[blasingame] spe 103204
TRANSCRIPT
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Copyright 2006, Society of Petroleum Engineers
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Abstract
The proposed work provides a new definition of the pressure-
derivative function [i.e., the -derivative function, pd(t)],
which is defined as:
p
tp
dt
pdt
ptd
pdtp dd
=
=
=
)(1
)ln(
)ln()(
(pd(t) is the "Bourdet" well testing derivative)
This formulation is based on the "power-law" concept (i.e., thederivative of the logarithm of pressure drop with respect to the
logarithm of time) this is not a trivial definition, but rathera definition that provides a unique characterization of "power-law" flow regimes.
The "power-law" flow regimes uniquely defined by the pd(t)
function are: [i.e., a constant pd(t) behavior]
Case pd(t)
Wellbore storage domination: 1
Reservoir boundaries: Closed reservoir (circle, rectangle, etc.).
2-Parallel faults (large time). 3-Perpendicular faults (large time).
1
1/21/2
Fractured wells:
Infinite conductivity vertical fracture. Finite conductivity vertical fracture. 1/21/4Horizontal wells:
Formation linear flow. 1/2
In addition, the pd(t) function provides unique characteristic
responses for cases of dual porosity (naturally-fractured) reser-
voirs.
The pd(t) function represents a new application of the tradi-
tional pressure derivative function, the "power-law"
differentiation method (i.e., computing the dln(p)/dln(t) deri-
vative) provides an accurate and consistent mechanism for
computing the primary pressure derivative (i.e., the Cartesian
derivative, dp/dt) as well as the "Bourdet" well testing
derivative [i.e., the "semilog" derivative, pd(t)=dp/dln(t)]
The Cartesian and semilog derivatives can be extracted direct
ly from the power-law derivative (and vice-versa) using the
definition given above.
Objectives
The following objectives are proposed for this work:
To develop the analytical solutions in dimensionless form awell as graphical presentations (type curves) of the -derivativefunctions for the following cases:
Wellbore storage domination. Reservoir boundaries (homogeneous reservoirs). Unfractured wells (homogeneous and dual porosity reser
voirs). Fractured wells (homogeneous and dual porosity reservoirs) Horizontal wells (homogeneous reservoirs).
To demonstrate the new -derivative functions using typecurves applied to field data cases using pressure drawdown/buildup and injection/falloff test data.
Introduction
The well testing pressure derivative function,1pd(t), is known
to be a powerful mechanism for interpreting well test behavior it is, in fact, perhaps the most significant single deve
lopment in the history of well test analysis. The pd(t) function as defined by Bourdet et al.[i.e., pd(t)=dp/dln(t)] provides a constant value for the case of a well producing at a
constant rate in an infinite-acting homogeneous reservoirThat is, pd(t) = constantduring infinite-acting radial flow behavior.
This single observation has made the Bourdet derivativepd(t), the most used diagnostic in pressure transient analysis
but what about cases where the reservoir model is not in
finite-acting radial flow? Of what value then is the pd(tfunction?
The answer is somewhat complicated in light of the fact tha
the Bourdet derivative function has almost certainly been
generated for every reservoir model in existence. Reservoi
engineers have come to use the characteristic shapes in theBourdet derivative for the diagnosis and analysis of wellbore
storage, boundary effects, fractured wells, horizontal wells
and heterogeneous reservoirs. For this work we prepare the
derivative for all of those cases but for heterogeneous re-servoirs, we only consider the case of a dual porosity reservoir
with pseudosteady-state interporosity flow.
The challenge is to actually define a flow regime with a
SPE 103204
The Pressure Derivative Revisited Improved Formulations and ApplicationsN. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame, SPE, Texas A&M University
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2 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
particular plotting function. For example, a derivative-basedplotting function that could classify a fractured well by a
unique signature would be of significant value as would be
such functions which could be used for wellbore storage,
boundary effects, horizontal wells, and heterogeneous reser-voir systems.
The purpose of this work is to demonstrate that the "power-
law" -derivative formulation does just that it provides a
single plotting function which can be used (in isolation) as amechanism to interpret pressure performance behavior for sys-
tems with wellbore storage, boundary effects, fractured wells,
horizontal wells.
The power-law derivative formulation is given by:
p
tp
dt
pdt
ptd
pdtp dd
=
=
=
)(1
)ln(
)ln()( .................... (1)
where pd(t) is the "Bourdet" well testing derivative.
In Appendix Awe provide the definitions of the power-law-
derivative function for various reservoir models as shownbelow. The graphical solution (or "type curve") for each case
of interest is shown in Appendix B, and categorized as shown
below.
Specific pd(t) Case: pd(t)App. A
Table
App. B
Figs.
Wellbore storage domination: 1 A-1 4,11-20
Reservoir boundaries:
Closed reservoir
Infinite-acting (incl. WBS)
2-Parallel faults3-Perpendicular faults
1
---1/21/2
A-2
A-3A-4A-4
1,2,25
433
Fractured wells:
Infinite cond. vert. fracture
Finite cond. vert. fracture1/21/4
A-5A-6
10,1110,12-14
Dual porosity reservoirs:
Unfractured well
Fractured well---
---
A-7
A-8
5-9
15-20
Horizontal wells:
Formation linear flow 1/2 A-9 21-24
The origin of the -derivative formulation pd(t) was aneffort by Sowers2to demonstrate that this formulation would
provide a consistently better estimate of the Bourdet derivative
function, pd(t), than the either the "Cartesian" or the "semi-log" formulations. For orientation, we present the definition
of each derivative formulation below:
The "Cartesian" pressure derivative is defined as:
dt
pdtpPd
= )( ..............................................................(2)
where pPd(t) is also known as the "primary pressure deri-vative" [ref. 3 (Mattar)].
The "semilog" or "Bourdet" pressure derivative is defined as:
dt
pdttpd
= )( ............................................................... (3)
Recalling that the " " pressure derivative is defined as:
p
tp
dt
pdt
ptd
pdtp dd
=
=
=
)(1
)ln(
)ln()( .................... (1)
solving for the "Cartesian" or "primary pressure derivative,"
)(tpt
p
dt
pdd
=
........................................................ (4
solving for the "semilog" or "Bourdet" pressure derivative,
)()( tpptp dd = .................................................... (5
Now the discussion turns to the calculation of these deri
vatives what approach is best? Our options are:1. A simple finite-difference estimate of the "Cartesian" (or "pri
mary") pressure derivative [pPd(t)=dp/dt].2. A simple finite-difference estimate of the "semilog" (or "Bour
det") pressure derivative [pd(t)=dp/dln(t)].3. Some type of weighted finite-difference or central difference es
timate of either the "Cartesian" or "semilog" pressure derivativefunctions. This is the approach of Bourdet et al.1and Clark andvan Golf-Racht4 this formulation is by far the most populartechnique used to compute pressure derivative functions for the
purpose of well test analysis, and will be presented in detail inthe next section.
4. Other more elegant and more statistical sophisticated algorithmhave been proposed for use in pressure transient (or well test)analysis, but the Bourdet et al. algorithm (and its variationscontinue to be the most popular approach, most likely due to the
simplicity and consistency of this algorithm. To be certain, theBourdet et al. algorithm does not provide the most accurate estimates of the derivative functions, but the predictability of thealgorithm is very good, and the purpose of the derivative is as adiagnostic function, not a function used to provide an exact estimate.
Some of the other algorithms proposed for estimating the various pressure derivative functions are summarized below:
Moving polynomial or another type of moving regressionfunction. This is generally referred to as a "window" approach (or "windowing").
Spline approximation by Lane et al.5is a powerful approach
but as pointed out in a general assessment of the computationof the pressure derivative (Escobar et al.6), the spline approximation requires considerable user input to obtain the "besfit" of the data, and for that reason, the method is less desi
rable than the traditional (i.e., Bourdet et al.1
) formulation. Gonzalez et al.7 applied a combination of power-law and
logarithmic functions to represent the characteristic signal andregression was used to find the "best-fit" to the data over aspecified window.
Cheng et al.8 utilized the fast Fourier transform and fre-quency-domain constraints to improve Bourdet algorithm byoptimizing the size of search window and they also used aGaussian filter to denoise the pressure derivative data. Thiresulted in an adaptive smoothing procedure that uses recursive differentiation and integration.
Calculation of the
-Derivative Function
To minimize the effect of truncation error, Bourdet et al.1in
troduced a weighted central-difference derivative formula:
R
R
RL
L
L
L
RL
R
t
p
tt
t
t
p
tt
t
td
pd
++
+=
)]ln([.......(6a
where:
tL = ln(tcalc) ln(tleft)........................................... (6b
tR = ln(tright) ln(tcalc)..........................................(6c
pL= pcalc pleft.................................................... (6d
pR= pright pcalc....................................................(6e
The left- and right-hand subscripts represent the "left" and"right" neighbor points located a specified distance (L) from
the objective point. The calcsubscript represents the point of
interest at which the derivative is to be computed. As for the
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 3
L-value, Bourdet gives only general guidance as to its select-ion, but we have long used a formulation where Lis the frac-
tional proportion of a log-cycle (log10base). Therefore,L=0.2
would translate into a "search window" of 20 percent of a log-
cycle from the point in question.
This search window approach (i.e.,L) helps to reduce the in-
fluence of data noise on the derivative calculation. However,
choosing a "small" L-value will cause Eq. 6a to revert to a
simple central-difference between a point and its nearestneighbors, and data noise will be amplified. On the contrary,
choosing a "large"L-value will cause Eq. 6a to provide a cen-
tral-difference derivative over a very great distance whichwill yield a poor estimate for the derivative, and this will tend
to "smooth" the derivative response (perhaps over-smoothing
the derivative). The common range for the search window is
between 10 and 50 percent of a log-cycle (0.10 < L< 0.5) where we prefer a startingL-value of 0.2 [20 percent of a log-
cycle (recall that log is the log10function)].
Sowers2proposed the "power-law" formulation of the weigh-
ted central difference as a method that he believed would pro-
vide a better representation of the pressure derivative than theoriginal Bourdet formulation. In particular, Sowers provides
the following definition of the power-law (or "") derivative
formulation:
[ ]
R
R
RL
L
L
L
RL
R
t
p
tt
t
t
p
tt
t
td
pd
+
+
+
=
)]ln([
)ln(..... (7a)
where:
tL = ln(tcalc) ln(tleft) ........................................... (6b)
tR = ln(tright) ln(tcalc) ......................................... (6c)
pL= ln(pcalc) ln(pleft) .........................................(7d)
pR= ln(pright) ln(pcalc) ....................................... (7e)
Multiplying the right-handside of Eq. 7a by pcalc(recall that
pcalc is the pressure drop at the point of interest), will yield
the "well-testing pressure derivative" function (i.e., the typical
"Bourdet" derivative definition). Sowers2 provides an ex-
haustive evaluation of the "power-law" derivative formulation
using various levels of noise in the pfunction and found that
the power-law (or ) derivative formulation always showed
improved accuracy of the well testing pressure derivative [i.e.,
the Bourdet derivative function, pd(t)].
In addition, Sowers found that the -derivative formulation
was less sensitive to theL-value than the original Bourdet for-
mulation which is a product of how well the power-law
relation represents the pressure drop over a specific period.
Sowers did notpursue the specific application of the -deri-vative function [pd(t)=d ln(p)/dln(t)] as a diagnostic plot-
ting function, as we have this work.
Type Curves Using the
-Derivative Function
Background: Without question, the Bourdet definition of the
pressure derivative function is the standard for all well test
analysis applications from hand methods to sophisticatedinterpretation/analysis/modeling software. The advent of the
-derivative function as proposed in this paper is not expectedto replace the Bourdet derivative (nor should this happen).
The -derivative function is proposed simply to serve as a
better interpretation device for certain flow regimes in particular, those flow regimes which are represented by power
law functions (e.g., wellbore storage domination, closed boun
dary effects, fractured wells, horizontal wells, etc.).
In the development of the models and type curves for the
derivative function, we reviewed numerous literature articles
which proposed plotting functions based on the Bourdet pres-
sure derivative or related functions (e.g., the primary pressurederivative (ref. 3)). In the late 1980's the "pressure derivative
ratio" was proposed (refs. 9 and 10), where this function was
defined as the pressure derivative divided by the pressure drop(or 2p in radial flow applications)) this ratio was (obviously) a dimensionless quantity. In particular, the pressure
derivative ratio was applied as an interpretation device as iis a dimensionless quantity, the type curve match consisted of
a vertical axis overlay (which is fixed) and a floating hori-
zontal axis (which is typically used to find the end of wellbore
storage distortion effects). The pressure derivative ratio has
found most utility in such interpretations.
In the present work, we have formulated a series of "type
curves" which are presented in Appendix B, developed fromthe-derivative solutions given in Appendix A.
The primary utility of the -derivative is the resolution tha
this function provides for cases where the pressure drop can be
represented by a power law function again, fractured wells
horizontal wells, and boundary-influenced (faults) and boundary-dominated (closed boundaries) are good candidates for
the-derivative.
10-2
10-1
100
101
102
103
10-5
10-4
10-3
10-2
10-1
100
101
10
Dimensionless Time, tD(model-dependent)
Legend: (pDd ) (pDd )
Unfractured Well (Radial Flow) Fractured Well (Infinite Fracture Conductivity) Fractured Well (Finite Fracture Conductivity) Horizontal Well (Full Penetration, Thin Reservoir)
Transient FlowRegion
Schematic of Dimensionless Pressure Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)
DIAGNOSTIC plot for Well Test Data (pDdand pD
d)
Bourdet"WellTest"DimensionlessPressureDerivativeFunction,pDd
"PowerLaw
"DimensionlessPressureDerivativeFu
nction,pD
d
Boundary-Dominated
Flow Region
pD
d= 0.5
(linear flow)
pD
d= 0.25
(bilinear flow)
pDd= 1
(boundarydominated flow)
1
1
1
2
41
2
1
Unfractured Well ina Bounded Circular
Reservoir
Fractured Well ina Bounded Circular
Reservoir(InfiniteConductivity
Vertical Fracture)
Horizontal Well in aBounded Square
Reservoir:(Full Penetration,Thin Reservoir)
Fractured Well ina Bounded Circular
Reservoir(FiniteConductivityVertical Fracture)
( )( )
( )( )
( )( )
( )( )
NO Wellbore Storage
or Skin Effects
Figure 1 Schematic of pDdand pD
dvs. tD Various reser-voir models and well configurations (no well-bore storage or skin effects).
Infinite-acting radial flow the "utility" case for the Bourde(semilog) derivative function is not a good candidate for inter
pretation using the-derivative as the radial flow regime is re-
presented by a logarithmic approximation which can not be
further approximated by a power-law model.
Schematic Case: In Fig. 1we present a schematic plot created
for illustrative purposes to represent the character of the
derivative for several distinctly different cases. Presented are
the-derivative profiles (in schematic form) for an unfractured
well (infinite-acting radial flow), 2 fractured well cases, and a
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4 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
horizontal well case. We note immediately the strong charac-
ter of the fractured well responses (pDd= 1/2 for the infinite
conductivity fracture case and 1/4 for the finite conductivity
fracture case). Interestingly, the horizontal well case shows a
pDd slope of approximately 1/2, but the pDd function never
achieves the expected 1/2 value, perhaps due to the "thin"
reservoir configuration that was specified for this particular
horizontal well case. We also note that, for all cases of boun-
dary-dominated flow, thepDdfunction yields a constant value
of unity, as expected. This observation suggests that thepDd
function (or an auxiliary function based on thepDdform) may
be of value for the analysis of production data. For reference,
Fig. 1is presented in a larger format in Appendix B(Fig. B-
1).
Infinite-Acting Radial Flow: The -derivative function for a
single well producing at a constant flowrate in an infinite-act-
ing homogeneous reservoir was computed using the cylin-
drical source solution given in ref. 11. For emphasis, we have
generated the-derivative solution (Fig. 2) with wellbore sto-rage and skin effects, as this is the typical configuration used
for well test analysis. As mentioned earlier, the-derivativefunction does not demonstrate a constant behavior for the ra-
dial flow case, but as noted in Appendix A, the -derivativefunction for the wellbore storage domination flow regime
yieldspDd= 1.
10-3
10-2
10-1
100
101
102
103
pD,pDda
ndpDbd
10-2
10-1
100
101
102
103
104
105
106
tD/CD
CDe2s
=110-3
310-3
110-2 310
-210
-1
1
101
102
103
104
106
108
1010
1020
1030
1015
1040
10100
1060
1080
1050
10100
1080
1060
1050
1040
1030
1020
1015
1010
108
106
104
103
102
101
310-2
110-2
CDe2s
=110-3
Type Curve for an Unfractured Well in an Infinite-Acting Homogeneous Reservoirwith Wellbore Storage and Skin Effects.
3
3
10100
CDe2s
=110-3
Legend: Radial Flow Type Curves p
DSolution
pDd Solution
pDd Solution
Radial Flow Region
Wellbore StorageDomination Region
Wellbore StorageDistortion Region
Figure 2 pD, pDd, and pDd vs. tD/CD solutions for anunfractured well in an infinite-acting homo-geneous reservoir wellbore storage and skineffects included (various CDvalues).
Sealing Faults: Ref. 12 provides pDd-format (Bourdet) typecurves for cases of a single well producing at a constant flow-
rate in an infinite-acting homogeneous reservoir with single,
double, and triple-sealing faults oriented some distance from
the well. This case provides an opportunity to illustrate the-
derivative function where the pDdfunctions show interesting
characteristics, as well as the 2-parallel sealing faults and 3-
perpendicular fault cases, which prove thatpDd= 1/2 at very
long times (see Fig. 3).
10-3
10-2
10-1
100
101
102
103
104
-PressureDerivativeFunction,pD
d
=(tD/pD
)d/dtD
(pD)
10-3
10-2
10-1
100
101
102
103
104
105
106
107
tD/LD2(LD= Lfault/rw)
Legend: -Pressure Derivative Function
Single Fault Case2 Perpendicular Faults (2 at 90 Degrees)2 Parallel Faults (2 at 180 Degrees)3 Perpendicular Faults (3 at 90 Degrees)
Single
Fault
2 PerpendicularFaults
3 PerpendicularFaults
2 ParallelFaults
Dimensionless Pressure Derivative Type Curves for Sealing Faults(Inifinite-Acting Homogeneous Reservoir)
Undistorted
Radial Flow Behavior
2 ParallelFaults
2 PerpendicularFaults
pD
d= (tD/pD)dpD/dtD
pDd= tDdpD/dtD
Legend: "Bourdet" Well Test Pressure Derivative
Single Fault Case2 Perpendicular Faults (2 at 90 Degrees)2 Parallel Faults (2 at 180 Degrees)3 Perpendicular Faults (3 at 90 Degrees)
" Bor de
t " W
e l l T
es
t P
ressre
Der ia
t iep
= t
dp
/ d t
SingleFault
3 PerpendicularFaults
Figure 3 pDd and pD
d vs. tD/LD2 various sealing faults
configurations (no wellbore storage or skineffects).
Unfractured Well in a Dual Porosity System: We used the
pseudosteady-state interporosity model13 to produce the
derivative type curves for a single well in an infinite-actingdual porosity reservoir with or without wellbore storage and
skin effects. For these cases, we chose to present our case
(which include wellbore storage) using the type curve formaof ref. 14 (the family parameters for the type curves are the
and -parameters, where = CD).
In Fig. 4we present a general set of cases (= 1x10-1, 1x10-2
and 1x10-3 and = 5x10-9, 5x10-6, and 5x10-3) with no well-bore storage or skin effects. Fig. 4shows the unique signature
of thepDdfunctions for this case, but we can also argue that
since this model is tied to infinite-acting radial flow, the pDdfunctions can, at best, be used as a diagnostic to view ideal
ized behavior.
10-4
10-3
10-2
10-1
100
101
102
pDa
ndpD
d
10-1
100
101
102
103
104
105
106
107
108
109
tD
Type Curve for an Unfractured Well i n an Infinite-Acting Dual Porosity Reservoir(Pseudosteady-State Interporosity Flow) No Wellbore Storage or Skin Effects.
Legend:pD Solution
pDdSolution
= 110-1
= 110-1= 110
-1
= 110-2
= 110-3
= 110-2
= 110-2
= 110-3
= 110-3
= 110-1
110-2
110-3
pDd(= 5 10-9
) pDd(= 5 10
-6)
pDd(= 5 10-3
)
Figure 4 pDand pD
dvs. tD solutions for an unfracturedwell in an infinite-acting dual porosity system
no wellbore storage or skin effects (various
andvalues).
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 5
In Fig. 5we present cases where = 110-1and = CD=
110-4for 110-4< CD< 110100. As with the results for the
pDdfunctions shown in ref. 14, thesepDdfunctions do provide
some insight into the form and character of the behavior for
the case of a well producing at infinite-acting flow conditionsin a dual porosity/naturally fractured reservoir system.
10-3
10-2
10-1
100
101
102
103
pDa
ndpDd
10-2
10-1
100
101
102
103
104
105
106
107
tD/CD
CDe2s
=110-3
310-3
110-2310
-2
10-1
110
1
102
103
104
106
108
1010
1020
1030
1015
1040
10100
1060
1080
1050
10100
1080
1060
1050
1040
1030
1020
1015
1010
108
106
104
103
102
101
110
-1
310-2
110-2 310
-3
CDe2s
=110-3
Type Curve for an Unfractured Well in an Infinite-Acting Dual P orosity Reservoir(Pseudosteady-State InterporosityFlow) with Wellbore Storage and Skin Effects.
( = CD = 110-4, = 110-1)
Legend: = CD = 110-4
, = 110-1
pD
Solution
pDd Solution
10100
CDe2s
=110-3
Wellbore StorageDomination Region
Radial Flow Region
Wellbore StorageDistortion Region
Figure 5 pDand pDdvs. tD/CD = 110-1, = CD=
110-4(dual porosity case includes wellbore
storage and skin effects).
Hydraulically Fractured Vertical Wells: In this section we
consider the case of a well with a finiteconductivity vertical
fracture where the -derivative type curves were generatedusing the Cinco and Meng15solution. In addition, we used the
Ozkan solution (ref. 16) to model the case of a well with an
infinite conductivity vertical fracture. The pD, pDd, and pDdfunctions for the case of no wellbore storage are shown in Fig.
6. We note clear evidence of the bilinear and linear flow re-
gimes where these regimes appear as horizontal lines onthe -derivative plot (bilinear flow: pDd = 1/4, linear flow:
pDd= 1/2).
10-3
10-2
10-1
100
101
pD,pDd
an
dpD
d
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
tDxf
CfD=0.250.5
1
CfD=1104
Type Curve for a Well with a Fi nite Conductivity VerticalFractured in an Infinite-Acting Homogeneous Reservoir
(CfD = (wk f)/(kx f) = 0.25, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 10000)
Legend: pD Solution
pDd Solution
pDdSolution
Radial Flow Region
CfD=0.25
CfD=1104
1
5
2
0.51
2110
3
500
Figure 6 pD, pDd, and pDd vs. tDxf solutions for anfractured well in an infinite-acting homogene-ous reservoir no wellbore storage or skin ef-fects (various CfDvalues).
In Fig. 7we present the case of a single well with a finite con-
ductivity vertical fracture (CfD= 10) producing at a constantrate in an infinite-acting homogeneous reservoir, with well-
bore storage effects included. We observe the characteristic
wellbore storage domination behavior (pDd= 1), as well as the
effect of bilinear (fracture and formation) flow (pDd = 1/4)
We believe that the pDd function (i.e., the -derivative) wilsubstantially improve the diagnosis of flow regimes in
hydraulically fractured wells.
10-4
10-3
10-2
10-1
100
101
pDa
ndpD
d
10-4
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
CDf=110-6
110-2
Type Curve for a Well with Finite Conductivity Vertical Fracture in an Infinite-ActingHomogeneous Reservoir with Wellbore Storage Effects CfD = (wk f)/(kx f)= 10
110-5
110-5
CDf=110-6
110-4 110
-3
1100
1101
1102
1102
1101 1100
110-1 110
-2
110-3
110-4
Legend: CfD = (wk f)/(kx f)= 10 pD Solution
pDd Solution
Wellbore StorageDomination Region
Wellbore StorageDistortion Region
Radial Flow Region
1102
Figure 7 pDand pDdvs. tDxf/CDfCfD= 10 (fractured wellcase includes wellbore storage effects).
Horizontal Wells: Ozkan16 created a line-source solution for
modeling horizontal well performance we used this solu-tion to generate-derivative type-curves for the case of a hori
zontal well, where the well is vertically-centeredwithin an in-
finite-acting, homogeneous (and isotropic) reservoir.
10-3
10-2
10-1
100
101
102
pD,pDd
an
dpD
d
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
1
tDL
0.125
0.25
0.5 1
5
10
25
50
100
Infinite ConductivityVertical Fracture
L=0.10.125
0.25
0.5
1
51025
50
100
Infinite Conductivity
Vertical Fracture
Type Curve for a Infinite Conductivity Horizontal Well in an Infinite-ActingHomogeneous Reservoir (L
D= 0.1, 0.125, 0.25, 0.5, 1, 5, 10, 25, 50, 100).
Legend:
pD Solution
pDd
Solution
pDd
Solution
50
25
LD= 0.1
0.125LD= 0.1
0.25
0.5
Figure 8 pD, pDd, and pD
dvs. tDLsolutions for an infiniteconductivity horizontal well in an infinite-actinghomogeneous reservoir no wellbore storageor skin effects (various LDvalues).
In Fig. 8we present thepD,pDd, andpDdsolutions for the caseof a horizontal well with no wellbore storage or skin effects
only the influence of theLDparameter (i.e.,LD= L/2h) inclu
ded in order to illustrate the performance of horizontal wells
with respect to reservoir thickness [thick reservoir (low LD)
thin reservoir (high LD)]. While we do not observe any fea
tures where the pDdfunction is constant, we do observe uni
que characteristic behavior in the pDdfunction, which should
be of value in the diagnostic interpretation of pressure transient test data obtained from horizontal wells.
The pDd and pDd solutions for the case of a horizontal welwith wellbore storage effects are shown in Fig. 9 (LD=100
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6 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
i.e., a thin reservoir). As expected, we do observe the strong
signature of the pDd function for the wellbore storage domi-
nationregime (i.e.,pDd= 1). We also note an apparentfor-mation linear flow regime for low values of the wellbore sto-
rage coefficient (i.e., CDL< 1x10-2). We believe that this is a
transition from the wellbore storage influence to linear flow
(which is brief for this case), then on through the transition
regime towards pseudo-radial flow.
10-3
10-2
10-1
100
101
102
pDandpD
d
10-2
10-1
100
101
102
103
104
105
106
107
tDL/CDL
CDL=110-6
110-2
Type Curve for an Infinite Conductivity Horizontal Well in an Infinite-ActingHomogeneous Reservoir with Wellbore Storage Effects (LD = 100).
110-5
110-4
110-3
1102
1102
Legend: LD = 100
pD Solution
pDd Solution
Wellbore StorageDomination Region
Wellbore StorageDistortion Region
Radial Flow Region
CDL=110-6
1110
1 110-1
1101
1 110-1
110-2
110-3
110-4
110-5
Figure 9 pDand pD
dvs. tDL/CDL LD=100 (horizontal wellcase includes wellbore storage effects).
Wellbore Storage and Boundary Effects: In Fig. 10we presentthe unique case of wellbore storage combined with closed
circular boundary effects (see ref. 17) as a means to demon-
strate that these two influences have the same effect (i.e.,pDd= 1).
10-3
10-2
10-1
100
101
102
103
pD,pDdandpD
d
10-2
10-1
100
101
102
103
104
105
106
107
tD/CD
CDe2s
=110-3
310-3
110-2
310-210
-1
1101
106
10
4
10
3
10
2
101
1 10-1
310-2
110-2
310-3
CDe2s
=110-3
Type Curve for an Unfractured Well i n a Bounded Homogeneous Reservoirwith Wellbore Storage and Skin Effects (reD= 100)
Legend: Bounded Resevoir reD= 100
pD Solution
pDd Solution
pDd
Solution
CDe2s
=110-3
Wellbore StorageDomination Region
Boundary DominatedFlow
Wellbore StorageDistortion Region
106
Figure 10 pD and pDd vs. tD/CD reD =100, boundedcircular reservoir case includes wellbore sto-rage and skin effects. Illustrates combined in-fluence of wellbore storage and boundary ef-fects.
Another aspect of this particular case is that we show the
plausibility of using the -derivative for the analysis of the
boundary-dominated flow regime i.e., the -derivative (or
another auxiliary form) may be a good diagnostic for the ana-
lysis of production data. In particular, the-derivative may beless influenced by data errors that lead to artifacts in the con-
ventional pressure derivative function (i.e., the Bourdet (or
"semilog") form of the pressure derivative).
Application Procedure for
-Derivative Type Curves
The -derivative is a ratio function the dimensionless for-
mulation of the -derivative (pDd) is the exactly the same
function as the "data" formulation of the-derivative [pd(t)]Therefore, when we plot the pd(t) (data) function onto thegrid of thepDdfunction (i.e., the type curve match) they-axis
functions are identical. As such, the vertical "match" is not a
match at all but rather, the model and the data functions aredefined to be the same so the vertical "match" is fixed.
At this point, the time axis match is the only remaining task,
so the pd(t) data function is shifted on top of thepDdfunction, only in the horizontal direction. The time (or horizontal
match is then used to diagnose the flow regimes and provide
an auxiliary match of the time axis. When the pd(t) functionis plotted with the p(t) and the pd(t) functions, we achieve a"harmony" in that the 3 functions are matched simultaneously
and one portion of the match (i.e., pd(t) pDd) is fixed.
The procedures for type curve matching the -derivative data
and models are essentially identical the process given for the
pressure derivative ratio functions in refs. 9 and 10. As with
the "pressure derivative ratio" function (refs. 9 and 10), the
pd(t) pDd is fixed, which then fixes the p(t) and thepd(t) functions, and only the x-axis needs to be resolved exactly like any other type curve for that particular case. Itype curves are not used, and some sort of software-driven
model-based matching procedure is used (i.e., event/history
matching), then the pd(t) andpDdfunctions are matched si-multaneously, in the same manner that the dimensionless pressure/derivative functions would be matched.
Examples Using the
-Derivative Function
To demonstrate/validate the -derivative function we presen
the results of 12 field examples obtained from the literature
(refs. 1, 18-22). The table below provides orientation for ou
examples.
CaseField
Example
Fig. ref.
[oil] Unfractured well (buildup) 1 11 18
[oil] Unfractured well (buildup) 2 12 1
[oil] Dual porosity (drawdown) 3 13 19
[oil] Dual porosity (buildup) 4 14 20
[gas] Fractured well (buildup 5 15 21
[gas] Fractured well (buildup) 6 16 21
[gas] Fractured well (buildup) 7 17 21
[water]Fractured well (falloff) 8 18 22
[water]Fractured well (falloff) 9 19 22
[water]Fractured well (falloff) 10 20 22
[water]Fractured well (falloff) 11 21 22
[water]Fractured well (falloff) 12 22 22
In all of the example cases we were able to successfully inter
pret and analyze the well test data objectively by using the
derivative function [pd(t)] in conjunction with the p(t) and
pd(t) functions. As a comment, for all of the example case
we considered, the -derivative function [pd(t)] provided a
direct analysis (i.e., the "match" was obvious using the pd(t
function the vertical axis match was fixed, and only hori-
zontal shifting was required). These examples and the model
based type curves validate the theory and application of the
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 7
derivative function.
Example 1 is presented in Fig. 11(from ref. 18) and shows the
field data and model matches for the p(t), pd(t), and pd(t)
functions in dimensionless format (i.e., the pD, pDd, and pDd
"data" functions are given as symbols), along with the corres-
ponding dimensionless solution functions (i.e., pD, pDd, and
pDd "model" functions given by the solid lines). This is the
common format used to view the example cases in this work.As noted in ref. 18, in this case wellbore storage effects are
evident, and for the purpose of demonstrating a variable-rate
procedure, downhole rates were measured. In Fig. 11we note
a strong wellbore storage signature, and we find that thepDd
data function (squares) does yield the required value of unity.
The pDd data function does not yield a quantitative inter-
pretation other than the wellbore storage domination region
(pDd= 1), but this function does also provide some resolution
for the data in the transition region from wellbore storage and
infinite-acting radial flow.
10-3
10-2
10-1
100
101
102
pD,pDd
andpD
d
10-2
10-1
100
101
102
103
104
105
tD/CD
Type Curve Analysis Results SPE 11463 (Buildup Case)
(Well in an Infinite-Acting Homogeneous Reservoir)
Legend: Radial Flow Type Curve p
DSolution
pDd
Solution
pDd
Solution
Legend:p
DData
pDd
Data
pDd
Data
Match Results and Parameter Estimates:
[pD/
p]match
=0.02 psi-1
, CDe
2s= 10
6(dim-less)
[(tD/C
D)/t]
match=38 hours
-1, k =399.481 md
Cs=0.25 bbl/psi, s = 1.91 (dim-less)
pDd
= 1
pDd
= 1/2
Reservoir and Fluid Properties:rw=0.3 ft, h= 100 ft,
ct= 1.110-5
psi-1
, =0.27 (fraction)
o= 1.24 cp, B
o= 1.002 RB/STB
Production Parameters:q
ref=9200 STB/D, p
wf(t=0)= 1844.65 psia
Figure 11 Field example 1 type curve match SPE 11463(ref. 18 Meunier) (pressure buildup case).
In Fig. 12we consider the initial literature case regarding well
test analysis using the Bourdet pressure derivative function
(pd) as shown in ref. 1. This is a pressure buildup test wherethe appropriate rate history superposition is used for the timefunction axis. This result is an excellent match of all func-
tions, but in particular, the -derivative function (pDd) is anexcellent diagnostic function for the wellbore storage and tran-
sition flow regimes.
Particular to this case is the fact that the pressure buildup por-
tion of the data was almost twice as long as the reported pres-
sure drawdown portion of the data. We note this issue be-
cause we believe that in order to validate the use of the-deri-
vative function (pDd), we must ensure that the analyst recog-nizes that this function will be affected by all of the same phe-
nomena which affect the "Bourdet" derivative function in
particular, the rate history must be accounted for, most likelyusing the effective time concept where a radial flow super-
position function is used for the time axis.
10-3
10-2
10-1
100
101
102
pD,pDdan
dpD
d
10-2
10-1
100
101
102
103
104
105
tD/CD
Type Curve Analysis SPE 12777 (Buildup Case)(Well in an Infinite-Acting Homogeneous Reservoir)
Legend: Radial Flow Type Curve p
DSolution
pDd
Solution
pD d Solution
Legend:pD Data
pDd Data
pDdData
Reservoir and Fluid Properties:rw=0.29 ft, h= 107 ft,
ct= 4.210-6
psi-1
, =0.25 (fraction)
o= 2.5 cp, B
o= 1.06 RB/STB
Production Parameters:qref=174 STB/D
Match Results and Parameter Estimates:
[pD/p]match=0.018 psi-1
, CDe2s
= 1010
(dim-less)
[(tD/C
D)/t]
match=15 hours
-1, k =10.95 md
Cs=0.0092 bbl/psi, s = 8.13 (dim-less)
pDd= 1
pDd= 1/2
Figure 12 Field example 2 type curve match SPE 12777(ref. 1 Bourdet) (pressure buildup case).
The next example case shown in Fig. 13is taken from a wel
in a known dual porosity/naturally fractured reservoir. As wenote in Fig. 13, the "late" portion of the data is not matchedexactly with the specified reservoir model (infinite-acting ra-
dial flow with dual porosity effects). We contend that part o
the less-than-perfect late time data match may be due to ratehistory effects (only a single production was reported it is
unlikely that the rate remained constant during the entire test
sequence).
However, we believe that this example illustrates the chal-lenges typical of what an analyst faces in practice, and as
such, we believe the -derivative function to be of significant
practical value. We note that the-derivative provides a clear
match of the wellbore storage domination/distortion period
and the function also works well in the transition to system ra-dial flow.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-2
10-1
100
101
102
103
104
105
tD/CD
Type Curve Analysis SPE 13054 Well MACH X3 (Drawdown Case)
(Well in a Dual Porosity System (pss ) = 110-2
, = 110-1
)
Legend:pD Data
pDd Data
pDd
Data
Legend:
=110-2
, = 110-1
pD Solution
pDd
Solution
pDd
Solution
Reservoir and Fluid Properties:rw=0.2917 ft, h= 65 ft,
ct= 24.510-6
psi-1
, =0.048 (fraction)
o= 0.362 cp, B
o= 1.8235 RB/STB
Production Parameters:qref=3224 STB/D, pwf(t=0)= 9670 psia
Match Results and Parameter Estimates:
[pD/p]match=0.000078 psi-1, CDe2s= 1 (dim-less)
[(tD/C
D)/t]
match=0.17 hours
-1, k =0.361 md
Cs=0.1124 bbl/psi, s = -4.82 (dim-less)
= 0.01 (dim-less), = CD
= 0.01(dim-less)
= 6.4510-6
(dim-less)
Figure 13 Field example 3 type curve match SPE 13054(ref. 19 DaPrat) (pressure drawdown case).
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8 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
Our next case (Example 4) also considers well performance ina dual porosity/naturally fractured reservoir (see Fig. 14).
From these data we again note a very strong performance of
the -derivative function particularly in the region defined
by transition from wellbore storage to transient interporosityflow. Cases such as these validate the application of the -
derivative for the interpretation of well test data obtained from
dual porosity/naturally fractured reservoirs.
10-3
10-2
10-1
100
101
102
pD,pDd
andpD
d
10-2
10-1
100
101
102
103
104
105
tD/CD
Type Curve Analysis SPE 18160 (Buildup Case)
(Well in an Infinite-Acting Dual-Porosity Reservoir (trn )
= 0.237, = 110-3
)
Legend:
= 0.237, = 110-3
pD Solution
pDd Solution
pDd Solution
Legend:pD Data
pDd Data
pDd
Data
Reservoir and Fluid Properties:rw=0.29 ft, h= 7 ft,
ct= 210-5
psi-1
,
=0.05 (fraction)
o= 0.3 cp, Bo= 1.5 RB/STBProduction Parameters:
qref
=830 Mscf/D
Match Results and Parameter Estimates:
[pD/p]match=0.09 psi-1
, CDe2s
= 1 (dim-less)
[(tD/CD)/t]match=150 hours-1
, k =678 md
Cs=0.0311 bbl/psi, s = -1.93 (dim-less)
= 0.237 (dim-less), = CD= 0.001(dim-less)
= 2.1310-8
(dim-less)
pDd= 1/2
pDd
= 1
Figure 14 Field example 4 type curve match SPE 18160(ref. 20 Allain) (pressure buildup case).
In Fig. 15we investigate the use of the -derivative function
for the case of a well in a low permeability gas reservoir with
an apparent infinite conductivity vertical fracture (Well 5
from ref. 21). This is the type of case where the-derivative
function provides a unique interpretation for a difficult case.
Most importantly, the-derivative function supports the exis-
tence (and influence) of the hydraulic fracture.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDxf
Type Curve Analysis SPE 9975 Well 5 (Buildup Case)(Well with Infinite Conductivity Hydraulic Fractured )
Legend: Infinite Conductivity Fracture pD Solution
pDd
Solution
pDd Solution
Legend:p
DData
pDd
Data
pDdData
Reservoir and Fluid Properties:rw=0.33 ft, h= 30 ft,
ct= 6.3710-5
psi-1
, =0.05 (fraction)
gi= 0.0297 cp, B
gi= 0.5755 RB/Mscf
Production Parameters:q
ref=1500 Mscf/D
Match Results and Parameter Estimates:
[pD
/
p]match
=0.000021 psi-1
, CDf
= 0.01 (dim-less)
[(tDxf/CDf)/t]match=0.15 hours-1
, k =0.0253 md
CfD
= 1000 (dim-less), xf= 279.96 ft
pDd= 1/2
pDd
= 1/2
Figure 15 Field example 5 type curve match SPE 9975Well 5 (ref. 21 Lee) (pressure buildup case).
Another application of the -derivative function is also to
prove when a flow regime does not (or at least probably does
not) exist the example shown in Fig. 16is just such a case.
In ref. 21 "Well 10" is designated as a hydraulically fractured
well in a gas reservoir and in Fig. 16we observe no evidence
of a hydraulic fracture treatment from any of the dimension-
less plotting functions, in particular, the-derivative function
shows no evidence of a hydraulic fracture. The well is eithe
poorly fracture-stimulated, or a "skin effect" has obscured any
evidence of a fracture treatment in either case, the perfor-
mance of the well is significantly impaired.
10-3
10-2
10-1
100
101
102
pD,pDd
andpD
d
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
Type Curve Analysis SPE 9975 Well 10 (Buildup Case)
(Well with Finite Conductivity Hydraulic Fracture CfD= 2 )
Legend:pD Data
pDd
Data
pDdData
Legend: CfD= 2
pD
Solution
pDd Solution
pDd
Solution
Reservoir and Fluid Properties:rw=0.33 ft, h= 27 ft,
ct= 5.1010
-5psi
-1,
=0.057 (fraction)
gi= 0.0317 cp, Bgi= 0.5282 RB/Mscf
Production Parameters:qref=1300 Mscf/D
Match Results and Parameter Estimates:
[pD/p]match=0.0012 psi-1
, CDf= 100 (dim-less)
[(tDxf/CDf)/t]match=7.5 hours-1
, k =0.137 md
CfD= 2 (dim-less),xf= 0.732 ft
pDd
= 1
pDd
= 1/2
Figure 16 Field example 6 type curve match SPE 9975Well 10 (ref. 21 Lee) (pressure buildup case).
Fig. 17is also taken from ref. 21 "Well 12" is also design-
nated as a hydraulically fractured well in a gas reservoir, and
although there is no absolute signature given by the -deri
vative function (i.e., we do not observe pDd = 1/2 (infinite
fracture conductivity) nor pDd = 1/4 (finite fracture con
ductivity)). We do note that pDd = 1 at early times, which
confirms the wellbore storage domination regime. The pDand pDdsignatures during mid-to-late times confirm the wel
is highly stimulated and the infinite fracture conductivityvertical fracture model is used for analysis and interpretation
in this case.
10-3
10-2
10-1
100
101
102
pD
,pDd
andpD
d
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
Type Curve Analysis SPE 9975 Well 12 (Buildup Case)(Well with Infinite Conductivity Hydraulic Fracture )
Legend:pD Data
pDd
Data
pDdData
Legend: Infinite Conductivity Fracture pD Solution
pDd Solution
pDd Solution
Reservoir and Fluid Properties:rw=0.33 ft, h= 45 ft,
ct= 4.6410-4
psi-1
, =0.057 (fraction)
gi= 0.0174 cp, Bgi= 1.2601 RB/Mscf
Production Parameters:q
ref=325 Mscf/D
Match Results and Parameter Estimates:
[pD/
p]match
=0.0034 psi-1
, CDf
= 0.1 (dim-less)
[(tDxf
/CDf
)/t]match
=37 hours-1
, k =0.076 md
CfD
= 1000 (dim-less), xf= 3.681 ft
pDd
= 1 pDd= 1/2
Figure 17 Field example 7 type curve match SPE 9975Well 12 (ref. 21 Lee) (pressure buildup case).
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 9
In Fig. 18 we present Well 207 from ref. 22, another hy-
draulically fractured well case this time the well is a water
injection well in an oil field, and a "falloff test" is conducted.
In this case there are no data at very early times so we cannot
confirm the wellbore storage domination flow regime. How-
ever; we can use the-derivative function to confirm the exis-
tence of an infinite conductivity vertical fracture for this case,
which is an important diagnostic.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-4
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
Type Curve Analysis Well 207 (Pressure Falloff Case)(Well with Infinite Conductivity Hydraulic Fracture)
Legend: Infinite Conductivity Fracture pD Solution
pDd Solution
pDd Solution
Legend:pD Data
pDd DatapDdData
Reservoir and Fluid Properties:rw=0.3 ft, h= 103 ft,
ct= 7.710-6
psi-1
, =0.11 (fraction)
w= 0.92 cp, Bw= 1 RB/STB
Production Parameters:
qref=1053 STB/D, pwf( t=0)= 3119.41 psia
Match Results and Parameter Estimates:
[pD/p]match=0.009 psi-1
, CDf= 0.001 (dim-less)
[(tDxf/CDf)/t]match=150 hours-1
, k =11.95 md
CfD= 1000 (dim-less), xf= 164.22 ft
pDd= 1
pDd= 1/2
Figure 18 Field example 8 type curve match Well 207(ref. 22 Samad) (pressure falloff case).
In Fig. 19we present Well 3294 from ref. 22, where the data
for this case are somewhat erratic due to acquisition at the sur-
face (i.e., only surface pressures are used). Using the-deri-
vative function we can identify the wellbore storage domina-
tion regime (i.e.,pDd= 1) and we can also reasonably confirm
the existence of an infinite fracture conductivity vertical frac-
ture (pDd= 1/2). The quality of these data impairs our ability
to define the reservoir model uniquely, but we can presumethat our assessment of the flow regimes is reasonable, based
on the character of the-derivative function.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-4
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
Type Curve Analysis Well 3294 (Pressure Falloff Case)(Well with Infinite Conductivity Hydraulic Fracture)
Legend: Infinite Conductivity Fracture pD Solution
pDd Solution
pDd Solution
Legend:pD Data
pDd Data
pDdData
Reservoir and Fluid Properties:
rw=0.3 ft, h= 200 ft,
ct= 7.2610-6
psi-1
, =0.06 (fraction)
w= 0.87 cp, Bw= 1.002 RB/STB
Production Parameters:qref=15 STB/D, pwf(t=0)= 4548.48 psia
Match Results and Parameter Estimates:
[pD/p]match=0.008 psi-1
, CDf= 0.1 (dim-less)
[(tDxf/CDf)/t]match=0.013 hours-1
, k =0.0739 md
CfD= 1000 (dim-less), xf= 198.90 ft
pDd= 1pDd= 1/2
Figure 19 Field example 9 type curve match Well 3294(ref. 22 Samad) (pressure falloff case).
The data for Well 203, taken from ref. 22 are presented in Fig.
20. The signature given by the pD, pDd, and pDd functions
does not appear to be that of a high conductivity vertical frac-
ture. In this case thepDandpDdfunctions suggest a finite con-
ductivity vertical fracture (note that these functions are less
than 1/2 slope). The analysis of these data yields a fairly low
estimate for the fracture conductivity (i.e., CfD= 2), where this
result could suggest that the injection process is not continuing
to propagate the fracture.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-4
10-3
10-2
10-1
100
101
102
103
tDxf/CDf
Type Curve Analysis Well 203 (Pressure Falloff Case)(Well with Finite Conductivity Hydraulic Fracture CfD=2)
Legend: CfD=2
pD Solution
pDd Solution
pDd Solution
Legend:pD Data
pDd Data
pDdData
Reservoir and Fluid Properties:rw=0.198 ft, h= 235 ft,
ct= 6.5310-6
psi-1
, =0.18 (fraction
w= 0.87 cp, Bw= 1.002 RB/STB
Production Parameters:qref=334 STB/D, pwf(t=0)= 2334.1 ps
Match Results and Parameter Estimates:
[pD/p]match=0.0036 psi-1
, CDf= 0.01 (dim-less)
[(tDxf/CDf)/t]match=9 hours-1
, k =0.676 md
CfD= 2 (dim-less), xf= 42.479 ft
pDd= 1 pDd= 1/2
Figure 20 Field example 10 type curve match Well 203
(ref. 22 Samad) (pressure falloff case).
In Fig. 21 we present the data for Well 5408, a pressure fallof
test obtained from ref. 22. This case also exhibits no unique
character in the pD, pDd, and pDd functions, other than well
bore storage domination (pDd= 1) and infinite-acting radia
flow (pDd=1/2). Based on the given data, we know that thi
well was hydraulically fractured and again, based on the in
jection history, we can conclude that this well exhibits the be-
havior of a well with an infinite conductivity vertical fracture
where wellbore storage domination and radial flow exists
These observations are relevant and valuable.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-4
10-3
10-2
10-1
100
101
102
103
10tDxf/CDf
Type Curve Analysis Well 5408 (Pressure Falloff Case)(Well with Infinite Conductivity Hydraulic Fracture)
Legend: Infinite Conductivity Fracture pD Solution
pDd Solution
pDd Solution
Legend:pD Data
pDd Data
pDdData
Reservoir and Fluid Properties:rw=0.198 ft, h= 196 ft,
ct= 6.5310-6
psi-1
, =0.18 (fraction)
w= 0.9344 cp, Bw= 1.002 RB/STB
Production Parameters:qref=350 STB/D, pwf(
t=0)= 2518.1 psia
Match Results and Parameter Estimates:
[pD/p]match =0.0045 psi-1
, CDf= 0.1 (dim-less)
[(tDxf/CDf)/t]match=3 hours-1
, k =1.06 md
CfD= 1000 (dim-less), xf= 29.13 ft
pDd= 1pDd= 1/2
Figure 21 Field example 11 type curve match Well 5408(ref. 22 Samad) (pressure falloff case).
Our last field example is a pressure falloff test performed on
Well 2403, also taken from ref. 22. These data are presentedin Fig. 22and we observe the flow regimes for wellbore sto
rage domination (pDd= 1), and the infinite-acting radial (pDd=1/2).
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10 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
As for characterization of the well efficiency, we can only
conclude that the signature appears to be that of a well with a
high conductivity vertical fracture, hence our match using the
model for a well with an infinite conductivity vertical fracture.
10-4
10-3
10-2
10-1
100
101
pD,pDd
andpD
d
10-4
10-3
10-2
10-1
100
101
102
103
104
tDxf/CDf
Legend: DatapD Data
pDd Data
pDdData
Legend: Infinite Conductivity Fracture pD Solution
pDd Solution
pDd Solution
Type Curve Analysis Well 2403 (Pressure Falloff Case)(Well with Infinite Conductivity Hydraulic Fracture)
Reservoir and Fluid Properties:rw=0.3 ft, h= 102 ft,
ct= 7.2110-6
psi-1
, =0.11 (fraction)
w= 0.85 cp, Bw= 1.002 RB/STB
Production Parameters:qref=73 STB/D, pwf(t= 0)= 2630.89 psia
Match Results and Parameter Estimates:
[pD/p]match=0.18 psi-1
, CDf= 1 (dim-less)
[(tDxf/CDf)/t]match=2 hours-1
, k =12.85 md
CfD= 1000 (dim-less), xf= 50.13 6 ft
pDd= 1/2pDd= 1
Figure 22 Field example 12 type curve match Well 2403
(ref. 22 Samad) (pressure falloff case).
In closing this section on the example application of the -
derivative function, we conclude that the-derivative can pro-
vide unique insight, particularly for pressure transient data
from fractured wells, pressure transient data which is in-
fluenced by wellbore storage, and pressure transient data (and
likely production data) which are influenced by closed boun-
dary effects. In addition, the -derivative function exhibits
some diagnostic character for the pressure transient behavior
of dual porosity/naturally fractured reservoir systems, al-
though these diagnostics are less quantitative in such cases
[i.e., the pd(t) and pDd functions do not exhibit "constant"behavior as with other cases (e.g., wellbore storage, fracture
flow regimes, and boundary-dominated flow)].
We believe that these examples confirm the utility and rele-
vance of the -derivative function and we expect the -
derivative to find considerable practical application in the
analysis/interpretation of pressure transient test data and
(eventually) production data.
Summary
The primary purpose of this paper is the presentation of thenew power-law or -derivative formulation which is given
by:
p
tp
dt
pdt
ptd
pdtp dd
=
=
=)(1
)ln(
)ln()( .................... (1)
This function can be computed directly from data using:
pd(t) = dln(p)/dln(t) (-derivative definition) ........... (8)
pd(t) = pd(t)/p (Bourdet derivative definition) ..(9)
The work of Sowers (ref. 2) shows that using the -derivative
definition (Eq. 8) does provide a slightly more accurate
derivative function than extracting the pd(t) function fromthepd(t) functionas defined in Eq. 9. However, the benefit
derived from using Eq. 8 is likely to be outweighed by the
popularity (and availability) of the Bourdet (or semilog)
pressure derivative function [pd(t)]. In short, if a derivative
computation module is being developed from nothing, Eq. 8
should be used. Otherwise, the "Bourdet" derivative function
[pd(t)] should be adequate to "extract" the -derivative func
tion [pd(t)] via Eq. 7.
Our goal in this work is the presentation of the -derivative
formulation. We have prepared the -derivative solutions for
some of the most popular well test analysis cases (see
Appendix A), as well as graphical representations of thesesolutions in the form of "type curves" (see Appendix B). The
-derivative has been shown to provide much improved
resolution for certain well test analysis cases in particular
the -derivative yields a constant value (i.e., pd(t) = constant) for the following cases:
Case pd(t)
Wellbore storage domination: 1
Reservoir boundaries: Closed reservoir (circle, rectangle, etc.). 2-Parallel faults (large time).
3-Perpendicular faults (large time).
11/2
1/2Fractured wells:
Infinite conductivity vertical fracture. Finite conductivity vertical fracture.
1/21/4
Horizontal wells: Formation linear flow. 1/2
In addition, the-derivative also provides a unique characterization of well test behavior in dual porosity reservoirs (al-
though the -derivative is never constant for these casesexcept for the possibility of a rare fractured or horizontal wel
case).
Finally, we have provided aschematic"diagnosis worksheet"
for the interpretation of the -derivative function (see
Appendix C).
Recommendations for Future WorkThe future work on this topic should focus on the applicationof the-derivative concept for production data analysis.
Acknowledgements
The authors wish to acknowledge the work of Mr. Steven FSowers (ExxonMobil) for access to his computation
routines, and for his efforts to lay the groundwork for this
study via his investigations of the -derivative function as a
statistically enhanced formulation for computing the Bourdet
derivative.
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 11
Nomenclature
Variables
bpss = Pseudosteady-state constant, dimensionlessB = FVF, RB/STB
ct = total system compressibility, psi-1
CA = shape factor, dimensionless
Cs = wellbore storage coefficient, bbl/psiCD =
dim-less wellbore storage coef. unfractured
well
CDf = dim-less wellbore storage coef. horizontal
well
CDL = dim-less wellbore storage coef. fractured wellCfD = fracture conductivity, dimensionless
h = pay thickness, ft
hma = matrix height, ft
k = permeability, md
kf = fracture permeability, mdkfb = dual porosity fracture permeability, md
kma = matrix permeability, md
L = horizontal well length, ftLD = dimensionless horizontal well length
LDf = dimensionless distance from fault
n = positive integer
p = pressure, psi
pD = dimensionless pressure
pDd = dimensionless pressure derivative
pDd = dimensionless-pressure derivative
pi = initial reservoir pressure, psi
pwf = well flowing pressure, psipwfd = well flowing pressure derivative, psi
pwfd =well flowingpressure derivative,
dimensionless
pws = well shut-in pressure, psipwsd = well shut-in pressure derivative, psi
pwsd = well shut-inpressure derivative, dimensionless
q = flow rate, STB/Day
re = reservoir outer boundary radius, ftreD = outer reservoir boundary radius, dimensionless
rw = wellbore radius, ft
rwD = dimensionless wellbore radiusrwzD = dimensionless wellbore radius
t = time, hr
tD = dimensionless time
tDA = dimensionless time with respect to drainage area
tDL = dimensionless time in horizontal well case
tDxf
= dimensionless time in fractured well case
x = distance from wellbore along fracture, ft
xD = dimensionless distance along fracture, ft
xf = fracture length, ft
z = distance in z direction, ft
zD = dimensionless distance in z direction
zw = well location, ftzwD = dimensionless well location
Greek Symbols
= porosity, fraction
f = fracture porosity, fraction
ma = matrix porosity, fraction
= Euler's constant, 0.577216
fD = hydraulic diffusivity, dimensionless
= viscosity, cp
= interporosity flow parameter
= storativity parameter
Subscript
g = gas
o = oilw water
wbs = wellbore storagepss = pseudosteady-state
References
1. Bourdet, D., Ayoub, J.A., and Pirad, Y.M.: "Use of PressureDerivative in Well-Test Interpretation," SPEFE(June 1989) 293302 (SPE 12777).
2. Sowers, S.: The Bourdet Derivative Algorithm Revisited Introduction and Validation of the Power-Law Derivative Algorithm
for Applications in Well-Test Analysis, (internal) B.S. ReportTexas A&M U., College Station, Texas (2005).
3. Mattar, L. and Zaoral, K.: "The Primary Pressure Derivative
(PPD) A New Diagnostic Tool in Well Test Interpretation,JCPT, (April 1992), 63-70.
4. Clark, D.G and van Golf-Racht, T.D.: "Pressure-Derivative Ap-proach to Transient Test Analysis: A High-Permeability NorthSea Reservoir Example,"JPT(Nov. 1985) 2023-2039.
5. Lane, H.S., Lee, J.W., and Watson, A.T.: "An Algorithm forDetermining Smooth, Continuous Pressure Derivatives from WelTest Data,"SPEFE(December 1991) 493-499.
6. Escobar, F.H., Navarrete, J.M., and Losada, H.D.: "Evaluation oPressure Derivative Algorithms for Well-Test Analysis," paperSPE 86936 presented at the 2004 SPE International ThermaOperations and Heavy Oil Symposium and Western Regiona
Meeting, Bakersfield, California, 16-18 March 2004.7. Gonzales-Tamez, F., Camacho-Velazquez, R. and Escalante
Ramirez, B.: "Truncation Denoising in Transient Pressure Tests,"
SPE 56422 presented at the 1999 SPE Annual Technical Con-ference and .Exhibition, Houston, Texas, 3-6 October 1999.
8. Cheng, Y., Lee, J.W., and McVay, D.A.: "Determination ofOptimal Window Size in Pressure-Derivative Computation UsingFrequency-Domain Constraints," SPE 96026 presented at the2005 SPE Annual Technical Conference and .Exhibition, DallasTexas, 9-12 October 2005.
9. Onur, M. and Reynolds, A.C.: "A New Approach forConstructing Derivative Type Curves for Well Test Analysis,"SPEFE(March 1988) 197-206; Trans., AIME, 285.
10.Doung, A.N.: "A New Set of Type Curves for Well Test Inter-pretation with the Pressure/Pressure-Derivative Ratio," SPEFE(June 1989) 264-72.
11.van Everdingen, A.F. and Hurst, W.: "The Application of theLaplace Transformation to Flow Problems in Reservoirs," Trans.
AIME (1949) 186, 305-324.12.Stewart, G., Gupta, A., and Westaway, P.: "The Interpretation of
Interference Tests in a Reservoir with Sealing and Partially Communicating Faults," paper SPE 12967 presented at the 1984 Euro
pean Petroleum Conf. held in London, England, 25-28 Oct. 1984.13.Warren, J.E. and Root, P.J.: "The Behavior of Naturally Fracture
reservoirs," SPEJ(September 1963) 245-55; Trans., AIME, 228.14.Angel, J.A.: Type Curve Analysis for Naturally Fractures
Reservoir (Infinite-Acting Reservoir Case) A New ApproachM.S. Thesis, Texas A&M U., College Station, Texas (2000).
15.Cinco-Ley, H. and Meng, H.-Z.: "Pressure Transient Analysis oWells with Finite Conductivity Vertical Fractures in DuaPorosity Reservoirs," SPE 18172 presented at the 1989 SPE
-
7/25/2019 [Blasingame] SPE 103204
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12 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
Annual Technical Conference and Exhibition, Houston, Texas, 2-5 October 1989.
16.Ozkan, E.: Performance of Horizontal Wells, Ph.D. Dissertation,U. of Tulsa, Tulsa, Oklahoma (1988)
17.Blasingame, T.A.: "Semi-Analytical Solutions for a Bounded Cir-cular Reservoir-No Flow and Constant Pressure Outer BoundaryConditions: Unfractured Well Case," SPE 25479 presented at the1993 SPE Production Operations Symposium, Oklahoma City,
OK, 21-23 March 1993.
18.Meunier, D., Wittmann, M.J., and Stewart, G.: "Interpretation ofPressure Buildup Test Using In-Situ Measurement of Afterflow,"
JPT(January 1985) 143 (SPE 11463).19.DaPrat, G.D. et al.: "Use of Pressure Transient Testing to
Evaluate Fractured Reservoirs in Western Venezuela," SPE13054 presented at the 1984 SPE Annual Technical Conference
and Exhibition, Houston, Texas, 16-19 September 1984.20.Allain, O.F. and Horne R.N.: "Use of Artificial Intelligence in
Well-Test Interpretation,"JPT(March 1990) 342.21.Lee, W.J. and Holditch, S.A.: "Fracture Evaluation with Pressure
Transient Testing in Low-Permeability Gas Reservoirs," JPT(September 1981) 1776.
22.Samad, Z.: Application of Pressure and Pressure IntegralFunctions for the Analysis of Well Test Data, M.S. Thesis, TexasA&M U., College Station, Texas (1994).
23.Gringarten, A.C., Ramey, H.J., Jr., and Raghavan, R.: "Unsteady-State Pressure Distributions Created by a Well with a SingleInfinite-Conductivity Vertical Fracture," SPEJ. (August 1974)347-360.
24.Cinco-Ley, H. and Samaniego-V., F.: "Transient PressureAnalysis for Fractured Wells,"JPT(September 1981) 1749.
25.van Golf-Racht, T.D.: Fundamentals of Fractured ReservoirEngineering, Elsevier, New York, NY (1982)
26.Blasingame, T.A., Johnston, J.L., and Lee, W.J.: "Advances inthe Use of Convolution Methods in Well Test Analysis," paper
SPE 21826 presented at the 1991 Joint Rocky MountainRegional/Low Permeability Reservoirs Symposium, Denver, CO,15-17 April 1991.
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 13
Appendix A Table of solutions for pD, pDd, and pD
d(conditions/flow regimes as specified).
Table A-1 Solutions for the wellbore storage domination flow regime.
Variable Solution Relation
wbsp twbswbs mp = ................................................................................................................................................(A.1.1)
wbsdp ,
twbswbsd mp = ,.............................................................................................................................................(A.1.2)
swbdp , 1
,
=swbd
p
...................................................................................................................................................(A.1.3)
Definitions: (field units)
24
1
swbs
C
qBm =
................................................................................................................................................................................................... (A.1.4)
Table A-2 Solutions for a well in a finite-acting, homogeneous reservoir (closed system, anywell/reservoir configuration).
Description Relation
Dp 214
ln2
12)(
2 pssDAAw
DADAD btsCr
A
ettp +=+
+=
..............................................................................(A.2.1)
Ddp DADADd ttp 2)( = ........................................................................................................................................(A.2.2)
)/( DDddD ppp = 1)2/(1
1)(
+=
DApssDAdD
tbtp
(large-time) ..................................................................................................................................(A.2.3)
Definitions: (field units)
tAc
kt
tDA 10637.2
4
=................................................................................................................................................................................... (A.2.4)
)(2.141
1wfiD pp
qB
khp =
................................................................................................................................................................................. (A.2.5)
sCr
A
eb
Awpss +
=
14ln
2
1
2
................................................................................................................................................................................... (A.2.6)
Table A-3 Solutions for an unfractured well in an infinite-acting, homogeneous reservoir (radial flow).
Description Relation
Dp
=
DDD
ttp 4
1E2
1)( 1
( )10>Dt .........................................................................................................................................(A.3.1)
Ddp
=
DDDd
ttp
4
1exp
2
1)(
( )10>Dt .........................................................................................................................................(A.3.2)
)/( DDddD ppp =
=
DDDdD
tttp
4
1E
4
1exp)( 1
( )10>Dt .........................................................................................................................................(A.3.3)
Definitions: (field units)
2
410637.2
wt
Drc
ktt
=................................................................................................................................................................................... (A.3.4)
)(2.141
1
wfiD ppqB
kh
p =
................................................................................................................................................................................. (A.3.5)
2
8936.0
wt
sD
rhc
CC
=
.................................................................................................................................................................................................. (A.3.6)
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14 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
Table A-4 Solutions for a single well in an infinite-acting homogeneous reservoir system with a single ormultiple sealing faults.
Description Relation
Dp
+
=
D
Df
DDD
t
L
ttp
2
11 E4
1E
2
1)(
(single fault)...................................................................................................................................(A.4.1)
+
+
=
D
Df
D
Df
DDD
t
L
t
L
ttp
2
1
2
11
2
EE24
1E21)(
(two perpendicular faults)..............................................................................................................(A.4.2)
+
=
=1
2
11 E24
1E
2
1)(
iD
Df
DDD
t
iL
ttp
(two parallel faults)........................................................................................................................(A.4.3)
+
+
++
=
=
=1
2
1
1
2
1
22
11 EE2)1(
E24
1E
2
1)(
iD
Df
iD
Df
D
Df
DDD
t
L
t
iL
t
Li
ttp
(three perpendicular faults)............................................................................................................(A.4.4)
Ddp
12
1
2
1)(
/4/12
+= DDfD
tLtDDd eetp
(single fault, complete solution and large-time approximation)...................................................(A.4.5)
221
21)( /2/4/1
22
++= DDfDDfD tLtLtDDd eeetp (two perpendicular faults, complete solution and large-time approximation)..............................(A.4.6)
=
+=
1
/4/12
2
1)(
i
tiLtDDd
DDfD eetp
(two parallel faults, complete solution and large-time approximation)........................................(A.4.7)
=
=
+ +++=
1
/
1
//)1(4/12222
2
1
2
1)(
i
tL
i
tiLtLitDDd
DDfDDfDDfD eeeetp
(three perpendicular faults)............................................................................................................(A.4.8)
)/( DDddD ppp =
+
+
+=
D
Df
DD
Df
D
tLt
DdD
t
L
tt
L
t
eetp
DDfD
2
11
2
11
/4/1
E4
1E
2
E4
1E
)(
2
(single fault, complete solution and large-time approximation)...................................................(A.4.9)
+
+
+
+
++=
D
Df
D
Df
DD
Df
D
Df
D
tLtLt
DdD
t
L
t
L
tt
L
t
L
t
eeetp
DDfDDfD
2
1
2
11
2
1
2
11
/2/4/1
2EE2
4
1E
4
2EE2
4
1E
2)(
22
(two perpendicular faults, complete solution and large-time approximation)............................(A.4.10)
2
1
E24
1E
2
)(
1
2
11
1
/4/12
+
+
=
=
=
iD
Df
D
i
tiLt
DdD
t
iL
t
ee
tp
DDfD
(two parallel faults, complete solution and large-time approximation)......................................(A.4.11)
2
1
EE2)1(E24
1E
22
)(
1
2
1
1
2
1
22
11
1
/
1
//)1(4/12222
+
+
++
+++
=
=
=
=
=
+
iD
Df
iD
Df
D
Df
D
i
tL
i
tiLtLit
DdD
tL
tiL
tLi
t
eeee
tp
DDfDDfDDfD
(three perpendicular faults, complete solution and large-time approximation)..........................(A.4.12)
Definitions: (field units)
2
410637.2
wt
Drc
ktt
=.................................................................................................................................................................................... (A.4.13)
)(2.141
1wfiD pp
qB
khp =
................................................................................................................................................................................. (A.4.14)
w
faultDf
r
LL =
.................................................................................................................................................................................................. .(A.4.15)
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SPE 103204 The Pressure Derivative Revisited Improved Formulations and Applications 15
Table A-5 Solutions for a hydraulically fractured well with an infinite conductivity fracture in an infinite-acting reservoir.
Description Relation
Dp
( ) ( ) ( )
+++
+
++
=
Dxf
DD
Dxf
DD
Dxf
D
Dxf
DDxfDxfD
t
xx
t
xx
t
x
t
xttp
4
1E
4
1
4
1E
4
)1(
2
1erf
2
1erf
2)(
2
1
2
1
(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.1)
DxfDxfD ttp =)(
(early time, linear flow) .................................................................................................................(A.5.2)
]80907.2)[ln(2
1)( += DxfDxfD ttp
(late time, uniform flux fracture)...................................................................................................(A.5.3)
]20000.2)[ln(2
1)( += DxfDxfD ttp
(late time, infinite conductivity fracture) ......................................................................................(A.5.4)
Ddp
++
=
Dxf
D
Dxf
DDxfDxfDd
t
x
t
xttp
2
1erf
2
1erf
4)(
(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.5)
4)(
DxfDxfDd
ttp
=
(early time, linear flow) .................................................................................................................(A.5.6)
5.0)( =DxfDd tp
(late time) .......................................................................................................................................(A.5.7)
)/( DDddD ppp =
+++
+
++
++
=
Dxf
DD
Dxf
DD
Dxf
D
Dxf
DDxf
Dxf
D
Dxf
DDxfDxfdD
t
xx
t
xx
t
x
t
xt
t
x
t
xttp
4
)1(E
4
)1(
4
)1(E
4
)1(
2
1erf
2
1erf
2
2
1erf
2
1erf
4)(
2
1
2
1
(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.8)5.0)( =DxfdD tp
(early time, linear flow) .................................................................................................................(A.5.9)
80907.2)ln(
1)(
+=
DxfDxfdD
ttp
(late time, uniform flux fracture).................................................................................................(A.5.10)
20000.2)ln(
1)(
+=
DxfDxfdD
ttp
(late time, infinite conductivity fracture) ....................................................................................(A.5.11)
Definitions: (field units)
2
410637.2
ft
Dxfxc
ktt
=.............................................................................................................................................................................. (A.5.12)
)(2.141
1wfiD pp
qB
khp =
............................................................................................................................................................................... (A.5.13)
fD xxx /=....................................................................................................................................................................................................... .(A.5.14)
2
8936.0
ft
sDf
xhc
CC
=
............................................................................................................................................................................................. .(A.5.15)
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16 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204
Table A-6 Early time solutions for a hydraulically fractured well with a finite conductivity fracture infinite-acting homogeneous reservoir (includes wellbore storage effects).
Description Relation
Dp
=
Dxft DfxfD
fD
fD
fD
DxfD dzz
ztC
z
Ctp
0
5.0)(
erfc
)(
(General solution) ..........................................................................................................................(A.6.1)
DxffDfD
DxfD tC
tp 2
)( =
(Short-time approximation),
2
201.0
fD
fDDxf
Ct
................................................................................(A.6.2)
( )4
1
225.1)( Dxf
fDDxfD t
Ctp
=
(Large-time approximation),
6.15.2
55.4
36.1)5.1(0205.0
31.0
4
2
53.1
fDfD
Dxf
fDfDDxf
fD
fD
Dxf
CCt
CCt
C
C
t
...........................(A.6.3)
Ddp
fD
DxffDDxfDd
C
ttp
=)(
(Short-time approximation)...........................................................................................................(A.6.4)
4
1612708.0)( Dxf
fDDxfDd t
Ctp =
(Large-time approximation) ..........................................................................................................(A.6.5)
)/( DDddD ppp =
2
1)( =DxfdD tp
(Short-time approximation)...........................................................................................................(A.6.6)
4
1)( =DxfdD tp
(Large-time approximation) ..........................................................................................................(A.6.7)
Definitions: (field units)
2
410637.2
ft
Dxfxc
ktt
=................................................................................................................................................................................ (A.6.8)
)(2.141
1wfiD pp
q