 ....................... (B-1) 
aXD xD=1 
Using Eq. B-1 instead of Eq. A-4 and considering 
incompressible flow within the fracture, the Laplace 
transformation solution for the problem stated in 
Appendix A is 
pWD=(-7r+ E 27rexp[-2n 
n=O 
[~ (kfb~)D2 tJ )/(S(kl'tln 
.[~ (kfb~)D2 t) ............ (B-2) 
or 
PWD (kfbf)D3~( -<+ 00 E 27r eXPt -2n 
n=O 
Hence, 
1 [ S ~] PWD(kfbf}D = (k
fb
f}D2! (k
fb
f}D2 ... (B-4) 
From Laplace inversion tables, 
PWD (kfbf) D =F[tDx/ (kfbf) D2]. . ..... (B-6) 
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Hence, a graph of Pw (kfbf)D vs. tDx (kfbf)2 . ld . I D f Yle s a smg e curve. 
For small values of dimensionless time, Eq. B-3 
yields Eq. A-24, corresponding to bilinear flow. 
The long-time approximation of Eq. B-3 yields 
7r P WD ~ 2s312 ' .••••••••••••••••••••••• (B-7) 
which is the Laplace transform of the dimensionless 
pressure for the formation linear flow. 
APPENDIXC 
Derivation of Correlation Parameters 
for Bilinear Flow with 
Wellbore Storage Effects 
When well bore storage effects are considered in the 
flow problem defined in Appendix A, the boundary 
condition at x D = 0 becomes 
apfD I 7r (1-CDf apWD), . (C-l) 
aXD xD=O (kfbf) D atDxf 
where 
C = C Df 27rA.c hx 2' ................... (C-2) 
'I' t f 
and C is the wellbore storage coefficient. 
If flow within the fracture is considered in­
compressible and Eq. C-1 is taken instead of Eq. A-
3, the solution for the wellbore pressure in the 
Laplace space is given by 
7r 
PwD = -V2(kfbf) D s5/4 + 7rCDfS2' 
which can be written as


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7r 
PwD = -V2(kfbf) D s5/4 + 7rCDfS2' 
which can be written as 
1766 ...... (C-3) 4/3 C 4/3 s 5/4 
(kfbf) D - _ ~Y2[ Df ] 
C 5/3 PwD-7r (k b ) 2/3 Df f f D 
CD//3s ~ 
+7r[ (kfbf)D2/3] j" .. (C-4) 
From properties of the Laplace transformation, 
-I _1 [t] L !(as) -~F ~ ................... (C-5) 
Hence, Eq. C-4 can be written as 
(kfbf)D2I3 =F[ (kfbf)D2I3 t ] 
(CDf) 113 PwD (CDf) 4/3 DXf' 
................... (C-6) 
This means that the bilinear flow solutions including 
well bore storage are represented by a single curve 
when graphed in terms of 
(k b ) 213 (k b ) 2/3 
--""f'--Lf'I::;'~;--PWD vs. f f ~/3 tDx' Cw (Cw) f 
SI Metric Conversion Factors 
cp x 1.0* E-03 = Pa·s 
ft x 3.048* E-Ol = m 
psi x 6.894 757 E + 00 = kPa 
·Conversion factor is exact. JPT 
Original manuscript received in Society of Petroleum Engineers office Aug. 
10, 1978. Paper accepted for publication June 4, 1979. Revised manuscript 
received July 6,1981. Paper (SPE 7490) first presented at the SPE 53rd Annual


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Technical Conference and Exhibition, held in Houston, Oct. 1-3, 1978. 
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Citation: Gabry, M.A.; Eltaleb, I.;
Soliman, M.Y.; Farouq-Ali, S.M. A
New Technique for Estimating Stress
from Fracture Injection Tests Using
Continuous Wavelet Transform.
Energies 2023 ,16, 764. https://
doi.org/10.3390/en16020764
Academic Editor: Maria Grazia
De Giorgi
Received: 2 December 2022
Revised: 23 December 2022
Accepted: 27 December 2022
Published: 9 January 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
A New Technique for Estimating Stress from Fracture Injection
Tests Using Continuous Wavelet Transform†
Mohamed Adel Gabry *
 , Ibrahim Eltaleb
 , Mohamed Y. Soliman
 and Syed M. Farouq-Ali
Department of Petroleum Engineering, The University of Houston, Houston, TX 77023, USA
*Correspondence: magabry@cougarnet.uh.edu


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*Correspondence: magabry@cougarnet.uh.edu
† Society of Petroleum Engineers Technical Conference and Exhibition, Houston, TX, USA, 3–5 October 2022.
Abstract: The diagnostic fracture injection test (DFIT) is widely used to obtain the fracture closure
pressure, reservoir permeability, and reservoir pressure. Conventional methods for analyzing DFIT
are based on the assumption that a vertical well is drilled in ultra-low permeability reservoirs with
potential multiple closures but fails to consider horizontal wells. There is still signiﬁcant debate about
the rigorousness and validity of these techniques due to the complexity of the hydraulic fracture
opening and closure process and assumptions of conventional fracture detection methods. The
paper introduces a new method for detecting fracture closure pressure using the continuous wavelet
transform (CWT). The new method aims to decompose the pressure fall-off signal into multiple levels


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with different frequencies using the CWT. This “short wavy” function is stretched or compressed and
placed at many positions along the signal to be analyzed. The wavelet then convoluted the signal
yielding a wavelet coefﬁcient value. The signal energy is observed during the fracture closure process
(pressure fall-off) and the fracture closure event is identiﬁed when the signal energy stabilizes to
a minimum level. A predeﬁned simple commercial fracture simulation case with known fracture
closure, ﬂow regime modeling, and actual ﬁeld cases was used to validate the new methodology.
Keywords: continuous wavelet transform; diagnostic fracture injection test; hydraulic fracture;
minimum horizontal stress; geothermal reservoirs
1. Introduction
1.1. Scope of Study
Estimating the minimum horizontal stress is a critical part of building formation
geomechanical models, designing hydraulic fractures, designing drilling schedules, etc.


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Hydraulic fracturing design and execution need to calibrate the mechanical Earth model
with a reliable method. This is performed to identify the formation stresses. Classical
methods for obtaining the minimum horizontal stress of the formation relies on the open-
hole well logs (Eaton (1969) [ 1]; Blanton and Olson (1999) [ 2]). These methods can be
categorized into two main techniques. The ﬁrst technique relies on compressional and
shear sonic waves to estimate dynamic geo-mechanical properties. The second technique
uses correlations to calculate the dynamic properties of the rock from conventional well-
logging parameters, then calculates the minimum horizontal stress [3].
The actual in situ minimum principal stress of the formation is calculated by calibrating
against the closure pressure from diagnostic pumping test [ 4]. That is performed by
pumping a small volume of non-wall building ﬂuids to create a small fracture and then


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analyze the pressure decline to know the pressure at which that fracture closes. That
fracture closure pressure should be the average of the minimum principal stress for the
area covered by the created fracture.
As no possible physical observation of the fracture propagation and closure in com-
plicated subsurface formations, there are signiﬁcant debates surrounding the various
methodologies used to determine closure pressure. This is because of several reasons. The
Energies 2023 ,16, 764. https://doi.org/10.3390/en16020764 https://www.mdpi.com/journal/energies


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Energies 2023 ,16, 764 2 of 29
ﬁrst reason is that the hydraulic fracture propagation and closure process are fairly compli-
cated due to the formation stress state, the presence of natural fractures, heterogeneity, and
the anisotropy of the subsurface formations’ properties. Conventional closure detection
methods have pre-set assumptions to simplify the state of the subsurface formation. Some
of them may simplify the leak off to carter leak off, whilst other methods assume the
constant compliance or make certain assumptions regarding the ﬂow regimes or the state
of injection. Many tests may deviate from the assumed pre-set conditions. The detected
closure be confused by the formation pressure response after the fracture closure. The
pressure change propagates in the formation and produces behavior that may be confused
with fracture closure detection.
Although fracture closure is deﬁned as the pressure at which a fracture closes, there


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is an attempt to detect the contact pressure when the two fracture faces ﬁrst come into
contact with one another. Assuming that closure pressure does not really exist, there is an
aperture between fracture faces. Complications regarding how the fracture propagates and
closes cause the interpreted fracture closure pressure debatable. In most cases, the industry
workers try various methods to determine the range of closure pressure. To date, there is no
global technique that can detect closure without pre-assumptions. Each of the conventional
methods detects the closure from a speciﬁc point of view with certain pre-assumptions.
This paper introduces an innovative method to ﬁnd the closure pressure using the wavelet
transform technique as a general technique without any pre-assumptions. It uses an
independent mathematical microscope to ﬁnd the fracture closure pressure in the pressure
fall-off signal. The new technique is validated using different ways: using simulated


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synthetic pressure data from a fracture simulator and using ﬂow regime modeling by
reverse calculating a pressure decay signal assumed to be both before fracture closure and
after fracture closure ﬂow regimes. The main advantage of this technique is that there is no
need to know formation parameters in advance or have a pre-prescribed testing procedure.
Since the beginning of the 1990s, the science and engineering applications of wavelet
transform have increased at a remarkable growth rate. Thousands of peer-reviewed journals
are now interested in wavelet transform applications covering numerous disciplines such
as ﬂuids, medicine, and fractal geometry. However, its use in the oil and gas industry
and geothermal energy has been limited to date. A lot of the data recorded in multiple
disciplines are considered as signals to be analyzed to understand the physics-based
models covering the many applications. For example, Germán-Salló et al. (2018) [ 5]


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introduced a method for crack detection in manufacturing systems using digital signal
processing methods.
Signal processing is widely used in the medical ﬁeld to detect heart problems by
analyzing electrocardiography (ECG) signals. Research in the medical ﬁeld deploys wavelet
transform to compare the ECG signals of a healthy heart to another ECG signal associated
with a diseased heart. The same methodology is adopted in the industrial application
of signal processing to predict motor failure. For instance, the analysis of the vibration
signal of a motor can help detect bearing failure when the recorded signal is compared to a
functional motor.
With the ongoing digitalization of the modern oil and gas industry and geothermal
energy industry, the same technique can be used in the analysis of hydraulic fracturing. In
this case, changes in pressure and rate over time are recorded during the fracture propa-
gation stages, proppant placement, and fracture closure to determine what is happening


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thousands of feet below the Earth’s surface during hydraulic fracturing treatment. The
most common methodology used by researchers for engineering wavelets applications is
to try to ﬁnd a correlation between the features detected from the signals and the actual
physical response of the system. The main problem with this methodology in hydraulic
fracturing is the complexity of hydraulic fracturing propagation and closure due to fracture
face roughness or natural fracture intensity and directions and a lot of other factors. This
may be complex not only because of the heterogeneity of the reservoir, but also because of
the interaction of the hydraulic fracture with natural fractures, ﬁssures, and other geologic


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Energies 2023 ,16, 764 3 of 29
features. Thus, all the complicating factors should be removed from the system to ﬁnd a
signal representing a closure event in a homogeneous reservoir with no geologic complexi-
ties. In order to understand the CWT behavior during fracture closure, a numerical fracture
simulator is used in this study.
The application of wavelet analysis in the oil industry was pioneered twenty years
ago by Soliman et al. (2003) [ 6] as an application for well-test analysis and fracturing. The
wavelet transform technique is a data transform technique that convolves the pressure
and/or temperature data using a short wavy signal called “wavelet”. This convolution can
be performed at discrete points in the discrete wavelet transform (DWT) or continuously
using the CWT. The wavelet transform provides a representation of the pressure signal
by letting the translation and scales of the wavelets vary continuously. This enables the


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analyst to ﬁnd the details of the pressure data by observing the wavelet energy spectrum
for the monitored signal (pressure and/or temperature).
As the DWT is computationally cheaper, it was used as the ﬁrst step to detect fracture
closure. The main disadvantage of using the DWT is its requirement of a uniform sampling
rate and the detected closure pressure changes with an increasing level of decomposition.
Soliman, Unal, Siddiqui, Rezaei, and Eltaleb (2019) [7,8] initiated a methodology to detect
the fracture closure pressure using the DWT. Determining the closure pressure in this
methodology is based on analyzing the detail levels to pick the time when the variance
changes. Eltaleb et al. (2020) [ 9] and Eltaleb et al. (2021) [ 10] improved the rigor of the
technique using an energy density plot. The technique was compared to the G-function
technique and showed good agreement. The DWT technique involves a multiresolution


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wavelet decomposition that breaks the (recorder pressure) signal into high pass (noise) and
low pass (approximation) components at various levels. The plot of energy distribution
in time is constructed using the energy of the high pass component (noise) at different
decomposition levels. These plots provide means of capturing the physical changes in
the fracture system through pressure fall-off. They represented the noise energy in the
recorded pressure at different frequency levels. The closure pressure in this methodology
was determined based on changes in the noise energy of the recorded pressure and where
it drops to a minimum stabilized level.
This paper introduces a closure detection technique similar to that introduced by
Eltaleb et al. (2021) [ 10], however, using the CWT. Using the CWT does not need a
uniform sampling rate and captures all signal details. CWT can be used as a method for
detecting the fractal dimension which is one of the key features of complex wavelets that


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is sometimes referred to as a “mathematical microscope” to detect the features. Using
CWT was recommended by Fatmir (2020) through personal communication. CWT can
be used as a ﬁlter to detect the signal features. It has the ﬂexibility to be used with or
without a uniform sampling rate. Thus, it could be validated using any type of data. This
new method was applied to both the fracture simulator data and actual ﬁeld data. The
synthetic data were produced using a commercially available fracture simulator developed
by Barree (1983) [ 11] based on fracture propagation and closure simulation principles with
predeﬁned fracture closure. Azlinda et al. (2010) [ 12] showed good agreement between the
used fracture simulator with down-hole events in a stacked ﬂuvial pay system. The CWT
technique showed success with the actual real ﬁeld data conducted on one of the oil and
gas ﬁelds and in another case in a geothermal reservoir.
1.2. History of Closure Pressure Detection Techniques


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1.2. History of Closure Pressure Detection Techniques
The basic concept of classical methods of closure event detection from the pressure
decay of the diagnostic fracture injection test is combining the basic concepts of well
testing and hydraulic fracture analysis. After hydraulic fracturing was ﬁrst introduced by
Clark (1949) [ 13], it was recognized that hydraulic fracturing could be used as a method
to estimate the magnitude of minimum principal stress (Hubbert and Willis (1957) [ 14];
Godbey and Hodges (1958) [ 15]; Kehle (1964) [ 16]; Haimson and Fairhurst (1967) [ 17];
Hickman and Zoback (1983) [ 18]). The main idea was to pump ﬂuid at a pressure higher


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Energies 2023 ,16, 764 4 of 29
than the fracturing pressure, to initiate a hydraulic fracture and propagate it through the
formation. The minimum principal stress can be estimated from pressure measurements
during and after injection. Nolte (1979) [ 19] extended the applicability of fracturing tests
by developing analytical techniques that can be used to estimate ﬂuid leak-off parameters.
These techniques can be used to obtain the leak-off rate of complex ﬂuids such as linear gel
or cross-linked gel. These tests are typically called mini-frac tests (McLennan and Roegiers,
(1982) [ 20]) or fracture calibration tests (Mayerhofer and Economides (1993) [ 21]). This is
performed to estimate the efﬁciency of the fracturing ﬂuid after the interpretation of the
minimum horizontal stress. One of the most widely used functions to obtain the closure
pressure is G-function (Nolte (1979) [ 19]). Several techniques have subsequently been


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published as essential parts of the analysis, including those by Castillo (1987) [ 22] as well
as Barree and Mukherjee (1996) [ 23]. Further modiﬁcations for the analysis were added,
for example, by Mayerhofer and Economides (1993) [ 21] and Mayerhofer et al. (1995) [ 24]
showed how to remove the ﬁlter cake effect and its effect on the ﬂuid leak-off factor from
the analysis. The recent development of the shale gas reservoirs came out with a new test
that uses the non-wall building ﬂuid instead of the cross-linked gel or linear gel. It is called
the “diagnostic fracture injection test” (DFIT). It is routinely performed in unconventional
reservoirs due to the incapability to ﬂow the well to do a pressure transient analysis. The
DFIT is used to obtain the best estimates for stress, permeability, and pore pressure (Craig
and Brown (1999) [25]. The analysis of pressure decay data has two parts.
The ﬁrst part of the DFIT analysis is called “Pre-closure analysis methods” which


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are usually based on techniques from Nolte (1979) [ 19] and Mayerhofer and Economides
(1993) [ 21], Mayerhofer et al. (1995) [ 24], Barree and Mukherjee (1996) [ 23], and Barree
et al. (2009) [ 26]. These methods are used to estimate the closure pressure and calibrate
the minimum principal stress. The second part is called ‘Post-closure’ analysis methods
which are usually based on techniques from conventional well testing (Gu et al. (1993) [ 27];
Soliman et al. (2005) [ 28]; Craig and Blasingame (2006) [ 29]) or the ‘linear ﬂow time
function’ technique developed by Nolte et al. (1997) [ 30]. Due the development of uncon-
ventional reservoirs, multiple DFIT interpretation techniques have recently been developed
by Marongiu-Porcu et al. (2011; 2014) [ 31,32], Soliman and Kabir (2012) [ 33], Soliman and
Gamadi (2012) [ 33], Cramer and Nguyen (2013) [ 34], Padmakar (2013) [ 35], Wallace et al.
(2014) [ 36], Meng et al. (2014) [ 37], Craig (2014) [ 38], Zanganeh et al. (2018) [ 39], and
Hawkes et al. (2018) [40].


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Hawkes et al. (2018) [40].
One of the noticeable techniques is the compliance method that was recently intro-
duced by McClure et al. (2014; 2016) [ 41,42] and Jung et al. (2016) [ 43]. This technique
usually leads to an earlier and higher stress estimate in contrast to the commonly used
tangent method (Barree et al. (2009) [ 26]) which is considered a modiﬁcations to Nolte’s
(1979) [ 19] technique. The technique introduced a new concept which is contact pressure
instead of closure pressure. McClure et al. (2016) [ 42] and Jung et al. (2016) [ 43] per-
formed detailed DFIT simulations considering the effect of a residual aperture fracture
after the contact between the fracture walls. Simulations indicate that, in low permeabil-
ity formations, the contacting of the fracture walls causes the magnitude of the pressure
derivative to increase, resulting in a pressure signal that was previously interpreted as
the height recession or closure of transverse fractures (Barree et al. (2009) [ 26]. Wang


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and Sharma (2017) [ 44] reproduced these ﬁndings and proposed a modiﬁed approach for
estimating stress called the ‘variable compliance method’. Currently, there are two main
closure detection techniques that are used by the industry, the tangent method, and the
compliance method.
The initial technique of the G function introduced by Nolte (1979) [ 19] relied on several
assumptions for the pumping and shut-in periods with some simpliﬁed concepts of how a
fracture propagates and closes. Nolte (1979) [ 19] provided the G time function to detect
the changes in the pressure during the shut-in period of the diagnostic pumping. The
second derivative (GdP/dG) is used to detect the closure. The closure deﬁned by this
technique is the pressure at a time when the deviation from the straight line starting from


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Energies 2023 ,16, 764 5 of 29
zero in GdP/dG occurs. The detailed equations for the G time function were discussed
by Nolte (1979) [ 19]. Castillo (1987) [ 22] added the condition that dP/dG should be with
zero slopes at the same point. The tangent method was introduced to cope with the
complicated fracture closure as a modiﬁcation for the Nolte (1979) [ 19] technique. The
tangent method proposed four types of leak-off in the formation during the closure, normal
leak-off, pressure-dependent leak-off, transverse storage leak-off or height recessions, and
tip extension leak-off. It is taken from a plot of GdP/dG versus G. A straight line is
drawn from the origin to the tangent to the curve. The closure pressure is deﬁned as the
pressure when the GdP/dG starts to deviate downward. The pressure at that point is
taken as the tangent method stress estimate. The tangent method was developed to match
numerical simulations by Barree and Mukherjee (1996) [ 23] and Barree et al. (2009) [ 26].


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Their work introduced a G-function signature for each type of leak-off. Each type of leak-
off can characterize speciﬁc reservoir properties such as the presence of natural fractures,
the presence of lower permeability streaks or weak barriers surrounding the pay zone.
The tangent method in many cases shows a monotonic increase which makes the closure
pressure too low to be identiﬁed. The tangent method is affected by the formation response
after the fracture closure. It is normally associated with the square root of time and log–log
analysis, as recommended by Baree et al. (2009) [ 26]. The pre-closure ﬂow regime and
post-closure ﬂow regimes should be identiﬁed to support the closure detected using the
tangent method. The second derivative of pressure to the square root of time should show
a local peak at the closure. The relations between the log–log, square root of time, and
tangent method were discussed in detail by Baree et al. (2009) [ 26]. The log–log technique


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and square root of time method can be combined as the “Holistic Fracture Diagnostics” as
described by Baree et al. (2007) [45].
The compliance method for estimating stress was developed from mathematical
solutions of the fracture closure process (McClure et al. (2016) [ 42]). As per the compliance
method, the very early part of the shut-in is ignored as the pressure drops very rapidly.
This part of the transient is ignored because it is caused by the dissipation of near-wellbore
tortuosity related to the initiation of a transverse hydraulic fracture from a horizontal
well. The straight section of the pressure versus G-time plot can be extrapolated back to
the y-intercept to estimate the pressure in the far-ﬁeld fracture at shut-in (the so-called
effective initial shut-in pressure). After the initial period, the pressure curve settles into
an approximately straight line. The ‘compliance method’ of estimating stress identiﬁes
the contact pressure at the point where the derivative increased from the minimum and


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subtracts 75 psi to account for a stress shadow from the residual aperture (McClure et al.
(2019) [ 46]). The 75 psi adjustment is based on a numerical simulation that matches the
ﬁeld DFITs (McClure et al. (2019) [46]).
When the fracture walls contact, the stiffness increases, and the storage coefﬁcient
drops as per Sneddon (1946) [ 47]. This causes the dP/dG to increase when the fracture walls
come into contact. McClure et al. (2016 [ 42], 2019 [ 46]) note that, because of the roughness
of the fracture walls, the ﬂuid pressure may be higher than the minimum principal stress
when the walls come into contact. Thus, it is recommended to estimate the closure pressure
as 75 psi lower than the contact pressure based on numerical simulations. The contact
pressure is estimated by plotting the magnitude of dP/dG and identifying the point where
this derivative reaches a minimum and begins to increase. This is the so-called ‘compliance’


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method stress estimate. It is noticeable that the ‘compliance method’ is close to the classical
method by Nolte (1979) [ 19]. However, an upward deﬂection in dP/dG is not always seen
in the shut-in transients.
In a lot of cases, dP/dG continuously decreases during the shut-in. McClure et al.
(2019) [ 46] discussed the possible causes of the lack of an upward deﬂection. That may
be due to the excessive near-wellbore tortuosity as the fracture closes almost immediately
after shut-in. The monotonic dP/dG might arise from the continued fracture propagation
after shut-in. One of the key assumptions of G-function is that the fracture does not
propagate after shut-in. This makes the G-function not accurate in many cases. McClure


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Energies 2023 ,16, 764 6 of 29
et al. (2019) [ 46] recommended interpreting tests with a monotonic dP/dG as indicating
‘rapid closure’ in vertical wells where the near wellbore tortuosity is usually minimal and
supplementing dP/dG plots with plots of ‘relative stiffness’ versus pressure. The relative
stiffness is not a normalized value calculated to be proportional to the system stiffness.
In tests where near-wellbore tortuosity is possible, McClure et al. (2019 [ 46]; 2022 [ 48])
recommended declining to provide a stress estimate and labeling the stress uninterpretable.
McClure et al. (2022) [ 48] found that ﬁfty-nine percent of cases showed a clear upward
deﬂection; twenty-ﬁve percent of cases showed an adequate upward deﬂection; and sixteen
percent of cases showed no indication of an upward deﬂection (i.e., a monotonic decrease
in dP/dG).
From the previous discussions, closure detection is still open for research and needs


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more studies to understand the complications of fracture closure without pre-assumptions
and fewer data requirements, which is the main advantage of wavelet transform techniques.
2. Methodology
2.1. Wavelet Transform
Wavelet transform is a mathematical transformation applied to signals to obtain
information that is not readily available. This signal can be pressure or any parameter
recorded during ﬁeld operations. In practice, most signals are in the time domain in their
raw format and plotted as a time–amplitude representation of the signal. As a result,
this is not always the best representation of the signal for most signal processing-related
applications; hence, the most valuable features are usually hidden in the signal frequency.
Several transformations may be used to analyze the acquired signals. Fourier trans-
forms (FTs) are the most widely used signal processing technique that gives signal frequency
information. However, it does not provide timely information on which those frequency


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components exist. FT works best in the analysis of stationary signals whose time domain
is periodic; then, the time domain signal can be converted into a combination of sine and
cosine waves to inﬁnity duration (Rioul and Vetidsaerli (1991) [49]).
One modiﬁcation for the Fourier transform is the windowed Fourier transform (WFT).
It is a tool for extracting local-frequency information from the non-stationary signal. The
signal is divided into small stationary segments through window functions whose width
must equal the part of the signal where it is stationary. As discussed by Kaiser (1994) [ 50],
WFT is prone to inaccurate results due to the aliasing of high- and low-frequency compo-
nents that do not fall within the window function proposed by the technique. Wavelet
transform (WT) has a dynamical frequency spectrum and can represent the signals in
both the time and frequency domains. Figure 1 summarizes the difference between each


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signal representation. As the wavelets analysis has a high dynamical frequency and a high
resolution in both frequency and time domains, it is preferred for signals with a wide range
of dominant frequencies and time–frequency localization independence.
Wavelet transform analysis uses small wavelike functions known as wavelets. Math-
ematically, it can be deﬁned as the convolution of the signal with the wavelet function.
Figure 2 shows a few of the most common wavelet types.
Wavelets can be manipulated through translation (i.e., moving the wavelets along
the time axis) and scale (i.e., stretching and squeezing the wavelet at a particular point in
time). The scale and translation of wavelets govern parameters (b) and (a), respectively, in
Equation (1).
T(a,b)=1√a∫∞
−∞x(t)ψ∗(t−b
a)
dt (1)
The Equation (1) comprised both the signal x(t), which could represent the pressure,
temperature, or any time series data, as well as the ( ψ∗(t−b)/a) mother wavelet function.


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The wavelet transform works by superimposing the wavelet function on an arbitrary signal,
as illustrated in Figure 3. Time segments A and B, where both the wavelet and original
signal agree (positive or negative), will result in a large positive transformation value of


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Energies 2023 ,16, 764 7 of 29
the integral given by Equation (1). On the contrary, in regions C, D, and E, when the
wavelet and the signal have opposite signs, the overall transformation results in a negative
contribution to T(a,b).
Figure 1. Comparison between different signal representations.
Figure 2. Shapes of different wavelet transform families.
Figure 3. Mechanism of wavelet transform [51].


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Energies 2023 ,16, 764 8 of 29
When using complex wavelets, the resulting coefﬁcient consists of both real and
imaginary parts. Fourier transforms of complex wavelets are zero for negative frequencies.
Therefore, we can separate the phase and frequency components within the signal using
complex wavelets. For instance, taking Fourier to transform the Mexican hat wavelet
and then performing an inverse Fourier transform after neglecting the zero components
in the Fourier transform will create a complex wavelet. It produces a less oscillatory
transform than in the case of real wavelets because a complex wavelet only responds to
the non-negative frequencies of the particular real signal. Therefore, it outperforms real
wavelets in detecting and tracking instantaneous frequencies (Guerrero, A.P ., and Paredes,
G. E. (2018) [52]). Complex wavelet transform is used in our analysis, which is deﬁned by
Equation (2) where the ﬁrst part is the normalization factor, the second part is the complex


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sinusoid, and the ﬁnal part is the Gaussian bell curve.
ψ(t)=1
π1/4ei2πf0te−t2/2(2)
To inspect the multi-scale dimensional characteristics of a signal using complex
wavelets, the Wavelet Transform Modulus (WTM) can be calculated using Equation (3).
T(a,b)=√
[Re(T(a,b))]+[Im(T(a,b))] (3)
One of the main features of the signals that can be observed and analyzed to identify
the system properties is the analysis of signal energy. For any signal, the total energy
contained in a signal, x(t), is deﬁned as its integrated squared magnitude as in Equation (4)
with the condition that the signal must contain ﬁnite energy.
E=∫∞
−∞|x(t)|2dt (4)
The relative contribution of the signal energy contained at a speciﬁc scale (a) and
location (b) is given by the two-dimensional wavelet energy density function:
E(a,b)=|T(a,b)|2(5)
The plot E(a)versus time (different location parameters b values) at different dilation
parameter values (scale) (a) can be plotted in a plot called a scalogram. From the scalogram,


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the location and scale of dominant energetic features within the signal can be detected from
the scale-dependent wavelet energy spectrum of signal E(a)at a speciﬁc scale. A plot of
E(a,b)as a function of time and frequency is known as a scalogram. Scalograms are usually
plotted with a logarithm and a scale axis. For example, Figure 4 shows a scalogram for The
Nino3 SST monthly climate time data series analyzed using Morlet wavelets (Torrence and
Compo (1998) [53]), as discussed by Addison, P .S. (2016) [51].
The features that are detected at small scales, the compressed wavelet, are the rapidly
changing details (ﬁne features) that cause high frequency in the signal while the features
that are detected at large scales (stretched wavelet) are the slowly changing details (coarse
features) that cause high frequency in the signal.


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Energies 2023 ,16, 764 9 of 29
Figure 4. Example for the wavelet transform analysis of the monthly climate temperature (Nino3 SST)
time series data: ( a) The Niño3 SST time series data; and ( b) the local wavelet power spectrum using
the wavelet transform analysis of ( a) using the Morlet wavelet (Torrence and Compo (1998) [53]).
2.2. CWT Fracture Closure Detection Technique
In this study, a new innovative technique to ﬁnd a closure pressure using the CWT
technique is introduced. The main idea behind estimating the minimum stress using the
diagnostic fracture injection test is to create a small fracture at a constant rate and then
shut down the pumps and observe the pressure at which the fracture closes. The closure
pressure is the minimum average of the minimum stress for the area covered by the fracture
created by the DFIT. The pressure during the shut-in and the observation period represents
the closure process of the formation. The main goal is to detect and magnify the pressure


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changes that reﬂect the fracture’s closure event. The pressure signal during the shut-in
period after DFIT is analyzed using the CWT technique with a complex Morlet wavelet to
obtain the signal energy. The complete workﬂow for the CWT closure detection technique
is shown in Figure 5. The technique starts by getting the coefﬁcient of CWT using the
complex Morlet wavelets at multiple scales (from zero up to a scale of 256) and then getting
the wavelet transform modulus (WTM) using Equation (3) as a scalogram. The scalogram
of signal energy is plotted using a log scale. The scalogram can be averaged to the average
of the log signal energy of all scales. Fracture closure as a dominant feature can be identiﬁed
by the average of all log signal energies at several wavelet scales. The average of the log of
signal energy can be plotted with time. The plot can be used to detect the closure event as
seen in Figure 5.
Fracture closure can be identiﬁed with the two following features: the fracture closure


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starts with a peak in the average of the log signal energy, which represents the fracture walls
coming into contact, then the drop in log signal energy level to a minimum stabilized level,
which represents reaching the equilibrium state. The characteristics of fracture closure
can be seen in Figure 6. When the real ﬁeld data are analyzed, high oscillatory signal
energy may be observed due to the noise of real ﬁeld data. That may be due to several
reasons, and the fracture faces are not smooth faces and the closure does not happen in
subsurface formation instantaneously, or there is noise in the pressure data, or due to that,
the equilibrium state is still disturbed with the DFIT.


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Figure 5. Fracture closure detection workﬂow using the CWT closure detection technique.
Figure 6. Fracture closure pressure using CWT closure detection technique.
2.3. Summary
The new CWT fracture closure detection technique can be conducted as the following
steps: The ﬁrst step is to apply the CWT using a complex Morlet wavelet for the pressure
signal starting from the start of DFIT pumping to obtain the CWT coefﬁcient at different
scales up to (a = 256) as per Equation (1) and then obtain the wavelet transform modulus
(WTM) as per Equation (4) from the complex continuous wavelet coefﬁcient. This can
be followed by the calculation of signal energy using Equation (2). Considering fracture
closure is a dominant feature so it can be detected by averaging the log of signal energy
values at each time point for different scales (up to scale (a) = 256). From the plotted signal
average log energy versus time, the start and the end of the fracture closure event can


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be identiﬁed with the following characteristics. The start of the fracture closure which
represents the contact of two fracture faces can be recognized by a peak in average signal
log energy at the start of the fracture closure. The fracture closure can be identiﬁed by the


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Energies 2023 ,16, 764 11 of 29
drop in signal average log energy level to a minimum stabilized level. The start of the
minimum stabilized level characterizes the end of the fracture closure event.
3. Results and Discussion
3.1. Validation Using Fracture Simulation
The numerical simulations in this study were performed using the commercial pla-
nar 3D fracture simulator introduced by Barree (1983). The simulator integrates a ﬁnite
difference formulation for the ﬂuid ﬂow calculation within the fracture with an integral
equation for fracture width. Fluid pressure and fracture width solutions are sequentially
coupled based on the linear-elastic solution of the deformation for an inﬁnite half-space
with a concentrated load. This relies on Poiseuille’s equation along with the continuity
condition. It uses a pressure-dependent leak-off model for the effect of natural fractures.
This simulation is solved using ﬁnite differences with a regular grid. The shear-rate and


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time-dependent laboratory ﬂuid test data are used to model the ﬂuid rheology. Proppant
transport is coupled with ﬂuid ﬂow and iteratively solved for proppant distribution in
fractures. The simulator is widely used by the oil and gas industry and its fracture geometry
is validated by downhole events in a stacked ﬂuvial pay system as per Nur. Azlinda et al.
(2010) [12].
To test the new CWT closure detection technique using fracture simulation, grids for
the synthetic model are created with ideal properties; the idealized reservoir model has
25 ft. of the sand body between two shale barriers with a sharp stress contrast and constant
values for reservoir characteristics and geo-mechanical parameters. High-stress contrast
between the pay zone and the surrounding shale zones is assumed. It is assumed that
the reservoir is tight (the formation permeability is 5 micro Darcy) and a simulation run
was conducted with a ﬂuid volume (10,000 gallons of KCL water) of diagnostic fracture


