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New Model for DFIT Fracture Injection and Falloff Pressure Match
G. Liu and C. Ehlig-Economides, University of Houston
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
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Analysts regard the fracture injection and falloff test known as the diagnostic fracture injection test (DFIT) as
a reliable tool to quantify formation closure stress, leakoff coefficient, formation permeability, and pressure.
The recent analytical DFIT model for before and after closure enables matching the pressure falloff for
abnormal leakoff behaviors, and quantification of many more formation parameters than traditional DFIT
models. However, the model design addressed only the falloff data after shut-in, and many express concerns
that net pressure implied by the falloff is inconsistent with the injection pressure behavior. This paper
provides a model capable of matching both injection and falloff pressure behavior.
The pressure falloff model is capable of quantifying essential pressure values including, in order of
occurrence, instantaneous shut-in pressure (ISIP), minimum fracture propagation pressure, one or more
closure stress values, minimum stress, and pore pressure. The early pressure response represents the
dissipation of three kinds of friction, wellbore, perforation, and near-wellbore friction, each of which are
quantified, and which together comprise the difference between the pressure at the end of injection and
the ISIP. Presence of tip extension enables quantification of the minimum fracture propagation pressure.
The minimum stress is consistent with the final closure stress. Subtracting the closure pressure and friction
pressure losses from the recorded or calculated bottomhole pressure provides the fracture net pressure. The
model match for injection pressure behavior incorporates the same pressures and consistent values for 2D
fracture geometry and leakoff coefficient.
The global match confirms not only the estimation of formation and fracture properties from the pressure
falloff analysis, but also the pressure distribution along the wellbore, through the perforations and the near
wellbore, and in the fracture during and after injection. In particular, by matching both injection and falloff,
the model incorporates friction pressure losses that can explain apparent excessive net pressure.
The match with both injection and falloff pressure variation addresses concerns that the net pressure
implied by the falloff model match cannot be consistent with observed injection behavior. When the pressure
difference between the final DFIT pick for closure stress and the pressure at the end of injection is very
large, the reason may be large friction pressure losses and/or tip extension. Design for the main fracture
treatments could consider an altered perforation strategy to reduce friction losses and/or reduce cluster
spacing to address very low effective permeability likely consistent with tip extension and confirmed by
the after closure match.
For a DFIT test, a relatively small amount of fluid is pumped into formation to create a fracture in the target
layer, and then the well is shut in to allow the injected fluid leak off into formation (Feng and Gray 2016;
Feng et al. 2015). Liu and Ehlig-Economides 2015, 2016 described available models for DFIT analysis.
The Liu and Ehlig-Economides 2015, 2016 model is designed to match the entire falloff, including such
abnormal leakoff behaviors as perforation and near wellbore friction, tip extension, pressure dependent
leakoff area, height recession, variable compliance, and multiple closure events. The focus of this paper is
matching the pressure injection during pumping with the model that matches the falloff behavior.
Three two-dimensional (2D) fracture models still widely used in hydraulic fracturing analysis and design
are known as PKN (Perkins and Kern 1961; Nordgren 1972), KGD (Khristianovic and Zheltov 1955;
Geertsma and de Klerk 1969) and radial (Geertsma and de Klerk 1969). All of these models assume that the
formation rock is continuous and homogeneous, and that it follows the linear elastic relationship between
stress (net pressure) and strain (half of fracture width). Fracture width is a key parameter because it is
the bridge connecting pressure change and material balance. Mack and Warpinski 2000 explained that net
pressure at the tip of all three models is assumed to be zero implies that the fracture shape is elliptical
during propagation. Therefore, fracture width varies with a maximum value at wellbore and zero at tip.
For convenience, the average width is used for material balance calculations. The geometric factor, γ is
introduced to denote the relationship between the average ( ) and the maximum (wmax) fracture width for
three fundamental models, and given as,
The geometric factor, γ, equals to 0.628 for PKN model, 1 for KGD and 0.8 for radial fracture model.
Besides the nonuniform pressure distribution along fracture, all three models assume constant rate
injection. However, field data frequently show that the surface injection rate is not constant during the test.
Some operators deliberately vary injection rate before the end of pumping with a step rate test (SRT) also
known as a step-down test, designed to measure friction pressure loss under different injection rates (Barree
et al. 2014, Naidu et al. 2015).
The commonly used 2D fracture models also assume that fluid leakoff is negligible during pumping. The
PKN model in Nordgren 1972 obtains analytical solutions for two limiting cases: one with large leakoff rate
and the other with zero. Typically, the analytical PKN solution with zero leakoff is preferred, especially in
the tight formations. For the other two models, KGD and radial fracture model, fluid leakoff is neglected.
At the end, the width change, or correspondingly, the fracture propagation rate, can be expressed in a simple
power law function.
where Af is fracture one-side surface area at any time t during injection; Af0 is the fracture one-side surface
area at shut-in, or at t = tp, and tp is the injection time; α is the area exponent. According to Nolte et al. 1993,
the area exponent is close to 1 for low leakoff, and ½ for high leakoff. In particular, for the Newtonian fluid
with no leakoff, area exponents for PKN, KGD and radial fracture model are 4/5, 2/3 and 8/9, respectively.
In this paper, we construct a consistent model that can match the injection profile using the parameters
quantified from the falloff data. We identify inconsistencies between injection and falloff modelling and
explain how to resolve them. Finally, the new model is used in two field cases to demonstrate that one model
is capable to match both injection and falloff data.
Inconsistencies between Injection and Falloff Modelling
As two successive processes, the injection and falloff should be consistent in following parameters: fracture
geometry at shut-in, formation leakoff coefficient, natural fracture surface area if its effect is present, closure
stress and rate dependent friction. These parameters are quantified via falloff analysis, and should be used
as input parameters in the model to fit the injection pressure.
Although the injection and falloff occur in the same fracture and formation setting, the pressure response
and fracture behavior during these two periods may not be exactly same. In this discussion, we are to
investigate the gap between pressure falloff modelling and the fracturing propagation modelling, and then
try to build a more general model working for both.
Wellbore storage and variable bottomhole injection rate
In modelling fracture initiation and propagation, the injection rate refers to bottomhole (BH) rate into the
created fracture, not the recorded rate at wellhead. Estimation of the bottomhole rate requires consideration
of the wellbore storage effect.
The volume change of fluid storage in the wellbore (ΔVfiuid) with pressure can be written as,
where cw is water compressibility, and Vw is the total wellbore volume; Δpw(t) is the treating pressure change.
As suggested by Barree et al. 2014, besides fluid volume expansion, the expansion of the injection string
should also be considered, and its diameter change with pressure is described as,
where Δp is the internal pressure change; d is the string diameter and Δd is its change; δ is the string thickness,
and E is the steel modulus. Then, the volume change with pressure due to string expansion, ΔVstring, can
be derived as,
The total volume change due to WBS (ΔVWBS) then can be determined as,
For a casing with outer diameter (OD) between 4.5 inches to 7 inches, the ratio between (
) to cw
lies between 11% to 28%, which means the string volume change is relatively small comparing with fluid
volume change due to the large magnitude of the steel modulus, E, which is around 30 ×106 psi. These two
factors are combined in Eq. (7) as one with one single effective WBS compressibility (cWBS) that is 11% to
28% larger than the water compressibility for casings with OD ranging from 4.5 to 7 inches.
The bottomhole (BH) rate is calculated by considering the compressibility of fluid and injection string,
where qsurf,i and qBH,i are surface and bottomhole rate during a time interval, Δti; pw(ti) is the treating pressure,
and pw,start is the pressure level at the start of injection, or pw,start = Pw(t = 0).
If there is no pre-existing fracture, initially all injected fluid is pressurizing the wellbore with little flowing
out of wellbore until the formation breaks down. During this period, the pressure rises linearly with the
injection volume, and time when the injection rate remains constant. This feature can be used to estimate
the wellbore compressibility defined in Eq. (7).
Frequently DFIT injection has a varying injection rate. Some operators include a step rate test (SRT),
also called a step-down test, to estimate a relationship between friction pressure loss and injection rate. In
such cases, the WBS effect distorts changes in bottomhole injection rate compared to step rate changes at
the surface. There are two possible ways to determine the friction-rate relationship from an SRT: either
maintain each rate step for a relatively long time to see a flat pressure level so that the bottomhole rate will
be similar to the surface, or correct the bottomhole rate with Eq. (8) so that friction loss can be estimated
using the corrected bottomhole rate values.
Another impact of WBS on pressure response is after shut-in. The bottomhole pressure (BHP) after
surface shut-in does not drop to zero immediately, some stored fluid continues flowing into the fracture
with the declining pressure. Liu and Ehlig-Economides 2016 modelled this behavior by coupling WBS with
friction dissipation.
Two important parameters for the DFIT analysis are injection time and total injected volume. Injection
volume is the basis for the material balance, which must be strictly obeyed for DFIT analysis. Variation
in the injection rate has little explicit impact on the pressure falloff behavior, and it usually assumed to be
Although the injection rate is a trivial factor for falloff analysis, it has significant impact on the pressure
response during pumping. Firstly, friction losses are rate dependent, and at high rate friction losses often
account for much of the total pressure increases during pumping. As a result, pressure and rate behavior
is highly correlated. Secondly, increasing the pumping rate increases the bottomhole rate into the created
fracture, which increases the net pressure in the fracture and should accelerate fracture propagation.
However, the commonly used fracture propagation model described in Eq. (2), which is derived by assuming
constant injection rate, implies that the fracture propagation rate is independent of injection rate and only
dependent on time. To honor the rate effect and material balance, we modify Eq. (2) by replacing time on
the right-hand side with the injection volume, giving,
This equation uses the bottomhole rate to calculate the effective injection volume into the fracture, and
VBH,inj is the total volume flowing into fracture before shut-in, which can be calculated as,
where, Vsurf,inj is the total volume injected at surface, and pws is the treating pressure at shut-in, which is
not equal to the instantaneous shut-in pressure (ISIP). The relation between the pressure before the start of
injection and the shut-in pressure depends on operating procedures. For example, after plug and perf, the
wellbore pressure may be low before the start of injection. In such cases, there may be a sharp linear rise
in pressure associated with pressurizing the wellbore, and the total fluid volume into fracture at shut- in is
less than the surface injection volume because part of it is used to pressurize the wellbore. In contrast, we
have seen field data in which the hydraulic pressure required to open a sleeve exceeded the shut-in pressure,
so the total fluid flowing into fracture at shut-in may be larger than the surface designed volume due to
fluid expansion.
Fracture Internal Pressure Distribution
The DFIT model commonly assumes that the fracture internal pressure is uniform. This has two
implications. First, this assumption implies that there is no friction along fracture, and that the fracture
width is also uniform, which further implies that the geometric factor in Eq. (1) is 1 for all mentioned
2D fracture models. Secondly, it implies that the net pressure is uniform and equals to the product of the
fracture compliance and the fracture maximum width, which is equal to the average width. The assumption is
acceptable for falloff analysis because rate dependent fracture internal friction is negligible for rate near zero.
However, the assumption of uniform fracture internal pressure is incorrect for fracture propagation models,
in which net pressure is assumed to be maximum close to the wellbore and zero at tip. The error in assuming
uniform fracture internal pressure or neglecting internal fracture friction depends on the bottomhole flow
rate into fracture. If the rate is maintained in a high level, flow friction magnifies the internal fracture
pressure gradient and the net pressure estimate from the maximum fracture width close to wellbore will be
greater than the pressure at any other location in the fracture. In contrast, for small bottomhole injection
rate or after-flow rate, the internal fracture pressure gradient is small and there is less error in assuming
uniform net pressure in the fracture. Thus, the geometry factor, γ, in Eq. (1) is not constant, but should be
a function of injection rate.
To address internal fracture friction, we assume γ to correspond to the known reference values for different
2D fracture models when the bottomhole rate equals to the average bottomhole injection rate, and allow it
to linearly increase to 1 when the rate drops to 0. The newly constructed geometric factor function, given
in Eq. (11), can be applied in both injection with variable bottomhole flow rate, and falloff with dissipating
after flow rate.
where, γ0 is the reference value for 2D fracture model, equal, as before, to 0.628 for PKN, 1 for KGD and
0.8 for radial fracture model; q is bottomhole flow rate, and qBH,inj is the average bottomhole injection rate.
Leakoff during Fracture Propagation
As discussed in the Introduction, leakoff is neglected in the three commonly used 2D fracture propagation
models. Accordingly, the area exponent in Eq. (2) can be set to a value close to 1. However, this assumption
is incomplete and questionable, even for tests in tight formations. It has been well known that the natural
fractures in tight formations are likely reactivated during pumping, and a large part of injected fluid volume
flows into these natural tissues and leakoff through their surface into matrix. Because the total leakoff area
could be much larger than the hydraulic fracture, the leakoff rate may not be negligible, so Eq. (2) or (9)
should be further accommodated by considering the possible fluid leakoff loss during injection. According
to Nolte et al. 1993, the leakoff loss, or fluid efficiency, is represented by the area exponent, a, in Eq. (2),
and its values are anchored to 4/5, 2/3 and 8/9 for PKN, KGD and radial fracture model by assuming zero
leakoff. Depending on fluid efficiency, the area exponent could be ranging from ½ for high leakoff to 1
for low. Therefore, it is reasonable to vary the area exponent for each 2D fracture model shown in Eq. (2)
and (9). Successfully matching injection pressure data in several field tests suggests that using the fluid
efficiency at shut-in as the initial guess for the area exponent during injection results in error less than 10%.
This finding is consistent with the conclusion of Nolte et al. 1993 that area exponent has positive correlation
with the fluid efficiency.
Case Studies
In this paper, a DFIT field cases demonstrate the ability to match both injection and falloff pressure data
with one consistent model. Liu and Ehlig-Economides 2015, 2016 provided before-closure model matches
for this test.
Well A
The DFIT for Well A was performed in the toe stage of a horizontal well using a single perforation cluster.
About 31.45 bbl fresh water was injected into the well in 7.13 minutes with an average rate at 4.41 bbl/min,
as shown in Figure 1. It should be pointed out that the rate profile is missing, and we assume that the rate
is constant during injection at the value given by the quotient of the injected volume and the injection rate.
Figure 1—injection profile of Well A
Fluid was pumped into the formation through casing with inner diameter (ID) of 121.4 mm (4.78 inch).
The measure depth (MD) of perforation is at 4644 m (15,236 ft). From this information, the wellbore volume
is estimated at about 338 bbl, which is much larger than the injected fluid volume and sufficiently large to
note that the WBS effect cannot be neglected. Indeed, the WBS effect is apparent from the straight line in
Fig. 15 marked by the dashed green line during the early injection. Assuming all injected volume is stored
in the wellbore, Eq. (6) enables estimation of the WBS fluid and pipe compressibility defined by Eq. (7) at
2.92 × 10-6 psi, which corresponds to the water and pipe compressibility at the test temperature and pressure
conditions, and the observed behavior also confirms that the flow rate is constant.
Figure 2 shows the analysis presented previously by Liu and Economides 2015. The left graph in Fig.
2 is the Bourdet derivative log-log plot with markings indicating slope trends of interest, and the right
shows the composite G-function plot displaying exactly the same closing trends. Briefly, the first behavior
indicated by the green unit slope (marked as circle 1) in the Bourdet log-log plot is identified as WBS with
friction dissipation. The next relatively flat derivative trend (circle 2) could be the tip extension after shutin coupled with near-wellbore friction dissipation. The minimum fracture propagation pressure at the end
of tip-extension is 6640 psi. Then, the unit slope ending at around Δt = tp (circle 3) is identified as the first
constant area closure trend. The following four 3/2-slopes (circle 4 – 7) are successive additional closure
trends, giving five successive closures in total. Mohamed et al. 2011 showed a similar multiple-closure field
case from a different formation.
Figure 2—Bourdet derivative plot (Left) and composite G-function plot (Right) for Well A
Liu and Ehlig-Economides 2016 previously quantified a different leakoff coefficient from each apparent
closure trend. In our recent work which is still under peer review, we assume all leakoff trends relate to
the same formation and, therefore, are governed by one constant leakoff coefficient. Different leakoff rates,
indicated by unit and 3/2-slope in the Bourdet derivative or extrapolated Gdp/dG slopes in the G-function
plot, are caused by different leakoff areas. The leakoff area decreases from its highest value associated with
the first closure trend to the lowest with the last final closure.
A possible interpretation for the closure trends is that each represents a different secondary fracture set,
and a possible explanation for the distinct closing pressures may relate to the angle of the fracture planes
in a given fracture set. Table 1 lists the interpretation parameter values for the final model match. Figure
3 shows the leakoff area for each closure trend associated with the pressure at the end of the constant area
leakoff. By subtraction, each closure trend has an associated leakoff area. Figure 3 starts from the second
closure event because the area associated with the first closure is extremely large and of very short duration.
This might be related to near wellbore fracture initiations under local drilling-modified stress, and the higher
leakoff rate may be partly explained by a higher leakoff coefficient through a smaller apparent leakoff area.
Table 1—Interpretation result of Well A
Figure 3—The decline of total (one side) fracture surface area with pressure during falloff.
The apparent ½-slope (circle 8) observed after the final closure is usually interpreted as formation
linear flow. From this behavior, we estimate formation permeability as 336 nano-Darcy using the method
introduced by Liu et al. 2016 and estimate the formation pressure at 4,065 psi. The derivative curve seems
to bend to flat at the very end (circle 9). Permeability estimated by identifying this trend as pseudo-radial
flow results in an unlikely pore pressure estimate.
Figures 4 and 5 show matches for both falloff and injection models achieved through slight adjustments
in the previously published friction and area exponent estimates based only on the falloff data. Several
interesting points merit discussions. First, the total fluid volume flowing into formation before shut-in is
25.6 bbl, which is almost 6 bbl less than the surface injected volume because of WBS. Further, about 9.6 bbl
out of the injected 25.6 bbl leaks off into the formation during injection. It means that the fluid efficiency
during injection is relatively low (about 0.625) even the well is drilled in shale. As mentioned above, initial
injection may be initiating multiple fractures under drilling-modified stress near the wellbore that close at
high leakoff rate opposite high stress at the start of the falloff.
Figure 4—Model match for falloff pressure behavior of the Well A.
Figure 5—Model match for injection and falloff pressure of the Well A.
Figure 6 (A) presents the comparison between surface injection rate and the bottomhole flow rate at the
formation face. At the very early injection time, injected fluid only pressurizes the wellbore with no fracture
initiation. After breakdown, the bottomhole rate rises quickly to the wellhead rate level. The surface rate
drops to zero immediately after shut-in, while the bottomhole rate continues flowing at a low rate for some
time as the after-flow effect. The rate dependent geometric factor, γ, defined in Eq. (11) varies between
its reference value (0.8 for radial fracture model) and 1. Figure 6 (B) shows the fracture internal pressure,
bottomhole shale face pressure and its components: friction occurring in wellbore & perforation and near
wellbore tortuosity, net pressure at the wellbore which is the maximum net pressure along the fracture due
to fracture internal friction, and the average net pressure. The gap between net pressure at the wellbore and
the average net pressure can be taken as the fracture internal friction, which is great during injection, but
dissipates soon after shut-in. The fracture internal pressure is calculated by adding average net pressure
to the closure stress quantified from closure stress of the last closure. It is clear that the fracture internal
pressure (red line) has much less pressure drop after shut-in compared with the bottomhole pressure, so the
Carter 1957 leakoff model is justified.
Figure 6—(A) Surface injection rate, bottomhole flow rate and the rate dependent geometric
factor of the Well A; (B) Bottomhole pressure and fracture internal pressure, friction, net
pressure at wellbore and average net pressure during injection and falloff of the Well A.
1. Reduced or eliminated need for step-down test.
To measure directly the relationship between flowing pressure and injection rate related
functionally to friction pressure loss, some operators perform a step-down test just before stopping
pumping for the DFIT injection falloff. As a consequence of wellbore storage, there is a delay between
a change in the surface injection rate and the corresponding change in bottomhole flow rate. Therefore,
it is essential to use the measured or calculated bottomhole rate to estimate rate-dependent friction, not
the wellhead surface rate. A down side to the step-down test is that it compromises either the injection
rate or the injection volume. To ensure fracture closure in a reasonable time frame, the DFIT design
calls for minimizing the injection volume. If some of the volume is used to perform the step-down
test, the remaining injection must be done at lower rate. This, in turn, may fail to activate main or
secondary fracture behaviors that occur at higher rate. The Liu and Ehlig-Economides 2016 model for
rate-dependent friction enables quantifying friction loss. When a step-down test has been performed, it
should confirm the falloff model. Otherwise, the injection and falloff pressure match should eliminate
the need for a step-down test.
2. Net pressure magnitude
Large pressure drop observed after shut-in could be caused by friction in the wellbore, perforation,
near wellbore and/or fracture internal flow friction. Usually, the net pressure in most parts of the
fracture is relatively small, and sufficiently large in-situ net pressure will trigger tip extension, which
serves to dissipate high net pressure.
3. Uniformity of the fracture internal pressure
In tests with high closure stress and relatively small net pressure, the fracture internal pressure
variation can be viewed as negligible or constant, and the Carter 1957 leakoff model remains valid.
Hence, while it may be useful to considering modeling leakoff as flow into a permeable formation,
there is no requirement to do so.
4. Implication of falloff behaviors on the injection pressure response
In this paper, all leakoff behaviors are quantified from the falloff analysis. Their behaviors are
simplified when they are used as input data in the injection modelling. For instance, the multiple
closure case shown in Well A can be explained as four sets of secondary fracture with different
directions, closure stresses and areas. Secondary fracture sets were opened and extended during
pumping by a combination of shear and tensile behaviors. A simplifying assumption is that they
propagate essentially in the same manner together with the hydraulic fracture. This assumption
facilities leakoff volume computation and fracture area estimation, and results in pressure calculated
at a stable level during injection, which fits the field data shown in this paper very well. However,
if these fractures are not propagating in the same manner, the observed treating pressure may not
remain at a constant level. Alternatively, if pressure during injection at constant rate shows an uneven
pattern, this suggests abnormal behavior that should also be observed in the falloff pressure response.
We anticipate that the matching both injection and falloff data will improve the model reliability and
provide additional insights valuable for fracture design.
5. Meaning of the minimum propagation pressure at the end of tip extension after shut-in.
The minimum propagation pressure picked at the end of tip extension after shut-in represents the
minimum pressure to create a static fracture in the formation. Fracture propagation will stop when
pressure drops below this pressure. The minimum propagation pressure should be less than the stable
injection pressure during injection even after friction is deducted because extra momentum or pressure
is required to maintain fracture growth at a high rate.
The paper shows a model capable of matching both injection and falloff data from a DFIT test.
Understanding the inconsistency between classic 2D fracture models and models used for the falloff analysis
reveals the way to construct a consistent model that is based on the falloff analysis and able to match the
entire injection and falloff pressure behavior. Following are key conclusions:
1. The bottomhole rate must be calculated in the injection and falloff analysis. In particular, wellbore
storage must always be considered when estimating rate dependent friction from a step-down test.
2. We use a rate dependent fracture geometric factor to estimate fracture internal pressure during
injection that satisfies both the constant fracture internal pressure gradient during injection and zero
internal fracture gradient during falloff.
3. The leakoff volume may not neglected during injection even for tight formation, and the match with
injection data must consider the resulting fluid volume loss.
4. Net pressure in the fracture could be much smaller than the observed pressure drop after shut-in. Extra
net pressure could be the driver for the tip extension, and then the net pressure in a static fracture
should be at a relatively low level. Therefore, the Carter leakoff model is justified for the fracture with
high closure stress but limited net pressure.
Af = fracture one side surface area, L2
Af0 = fracture one side surface area at shut-in, L2
BHP = bottomhole pressure, m/Lt2
pp, min
pw, start
= water compressibility, Lt2/m
= leakoff coefficient, L/√t
= wellbore diameter, L
= G-function, dimensionless
= steel modulus, m/Lt2
= instantaneous shut-in pressure, m/Lt2
= permeability, L2
= pressure, m/Lt2
= closure pressure, m/Lt2
= formation initial pressure, m/Lt2
= minimum fracture propagation pressure, m/Lt2
= bottomhole pressure, m/Lt2
= shut-in pressure, m/Lt2
= bottomhole pressure before injection, m/Lt2
= flow rate, L3/t
= after-flow rate into fracture due to fluid expansion in the wellbore, L3/t
= pumping rate, L3/t
= fracture radius in radial fracture model, L
= fracture radius at shut-in, L
= fracture radius at the end of tip extension, L
= time, t
= pumping time, t
= volume, L3
= leakoff volume, L3
= pumping volume, L3
= injection string or wellbore volume, L3
= fracture width, L
= area exponent, dimensionless
= difference, dimensionless
= string thickness, L
= geometric factor, dimensionless
= fluid efficiency, percentage
w, wb
= closure
= end of constant area
= fracture
= friction
= injection
= maximum
= perforation
= near-wellbore fracture tortuosity
= wellbore
= wellbore storage
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