SPE 103204 The Pressure Derivative Revisited — …Zonoozi)_Pressure_Derivative_Rev… · The Pressure Derivative Revisited — Improved Formulations and Applications ... (or well

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Abstract

The proposed work provides a new definition of the pressure-derivative function [i.e., the β-derivative function, Δpβd(t)], which is defined as:

ptp

dtpdt

ptdpdtp d

d ΔΔ

=Δ

Δ=

Δ=Δ

)(1)ln()ln()(β

(Δpd(t) is the "Bourdet" well testing derivative)

This formulation is based on the "power-law" concept (i.e., the derivative of the logarithm of pressure drop with respect to the logarithm of time) — this is not a trivial definition, but rather a definition that provides a unique characterization of "power-law" flow regimes.

The "power-law" flow regimes uniquely defined by the Δpβd(t) function are: [i.e., a constant Δpβd(t) behavior]

Case Δpβd(t)

Wellbore storage domination:

1

Reservoir boundaries: — Closed reservoir (circle, rectangle, etc.). — 2-Parallel faults (large time). — 3-Perpendicular faults (large time).

1 1/2 1/2

Fractured wells: — Infinite conductivity vertical fracture. — Finite conductivity vertical fracture.

1/2 1/4

Horizontal wells: — Formation linear flow.

1/2

In addition, the Δpβd(t) function provides unique characteristic responses for cases of dual porosity (naturally-fractured) reser-voirs.

The Δpβd(t) function represents a new application of the tradi-tional pressure derivative function, the "power-law" differentiation method (i.e., computing the dln(Δp)/dln(t) deri-vative) provides an accurate and consistent mechanism for computing the primary pressure derivative (i.e., the Cartesian

derivative, dΔp/dt) as well as the "Bourdet" well testing derivative [i.e., the "semilog" derivative, Δpd(t)=dΔp/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 as well as graphical presentations (type curves) of the β-derivative functions 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 β-derivative functions using type curves applied to field data cases using pressure drawdown-/buildup and injection/falloff test data.

Introduction

The well testing pressure derivative function,1 Δpd(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 Δpd(t) func-tion as defined by Bourdet et al. [i.e., Δpd(t)=dΔp/dln(t)] pro-vides a constant value for the case of a well producing at a constant rate in an infinite-acting homogeneous reservoir. That is, Δpd(t) = constant during infinite-acting radial flow be-havior.

This single observation has made the Bourdet derivative, Δpd(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 Δpd(t) function?

The answer is somewhat complicated in light of the fact that the Bourdet derivative function has almost certainly been generated for every reservoir model in existence. Reservoir engineers have come to use the characteristic shapes in the Bourdet derivative for the diagnosis and analysis of wellbore storage, boundary effects, fractured wells, horizontal wells, and heterogeneous reservoirs. For this work we prepare the β-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 Applications N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame, SPE, Texas A&M University

2 N. Hosseinpour-Zonoozi, D. Ilk, and T. A. Blasingame SPE 103204

particular plotting function. For example, a derivative-based plotting 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" β-derivative formulation does just that — it provides a single plotting function which can be used (in isolation) as a mechanism 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:

ptp

dtpdt

ptdpdtp d

d ΔΔ

=Δ

Δ=

Δ=Δ

)(1)ln()ln()(β .................... (1)

where Δpd(t) is the "Bourdet" well testing derivative.

In Appendix A we provide the definitions of the power-law β-derivative function for various reservoir models as shown below. The graphical solution (or "type curve") for each case of interest is shown in Appendix B, and categorized as shown below.

Specific Δpβd(t) Case:

Δpβd(t)

App. A Table

App. BFigs.


1

A-1

4,11-20

Reservoir boundaries: Closed reservoir Infinite-acting (incl. WBS) 2-Parallel faults 3-Perpendicular faults

1 --- 1/2 1/2

A-2 A-3 A-4 A-4

1,2,25

4 3 3

Fractured wells: Infinite cond. vert. fracture Finite cond. vert. fracture

1/2 1/4

A-5 A-6

10,11

10,12-14Dual 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 β-derivative formulation Δpβd(t) was an effort by Sowers2 to demonstrate that this formulation would provide a consistently better estimate of the Bourdet derivative function, Δpd(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:

dtpdtpPd

Δ=Δ )( .............................................................. (2)

where ΔpPd(t) is also known as the "primary pressure deri-vative" [ref. 3 (Mattar)].

The "semilog" or "Bourdet" pressure derivative is defined as:

dtpdttpd

Δ=Δ )( ............................................................... (3)

Recalling that the "β " pressure derivative is defined as:

ptp

dtpdt

ptdpdtp d

d ΔΔ

=Δ

Δ=

Δ=Δ

)(1)ln()ln()(β .................... (1)

solving for the "Cartesian" or "primary pressure derivative,"

)(tptp

dtpd

dβΔΔ

=Δ ........................................................ (4)

solving for the "semilog" or "Bourdet" pressure derivative,

)( )( tpptp dd βΔΔ=Δ .................................................... (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 [ΔpPd(t)=dΔp/dt].

2. A simple finite-difference estimate of the "semilog" (or "Bour-det") pressure derivative [Δpd(t)=dΔp/dln(t)].

3. Some type of weighted finite-difference or central difference es-timate of either the "Cartesian" or "semilog" pressure derivative functions. This is the approach of Bourdet et al.1 and Clark and van Golf-Racht4 — this formulation is by far the most popular technique used to compute pressure derivative functions for the purpose of well test analysis, and will be presented in detail in the next section.

4. Other more elegant and more statistical sophisticated algorithms have been proposed for use in pressure transient (or well test) analysis, but the Bourdet et al. algorithm (and its variations) continue to be the most popular approach, most likely due to the simplicity and consistency of this algorithm. To be certain, the Bourdet et al. algorithm does not provide the most accurate es-timates of the derivative functions, but the predictability of the algorithm is very good, and the purpose of the derivative is as a diagnostic function, not a function used to provide an exact esti-mate.

Some of the other algorithms proposed for estimating the va-rious pressure derivative functions are summarized below:

● Moving polynomial or another type of moving regression function. This is generally referred to as a "window" app-roach (or "windowing").

● Spline approximation by Lane et al.5 is a powerful approach, but as pointed out in a general assessment of the computation of the pressure derivative (Escobar et al.6), the spline appro-ximation requires considerable user input to obtain the "best fit" 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 and regression was used to find the "best-fit" to the data over a specified window.

● Cheng et al.8 utilized the fast Fourier transform and fre-quency-domain constraints to improve Bourdet algorithm by optimizing the size of search window and they also used a Gaussian filter to denoise the pressure derivative data. This resulted in an adaptive smoothing procedure that uses recur-sive differentiation and integration.

Calculation of the β-Derivative Function

To minimize the effect of truncation error, Bourdet et al.1 in-troduced a weighted central-difference derivative formula:

RR

RLL

LL

RLR

tp

ttt

tp

ttt

tdpd

ΔΔ

Δ+ΔΔ

+ΔΔ

Δ+ΔΔ

=Δ

Δ)]ln([

.......(6a)

where:

ΔtL = ln(Δtcalc) – ln(Δtleft)........................................... (6b)

ΔtR = ln(Δtright) – ln(Δtcalc) ..........................................(6c)

ΔpL = Δpcalc – Δpleft .................................................... (6d)

ΔpR = Δpright – Δpcalc....................................................(6e)

The left- and right-hand subscripts represent the "left" and "right" neighbor points located a specified distance (L) from the objective point. The calc subscript represents the point of interest at which the derivative is to be computed. As for the

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 L is the frac-tional proportion of a log-cycle (log10 base). 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 nearest neighbors, 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 — which will 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 starting L-value of 0.2 [20 percent of a log-cycle (recall that log is the log10 function)].

Sowers2 proposed 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 the original Bourdet formulation. In particular, Sowers provides the following definition of the power-law (or "β") derivative formulation:

[ ]RR

RLL

LL

RLR

tp

ttt

tp

ttt

tdpd

ΔΔ

Δ+ΔΔ

+ΔΔ

Δ+ΔΔ

=ΔΔ

)]ln([)ln( ..... (7a)

where:

ΔtL = ln(Δtcalc) – ln(Δtleft) ........................................... (6b)

ΔtR = ln(Δtright) – ln(Δtcalc) ......................................... (6c)

ΔpL = ln(Δpcalc) – ln(Δpleft) ......................................... (7d)

ΔpR = ln(Δpright) – ln(Δpcalc) ....................................... (7e)

Multiplying the right-handside of Eq. 7a by Δpcalc (recall that Δpcalc 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 Δp function and found that the power-law (or β) derivative formulation always showed improved accuracy of the well testing pressure derivative [i.e., the Bourdet derivative function, Δpd(t)].

In addition, Sowers found that the β-derivative formulation was less sensitive to the L-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 not pursue the specific application of the β-deri-vative function [Δpβd(t)=d ln(Δp)/dln(t)] as a diagnostic plot-ting function, as we have this work.

Type Curves Using the β-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 sophisticated interpretation/analysis/modeling software. The advent of the β-derivative function as proposed in this paper is not expected to replace the Bourdet derivative (nor should this happen). The β-derivative function is proposed simply to serve as a

better interpretation device for certain flow regimes — in par-ticular, 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 β-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 pressure derivative (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 2Δp in radial flow applications)) — this ratio was (ob-viously) a dimensionless quantity. In particular, the pressure derivative ratio was applied as an interpretation device — as it is 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 from the β-derivative solutions given in Appendix A.

The primary utility of the β-derivative is the resolution that 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 boun-dary-dominated (closed boundaries) are good candidates for the β-derivative.

10-2

10-1

100

101

102

103

10-5 10-4 10-3 10-2 10-1 100 101 102

Dimensionless Time, tD (model-dependent)

Legend: (pDd ) (pDβd ) Unfractured Well (Radial Flow) Fractured Well (Infinite Fracture Conductivity) Fractured Well (Finite Fracture Conductivity) Horizontal Well (Full Penetration, Thin Reservoir)

Transient Flow Region

Schematic of Dimensionless Pressure Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)

DIAGNOSTIC plot for Well Test Data (pDd and pDβd)

Bou

rdet

"W

ell T

est"

Dim

ensi

onle

ss P

ress

ure

Der

ivat

ive

Func

tion,

pD

d"P

ower

Law

— β

" D

imen

sion

less

Pre

ssur

e D

eriv

ativ

e Fu

nctio

n, p

Dβd

Boundary-Dominated

Flow Region

pDβd = 0.5(linear flow)

pDβd = 0.25(bilinear flow)

pDβd = 1(boundary

dominated flow)

1

1

1

2

41

2

1

Unfractured Well ina Bounded Circular

ReservoirFractured Well in

a Bounded CircularReservoir

(Infinite ConductivityVertical Fracture)

Horizontal Well in aBounded Square

Reservoir:(Full Penetration,Thin Reservoir)

Fractured Well ina Bounded Circular

Reservoir(Finite ConductivityVertical Fracture)

( )( )

( )( )

( )( )

( )( )

NO Wellbore Storage or Skin Effects

Figure 1 — Schematic of pDd and pDβd vs. tD — Various reser-voir models and well configurations (no well-bore storage or skin effects).

Infinite-acting radial flow — the "utility" case for the Bourdet (semilog) derivative function is not a good candidate for inter-pretation using the β-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. 1 we present a schematic plot created for illustrative purposes to represent the character of the β-derivative for several distinctly different cases. Presented are the β-derivative profiles (in schematic form) for an unfractured well (infinite-acting radial flow), 2 fractured well cases, and a


horizontal well case. We note immediately the strong charac-ter of the fractured well responses (pDβd = 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 pDβd 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, the pDβd function yields a constant value of unity, as expected. This observation suggests that the pDβd function (or an auxiliary function based on the pDβd form) may be of value for the analysis of production data. For reference, Fig. 1 is presented in a larger format in Appendix B (Fig. B-1).

Infinite-Acting Radial Flow: The β-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 β-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 β-derivative function does not demonstrate a constant behavior for the ra-dial flow case, but as noted in Appendix A, the β-derivative function for the wellbore storage domination flow regime yields pDβd = 1.

10-3

10-2

10-1

100

101

102

103

p D, p

Dd

and

p Db d

10-2 10-1 100 101 102 103 104 105 106

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2 10-1

1 101

102 103

104106

108

1010

1020 1030

1015

1040

10100

1060

1080 1050

10100

1080

1060 1050

1040

1030

1020 1015

1010

108106

104 103

102 101

3×10-2 1×10-2

CDe2s=1×10-3

Type Curve for an Unfractured Well in an Infinite-Acting Homogeneous Reservoir with Wellbore Storage and Skin Effects.

3

3

10100

CDe2s=1×10-3

Legend: Radial Flow Type Curves pD Solution pDd Solution pDβd Solution

Radial Flow Region

Wellbore StorageDomination Region

Wellbore StorageDistortion Region

Figure 2 — pD, pDd, and pDβd vs. tD/CD — solutions for an unfractured well in an infinite-acting homo-geneous reservoir — wellbore storage and skin effects included (various CD values).

Sealing Faults: Ref. 12 provides pDd-format (Bourdet) type curves 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 β-derivative function where the pDβd functions show interesting characteristics, as well as the 2-parallel sealing faults and 3-perpendicular fault cases, which prove that pDβd = 1/2 at very long times (see Fig. 3).

10-3

10-2

10-1

100

101

102

103

104

β-Pr

essu

re D

eriv

ativ

e Fu

nctio

n, p

Dβ d

= (t

D/p

D) d

/dt D

(pD

)

10-3 10-2 10-1 100 101 102 103 104 105 106 107

tD/LD2 (LD = Lfault/rw)

Legend: β-Pressure Derivative Function

Single Fault Case 2 Perpendicular Faults (2 at 90 Degrees) 2 Parallel Faults (2 at 180 Degrees) 3 Perpendicular Faults (3 at 90 Degrees)

SingleFault

2 Perpendicular Faults


2 Parallel Faults

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

UndistortedRadial Flow Behavior

2 Parallel Faults


pDβd = (tD/pD) dpD/dtD

pDd = tD dpD/dtD

Legend: "Bourdet" Well Test Pressure Derivative


"Bou

rdet

" W

ell T

est P

ress

ure

Der

ivat

ive,

pD

d =

t D d

p D/d

t D

SingleFault


Figure 3 — pDd and pDβd vs. tD/LD2 — various sealing faults

configurations (no wellbore storage or skin effects).

Unfractured Well in a Dual Porosity System: We used the pseudosteady-state interporosity model13 to produce the β-derivative type curves for a single well in an infinite-acting, dual porosity reservoir with or without wellbore storage and skin effects. For these cases, we chose to present our cases (which include wellbore storage) using the type curve format of ref. 14 (the family parameters for the type curves are the ω and α-parameters, where α = λCD).

In Fig. 4 we present a general set of cases (ω = 1x10-1, 1x10-2, and 1x10-3 and λ = 5x10-9, 5x10-6, and 5x10-3) with no well-bore storage or skin effects. Fig. 4 shows the unique signature of the pDβd functions for this case, but we can also argue that, since this model is tied to infinite-acting radial flow, the pDβd functions can, at best, be used as a diagnostic to view ideal-ized behavior.

10-4

10-3

10-2

10-1

100

101

102

p D a

nd p

Dβ d

10-1 100 101 102 103 104 105 106 107 108 109

tD

Type Curve for an Unfractured Well in an Infinite-Acting Dual Porosity Reservoir (Pseudosteady-State Interporosity Flow) — No Wellbore Storage or Skin Effects.

Legend: pD Solution pDβd Solution

ω = 1×10-1

ω = 1×10-1ω = 1×10-1

ω = 1×10-2

ω = 1×10-3

ω = 1×10-2

ω = 1×10-2

ω = 1×10-3ω = 1×10-3

ω = 1×10-11×10-2

1×10-3

pDβd (λ = 5 ×10-9) pDβd (λ = 5 ×10-6)

pDβd (λ = 5 ×10-3)

Figure 4 — pD and pDβd vs. tD — solutions for an unfractured well in an infinite-acting dual porosity system — no wellbore storage or skin effects (various λ and ω values).


In Fig. 5 we present cases where ω = 1×10-1 and α = λCD = 1×10-4 for 1×10-4 < CD < 1×10100. As with the results for the pDd functions shown in ref. 14, these pDβd functions do provide some insight into the form and character of the behavior for the case of a well producing at infinite-acting flow conditions in a dual porosity/naturally fractured reservoir system.

10-3

10-2

10-1

100

101

102

103

p D a

nd p

Dßd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2

10-11 101

102

103 104

106

108

1010

1020

1030 1015

1040

10100

1060

1080 1050

10100

1080

1060

1050

1040

1030

1020

1015

1010

108106

104

103102 101

1 10-1

3×10-2

1×10-2 3×10-3

CDe2s=1×10-3

Type Curve for an Unfractured Well in an Infinite-Acting Dual Porosity Reservoir (Pseudosteady-State InterporosityFlow) with Wellbore Storage and Skin Effects.

( α = λCD = 1×10-4, ω = 1×10-1)

Legend: α = λCD = 1×10-4, ω = 1×10-1

pD Solution pDβd Solution

10100

CDe2s=1×10-3


Radial Flow Region


Figure 5 — pD and pDβd vs. tD/CD — ω = 1×10-1, α = λCD = 1×10-4 (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 finite conductivity vertical fracture where the β-derivative type curves were generated using the Cinco and Meng15 solution. 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 pDβd functions 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 on the β-derivative plot (bilinear flow: pDβd = 1/4, linear flow: pDβd = 1/2).

10-3

10-2

10-1

100

101

p D, p

Dd

and

p Dβ d

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

tDxf

CfD=0.250.5

1

CfD=1×104

Type Curve for a Well with a Finite Conductivity VerticalFractured in an Infinite-Acting Homogeneous Reservoir

(CfD = (wkf)/(kxf) = 0.25, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 10000)

Legend: pD Solution pDd Solution pDβd Solution

Radial Flow Region

CfD=0.25

CfD=1×104

1

5

2

0.51

21×103

500

Figure 6 — pD, pDd, and pDβd vs. tDxf — solutions for an fractured well in an infinite-acting homogene-ous reservoir — no wellbore storage or skin ef-fects (various CfD values).

In Fig. 7 we present the case of a single well with a finite con-ductivity vertical fracture (CfD = 10) producing at a constant rate in an infinite-acting homogeneous reservoir, with well-bore storage effects included. We observe the characteristic

wellbore storage domination behavior (pDβd = 1), as well as the effect of bilinear (fracture and formation) flow (pDβd = 1/4). We believe that the pDβd function (i.e., the β-derivative) will substantially improve the diagnosis of flow regimes in hydraulically fractured wells.

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf

CDf=1×10-6

1×10-2

Type Curve for a Well with Finite Conductivity Vertical Fracture in an Infinite-Acting Homogeneous Reservoir with Wellbore Storage Effects CfD = (wkf)/(kxf)= 10

1×10-5

1×10-5

CDf=1×10-6

1×10-4 1×10-3

1×100 1×101

1×102

1×102 1×101 1×100

1×10-1 1×10-2

1×10-3

1×10-4

Legend: CfD = (wkf)/(kxf)= 10 pD Solution pDβd Solution



Radial Flow Region

1×102

Figure 7 — pD and pDβd vs. tDxf/CDf—CfD = 10 (fractured well case — includes wellbore storage effects).

Horizontal Wells: Ozkan16 created a line-source solution for modeling horizontal well performance — we used this solu-tion to generate β-derivative type-curves for the case of a hori-zontal well, where the well is vertically-centered within an in-finite-acting, homogeneous (and isotropic) reservoir.

10-3

10-2

10-1

100

101

102

p D, p

Dd

and

p Dβ d

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

tDL

0.125

0.25

0.5 1

5

10

25

50

100

Infinite Conductivity Vertical Fracture

L=0.10.125

0.25

0.5

1

5102550

100


Type Curve for a Infinite Conductivity Horizontal Well in an Infinite-Acting Homogeneous Reservoir (LD= 0.1, 0.125, 0.25, 0.5, 1, 5, 10, 25, 50, 100).


50

25

LD= 0.1

0.125LD= 0.1

0.25

0.5

Figure 8 — pD, pDd, and pDβd vs. tDL — solutions for an infinite conductivity horizontal well in an infinite-acting homogeneous reservoir — no wellbore storage or skin effects (various LD values).

In Fig. 8 we present the pD, pDd, and pDβd solutions for the case of a horizontal well with no wellbore storage or skin effects, only the influence of the LD parameter (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 pDβd function is constant, we do observe uni-que characteristic behavior in the pDβd function, which should be of value in the diagnostic interpretation of pressure tran-sient test data obtained from horizontal wells.

The pDd and pDβd solutions for the case of a horizontal well with wellbore storage effects are shown in Fig. 9 (LD=100,


i.e., a thin reservoir). As expected, we do observe the strong signature of the pDβd function for the wellbore storage domi-nation regime (i.e., pDβd = 1). We also note an apparent for-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

p D a

nd p

Dβ d

10-2 10-1 100 101 102 103 104 105 106 107

tDL/CDL

CDL=1×10-6

1×10-2

Type Curve for an Infinite Conductivity Horizontal Well in an Infinite-Acting Homogeneous Reservoir with Wellbore Storage Effects (LD = 100).

1×10-5

1×10-4

1×10-3

1×102

1×102

Legend: LD = 100 pD Solution pDβd Solution



Radial Flow Region

CDL=1×10-6

11×101 1×10-1

1×1011 1×10-1

1×10-21×10-3

1×10-4

1×10-5

Figure 9 — pD and pDβd vs. tDL/CDL — LD=100 (horizontal well case — includes wellbore storage effects).

Wellbore Storage and Boundary Effects: In Fig. 10 we present the 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., pDβd = 1).

10-3

10-2

10-1

100

101

102

103

p D, p

Dda

nd p

Dβd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3 1×10-2 3×10-2 10-1

1 101

106

104 103 102

101 1 10-1

3×10-2

1×10-2

3×10-3

CDe2s=1×10-3

Type Curve for an Unfractured Well in a Bounded Homogeneous Reservoir with Wellbore Storage and Skin Effects (reD= 100)

Legend: Bounded Resevoir reD= 100 pD Solution pDd Solution pDβd Solution

CDe2s=1×10-3


Boundary Dominated Flow


106

Figure 10 — pD and pDβd vs. tD/CD — reD =100, bounded circular 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 β-derivative for the analysis of the boundary-dominated flow regime — i.e., the β-derivative (or another auxiliary form) may be a good diagnostic for the ana-lysis of production data. In particular, the β-derivative may be less 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 β-Derivative Type Curves

The β-derivative is a ratio function — the dimensionless for-mulation of the β-derivative (pDβd) is the exactly the same function as the "data" formulation of the β-derivative [Δpβd(t)]. Therefore, when we plot the Δpβd(t) (data) function onto the grid of the pDβd function (i.e., the type curve match) the y-axis functions are identical. As such, the vertical "match" is not a match at all — but rather, the model and the data functions are defined to be the same — so the vertical "match" is fixed.

At this point, the time axis match is the only remaining task, so the Δpβd(t) data function is shifted on top of the pDβd func-tion, 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 Δpβd(t) function is plotted with the Δp(t) and the Δpd(t) functions, we achieve a "harmony" in that the 3 functions are matched simultaneously, and one portion of the match (i.e., Δpβd(t) — pDβd) is fixed.

The procedures for type curve matching the β-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 Δpβd(t) — pDβd is fixed, which then fixes the Δp(t) and the Δpd(t) functions, and only the x-axis needs to be resolved — exactly like any other type curve for that particular case. If type curves are not used, and some sort of software-driven, model-based matching procedure is used (i.e., event/history matching), then the Δpβd(t) and pDβd functions are matched si-multaneously, in the same manner that the dimensionless pres-sure/derivative functions would be matched.

Examples Using the β-Derivative Function

To demonstrate/validate the β-derivative function we present the results of 12 field examples obtained from the literature (refs. 1, 18-22). The table below provides orientation for our examples.

Case

Field 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 β-derivative function [Δpβd(t)] in conjunction with the Δp(t) and Δpd(t) functions. As a comment, for all of the example cases we considered, the β-derivative function [Δpβd(t)] provided a direct analysis (i.e., the "match" was obvious using the Δpβd(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 β-


derivative function.

Example 1 is presented in Fig. 11 (from ref. 18) and shows the field data and model matches for the Δp(t), Δpd(t), and Δpβd(t) functions in dimensionless format (i.e., the pD, pDd, and pDβd "data" functions are given as symbols), along with the corres-ponding dimensionless solution functions (i.e., pD, pDd, and pDβd "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. 11 we note a strong wellbore storage signature, and we find that the pDβd data function (squares) does yield the required value of unity. The pDβd data function does not yield a quantitative inter-pretation — other than the wellbore storage domination region (pDβd = 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

p D, p

Dd

and

p Dβd

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 pD Solution pDd Solution pDβd Solution

Legend:pD DatapDd DatapDβd Data

Match Results and Parameter Estimates:[pD/Δp]match = 0.02 psi-1, CDe2s= 106 (dim-less)

[(tD/CD)/t]match= 38 hours-1, k = 399.481 md Cs = 0.25 bbl/psi, s = 1.91 (dim-less)

pDβd = 1

pDd = 1/2

Reservoir and Fluid Properties:rw = 0.3 ft, h = 100 ft,

ct = 1.1×10-5 psi-1, φ = 0.27 (fraction)μo = 1.24 cp, Bo= 1.002 RB/STB

Production Parameters:qref = 9200 STB/D, pwf(Δt= 0)= 1844.65 psia

Figure 11 — Field example 1 type curve match — SPE 11463 (ref. 18 — Meunier) (pressure buildup case).

In Fig. 12 we consider the initial literature case regarding well test analysis using the Bourdet pressure derivative function (Δpd) as shown in ref. 1. This is a pressure buildup test where the appropriate rate history superposition is used for the time function axis. This result is an excellent match of all func-tions, but in particular, the β-derivative function (pDβd) is an excellent 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 β-deri-vative function (pDβd), 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 likely using 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

p D, p

Dd

and

p Dβ 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 pD Solution pDd Solution pDβ d Solution

Legend: pD DatapDd DatapDβd Data



Production Parameters:qref = 174 STB/D


[(tD/CD)/t]match= 15 hours-1, k = 10.95 md Cs = 0.0092 bbl/psi, s = 8.13 (dim-less)

pDβd = 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. 13 is taken from a well in a known dual porosity/naturally fractured reservoir. As we note in Fig. 13, the "late" portion of the data is not matched exactly with the specified reservoir model (infinite-acting ra-dial flow with dual porosity effects). We contend that part of the less-than-perfect late time data match may be due to rate history 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 β-derivative function to be of significant practical value. We note that the β-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

p D, p

Dd

and

p Dβ 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)— ω = 1×10-2, α = 1×10-1)


Legend: ω =1×10-2, α = 1×10-1

pD Solution pDd Solution pDβd Solution



Production Parameters:qref = 3224 STB/D, pwf(Δt= 0)= 9670 psia


[(tD/CD)/t]match= 0.17 hours-1, k = 0.361 md Cs = 0.1124 bbl/psi, s = -4.82 (dim-less)

ω = 0.01 (dim-less), α = CD×λ = 0.01(dim-less)

λ = 6.45×10-6(dim-less)

Figure 13 — Field example 3 type curve match — SPE 13054 (ref. 19 — DaPrat) (pressure drawdown case).


Our next case (Example 4) also considers well performance in a dual porosity/naturally fractured reservoir (see Fig. 14). From these data we again note a very strong performance of the β-derivative function — particularly in the region defined by transition from wellbore storage to transient interporosity flow. Cases such as these validate the application of the β-derivative for the interpretation of well test data obtained from dual porosity/naturally fractured reservoirs.

10-3

10-2

10-1

100

101

102

p D, p

Dd

and

p Dβ 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)— ω = 0.237, α = 1×10-3)

Legend: ω = 0.237, α = 1×10-3

pD Solution pDd Solution pDβd Solution



ct = 2×10-5 psi-1, φ = 0.05 (fraction)μo = 0.3 cp, Bo= 1.5 RB/STB

Production Parameters:qref = 830 Mscf/D


[(tD/CD)/t]match= 150 hours-1, k = 678 md Cs = 0.0311 bbl/psi, s = -1.93 (dim-less)

ω = 0.237 (dim-less), α = CD×λ = 0.001(dim-less)

λ = 2.13×10-8(dim-less)

pDd = 1/2

pDβd = 1

Figure 14 — Field example 4 type curve match — SPE 18160 (ref. 20 — Allain) (pressure buildup case).

In Fig. 15 we investigate the use of the β-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 β-derivative function provides a unique interpretation for a difficult case. Most importantly, the β-derivative function supports the exis-tence (and influence) of the hydraulic fracture.

10-4

10-3

10-2

10-1

100

101

p D, p

Dd

and

p Dβ 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 pDβd Solution



ct = 6.37×10-5 psi-1, φ = 0.05 (fraction)μgi = 0.0297 cp, Bgi= 0.5755 RB/Mscf


Match Results and Parameter Estimates:[pD/Δp]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

pDβd = 1/2

Figure 15 — Field example 5 type curve match — SPE 9975 Well 5 (ref. 21 — Lee) (pressure buildup case).

Another application of the β-derivative function is also to prove when a flow regime does not (or at least probably does not) exist — the example shown in Fig. 16 is just such a case. In ref. 21 "Well 10" is designated as a hydraulically fractured well in a gas reservoir and in Fig. 16 we observe no evidence

of a hydraulic fracture treatment from any of the dimension-less plotting functions, in particular, the β-derivative function shows no evidence of a hydraulic fracture. The well is either 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

p D, p

Dd

and

p Dβ 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: CfD= 2 pD Solution pDd Solution pDβd Solution




Match Results and Parameter Estimates:[pD/Δp]match = 0.0012 psi-1, CDf= 100 (dim-less)


pDβd = 1

pDd = 1/2


Fig. 17 is 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 β-deri-vative function (i.e., we do not observe pDβd = 1/2 (infinite fracture conductivity) nor pDβd = 1/4 (finite fracture con-ductivity)). We do note that pDβd = 1 at early times, which confirms the wellbore storage domination regime. The pDd and pDβd signatures during mid-to-late times confirm the well is highly stimulated — and the infinite fracture conductivity vertical fracture model is used for analysis and interpretation in this case.

10-3

10-2

10-1

100

101

102

p D, p

Dd

and

p Dβ 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 )







[(tDxf/CDf)/t]match= 37 hours-1, k = 0.076 md CfD = 1000 (dim-less), xf = 3.681 ft

pDβd = 1 pDd = 1/2



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 β-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

p D, p

Dd

and

p Dβ 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)




ct = 7.7×10-6 psi-1, φ = 0.11 (fraction)μw = 0.92 cp, Bw= 1 RB/STB




pDβd = 1pDβd = 1/2

Figure 18 — Field example 8 type curve match — Well 207 (ref. 22 — Samad) (pressure falloff case).

In Fig. 19 we 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 β-deri-vative function we can identify the wellbore storage domina-tion regime (i.e., pDβd = 1) and we can also reasonably confirm the existence of an infinite fracture conductivity vertical frac-ture (pDβd = 1/2). The quality of these data impairs our ability to define the reservoir model uniquely, but we can presume that our assessment of the flow regimes is reasonable, based on the character of the β-derivative function.

10-4

10-3

10-2

10-1

100

101

p D, p

Dd

and

p Dβ d

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf





ct = 7.26×10-6 psi-1, φ = 0.06 (fraction)μw = 0.87 cp, Bw= 1.002 RB/STB




pDβd = 1 pDd = 1/2


The data for Well 203, taken from ref. 22 are presented in Fig. 20. The signature given by the pD, pDd, and pDβd functions does not appear to be that of a high conductivity vertical frac-ture. In this case the pD and pDd functions 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

p D, p

Dd

and

p Dβ d

10-4 10-3 10-2 10-1 100 101 102 103 104

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 pDβd Solution







pDβd = 1 pDd = 1/2


In Fig. 21 we present the data for Well 5408, a pressure falloff test obtained from ref. 22. This case also exhibits no unique character in the pD, pDd, and pDβd functions, other than well-bore storage domination (pDβd = 1) and infinite-acting radial flow (pDd =1/2). Based on the given data, we know that this 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

p D, p

Dd

and

p Dβ d

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf









pDβd = 1 pDd = 1/2


Our last field example is a pressure falloff test performed on Well 2403, also taken from ref. 22. These data are presented in Fig. 22 and we observe the flow regimes for wellbore sto-rage domination (pDβd = 1), and the infinite-acting radial (pDd =1/2).


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

p D, p

Dd

and

p Dβ d

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf

Legend: DatapD DatapDd DatapDβd Data






Match Results and Parameter Estimates:[pD/Δp]match = 0.18 psi-1, CDf= 1 (dim-less)


pDd = 1/2pDβd = 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 β-derivative function, we conclude that the β-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 β-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 Δpβd(t) and pDβd 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 β-derivative function — and we expect the β-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 the new power-law or β-derivative formulation — which is given by:

ptp

dtpdt

ptdpdtp d

d ΔΔ

=Δ

Δ=

Δ=Δ

)(1)ln()ln()(β .................... (1)

This function can be computed directly from data using:

● Δpβd(t) = dln(Δp)/dln(t) (β-derivative definition) ........... (8)

● Δpβd(t) = Δpd(t)/Δp (Bourdet derivative definition) .. (9)

The work of Sowers (ref. 2) shows that using the β-derivative definition (Eq. 8) does provide a slightly more accurate derivative function than extracting the Δpβd(t) function from the Δpd(t) function as 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 [Δpd(t)]. In short, if a derivative

computation module is being developed from nothing, Eq. 8 should be used. Otherwise, the "Bourdet" derivative function [Δpd(t)] should be adequate to "extract" the β-derivative func-tion [Δpβd(t)] via Eq. 7.

Our goal in this work is the presentation of the β-derivative formulation. We have prepared the β-derivative solutions for some of the most popular well test analysis cases (see Appendix A), as well as graphical representations of these solutions in the form of "type curves" (see Appendix B). The β-derivative has been shown to provide much improved resolution for certain well test analysis cases — in particular, the β-derivative yields a constant value (i.e., Δpβd(t) = con-stant) for the following cases:

Case Δpβd(t)


1

Reservoir boundaries: — Closed reservoir (circle, rectangle, etc.). — 2-Parallel faults (large time). — 3-Perpendicular faults (large time).

1 1/2 1/2

Fractured wells: — Infinite conductivity vertical fracture. — Finite conductivity vertical fracture.

1/2 1/4

Horizontal wells: — Formation linear flow.

1/2

In addition, the β-derivative also provides a unique characteri-zation of well test behavior in dual porosity reservoirs (al-though the β-derivative is never constant for these cases, except for the possibility of a rare fractured or horizontal well case).

Finally, we have provided a schematic "diagnosis worksheet" for the interpretation of the β-derivative function (see Appendix C).

Recommendations for Future Work

The future work on this topic should focus on the application of the β-derivative concept for production data analysis.

Acknowledgements

The authors wish to acknowledge the work of Mr. Steven F. Sowers (ExxonMobil) — for access to his computation routines, and for his efforts to lay the groundwork for this study via his investigations of the β-derivative function as a statistically enhanced formulation for computing the Bourdet derivative.


Nomenclature

Variables

bpss = Pseudosteady-state constant, dimensionless B = FVF, RB/STB ct = total system compressibility, psi-1

CA = shape factor, dimensionless Cs = wellbore storage coefficient, bbl/psi

CD = dim-less wellbore storage coef. — unfractured well

CDf = dim-less wellbore storage coef. — horizontal well

CDL = dim-less wellbore storage coef. — fractured well CfD = fracture conductivity, dimensionless

h = pay thickness, ft hma = matrix height, ft

k = permeability, md kf = fracture permeability, md

kfb = dual porosity fracture permeability, md kma = matrix permeability, md

L = horizontal well length, ft LD = dimensionless horizontal well length LDf = dimensionless distance from fault

n = positive integer p = pressure, psi

pD = dimensionless pressure pDd = dimensionless pressure derivative

pDβd = dimensionless β-pressure derivative pi = initial reservoir pressure, psi

pwf = well flowing pressure, psi pwfd = well flowing pressure derivative, psi

pwfβd = well flowing β−pressure derivative, dimensionless

pws = well shut-in pressure, psi pwsd = well shut-in pressure derivative, psi

pwsβd = well shut-in β−pressure derivative, dimensionless q = flow rate, STB/Day re = reservoir outer boundary radius, ft

reD = outer reservoir boundary radius, dimensionless rw = wellbore radius, ft

rwD = dimensionless wellbore radius rwzD = 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, ft

zwD = dimensionless well location

Greek Symbols

φ = porosity, fraction φf = fracture porosity, fraction

φma = matrix porosity, fraction

γ = Euler's constant, 0.577216… ηfD = hydraulic diffusivity, dimensionless

μ = viscosity, cp λ = interporosity flow parameter ω = storativity parameter

Subscript

g = gas o = oil w water

wbs = wellbore storage pss = pseudosteady-state

References

1. Bourdet, D., Ayoub, J.A., and Pirad, Y.M.: "Use of Pressure Derivative in Well-Test Interpretation," SPEFE (June 1989) 293-302 (SPE 12777).

2. Sowers, S.: The Bourdet Derivative Algorithm Revisited — Intro-duction and Validation of the Power-Law Derivative Algorithm for Applications in Well-Test Analysis, (internal) B.S. Report, Texas 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 North Sea Reservoir Example," JPT (Nov. 1985) 2023-2039.

5. Lane, H.S., Lee, J.W., and Watson, A.T.: "An Algorithm for Determining Smooth, Continuous Pressure Derivatives from Well Test Data," SPEFE (December 1991) 493-499.

6. Escobar, F.H., Navarrete, J.M., and Losada, H.D.: "Evaluation of Pressure Derivative Algorithms for Well-Test Analysis," paper SPE 86936 presented at the 2004 SPE International Thermal Operations and Heavy Oil Symposium and Western Regional 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 of Optimal Window Size in Pressure-Derivative Computation Using Frequency-Domain Constraints," SPE 96026 presented at the 2005 SPE Annual Technical Conference and .Exhibition, Dallas, Texas, 9-12 October 2005.

9. Onur, M. and Reynolds, A.C.: "A New Approach for Constructing 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 the Laplace 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 Com-municating 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 Fractures 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 Approach, M.S. Thesis, Texas A&M U., College Station, Texas (2000).

15. Cinco-Ley, H. and Meng, H.-Z.: "Pressure Transient Analysis of Wells with Finite Conductivity Vertical Fractures in Dual Porosity Reservoirs," SPE 18172 presented at the 1989 SPE


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 Boundary Conditions: Unfractured Well Case," SPE 25479 presented at the 1993 SPE Production Operations Symposium, Oklahoma City, OK, 21-23 March 1993.

18. Meunier, D., Wittmann, M.J., and Stewart, G.: "Interpretation of Pressure 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," SPE 13054 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 Integral Functions for the Analysis of Well Test Data, M.S. Thesis, Texas A&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 Single Infinite-Conductivity Vertical Fracture," SPEJ. (August 1974) 347-360.

24. Cinco-Ley, H. and Samaniego-V., F.: "Transient Pressure Analysis for Fractured Wells," JPT (September 1981) 1749.

25. van Golf-Racht, T.D.: Fundamentals of Fractured Reservoir Engineering, Elsevier, New York, NY (1982)

26. Blasingame, T.A., Johnston, J.L., and Lee, W.J.: "Advances in the Use of Convolution Methods in Well Test Analysis," paper SPE 21826 presented at the 1991 Joint Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 15-17 April 1991.


Appendix A — Table of solutions for pD, pDd, and pDβd (conditions/flow regimes as specified).

Table A-1 — Solutions for the wellbore storage domination flow regime.

Variable Solution Relation

wbspΔ twbswbs mp =Δ ................................................................................................................................................(A.1.1)

wbsdp ,Δ twbswbsd mp =Δ , .............................................................................................................................................(A.1.2)

swbdp ,βΔ 1, =Δ swbdpβ ...................................................................................................................................................(A.1.3)

Definitions: (field units)

241

swbs C

qBm = ................................................................................................................................................................................................... (A.1.4)

Table A-2 — Solutions for a well in a finite-acting, homogeneous reservoir (closed system, any

well/reservoir configuration).

Description Relation

Dp 214ln

212)(

2 pssDAAw

DADAD btsCr

A

ettp +=+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+= ππ

γ..............................................................................(A.2.1)

Ddp DADADd ttp π2)( = ........................................................................................................................................(A.2.2)

)/( DDddD ppp =β 1)2/(1

1)( ≈+

=DApss

DAdD tbtp

πβ

(large-time) ..................................................................................................................................(A.2.3)


tAc

ktt

DA 10637.2 4φμ

−×= ................................................................................................................................................................................... (A.2.4)

)(2.141

1wfiD pp

qBkhp −=

μ................................................................................................................................................................................. (A.2.5)

sCr

A

eb

Awpss +

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

14ln21

2γ................................................................................................................................................................................... (A.2.6)

Table A-3 — Solutions for an unfractured well in an infinite-acting, homogeneous reservoir (radial flow).


Dp

⎥⎦

⎤⎢⎣

⎡=

DDD t

tp4

1E21)( 1

( )10>Dt .........................................................................................................................................(A.3.1)

Ddp

⎥⎦

⎤⎢⎣

⎡ −=

DDDd t

tp4

1exp21)(

( )10>Dt .........................................................................................................................................(A.3.2)

)/( DDddD ppp =β

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

DDDdD tt

tp4

1E4

1exp)( 1β

( )10>Dt .........................................................................................................................................(A.3.3)


2410637.2

wtD

rc

kttμφ

−×=................................................................................................................................................................................... (A.3.4)

)(2.141

1wfiD pp

qBkhp −=

μ................................................................................................................................................................................. (A.3.5)

28936.0

wt

sD

rhc

CC

φ=

.................................................................................................................................................................................................. (A.3.6)


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


Dp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

D

Df

DDD t

L

ttp

2

11 E4

1E21)(

(single fault)...................................................................................................................................(A.4.1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

D

Df

D

Df

DDD t

L

t

L

ttp

2

1

2

112

EE24

1E21)(

(two perpendicular faults)..............................................................................................................(A.4.2)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡= ∑

∞

=1

2

11 E24

1E21)(

i D

Df

DDD t

iL

ttp

(two parallel faults)........................................................................................................................(A.4.3)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ++⎥

⎦

⎤⎢⎣

⎡= ∑ ∑

∞

=

∞

=1

2

11

2

1

22

11 EE2)1(

E24

1E21)(

i D

Df

i D

Df

D

Df

DDD t

L

t

iL

t

Li

ttp

(three perpendicular faults)............................................................................................................(A.4.4)

Ddp

1

21

21)(

/4/12

≈+=−− DDfD

tLtDDd eetp

(single fault, complete solution and large-time approximation) ...................................................(A.4.5)

221

21)(

/2/4/122

≈++=−−− DDfDDfD

tLtLtDDd eeetp

(two perpendicular faults, complete solution and large-time approximation)..............................(A.4.6)

∑∞

=

−− +=1

/4/12

21)(

i

tiLtDDd

DDfD eetp

(two parallel faults, complete solution and large-time approximation)........................................(A.4.7)

∑ ∑∞

=

−∞

=

−+−− +++=1

/

1

//)1(4/12222

21

21)(

i

tL

i

tiLtLitDDd

DDfDDfDDfD eeeetp

(three perpendicular faults)............................................................................................................(A.4.8)

)/( DDddD ppp =β

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

≈

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

+=

−−

D

Df

DD

Df

D

tLtDdD

t

L

tt

L

t

eetpDDfD

2

11

2

11

/4/1

E4

1E

2

E4

1E

)(

2

β

(single fault, complete solution and large-time approximation) ...................................................(A.4.9)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

≈

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

++=

−−−

D

Df

D

Df

DD

Df

D

Df

D

tLtLtDdD

t

L

t

L

tt

L

t

L

t

eeetpDDfDDfD

2

1

2

11

2

1

2

11

/2/4/1

2EE2

41E

4

2EE2

41E

2)(

22

β

(two perpendicular faults, complete solution and large-time approximation)............................(A.4.10)

21

E24

1E

2

)(

1

2

11

1

/4/12

≈

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

+

=

∑

∑∞

=

∞

=

−−

i D

Df

D

i

tiLt

DdD

t

iL

t

ee

tp

DDfD

β

(two parallel faults, complete solution and large-time approximation)......................................(A.4.11)

21

EE2)1(

E24

1E

22

)(

1

2

11

2

1

22

11

1

/

1

//)1(4/12222

≈

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ++⎥

⎦

⎤⎢⎣

⎡

+++

=

∑ ∑

∑ ∑∞

=

∞

=

∞

=

−∞

=

−+−−

i D

Df

i D

Df

D

Df

D

i

tL

i

tiLtLit

DdD

t

L

t

iL

t

Li

t

eeee

tp

DDfDDfDDfD

β

(three perpendicular faults, complete solution and large-time approximation)..........................(A.4.12)


2410637.2

wtD

rc

kttμφ

−×=.................................................................................................................................................................................... (A.4.13)

)(2.141

1wfiD pp

qBkhp −=

μ................................................................................................................................................................................. (A.4.14)

w

faultDf r

LL =

.................................................................................................................................................................................................. .(A.4.15)


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


Dp

( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ +++

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

DxfDD

DxfDD

Dxf

D

Dxf

DDxfDxfD t

xxtxx

tx

txt

tp4

1E

41

41

E4

)1(21

erf21

erf2

)(2

12

1π

(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.1)DxfDxfD ttp π=)(

(early time, linear flow) .................................................................................................................(A.5.2)]80907.2)[ln(

21)( += DxfDxfD ttp

(late time, uniform flux fracture)...................................................................................................(A.5.3)]20000.2)[ln(

21)( += DxfDxfD ttp

(late time, infinite conductivity fracture) ......................................................................................(A.5.4)

Ddp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

Dxf

D

Dxf

DDxfDxfDd

tx

txt

tp21erf

21erf

4)(

π

(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.5)

4)(

DxfDxfDd

ttp

π=

(early time, linear flow) .................................................................................................................(A.5.6)5.0)( =DxfDd tp

(late time).......................................................................................................................................(A.5.7)

)/( DDddD ppp =β

⎥⎥

⎦

⎤

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +++

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−+

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

Dxf

DD

Dxf

DD

Dxf

D

Dxf

DDxf

Dxf

D

Dxf

DDxfDxfdD

txx

txx

t

x

t

xt

t

x

t

xttp

4)1(

E4

)1(4

)1(E

4)1(

2

1erf

2

1erf

2

2

1erf

2

1erf

4)(

21

21

π

πβ

(Uniform-flux (xD=0) or infinite conductivity(xD=0.732))............................................................(A.5.8)5.0)( =DxfdD tp β

(early time, linear flow) .................................................................................................................(A.5.9)

80907.2)ln(1)(+

=Dxf

DxfdD ttp β

(late time, uniform flux fracture).................................................................................................(A.5.10)

20000.2)ln(1)(+

=Dxf

DxfdD ttp β

(late time, infinite conductivity fracture) ....................................................................................(A.5.11)


2410637.2

ftDxf

xc

kttμφ

−×=.............................................................................................................................................................................. (A.5.12)

)(2.141

1wfiD pp

qBkhp −=

μ............................................................................................................................................................................... (A.5.13)

fD xxx /= ....................................................................................................................................................................................................... .(A.5.14)

28936.0

ft

sDf

xhc

CC

φ=

............................................................................................................................................................................................. .(A.5.15)


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

fDDxfD dz

z

ztC

z

Ctp

0

5.0)(erfc

)(

η

πη

(General solution) ..........................................................................................................................(A.6.1)

DxffDfD

DxfD tC

tp πη2)( =

(Short-time approximation), 2

201.0

fD

fDDxf

Ct

η≤

................................................................................(A.6.2)

( ) 41

225.1)( Dxf

fDDxfD t

Ctp

Γ=

π

(Large-time approximation),

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

≤⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−≤

≤≤−≤

≥≤

−

6.15.255.4

36.1)5.1(0205.0

31.0

4

2

53.1

fDfD

Dxf

fDfDDxf

fDfD

Dxf

CC

t

CCt

CC

t...........................(A.6.3)

Ddp

fD

DxffDDxfDd C

ttp

πη=)(

(Short-time approximation) ...........................................................................................................(A.6.4)

41612708.0)( Dxf

fDDxfDd t

Ctp =

(Large-time approximation) ..........................................................................................................(A.6.5)

)/( DDddD ppp =β

21)( =DxfdD tp β

(Short-time approximation) ...........................................................................................................(A.6.6)

41)( =DxfdD tp β

(Large-time approximation) ..........................................................................................................(A.6.7)


2410637.2

ftDxf

xc

kttμφ

−×=................................................................................................................................................................................ (A.6.8)

)(2.141

1wfiD pp

qBkhp −=

μ................................................................................................................................................................................. (A.6.9)

ftf

tffD ck

ck

φ

φη =

.................................................................................................................................................................................................. (A.6.10)

f

ffD kx

wkC =

...................................................................................................................................................................................................... (A.6.11)

28936.0

ft

sDf

xhc

CC

φ=

.............................................................................................................................................................................................. (A.6.12)


Table A-7 — Solutions for an unfractured well in an infinite-acting, dual porosity (naturally fractured) reservoir system (pseudosteady-state interporosity flow model).


Dp

⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡= DDDDD ttt

etp

)1(E

21

)1(E

214ln

21)( 11 ω

λωω

λγ

(logarithmic approximation)..........................................................................................................(A.7.1)

Ddp

⎥⎦

⎤⎢⎣

⎡−−

−⎥⎦

⎤⎢⎣

⎡−

−+= DDDDd tttp

)1(exp

21

)1(exp

21

21)(

ωλ

ωωλ ........................................................................................(A.7.2)

21)( ≈DDd tp

(large-time) ....................................................................................................................................(A.7.3)

)/( DDddD ppp =β

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−

−⎥⎦

⎤⎢⎣

⎡−

−+= DDDDDDdD ttt

etttp

)1(E

)1(E4ln

)1(exp

)1(exp1)( 11 ω

λωω

λωλ

ωωλ

γβ

........................................................................................................................................................(A.7.4)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡≈

DDD

DdDttt

e

tp

)1(E

)1(E4ln

1)(

11 ωλ

ωωλ

γ

β

(large-time) ....................................................................................................................................(A.7.5)


24

)(10637.2

wtmamatfbfbD

rcc

kttμφφ +

×= − ......................................................................................................................................................... (A.7.6)

)(2.141

1wfiD pp

qBkhp −=

μ................................................................................................................................................................................. (A.7.7)

tmamatfbfb

tfbfbcc

c

φφ

φω

+=

...................................................................................................................................................................................... .(A.7.8)

fb

ma

ma

wkk

h

r2

212=λ

................................................................................................................................................................................................ .(A.7.9)

28936.0

wt

sD

rhc

CC

φ=

.............................................................................................................................................................................................. .(A.7.10)


Table A-8 — Solutions for a hydraulically fractured well in an infinite-acting, dual porosity (naturally

fractured) reservoir system (pseudosteady-state interporosity flow model).


Dp

( )

41

225.1)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡

Γ=

ωπ Dxf

fDDxfD

t

Ctp

(fracture storage dominated flow period, early time)....................................................................(A.8.1)

fD

DxfDxfD C

ttp

3)( π

ω

π+=

(fracture storage dominated flow period, intermediate time) .......................................................(A.8.2)41

2)45()( Dxf

fDDxfD t

Ctp

Γ=

π

(total system compressibility dominated flow period, early time) ...............................................(A.8.3)

fDDxfDxfD C

ttp3

)( ππ +=

(total system compressibility dominated flow period, intermediate time) ...................................(A.8.4)

Ddp

41612708.0)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

ωDxf

fDDxfDd

t

Ctp


fD

DxfDxfDd C

ttp

3)( π

ω

π+=

(fracture storage dominated flow period, intermediate time) .......................................................(A.8.6)41

2)45()( Dxf

fDDxfDd t

Ctp

Γ=

π

(total system compressibility dominated flow period, early time) ...............................................(A.8.7)

fDDxfDxfDd C

ttp3

)( ππ +=

(total system compressibility dominated flow period, intermediate time) ...................................(A.8.8)

)/( DDddD ppp =β

41)( =DxfdD tp β


ωπβ

DxfDxffD

DxffDDxfdD

ttC

tCtp

26

3)(

+=

(fracture storage dominated flow period, intermediate time) .....................................................(A.8.10)

41)( =DxfdD tp β

(total system compressibility dominated flow period, early time) .............................................(A.8.11)

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

=

fDDxf

DxfDxfdD

Ct

ttp

32

)(ππ

πβ

(total system compressibility dominated flow period, intermediate time) .................................(A.8.12)


2410637.2

ftDxf

xc

kttμφ

−×=.............................................................................................................................................................................. (A.8.13)

)(2.141

1wfiD pp

qBkhp −=

μ............................................................................................................................................................................... (A.8.14)

tmamatfbfb

tfbfbcc

c

φφ

φω

+=

.................................................................................................................................................................................... .(A.8.15)

fb

ma

ma

wkk

h

r2

212=λ

.............................................................................................................................................................................................. .(A.8.16)

f

ffD kx

wkC =

..................................................................................................................................................................................................... .(A.8.17)

28936.0

ft

sDf

xhc

CC

φ=

............................................................................................................................................................................................. .(A.8.18)


Table A-9 — Solutions for an infinite conductivity horizontal well in an infinite-acting, homogeneous (and isotropic) reservoir system.


Dp

τ

τππτπττ

π dznznLntp wDn

DDt

DLDDL

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+×

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑∫

∞

=

coscos)exp(21134.0erf866.0erf4

)(1

2220

........................................................................................................................................................ (A.9.1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

D

DwDD

DDLD t

Lzz

Ltp

4

/)(E

41)(

22

1

(early time approximation) ............................................................................................................ (A.9.2)

( ) Fttp DLDLD ++= 509843.2ln21)(


Ddp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+×

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑

∞

=wD

nDDLD

DLDL

DLDLDd znzntLn

tt

ttp πππ

πcoscos)exp(21134.0erf866.0erf

4)(

1

222

........................................................................................................................................................ (A.9.4)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −−=

DLDwDD

DDLDd t

LzzL

tp4

/)(exp

41)(

22


21)( =DLD tp


)/( DDddD ppp =β

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+×

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+×

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

∑∫

∑∞

=

∞

=

τ

τππτπ

πππβ

dznznLntt

znzntLntt

ttp

wDn

DDt

DLDL

wDn

DDLDDLDL

DLDLdD

DLcoscos)exp(21134.0erf866.0erf

coscos)exp(21134.0erf866.0erf)(

1

2220

1

222

........................................................................................................................................ (A.9.7)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −−=

DL

DwDD

DL

DwDDDLdD t

Lzz

t

Lzztp

4

/)(E

4

/)(exp)(

22

1

22

β

(early time)..................................................................................................................................... (A.9.8)

Fttp

DLDLdD ++

=)509843.2(ln

1)(β

(late time) ....................................................................................................................................... (A.9.9)


ααπππ dnLKznznF DwDn

D ∫∑+

−

∞

=

−=1

12

01

])732.0([coscos................................................................................................................................ (A.9.10)

( )24

210637.2

Lc

kttt

DLμφ

−×=......................................................................................................................................................................... (A.9.11)

)(2.141

1wfiD pp

qBkhp −=

μ......................................................................................................................................................................... (A.9.12)

hzzD = ............................................................................................................................................................................................................ .(A.9.13)

wzDDwD rzz −= ............................................................................................................................................................................................ .(A.9.14)

Lr

r wwD

2= ...................................................................................................................................................................................................... .(A.9.15)

DwDwzD Lrr = ................................................................................................................................................................................................ .(A.9.16)

2)2/(

8936.0

Lhc

CC

t

sDL

φ=

......................................................................................................................................................................................... .(A.9.17)

hLLD 2

= ......................................................................................................................................................................................................... .(A.9.18)


Appendix B — Dimensionless "type curve" representations of the β-pressure derivative and various other pressure functions, selected reservoir/well configurations

10-2

10-1

100

101

102

103

10-5 10-4 10-3 10-2 10-1 100 101 102


Legend: (pDd ) (pDβd ) Unfractured Well (Radial Flow) Fractured Well (Infinite Fracture Conductivity) Fractured Well (Finite Fracture Conductivity) Horizontal Well (Full Penetration, Thin Reservoir)



DIAGNOSTIC plot for Well Test Data (pDd and pDβd)

Bou

rdet

"W

ell T

est"

Dim

ensi

onle

ss P

ress

ure

Der

ivat

ive

Func

tion,

pD

d"P

ower

Law

— β

" D

imen

sion

less

Pre

ssur

e D

eriv

ativ

e Fu

nctio

n, p

Dβd

Boundary-Dominated

Flow Region



pDβd = 1(boundary

dominated flow)

1

1

1

2

41

2

1









( )( )

( )( )

( )( )

( )( )


Figure B.1 — Schematic of pDd and pDβd vs. tD — Various reservoir models and well configurations (no wellbore storage or skin effects).

10-2

10-1

100

101

102

103

10-5 10-4 10-3 10-2 10-1 100 101 102


Legend: (pDβd ) Unfractured Well (Radial Flow) Fractured Well (Infinite Fracture Conductivity) Fractured Well (Finite Fracture Conductivity) Horizontal Well (Full Penetration, Thin Reservoir)



DIAGNOSTIC plot for Well Test Data (pDβd)

"Pow

er L

aw —

β"

Dim

ensi

onle

ss P

ress

ure

Der

ivat

ive

Func

tion,

pD

βd

Boundary-Dominated

Flow Region



pDβd = 1(boundary

dominated flow)









( )

( )

( )

( )


Figure B.2 — Schematic of pDβd vs. tD — Various reservoir models and well configurations (no wellbore storage or skin effects).


10-3

10-2

10-1

100

101

102

103

104

β-Pr

essu

re D

eriv

ativ

e Fu

nctio

n, p

Dβ d

= (t

D/p

D) d

/dt D

(pD

)

10-3 10-2 10-1 100 101 102 103 104 105 106 107

tD/LD2 (LD = Lfault/rw)

Legend: β-Pressure Derivative Function


SingleFault



2 Parallel Faults

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

UndistortedRadial Flow Behavior

2 Parallel Faults


pDβd = (tD/pD) dpD/dtD

pDd = tD dpD/dtD

Legend: "Bourdet" Well Test Pressure Derivative


"Bou

rdet

" W

ell T

est P

ress

ure

Der

ivat

ive,

pD

d =

t D d

p D/d

t D

SingleFault


Figure B.3 — pDd and pDβd vs. tD/LD2 — various sealing faults configurations (wellbore storage and skin effects are NOT included).

10-3

10-2

10-1

100

101

102

103

p D, p

Dd

and

p Db d

10-2 10-1 100 101 102 103 104 105 106

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2 10-1

1 101

102 103

104106

108

1010

1020 1030

1015

1040

10100

1060

1080 1050

10100

1080

1060 1050

1040

1030

1020 1015

1010

108106

104 103

102 101

3×10-2 1×10-2

CDe2s=1×10-3

Type Curve for an Unfractured Well in an Infinite-Acting Homogeneous Reservoir with Wellbore Storage and Skin Effects.

3

3

10100

CDe2s=1×10-3

Legend: Radial Flow Type Curves pD Solution pDd Solution pDβd Solution

Radial Flow Region



Figure B.4 — pD, pDd, and pDβd vs. tD/CD — solutions for an unfractured well in an infinite-acting homogeneous reservoir— wellbore storage and skin effects included (various CD values).


10-4

10-3

10-2

10-1

100

101

102

p D a

nd p

Dβ d

10-1 100 101 102 103 104 105 106 107 108 109

tD

Type Curve for an Unfractured Well in an Infinite-Acting Dual Porosity Reservoir (Pseudosteady-State Interporosity Flow) — No Wellbore Storage or Skin Effects.

Legend: pD Solution pDβd Solution

ω = 1×10-1

ω = 1×10-1ω = 1×10-1

ω = 1×10-2

ω = 1×10-3

ω = 1×10-2

ω = 1×10-2

ω = 1×10-3ω = 1×10-3

ω = 1×10-11×10-2

1×10-3

pDβd (λ = 5 ×10-9) pDβd (λ = 5 ×10-6)

pDβd (λ = 5 ×10-3)

Figure B.5 — pD and pDβd vs. tD — solutions for an unfractured well in an infinite-acting dual porosity system — no wellbore storage or skin effects (various λ and ω values).

10-3

10-2

10-1

100

101

102

103

p D a

nd p

Dßd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2 10-1

1 101

102 103

104106

108

1010

1020

1030

1015

1040

10100 1060

1080 1050

10100

1080 1060

1050

1040

1030

1020

1015

1010

108

106 104

103

102

1011

10-1 3×10-2

1×10-2 3×10-3

CDe2s=1×10-3


( α = λCD = 1×10-1, ω = 1×10-1)

Legend: α = λCD = 1×10-1, ω = 1×10-1


10100

CDe2s=1×10-3


Radial Flow Region


Figure B.6 — pD and pDβd vs. tD/CD — ω = 1×10-1, α = λCD = 1×10-1 (dual porosity case — includes wellbore storage and skin effects).

10-3

10-2

10-1

100

101

102

103

p D a

nd p

Dßd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2

10-11 101

102

103 104

106

108

1010

1020

1030 1015

1040

10100

1060

1080 1050

10100

1080

1060

1050

1040

1030

1020

1015

1010

108106

104

103102 101

1 10-1

3×10-2

1×10-2 3×10-3

CDe2s=1×10-3


( α = λCD = 1×10-4, ω = 1×10-1)

Legend: α = λCD = 1×10-4, ω = 1×10-1


10100

CDe2s=1×10-3


Radial Flow Region



10-3

10-2

10-1

100

101

102

103

p D a

nd p

Dβd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3

1×10-2 3×10-2 10-1

1 101

102 103

104106

108

1010

1020

1030

1015

1040

10100 1060

1080 1050

10100

1080 1060

1050

1040

1030

1020

1015

1010

108

106 104

103

102

1011

10-1 3×10-2

1×10-2 3×10-3

CDe2s=1×10-3


( α = λCD = 1×10-1, ω = 1×10-3)

Legend: α = λCD = 1×10-1, ω = 1×10-3


10100

CDe2s=1×10-3


Radial Flow Region



10-3

10-2

10-1

100

101

102

103

p D a

nd p

Dβd

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-31×10-2

3×10-2

10-1

1 101102

103

104106

108

1010

1020

10301015

1040

10100

1060

10801050

10100

1060

1050 1040

1030

1020

1015

1010

108

106 104

103102

101

110-1

3×10-2

1×10-2

3×10-3

CDe2s=1×10-3


( α = λCD = 1×10-4, ω = 1×10-3)

Legend: α = λCD = 1×10-4, ω = 1×10-3


10100

CDe2s=1×10-3


Radial Flow Region


1080



10-3

10-2

10-1

100

101

p D, p

Dd

and

p Dβ d

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

tDxf

CfD=0.250.5

1

CfD=1×104

Type Curve for a Well with a Finite Conductivity VerticalFractured in an Infinite-Acting Homogeneous Reservoir

(CfD = (wkf)/(kxf) = 0.25, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 10000)


Radial Flow Region

CfD=0.25

CfD=1×104

1

5

2

0.51

21×103

500

Figure B.10 — pD, pDd, and pDβd vs. tDxf — solutions for a fractured well in an infinite-acting homogeneous reservoir— no wellbore storage or

skin effects (various CfD values).

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-3 10-2 10-1 100 101 102 103 104 105

tDxf/CDf

CDf=1×10-6

1×10-2

Type Curve for a Well with Infinite Conductivity Vertical Fracture in an Infinite-Acting Homogeneous Reservoir with Wellbore Storage Effects

1×10-5

1×10-5

CDf=1×10-6

1×10-4 1×10-3

1×100 1×101

1×102

1×102 1×101

1×100

1×10-1

1×10-1 1×10-2

1×10-3

1×10-4

Legend: Infinite Conductivity pD Solution pDβd Solution



Radial Flow Region

1×102

CDf=1×10-6

Figure B.11 — pD and pDβd vs. tDxf/CDf — CfD = ∞ (fractured well case — includes wellbore storage effects).

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5 1×10-5

CDf=1×10-6 1×10-4

1×10-3

1×100 1×101

1×102

1×102 1×101 1×100

1×10-1

1×10-1 1×10-2

1×10-3 1×10-4




Radial Flow Region

1×102

Figure B.12 — pD and pDβd vs. tDxf/CDf — CfD = 1 (fractured well case — includes wellbore storage effects).

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-4 1×10-3

1×100 1×101

1×102

1×102 1×101 1×100

1×10-1 1×10-2

1×10-3

1×10-4




Radial Flow Region

1×102


10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-4 1×10-3

1×100 1×101

1×102

1×102 1×101

1×100

1×10-1

1×10-1 1×10-2

1×10-3

1×10-4




Radial Flow Region

1×102



10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2

Type Curve for a Well with Finite Conductivity Vertical Fracture in an Infinite-Acting Dual PorosityReservoir with Wellbore Storage Effects (CfD = (wkf)/(kxf) = 1, α = λCDf = 1×10-1, ω = 1×10-1).

1×10-5

1×10-5 CDf=1×10-6

1×10-4

1×10-3

1

1×101

1×102

1×102 1×101

1

1×10-1

1×10-1

1×10-2

1×10-3

1×10-4

Legend: CfD = 1, α = 1×10-1, ω = 1×10-1




Radial Flow Region

1×102

CDf=1×10-6

Figure B.15 — pD and pDβd vs. tDxf/CDf — CfD =1, ω = 1×10-1, α = λCDf = 1×10-1 (fractured well in dual porosity system case — includes wellbore storage effects).

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-41×10-3

11×101

1×102

1×102 1×101

1

1×10-1

1×10-1

1×10-2

1×10-3

1×10-4

Legend: CfD = 1, α = 1×10-5, ω = 1×10-1




Radial Flow Region

1×102

CDf=1×10-6


10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-41×10-3

1 1×101

1×102

1×102 1×101 1

1×10-1 1×10-2

1×10-3 1×10-4

Legend: CfD = 1, α = 1×10-5, ω = 1×10-2




Radial Flow Region

1×102

CDf=1×10-6


10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-4

1×10-3

11×101

1×102

1×102 1×101

1

1×10-1

1×10-1

1×10-2

1×10-3

1×10-4

Legend: CfD = 100, α = 1×10-1, ω = 1×10-1




Radial Flow Region

1×102

CDf=1×10-6


10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-41×10-3

1

1×101

1×102

1×102 1×101 1

1×10-1

1×10-1 1×10-2

1×10-3

1×10-4

Legend: CfD = 100, α = 1×10-5, ω = 1×10-1




Radial Flow Region

1×102

CDf=1×10-6

Figure B.19 — pD and pDβd vs. tDxf/CDf — CfD =100, ω = 1×10-1, α = λCD = 1×10-5 (fractured well in dual porosity system case — includes wellbore storage effects).

10-4

10-3

10-2

10-1

100

101

p D a

nd p

Dβd

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

tDxf/CDf

CDf=1×10-6

1×10-2


1×10-5

1×10-5

CDf=1×10-6

1×10-41×10-3

11×101

1×102

1×102 1×101

1

1×10-1

1×10-1 1×10-2

1×10-3

1×10-4

Legend: CfD = 100, α = 1×10-5, ω = 1×10-2




Radial Flow Region

1×102

CDf=1×10-6

Figure B.20 — pD and pDβd vs. tDxf/CDf — CfD =100, ω = 1×10-2, α = λCD = 1×10-5 (fractured well in dual porosity system case — includes wellbore storage effects).


10-3

10-2

10-1

100

101

102

p D, p

Dd

and

p Dβ d

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

tDL

0.125

0.25

0.5 1

5

10

25

50

100


L=0.10.125

0.25

0.5

1

5102550

100


Type Curve for a Infinite Conductivity Horizontal Well in an Infinite-Acting Homogeneous Reservoir (LD= 0.1, 0.125, 0.25, 0.5, 1, 5, 10, 25, 50, 100).


50

25

LD= 0.1

0.125LD= 0.1

0.25

0.5

Figure B.21 — pD, pDd, and pDβd vs. tDL — solutions for an infinite conductivity horizontal well in an infinite-acting homogeneous reservoir— no wellbore storage or skin effects (various LD values).

10-3

10-2

10-1

100

101

102

p D a

nd p

Dβ d

10-2 10-1 100 101 102 103 104 105 106 107

tDL/CDL

CDL=1×10-6

1×10-2


1×10-51×10-41×10-3

1×102

1×102




Radial Flow Region

CDL=1×10-6

1

1×10-11×101

Figure B.22 — pD and pDβd vs. tDL/CDL — LD=1 (horizontal well case — includes wellbore storage effects).

10-3

10-2

10-1

100

101

102

p D a

nd p

Dβ d

10-2 10-1 100 101 102 103 104 105 106 107

tDL/CDL

CDL=1×10-6

1×10-2


1×10-5

1×10-41×10-3

1×102

1×102




Radial Flow Region

CDL=1×10-6

11×101 1×10-1

1×101 1 1×10-1

1×10-21×10-3

1×10-4

1×10-5


10-3

10-2

10-1

100

101

102

p D a

nd p

Dβ d

10-2 10-1 100 101 102 103 104 105 106 107

tDL/CDL

CDL=1×10-6

1×10-2


1×10-5

1×10-4

1×10-3

1×102

1×102




Radial Flow Region

CDL=1×10-6

11×101 1×10-1

1×1011 1×10-1

1×10-21×10-3

1×10-4

1×10-5



10-3

10-2

10-1

100

101

102

103

p D, p

Dda

nd p

Dβ d

10-2 10-1 100 101 102 103 104 105 106 107

tD/CD

CDe2s=1×10-3

3×10-3 1×10-2 3×10-2 10-1

1 101

106

104 103 102

101 1 10-1

3×10-2

1×10-2

3×10-3

CDe2s=1×10-3

Type Curve for an Unfractured Well in a Bounded Homogeneous Reservoir with Wellbore Storage and Skin Effects (reD= 100)

Legend: Bounded Resevoir reD= 100 pD Solution pDd Solution pDβd Solution

CDe2s=1×10-3


Boundary Dominated Flow


106

Figure B.25 — pD and pDβd vs. tD/CD — reD =100, bounded circular reservoir case — includes wellbore storage and skin effects to illustrate both wellbore storage and boundary effects.


Appendix C — Diagnostic worksheet — A Summary of schematic well test responses for the β-derivative formulation.

Figure C.1 — Summary of schematic well test responses for the β-derivative formulation.

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