Decline-Curve Analysis Considering Nonlaminar Flow in Dual-Porosity Systems

Decline-Curve Analysis ConsideringNonlaminar Flow in Dual-Porosity Systems

J. Rodríguez-Román and R. Camacho-Velázquez, SPE, Pemex E&P

SummaryMost of the theoretical work on decline-curve analysis with non-laminar flow in the reservoir considers either a dry gas or liquidflow in a homogeneous system. The purpose of this work is toanalyze decline curves considering the non-Darcy flow effect inthe reservoir for a slightly-compressible-liquid flow in dual-porosity systems at constant wellbore pressure.

The transient and boundary-dominated flow periods are exam-ined by means of computer results generated with a finite-difference simulator. Analytical expressions of velocity and well-bore rate are presented for the first time in this paper. Theseequations contain the laminar-flow solution as a particular case. Amethod is proposed to identify the presence of inertial effects byplotting d[log(q)]/d[log(t)] against log(t).

The method for area calculation under laminar conditions pro-duces a good approximation for a reservoir under nonlaminar flow.However, the presence of inertial effects distorts the shape ofdecline curves, resulting in erroneous estimates of wellbore and res-ervoir parameters by type-curve analysis with laminar decline curves.

The utility of the methodology presented in this work is illus-trated with synthetic examples.

IntroductionMost of the existing techniques available in the literature for de-termining productivities and reservoir parameters are based on twomain assumptions: applicability of Darcy’s law and constant inner-boundary condition, either constant production rate or constantbottomhole flowing pressure. Neither of these assumptions is fre-quently fulfilled.1

In a naturally fractured formation, we may have wells initiallyproducing at a high rate and, in some cases, production declinesafter a few hours.2 Analyzing the transient flow-rate behavior willadd more information to produce a more complete evaluation.From an engineering viewpoint, the initial decline could be a keyfactor in deciding whether to complete or abandon a well. Inhomogeneous systems, this decline is the only one observed, butfor fractured reservoirs, the initial decline does not always repre-sent the final state of depletion.

At present, most of the reports on constant-bottomhole-pressuretests found in the literature assume that only one phase flows inporous media and that Darcy’s law is applicable to both homoge-neous and fractured reservoirs.3–7

In cases in which Darcy’s law is not valid, there are someworks that consider different kinds of tests as well as differentreservoir types.8–13 Refs. 10 and 11 showed that inertial effectscould be important in systems with slightly-compressible-liquidflow, providing analytical solutions for both transient and bound-ary-dominated flow. Nevertheless, there are no analytical expres-sions to analyze flow-rate responses for constant-bottomhole-pressure tests in naturally fractured reservoirs with non-Darcyflow. These expressions are relevant if we take into considerationthat inertial effects are expected to occur mainly in the fracturesystem. The objective of this paper is to present new analyticalexpressions for non-Darcy liquid flow in dual-porosity systemsproduced at constant bottomhole pressure.

Basic EquationsThe differential equations for the fracture and matrix systems aregiven by8

1rD

!

!rD !rD

2

1 +"1 + 4"D #!pfD

!rD#

!pfD

!rD$= !1 − #"

!pmD

!tD+ #

!pfD

!tD, . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

and !1 − #"!pmD

!tD= $!pfD − pmD". . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

These equations imply that non-Darcy flow is considered only inthe fracture system. The dimensionless variables are defined by

# =!%VC"f

!%VC"f + !%VC"m, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

$ = &km

kerw

2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

"D =ke'q

2(rwh), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

rD =r

rw, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

pfD =pi − p

pi − pwf, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

and tD =*1ket

)ct%frw2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

where *1 is a unit conversion constant and ke!kf %f.The Forchheimer equation9 is used to include the presence of

non-Darcy flow, given by

!pf

!r=

)

kvf + "'vf

2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

where " is a function of the porous-media properties. The corre-lation used in this work is the one developed by Geertsma14:

" =48511

%f5.5kf

0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

As is well known, there are several correlations for the non-Darcyflow coefficient available in the literature. For example, for vugu-lar carbonate rocks, the correlation of Ref. 15 is more appropriate.However, because Geertsma’s correlation is dimensionally correct,this correlation is used for the fracture system in the present study,although in practice it is necessary to develop an appropriate cor-relation for the specific rock type under consideration.

It is important to point out that the analytical solutions obtainedin this work are independent of the non-Darcy flow correla-tion used in their derivation. Thus, for the purpose of this study,there is no reason to delve into a justification for the use ofGeertsma’s correlation.

To solve the system of Eqs. 1 and 2, we transform it into thefollowing (see Appendix A):

!1 − #"!Vm

!tD= $%Vf&1 +

+Vf

2 ' − Vm( . . . . . . . . . . . . . . . . . . . (11)

Copyright © 2005 Society of Petroleum Engineers

This paper (SPE 74388) was first presented at the 2002 SPE International PetroleumConference and Exhibition in Mexico, Villahermosa, Mexico, 10–12 February, and revisedfor publication. Original manuscript received for review 1 April 2002. Revised manuscriptreceived 28 July 2005. Paper peer approved 13 September 2005.

478 December 2005 SPE Reservoir Evaluation & Engineering

rodolfo camacho

rodolfo camacho

1rD

!

!rD&rD

!Vf

!rD' −

Vf

rD2 = !1 + +Vf"#

!Vf

!tD+ !1 − #"

!Vm

!tD, . . . . (12)

where

+ = ,kf =2"'!kf"

2

). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

Vm =vm

km,Vf =

vf

kf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Stationary Flow SolutionThe development of the steady-state solution is shown in Appen-dix B, which is given in real variables as follows:

qo =)

2kf

%ln& re

rw'(

"'& 1rw

−1re'

× )*1 + 4!pi − pwf"kf

2"'

)2ln2& re

rw' & 1

rw−

1re' − 1+2(h. . . . (15)

Fig. 1 presents a comparison of this expression with the re-sponse of a simulation model10 in terms of dimensionless vari-ables. It is observed that the solution reproduces the behaviorduring the steady-state-flow period (tDA!tD*rw

2/A>0.1).Eq. 15 can be rearranged to obtain ":

" =4!pi − pwf"

'& 1rw

−1re' &

(h

qo'2

−2(h)%ln& re

rw'(

qokf'& 1rw

−1re' . . . . . . . . . . . . . . . (16)

Thus, if flow data are available during the steady-state period andestimates of the outer radius and effective permeability are given,then it is feasible to attain a value of ". However, assuming that

2!pi − pwf"!(h"

qo)--

%ln& re

rw'(

kf

and considering that re>>rw, we obtain the following approximation:

" ≈4rw !pi − pwf"

' &(h

qo'2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

Therefore, we can obtain an estimate of " with flow data duringthe steady-state period without requiring values of re and kf.

Transient and Boundary-Dominated FlowTo solve nonlinear Eqs. 11 and 12, a combination of perturba-tion,16,17 Laplace’s transform,18,19 and a variation of param-eter16,18,20 methods was used.

The perturbation16 method assumes that when an equation ischanged by a small amount, the solution also will change by asmall amount. The method produces a series of decreasing termsthat approximate the solution of the original differential equation.The application procedure consists of expanding the dependentvariable in a power series of the perturbation parameter of theproblem. To replace the original equation by this series, the initialand boundary conditions should be specified. To expand all theterms involved in the equation, it is necessary to equate terms withdifferent powers of the perturbation parameter; the solutions of theresulting equations are carried out sequentially, during which largepowers of the perturbation parameter are eliminated.

The application of the perturbation method is presented in Ap-pendix C, where the perturbation parameter is defined as

. =$+

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

with the following relationship:

Vf = V0 + .V1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)This equation was used in Eqs. 11 and 12. Using Laplace trans-formation and simplifying the resulting equations, we get

$d2V0

drD2 +

$

rD

dV0

drD= #$uV0 + !1 − #"$u

$V0

$ + u!1 − #"

+ $V0

rD2 for . ..0, . . . . . . . . . . . . . . . . . . (20)

and $d2V1

drD2 +

$

rD

dV1

drD= #$uV1 + #uV0

2 + !1 − #"$u$V1

$ + u!1 − #"

+ !1 − #"$uV0

2

$ + u!1 − #"

+ $V1

rD2 for . ..1. . . . . . . . . . . . . . . . . (21)

Using the solution of Eq. 20 in Eq. 21 and the method of variationof parameters, we obtain the following expressions (see Appendix D):

Fig. 1—Analytical vs. simulator solutions, non-Darcy stationary flow.

479December 2005 SPE Reservoir Evaluation & Engineering

V0 = A / I1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ B / K1%rD"u#u!1 − #" + $

$ + u!1 − #"( . . . . . . . . . . . . . . . . . . . (22)

and

V1 = C / I1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ D / K1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ I1%rD"u#u!1 − #" + $

$ + u!1 − #"(

× #rD

K1%x"u#u!1 − #" + $

$ + u!1 − #"( u

$

$ + #u!1 − #"

$ + u!1 − #"xV0

2dx

− K1%rD"u#u!1 − #" + $

$ + u!1 − #"(

× #rD

I1%x"u#u!1 − #" + $

$ + u!1 − #"( u

$

$ + #u!1 − #"

$ + u!1 − #"xV0

2dx.

. . . . . . . . . . . . . . . . . . . . . . . . . . (23)

Note that V0 corresponds to the laminar-flow solution.To find the value of the constants A, B, C, and D, we used the

following boundary conditions, corresponding to a closed cylin-drical reservoir:

V0|rD→reD= 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

V1|rD→reD= 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

pi − pwf

u= #

1

reD

rw)VodrD , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

0 = #1

reD

V1drD + #1

reDV0

2

$drD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)

For unbounded reservoirs, boundary conditions are the same, withreD→0.

Of course, the solutions presented in this work are valid whenthe perturbation parameter is much smaller than 1.

Laminar-Flow SolutionsAs mentioned previously, a particular solution to the non-Darcyflow problem may be the corresponding laminar solution. Forthe case of an infinite system, Da Prat6 presented the follow-ing expressions:

qD =" #

(tD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28)

and

qD =2

lntD + 0.80907, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

valid for early and late times during the transient period, with qD

defined by

qD!tD" =*2B)q!t"

2(keh!pi − pwf ",

where *2 is a unit-conversion constant.Solving the Warren and Root21 model using velocity as an

unknown variable, the authors of Ref. 22 obtained the followingexpression at late times during the transient period:

qD = −2 #0

0 % 4exp!21"

tD(x

2!x + 1"dx. . . . . . . . . . . . . . . . . . . . . . . . . . . (30)

Also, we can get new approximations for early and intermediatetimes given by the following (see Appendix E):

qD =" #

(tD+

38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)

qD = −2 #0

0 % 4#exp!21"

tD(x

2!x + 1"dx. . . . . . . . . . . . . . . . . . . . . . . . . (32)

As shown in Fig. 2, Eqs. 28 and 29 describe early- and late-time behaviors under laminar conditions. The expression obtainedby Rodríguez and Camacho22 at late times is slightly better thanEq. 29 at the beginning of the late-time period. Although for prac-tical purposes Eq. 29 is good enough, the use of Eq. 29 in theperturbation method makes it more difficult to obtain the corre-sponding nonlaminar solution compared to that obtained with theuse of Eq. 30. Also, the new integral equation, Eq. 30, is moreaccurate than the expression proposed by Da Prat, Eq. 29, as

Fig. 2—Da Prat6 and this paper, analytical solutions vs. simulator solution, transient period, laminar flow.


shown in Fig. 3, where the relative error obtained with both Eqs.29 and 30 with respect to the numerical solution is presented fortwo values of # and five values of $. We can observe that theproposed integral solution works better than the Da Prat expres-sion, especially at early times for high values of #. For these tworeasons, Eq. 30 is used to obtain the corresponding transient long-time nonlaminar approximation. Also, the behavior of Eqs. 31 and32 is presented in Fig. 2. Again, the approximation given by Eq. 31works better than the corresponding expression given by Da Prat,6

Eq. 28, and as expected, the intermediate-time approximation, Eq.32, works better at earlier times than the approximations given byEqs. 29 and 30.

In the case of a closed reservoir, Da Prat6 presented the fol-lowing expression for the boundary-dominated time period:

qD =reD

2 − 12

$ / exp&−$

1 − #tD'. . . . . . . . . . . . . . . . . . . . . . (33)

By considering more terms in the approximation series for Besselfunctions, Rodríguez and Camacho22 obtained the following equa-tion during the boundary-dominated flow period:

qD =

reD2 − 12

%reD2

2lnreD −

reD2!1 − 1"

2+

14(

×!a −r1"exp!−r1tD" − !a − r2"exp!−r2tD"

r2 − r1, . . . . . . . . . . . . . . (34)

where

−r1 =−b + $b2 − 4c

2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (35)

−r2 =−b − $b2 − 4c

2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36)

a =$

#!1 − #", . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (37)

b = a +1

#%reD2

2lnreD −

reD2!1 − 1"

2+

14(

, . . . . . . . . . . . . . . . . (38)

and c =a

%reD2

2lnreD −

reD2!1 − 1"

2+

14(

. . . . . . . . . . . . . . . . . . . . (39)

Fig. 4 presents the behavior of Eqs. 33 and 34 during the bound-ary-dominated period. It is observed that Eq. 34 has a greater rangeof applicability than the equation proposed by Da Prat,6 althoughthe data analysis is more complicated with Eq. 34.

Non-Darcy Flow SolutionsWe consider unbounded and closed fractured reservoirs. Foreach kind of reservoir, early-time and late-time approximationswere obtained.

Unbounded Reservoir Solutions. To obtain the transient rate so-lutions, we first obtain the V0 function; then, we obtain V1 to getthe porous-media velocity and the wellbore rate.

One way of detecting the presence of nonlaminar-flow effectsduring the transient period is by observing the behavior of thelogarithmic derivative. When nonlaminar effects are important, thelogarithmic derivative has a minimum value greater than –0.5 atearly times, whereas in the laminar-flow case, the minimum valueis smaller than –0.5, as shown in Fig. 5. Of course, the derivativecomputation with noisy rate data imposes certain problems thatmay be solved if some smoothing is applied while performing thederivative operation proposed by Horne.23 In this way, it will beeasier to use the proposed technique.

Because of the difficulty in transforming the obtained functionsinto Laplace space, we worked with approximations for interme-diate and late times.

At late times during the transient period, the following solutionis obtained (see Appendix F):

qD = −,2#0

0 % 4exp!21"

tD(x

2!x + 1"dx-

× ,1 −4"'kf

2

) %!pi − pwf"

rw) (#0

0 % 4exp!21"

tD(x

x

2!x + 1"dx- , . . (40)

which is an extension of the corresponding laminar-flow solution,Eq. 30. Fig. 6 shows the behavior of this solution, which at latetimes during the transient period is working properly.

Fig. 3—Relative error with late-time approximations, Eqs. 29 and 30, for an infinite reservoir, laminar flow.


The intermediate-time approximation is developed in AppendixG and is an extension of Eq. 40:

qD = −2#0

0 % 4#exp!21"

tD(x

2!x + 1"dx +

"'kf2

)F!at̃D"4%!pi − pwf"

rw) (

× #0

0 % 4#exp!21"

tD(x

x

2!x + 1"dx, . . . . . . . . . . . . . . . . . . . . . . . . . . (41)

where

F!at̃D" = #0

0 % 4#exp!21"

t̃D(x

2!x + 1"dx, . . . . . . . . . . . . . . . . . . . . . . . (42)

and t̃D can be approximated by the dimensionless time wheredlog(qD)/dlog(tD) presents a minimum during the transient period.This observation is empirical, but based on numerous numericalexperiments, it seems to yield a reasonable approximation. It canbe observed that when non-Darcy flow effects are neglected, thelaminar-flow solution, Eq. 32, is obtained.

Fig. 7 shows the behavior of this intermediate-time approxi-mation, where it can be observed that, during the transient period,it is working properly. Fig. 8 shows the computation of t̃D based onthe logarithmic derivative.

Closed Reservoir. During the boundary-dominated flow period,the same solution is obtained as for the laminar-flow case, asshown in Appendix H (i.e., Eq. 34 is also valid under non-Darcyflow conditions). Fig. 9 shows how at late times, both laminar andnon-laminar numerical solutions coincide, and it also illustrates thethe type of approximation done by Eq. 34.

Fig. 5—dlog(qD)/dlog(tD) in a closed reservoir with laminar and non-Darcy flow.

Fig. 4—Da Prat6 and Rodríguez et al.,20 analytical solutions vs. simulator solution for a closed-reservoir laminar flow.


The implication of the above result is that the computation ofthe drainage area presented in Ref. 7 for laminar-flow conditionsis also valid under nonlaminar conditions. We have found a maxi-mum error of 3% with this procedure. Fig. 10 presents two ex-ample calculations of the drainage area. In both cases, the proce-dure of Ref. 7 is working properly.

ConclusionsThe following conclusions can be drawn from the results presentedin this study:1. A solution for steady-state non-Darcy flow in dual-porosity sys-

tems is presented. This solution could be used to get an estimateof ".

2. New approximations for infinite and closed fractured systemsunder laminar flow are presented. The expression for closedsystems has a better performance than the corresponding equa-tion presented previously in the literature.

3. Intermediate- and late-time approximations for infinite fracturedreservoirs under non-Darcy flow are presented. Both approxi-mations reproduce the correct behavior.

4. If the value of the derivative, dlog(qD)/dlog(tD), is bigger than–0.5 at early times, then non-Darcy flow effects are important inthe dual-porosity system when the bottomhole pressure is keptconstant.

5. Non-Darcy flow effects decrease with time, and eventually,laminar flow will be reached at late times. During the boundary-dominated flow, when nonlaminar effects disappear, we canapproximate the behavior by the laminar solution, and themethod to compute drainage area yields good results.

Nomenclature

Bo ! formation volume factor, [L3/L3], dimensionlesscf ! fracture compressibility, [Lt2/m], psi–1

ct ! total compressibility, [Lt2/m], psi–1

h ! formation thickness, [L], ftk ! permeability, [L2], mdkf ! fracture permeability, [L2], md

km ! matrix permeability, [L2], md

Fig. 7—Intermediate-time approximations, Eqs. 32 and 41, for an infinite reservoir, laminar and non-Darcy flow.

Fig. 6—Analytical vs. simulator solutions, late-time transient flow.


p ! pressure, [m/Lt2], psipfD ! dimensionless fracture pressure

pmD ! dimensionless matrix pressurepi ! initial pressure, [m/Lt2], psi

pwf ! wellbore flowing pressure, [m/Lt2], psiqD ! dimensionless rateqo ! oil rate, [L3/t], STB/Dr ! radius, [L], ft

rD ! dimensionless radiusreD ! dimensionless outer-boundary radiusrw ! wellbore radius, [L], ft

t ! time, [t], day, hrtD ! dimensionless time

tDA ! dimensionless time based on drainage areau ! Laplace-space variablev ! flow velocity, [L/t]

Vm,f ! flow velocity divided by permeability, [L3/t]

" ! inertial coefficient given by Geertsma,14 [L–1], ft–1

1 ! Euler constant2(x) ! gamma function

. ! perturbation parameter% ! porosity, [L3/L3] dimensionless%f ! fracture porosity, [L3/L3], dimensionless

%m ! matrix porosity, [L3/L3], dimensionless$ ! dimensionless matrix-/fracture-permeability ratio

)o ! oil viscosity, [m/Lt], cp' ! fluid density, [m/L3], lbm/ft3

& ! shape factor, [L–2], ft–2

# ! dimensionless fracture storage

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Fig. 9—Analytical vs. simulator solution, boundary-dominated flow.

Fig. 8—t̃D calculation for intermediate-time non-Darcy flow.


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23. Horne, R.N.: Modern Well Test Analysis: A Computer-Aided Ap-proach, Petroway, Palo Alto, California (1995).

24. Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Func-tions, Dover, New York City (1964).

Appendix A—Development of Basic EquationsNonlaminar flow can be described with Forchheimer’s equation,9

given by

!p

!r=

)

kv + "'v2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

If we define

c =k

)"', . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

considering that non-Darcy flow is present in fractures,

vf =kf

)& !pf

!r

1 + cvf

', . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

and Darcy flow is present in the matrix blocks

vm =km

)

!pm

!r, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

the diffusivity equation is given as

1r

!

!r!rvf" = !%VC"f

!pf

!t+ !%VC"m

!pm

!t. . . . . . . . . . . . . . . . . (A-5)

Taking the derivative of Eq. A-5 with respect to r and defining

Fig. 10—Drainage-area calculation for non-Darcy flow in a two-porosity system (# = 0.001, S = 0, reD = 500, $ = 1×10–5).


, =2"'kf

), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)

# =!%VC"f

!%VC"f + !%VC"m, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7)

we obtain the following expression:

1rD

!

!rD&rD

!vf

!rD' −

vf

rD2 = !1 + ,vf"#

!vf

!tD+

kf

km!1 − #"

!vm

!tD.

. . . . . . . . . . . . . . . . . . . . . . . . (A-8)

Considering,2

!1 − #"!pm

!t=

$kf

%!%VC"f + !%VC"m& / )rw2 !pf − pm". . . . . . . . (A-9)

Obtaining the derivative of Eq. A-4 with respect to time, we obtain

!1 − #"

km

!vm

!tD= $%vf

kf&1 +

,vf

2 ' −vm

km(. . . . . . . . . . . . . . . . . . (A-10)

If we define

Vm =vm

kmand Vf =

vf

kf, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-11)

+ = ,kf =2"'!kf"

2

), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-12)

then we have the next system of equations:

!2Vf

!rD2 +

1rD

!Vf

!rD= #

!Vf

!tD+

#+

2!Vf

2

!tD+ !1 − #"

!Vm

!tD+

Vf

rD2 . . . . (A-13)

!1 − #"!Vm

!tD= $Vf +

$+

2Vf

2 − $Vm . . . . . . . . . . . . . . . . . . . . (A-14)

Appendix B—Stationary Flow SolutionThe diffusivity equation under steady-state conditions is given by

1r

!

!r!rvf" = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

Integrating this equation,

vf =A

r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2)

Substituting Eq. B-2 in Forchheimer’sequation9 using the appro-priate boundary conditions,

pf |re = pi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)

pf |rw = pwf, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)

we get

A =)

2kf

%ln& re

rw'(

"'& 1rw

−1re'

× )*1 + 4!pi − pwf"kf

2"'

)2ln2& re

rw' & 1

rw−

1re' − 1+. . . . . . . . (B-5)

Then, the wellbore flow rate is given as

qo =)

2kf

%ln& re

rw'(

"'& 1rw

−1re'

× )*1 + 4!pi − pwf"kf

2"'

)2ln2& re

rw' & 1

rw−

1re' − 1+2(h. . . . (B-6)

Appendix C—Perturbation MethodDefining the perturbation parameter, ., as

. =$+

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-1)

and substituting Eq. C-1 into Eqs. A-13 and A-14, we obtain

$!2Vf

!rD2 +

$

rD

!Vf

!rD= #$

!Vf

!tD+ #.

!Vf2

!tD+ !1 − #" $

!Vm

!tD+ $

Vf

rD2

. . . . . . . . . . . . . . . . . . . . . . . . . (C-2)

and !1 − #"!Vm

!tD= $Vf + .Vf

2 − $Vm. . . . . . . . . . . . . . . . . . . . . (C-3)

Defining Vf as

Vf = V0 + .V1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-4)substituting Eq. C-4 into Eqs. C-2 and C-3, applying the Laplacetransform, and neglecting powers of . greater than 2, we get thefollowing system of equations:

$d2V0

drD2 +

$

rD

dV0

drD= #$uV0 + !1 − #" $u

$V0

$ + u!1 − #"+ $

V0

rD2

. . . . . . . . . . . . . . . . . . . . . . . . . (C-5)

d2V1

drD2 +

1rD

dV1

drD= %u

#u!1 − #" + $

$ + u!1 − #"+

1

rD2( V1

+u

$

$ + #u!1 − #"

$ + u!1 − #"V0

2 . . . . . . . . . . . . . . . . . . . (C-6)

Solving Eqs. C-5 and C-6 simultaneously, the following ex-pressions are obtained:

V0 = A / I1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ B / K1%rD"u#u!1 − #" + $

$ + u!1 − #"( . . . . . . . . . . . . . . . . . . (C-7)

V1 = C / I1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ D / K1%rD"u#u!1 − #" + $

$ + u!1 − #"(

+ I1%rD"u#u!1 − #" + $

$ + u!1 − #"(E

− K1%rD"u#u!1 − #" + $

$ + u!1 − #"(F, . . . . . . . . . . . . . . . . . . . (C-8)

where

E = #rD

K1%x"u#u!1 − #" + $

$ + u!1 − #"( u

$

$ + #u!1 − #"

$ + u!1 − #"xV0

2 dx

. . . . . . . . . . . . . . . . . . . . . . . . . (C-9)

F = #rD

I1%x"u#u!1 − #" + $

$ + u!1 − #"( u

$

$ + #u!1 − #"

$ + u!1 − #"xV0

2 dx

. . . . . . . . . . . . . . . . . . . . . . . (C-10)


Appendix D—Boundary ConditionsFor a closed cylindrical reservoir, we have the following boundaryconditions for a constant-wellbore-pressure test:

Vf |rD→reD= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-1)

pi − pwf = #1

reD

rw)Vf drD + #1

reD

rw"'kf2Vf

2drD. . . . . . . . . . . . . . (D-2)

Let us define

"'kf2 =

).

$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-3)

Substituting Eq. C-4 into Eq. D-2, we obtain:

pi − pwf = #1

reD

rw)!V0 + .V1"drD

+ #1

reDrw)

$.!V0

2 + 2.V0V1 + .2V12"drD. . . . . . . . . . . (D-4)

Neglecting terms with . powers greater than 2, we get:

pi − pwf = #1

reD

rw)V0drD. . . . . for / .0 . . . . . . . . . . . . . . . . . . . (D-5)

#1

reDrw)

$V0

2drD = #1

reD

rw)V1drD . . . . . . for / .1, . . . . . . . . . . (D-6)

where

V0 |rD→reD= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-7)

V1 |rD→reD= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-8)

Eqs. D-5 through D-8 are the appropriate boundary conditions.

Appendix E—Short- and Intermediate-TimeApproximations for Laminar Flow in InfiniteFractured ReservoirsConsidering Eq. C-7 and the appropriate boundary conditions, thefollowing expression is obtained for laminar flow during the tran-sient period:

V0 =!pi − pwf"31K1!rD31"

)rwuK0!31", . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (E-1)

where

31 = u!1 − #"#u + $

!1 − #"u + $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (E-2)

Assuming that (1−#)u >> $ and using approximations for smallarguments for the Bessel functions, we obtain

V0 =!pi − pwf"2

)rwrDu ln!e21# u'4". . . . . . . . . . . . . . . . . . . . . . . . . . . . . (E-3)

Inverting Eq. E-3, the following expression is obtained at inter-mediate times:

V0 = −2!pi −pwf"

rDrw) #0

0 & 4

e21#tD'x

2!x + 1"dx, . . . . . . . . . . . . . . . . . . . (E-4)

which is similar to the late-time laminar solution given by Eq. F-1.For short times, Eq. E-1 can be written as follows:

V0 =pi −pwf

urw)

$u#

K0!$u#"/ K1!rD$u#". . . . . . . . . . . . . . . . . . (E-5)

If we consider the following approximations,

K0!z" ≈"(

2zexp!−z" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (E-6)

K1!z" ≈"(

2zexp!−z" / &1 +

38z', . . . . . . . . . . . . . . . . . . . . (E-7)

after simplifying and inverting, we obtain the following:

V0 =!pi − pwf"$#

$rDrw)

× ,exp%−#!rD − 1"2

4tD(

$(tD

+3

8rD$#erfc%$#!rD − 1"

2$tD(- .

. . . . . . . . . . . . . . . . . . . . . . . . . (E-8)Calculating qD, we obtain:

qD =" #

(tD+

38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (E-9)

Appendix F—Late-Time Solution for Non-DarcyFlow in Infinite Fractured ReservoirsAgain, solving the Warren and Root21 model, using velocity as anunknown variable, the authors of Ref. 22 obtained the followingexpression at late times during the transient period:

V0 =2!pi −pwf"

rDrw) #0

0 & 4

e21tD'x

2!x + 1"dx. . . . . . . . . . . . . . . . . . . . . . . . (F-1)

If we use the mean value theorem24 and the Laplace transform, thefollowing expression is obtained:

V02 = %2!pi − pwf"

rDrw) (2

F!at̃D"1

u ln%u

4exp!21"(, . . . . . . . . . . . (F-2)

where

F!at̃D" = #0

0 % 4exp!21"

t̃D(x

2!x + 1"dx. . . . . . . . . . . . . . . . . . . . . . . . . (F-3)

Using the following approximations,24

K1!x" ≈1x, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (F-4)

I1!x" ≈14

x, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (F-5)

"u#u!1 − #" + $

$ + u!1 − #"≈ $u, . . . . . . . . . . . . . . . . . . . . . . . . . . (F-6)

we obtain:

E = −1

$$u%2!pi − pwf"

rw) (2

F!at̃D"1

ln%u

4exp!21"(

1rD

. . . . . . (F-7)

F$u

41$ %2!pi − pwf"

rw) (2

F!at̃D"1

ln%u

4exp!21"( rD. . . . . . . . . (F-8)

Substituting these equations into Eq. C-8, we obtain

V1 = CI1%rD"u#u!1 − #" + $

u!1 − #" + $(

+ DK1%rD"u#u!1 − #" + $

u!1 − #" + $(. . . . . . . . . . . . . . . . . . . . (F-9)


Applying the appropriate boundary conditions, we get:

V1 = %2!pi − pwf"

rw) (2 F!at̃D"

$×

K1!rD$u"

$u ln%u

4exp!21"(K0!$u"

.

. . . . . . . . . . . . . . . . . . . . . . . . (F-10)

Inverting and simplifying, we obtain the following:

V1 = %2!pi − pwf"

rw) (2

2F!at̃D"

$

1rD2!2"

× #0

0 % 4exp!21"

t(x

x

2!x + 1"dx.

. . . . . . . . . . . . . . . . . . . . . . . . (F-11)

Thus,

Vf = −2!pi − pwf"

rDrw) #0

0 % 4exp!21"

tD(x

2!x + 1"dx

+"'kf

2

)

F!at̃D"

rD

22!2" %2!pi − pwf"

rw) (2

× #0

0 % 4exp!21"

t(x

x

2!x + 1"dx.

. . . . . . . . . . . . . . . . . . . . . . . . (F-12)

Additionally,

F!at̃D" = #0

0!at̃D"x

2!x + 1"dx = (

i=1

23 bi

lni!at̃D"→ F!at̃D" ≈ F!atD".

. . . . . . . . . . . . . . . . . . . . . . . . (F-13)

Then,

Vf = −,2!pi − pwf"

rDrw) #0

0 % 4exp!21"

tD(x

2!x + 1"dx-

× ,1 −"'kf

2

)

22!2" %2!pi − pwf"

rw) ( / #0

0 % 4exp!21"

t(x

x

2!x + 1"dx- ,

. . . . . . . . . . . . . . . . . . . . . . . . (F-14)

and qD can be written as

qD = −,2 #0

0 % 4exp!21"

tD(x

2!x + 1"dx-

× ,1 −"'kf

2

) %!pi − pwf"

rw) ( / #0

0 % 4exp!21"

t(x

x

2!x + 1"dx- .

. . . . . . . . . . . . . . . . . . . . . . . . (F-15)

Appendix G—Intermediate-Time Solution forNon-Darcy Flow in Infinite Fractured ReservoirsUsing Eq. E-4 and the same considerations used in Appendix F, weobtain the following expression:

Vf =2!pi − pwf"

rDrw) #0

0 % 4#exp!21"

tD(x

2!x + 1"dx

+"'kf

2

)

F!at̃D"

rD

22!2"%2!pi − pwf"

rw) (2

× #0

0 % 4#exp!21"

tD(x

x

2!x + 1"dx,

. . . . . . . . . . . . . . . . . . . . . . . . . (G-1)

where

F!at̃D" = #0

0 % 4#exp!21"

t̃D(x

2!x + 1"dx, . . . . . . . . . . . . . . . . . . . . . . (G-2)

and qD can be written as

qD = −2 #0

0 % 4#exp!21"

tD(x

2!x + 1"dx

+"'kf

2

)F!at̃D"4%!pi − pwf"

rw) ( × #0

0 % 4#exp!21"

tD(x

x

2!x + 1"dx.

. . . . . . . . . . . . . . . . . . . . . . . . (G-3)

Appendix H—Late-Time Solution for Non-DarcyFlow in Closed Fractured ReservoirsBy considering more terms in the approximation series for Besselfunctions, Rodriguez and Camacho22 obtained the following equa-tion during the boundary-dominated flow period:

V0 =pi − pwf

rw)

reD2 − rD

2

2rD

%reD2

2lnreD −

reD2!1 − 1"

2+

14(

×!a − r1"exp!−r1tD" − !a − r2 "exp!−r2tD"

r2 − r1, . . . . . . . . . . . (H-1)

where

a =$

#!1 − #". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-2)

b = a +1

#%reD2

2lnreD −

reD2!1 − 1"

2+

14(

. . . . . . . . . . . . . . . . (H-3)

c =a

%reD2

2lnreD −

reD2!1 − 1"

2+

14(

. . . . . . . . . . . . . . . . . . . . . (H-4)

−r1 =−b + $b2 − 4c

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-5)

−r2 =−b − $b2 − 4c

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-6)

Taking V1 from Eq. C-8, we can calculate V02 and obtain the

Laplace transform to get

V02 = A/ B& reD

4

4rD2 −

reD2

2+

rD2

4 ' . . . . . . . . . . . . . . . . . . . . . . . (H-7)

A =&pi − pwf

rw) '2

%reD2

2lnreD −

reD2!1 − 1"

2+

14(2

!r2 − r"12

. . . . . . . . . . . . (H-8)

B =

!% $

#!1 − #"− r(

1

2 1u + 2r1

−2.% $

#!1 − #"(2

− !r1 + r2"$

#!1 −#"+ r1r2/ 1

u + !r1 + r2"

+ % $

#!1 − #"− r2(2 1

u + 2r2

$ .

. . . . . . . . . . . . . . . . . . . . . . . . (H-9)


Defining x as

x = u#u!1 − #" + $

$ + u!1 − #", . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-10)

using the following approximation to Bessel functions,

I0!z" ≈ 1 +14

z2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-11)

I1!z" ≈12

z +14

z3, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-12)

K0!z" ≈ −&lnz

2+ 1' −

14

z2&lnz

2+ 1' +

14

z2, . . . . . . . . . (H-13)

K1!z" ≈1z

+z

2ln & z

2' +z2

16ln& z

2' −z

2!−21 + 1"

−z3

16 &−21 +52', . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-14)

and substituting into Eq. C-8 and simplifying, we obtain

V1 = C/ I1!rD$x" + D / K1!rD$x"

+A / B / x

2$&−

reD4

2−

reD2rD

2

3+

rD4

30'. . . . . . . . . . . . . . . . (H-15)

Applying the following boundary conditions,

V1|rD→reD= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H-16)

0 = #1

reD

V1drD + #1

reDV0

2

$drD, . . . . . . . . . . . . . . . . . . . . . . . . . . (H-17)

we obtain

V1 = −

A / B

$

G / rD

4

reD

2 &reD2 − 1

4reD−

reDlnreD

2 ' +25

/A / B / x / rD / reD3

$

+

A / B

$G

reD

2rD

&reD2 − 1

4reD−

reDlnreD

2 ' +A / B / x

2$&−

reD4

2−

reD2rD

2

3+

rD4

30'.

. . . . . . . . . . . . . . . . . . . . . . . (H-18)Transforming into real space,

V1 =)−

A

$

G / rD

4

reD

2 &reD2 − 1

4reD−

reDlnreD

2 ' +

A / B

$G

reD

2rD

&reD2 − 1

4reD−

reDlnreD

2 '+M

+%25

/A / rD / reD3

$+

A

2$&−

reD4

2−

reD2rD

2

3+

rD4

30'(N, . . . . (H-19)

and we can write qD as

qD =

reD2 − 12

%reD2

2lnreD −

reD2 !1 − 1"

2+

14(

×!a − r1"exp!−r1tD" − !a − r2"exp!−r2tD"

r2 − r1

+&pi − pwf

rw) ' "'kf

)

%reD2

2lnreD −

reD2!1 − 1"

2+

14(2

!r2 − r"12

× !,−

G

4%reD

2 &reD2 − 1

4reD−

reDlnreD

2 '(+

GreD

2&reD2 − 1

4reD−

reDlnreD

2 ' -M

+ %25

reD3 +

12 &−

reD4

2−

reD2

3+

130'(N .

$ ,

. . . . . . . . . . . . . . . . . . . . . . . (H-20)

where a, r1,and r2 are given by Eqs. H-2, H-5, and H-6, respec-tively. G, M, and N are given by

G = &−2reD

3

3+

reD4

4+

reD2

2−

112', . . . . . . . . . . . . . . . . . . . (H-21a)

M = % $

#!1 − #"− r1(2

exp!−2r1tD"

− 2, % $

#!1 − #"(2

− !r1 + r2"$

#!1 − #"+ r1r2

- exp

%−!r1 + r2" / tD& + % $

#!1 − #"− r2(2

exp!−2r2tD", . . . . (H-21b)

N = % $

#!1 − #"− r1(2 $

!1 − #"

× ,−2r1exp!−2r1 / tD"

$

!1 − #"− 2r1

+

$

!1 − #"exp%−

$

!1 − #"tD(

$

!1 − #"− 2r1

-− 2.% $

#!1 − #"(2

− !r1 + r2"$

#!1 − #"+ r1r2/ $

!1 − #"

× ,−

!r1 + r2"exp%−!r1 + r2"tD&

$

!1 − #"− !r1 + r2"

+$

!1 − #"exp%−

$

!1 − #"tD(

$

!1 − #"− !r1 + r2"

-+ % $

#!1 − #"− r2(2 $

!1 − #"

× ,−2r2exp!−2r2 / tD"

$

!1 − #"− 2r2

+

$

!1 − #"exp%−

$

!1 − #"tD(

$

!1 − #"− 2r2

- .

. . . . . . . . . . . . . . . . . . . . . . .(H-21c)

For large times, we can write qD as

qD =

reD2 − 12

%reD2

2lnreD −

reD2 !1 − 1"

2+

14(

×!a − r1"exp!−r1tD" − !a − r2"exp!−r2tD"

r2 − r1, . . . . . . . . . . (H-22)

which is the corresponding laminar-flow solution.


SI Metric Conversion Factorsbbl × 1.589 873 E–01 ! m3

cp × 1.0* E–03 ! Pa/sft × 3.048* E–01 ! m

ft2 × 9.290 304* E–02 ! m2

ft3 × 2.831 685 E–02 ! m3

lbm × 4.535 924 E–01 ! kgpsi × 6.894 757 E+00 ! kPa

*Conversion factor is exact.

Rodolfo Camacho-Velázquez is Manager of Technology De-velopment in E&P in the Operations Corporate Direction ofPemex. Previously, he was Well Productivity Manager in theTechnology and Professional Development Vice-Presidency

of Pemex E&P. He also has been the head of the PetroleumDept. in the Graduate Div. of the Natl. U. of México (UNAM)and a researcher at the Mexican Petroleum Inst. His profes-sional interests include well-test analysis, well productivity, andreservoir engineering. He has written several technical papers,most of them published by SPE. Camacho holds a BS degree ingeophysical engineering from the U. of Mexico and MS andPhD degrees in petroleum engineering from the U. of Tulsa. Hewas recognized as an Outstanding Technical Editor for SPE in1997 and 2003. From 1997 to 1999, he was a Review Chair forSPEREE, and he was the 2001–02 Chair of SPE’s Education andProfessionalism Committee. Jesús Rodríguez-Román is a reser-voir engineering specialist with Pemex E&P. He has 10 years ofreservoir experience, especially in naturally fractured reser-voirs. He was an assistant professor in the Earth Sciences Div.at the Natl. U. of Mexico (UNAM). His professional interests in-clude well-test analysis, well productivity, and reservoir engi-neering. He holds BS and MS degrees in petroleum engineeringfrom the UNAM.


Decline-Curve Analysis Considering Nonlaminar Flow in Dual-Porosity Systems

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Transcript of Decline-Curve Analysis Considering Nonlaminar Flow in Dual-Porosity Systems