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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


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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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


2 ParallelFaults

Dimensionless Pressure Derivative Type Curves for Sealing Faults(Inifinite-Acting Homogeneous Reservoir)

Undistorted

Radial Flow Behavior

2 ParallelFaults


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


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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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


Radial Flow 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



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

7/25/2019 [Blasingame] SPE 103204

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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



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


Boundary DominatedFlow


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


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





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

7/25/2019 [Blasingame] SPE 103204

7/27


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


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


[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


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


[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).

7/25/2019 [Blasingame] SPE 103204

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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


ct= 210-5

psi-1

,

=0.05 (fraction)

o= 0.3 cp, Bo= 1.5 RB/STBProduction Parameters:

qref

=830 Mscf/D



, 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


ct= 6.3710-5

psi-1

, =0.05 (fraction)

gi= 0.0297 cp, B

gi= 0.5755 RB/Mscf


ref=1500 Mscf/D


[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


ct= 5.1010

-5psi

-1,

=0.057 (fraction)

gi= 0.0317 cp, Bgi= 0.5282 RB/Mscf

Production Parameters:qref=1300 Mscf/D



, CDf= 100 (dim-less)


, k =0.137 md

CfD= 2 (dim-less),xf= 0.732 ft

pDd

= 1

pDd

= 1/2


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


pDd Solution

pDd Solution


ct= 4.6410-4

psi-1

, =0.057 (fraction)

gi= 0.0174 cp, Bgi= 1.2601 RB/Mscf


ref=325 Mscf/D


[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


7/25/2019 [Blasingame] SPE 103204

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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)


pDd Solution

pDd Solution

Legend:pD Data

pDd DatapDdData


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



, 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



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





, k =0.0739 md


pDd= 1pDd= 1/2


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


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





, k =0.676 md


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



pDd Solution

pDd Solution

Legend:pD Data

pDd Data

pDdData


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


[pD/p]match =0.0045 psi-1



, k =1.06 md


pDd= 1pDd= 1/2


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).

7/25/2019 [Blasingame] SPE 103204

10/27


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


pDd Solution

pDd Solution



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



, CDf= 1 (dim-less)


, 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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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

12/27


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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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)


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).


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)


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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Table A-4 Solutions for a single well in an infinite-acting homogeneous reservoir system with a single ormultiple sealing faults.


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)


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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Table A-5 Solutions for a hydraulically fractured well with an infinite conductivity fracture in an infinite-acting reservoir.


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

=


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


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)


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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Table A-6 Early time solutions for a hydraulically fractured well with a finite conductivity fracture infinite-acting homogeneous reservoir (includes wellbore storage effects).


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)


2

410637.2

ft

Dxfxc

ktt

=................................................................................................................................................................................ (A.6.8)

)(2.141

1wfiD pp

q

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