SPE 23668

A New Model for Pressure Transient Analysis in Stress Sensitive Naturally Fractured Reservoirs

In this paper, based on our analytical model, we analyze the transient pressure
response of a naturally fractured reservoir with pressure dependent rock properties.
V. Celis, R.Silva, M. Ramones, Intevep, S.A.,
J. Guerra, Universidad Central de Venezuela,
G. Da Prat, Schlumberger - Argentina Division
© 1994 Society of Petroleum Engineers

ABSTRACT Several authors have suggested theories to estimate the characteris-

tics of a naturally fractured reservoir. Among the first and most relevant
studies on the subject is the one by Warren and Root 1, their theory as-
The most commonly used methods to describe the transient flow be-
sumes an ortogonal network of fractures uniformly distributed that allow
havior in porous media are based on the assumption of constant rock
the flow between the matrix and the fracture under pseudosteady-state
properties. Nevertheless, these methods are not strictly applicable to
conditions. Based on these assumptions they developed the differential
reservoirs that undergo changes in the rock properties due to variation
equation for the problem that allowed to obtain analytical solutions for
in pore pressure. A frequent characteristic of fractured reservoirs is
sensitivity of permeability and porosity to effective stress. well tests analysis, characterizing the reservoir with only two parameters.
Later, Kazemi2 developed a numerical model for a finite reser-
This paper presents a new analytical model, for interpreting pres- voir, where the fracture is horizontal and considering radial symmetry.
sure transient test. The model considers the flow in a naturally fractured Kazemi's results are similar to those of Warren and Root, except for
stress sensitive reservoir, that is to say, the dependency of rock proper- a smooth transition zone as a consequence of an unsteady-state regime
ties on pressure is established (dependency of permeability on pore pres- at the matrix.
sure is measured by the permeability modulus parameter), and makes
that the flow equation be strongly non linear. Likewise, the theory is A. de Swaan3 proposed an analytical solution for the same physical
developed under unsteady- state or pseudosteady-state conditions, and problem working with a new term in the formulation of the equation that
considers that: the fracture is uniformly distributed, matrix geometry represents the flow transfer from the matrix to the fracture in unsteady-
is the so called stratum and flow contribution from the matrix to the state conditions. This model became the basis for the publication
fracture is described by the term source proposed by De Swaan. of Cinco-Ley and Samanieg04 who concluded that the behavior of a
dual-porosity reservoir could only be correlated by three dimensionless
The problem stated is solved through an analytical approach for parameters.
different boundary conditions and different permeability modulus values, A more complete study was presented by Olarewaju and Lee' in
thus obtaining type curves that can be used for the analysis of pressure which analytical solutions for naturally fractured reservoirs for different
buildup and drawdown tests.
conditions are given, using the same parameters of Cinco-Ley and
Samaniego.
All the above mentioned studies considered rock properties con-
INTRODUCTION stant. Some naturally fractured reservoirs present the characteristic of
permeability and porosity sensitivity of fractures to effective stresses.
Reservoirs usually have heterogeneities, and one of the most relevant The decline of pore pressure causes and increase of effective stress and
is caused by natural fractures, which are rock fragmentations as a thus the porous media can be compacted, reducing hydraulic permeabil-
consequence of geological factors. This type of reservoir is known ity and porosity.
as dual porosity reservoir, that is, it is considered a formation with This problem was solved by Pedrosa6 considering the diffusivity
two different porous media with different properties of permeability and equation in a homogeneous porous media (non fractured) and consid-
porosity. A first medium is made up of matrix blocks that contain the ering the permeability reduction by adding a new parameter called per-
majority of the fluid, which is usually called primary porosity and has a meability modulus; this mechanism makes permeability and porosity
low conductivity. The other one, is made up of a network of fractures, directly dependent on pressure.
it is said to have secondary porosity and acts as a conductor medium In this paper, based on our analytical model, we analyze the
for the fluid, since it has a high flow capacity, but the matrix has a low transient pressure response of a naturally fractured reservoir, where
flow capacity. rock properties depend on pressure and considering the sensitivity to
In some reservoirs of this type there can be a problem of produc- formation stresses.
tivity loss due to the plugging of fractures contacted by the producing
well, caused by a reduction of porosity and permeability of the fractured
media. The sensitivity of this type of reservoir to effective stresses can MODEL DESCRIPTION
be determined by pressure tests (usually variable flow rates are used).
The purpose of this work is to show the applicability of the model Physical model
developed using the response of transient pressure behavior in a natu- The idealization of a naturally fractured reservoir from a physical stand-
rally fractured reservoir, considering the rock properties dependent on point consists of a set of matrix horizontal layers, uniformly spaced,
pressure, taking into account the sensitivity of these types of reservoirs where fractures are the separations between each layer. The model ge-
to effective stresses. ometry is the so called stratum. Flow is radial and converges towards

126 SPE Advanced Technology Series, Vol. 2, No.1

the well that is centrally located in a circular reservoir, which in one- An initial condition, two inner boundary conditions and an outer
dimensional form corresponds to an interval whose middle point is the boundary condition are applied. These conditions are presented in
well and the edges are its boundaries. On the other hand, the math- dimensionless form as:
ematical statement of the problem must reflect the flow behavior in a Initial condition:
reservoir under the following assumptions: (1) isothermal and one-phase
laminar flow in a dual-porosity isotropic reservoir of uniform thickness; PD{rD,O) =0 for rD >0 (2)
(2) negligible gravity force; (3) uniform initial pressure throughout the
reservoir; (4) fluid production is obtained through the network of frac- Boundary conditions:
tures considering that the matrix blocks act as uniformly distributed a) External:
sources; and (5) the well is producing at a constant rate in finite or in- • Finite reservoir:
finite reservoir, taking into account the effects of damage and wellbore
storage. Since the formation is considered stress sensitive, pressure gra- OPD
-o-{reD, tD)
rD
=0 (3)
dients close to the well are very high, which makes that the assumption
of low pressure gradients cannot be applied. • Infinite reservoir:

Mathematical model lim PD{rD, tD) = 0 (4)

rD-oo
The diffusivity equation that describes the flow in the fracture
system for an oil reservoir is given by: b) Internal:

CD OPwD _ (e--rDPD OPD) IrD=l =1 (5)

OtD orD
-1 -0- ( rD--
OPD) - 'YD (OPD)2
--
rD orD orD orD
PwD = ( PD - SrDe _~,D PD __
OPD)
P OPD
[
e-rD D w OtD +(I-w)AfD / ~ OPD r
-----aPF{rm,tD -r)dr 1 orD rD=l
Let us introduce a dimensionless dependent variable U, related to
(1) dimensionless pressure according to:
These differential equations are derived in more detail in Appendix A.
1
PD = --In{1 - 'YDU) (6)
'YD
Definition or dimensionless variables: It may be noted that 'Y D U must lie between zero and unity. Making
The dimensionless variables used in equation (1) are defined below. use of Eqn. 6, it can be found that U satisfies the following partial
differential equation:
Dimensionless pressure, PD.

PD = k;fh{P; - Pf{r, t))

QqBjt 1 0 ( oU ) ( 1 ) oU
rD orD rD orD = 1 - 'YDU W otD +
Dimensionless time, t D. tD (7)
{I - w)AfD J(1 _~DU
o
) o~~r) F{'1/D, tD - r)dr

Dimensionless radial distance, rD. Subject to the initial condition and boundary conditions respectively
r
rD = - U{rD,O) = 0
rw
Dimensionless fractured storativity ratio, w a) External:
Finite reservoir:

au
- a (reD, tD)
rD
=0 (8)
Dimensionless Wellbore Storage, CD
Infinite reservoir:
CD = 0.894C.
ifJCt hr w 2
lim U{rD, tD) = 0 (9)
rD-oo
Dimensionless matrix hydraulic diffusivity, '1/D
b) Internal:

CD 1 oUw _ rD oU = 1
Dimensionless fractured area, Afd 1 - 'YDUw atD orD
1
--In{l - 'YDUw) = (10)
AfD = Afmhm 'YD

Dimensionless permeability module, 'YD -2-1n{1 - 'YDU)lrD=l _ S ( au )

'YD arD rD=l
An aproximate analytical solution for this problem was obtained
using a perturbation technique. To solve Eqn. 7, subject to the

SPE Advanced Technology Series, Vol. 2, No. I 127

conditions 8 to 10, U can be expanded as a power series in the parameter smooth, for the pseudosteady-state time. There is an abrupt increase
'YD. in the pressure, followed by the flat portion that has an approximate
duration of two cycles. Wellbore storage effects mask the transition
(11) period and does not allow to identify the flow regime dominated by
the fracture, independently of the matrix-fracture flow type. In relation
The perturbation technique leads to a sequence of linear bound- to outer boundary effect, it is seen that flow regime identification at
ary value problems. The derivation and solution of the systems are early and intermediate times becomes difficult as the size of reservoir
explained in more detail in Apendix A. becomes smaller. Bigger the adimensional external radius, later will
The solution for the dimensionless wellbore flowing pressure is appear the exponential shape in Figure 7.
presented. The dependency of fracture properties (permeability and
porosity) on pressure is considered. and it is measured by the parameter Figure 6 shows the dimensionless pressure response for several
called permeability modulus. The solution is obtained by considering values of the permeability modulus, with a fix dimensionless drainage
the dimensionless pressure of the well as a function of Uo by relation: radius (reD = 1(0), whereas Fig. 7 shows the approximate solution of
bounded reservoirs for different outer-boundary radii and a fixed value
PwD = __
1 In[1 - "YDL- 1 {Vow}] of "YD of 0.1. The solution shown a late times is typical of outer closed
(12)
"YD boundaries systems.
In the case of buildup test analysis the dual Laplace corresponding Kikani and Pedrosa7 showed that higher order terms for homoge-
to(t p+ ~t)D Y (~t)D , let us say and 81 and 82 are numerically neous reservoirs do not have significant effect on the pressure analytical
calculated, and then are evaluated in the following relation: solution. Consequently, only the first order term was used to approxi-
mate the solution in this paper.
(13)
Then buildup dimensionless wellbore pressure pressure is given by:
CONCLUSIONS
b
PwD = - -1I n [1- "YDL -1 { U
- b }]
ow (14)
"YD .
The analytical model that has been developed includes all the char-
acteristics of transient pressure behavior of a stress sensitive naturally
fractured oil reservoir.
DISCUSSION OF RESULTS
The model can handle the unsteady-state or pseudo steady -state
The numerical solutions obtained with the proposed model are in ex- flow from the matrix to the fracture. It incorporates the effects of dam-
cellent agreement with analytical solutions presented by Pedrosa 6 as age, storage, boundary and variations of rock properties with pressure.
shown in Figure 1. The zero order solution corresponds with accuracy to the case in
Figure 2 shows a semi-log graph of buildup pressure versus shut- which there is no stress sensitivity, that is to say, the model reproduces
in times, for a constant wellbore storage effect varying the permeability the pressure response of a naturally fractured reservoir by means of
modulus "YD. The effect of the permeability modulus is very pronounced a complete characterization of all flow regimes, allowing the use of
at early times and disappears at longer times. It can also be observed that different boundary and flow conditions at the matrix.
a reservoir with constant rock properties and a stress sensitive reservoir,
shows the same buildup response at late times. This same response was In a well with low wellbore storage it is difficult to conclude if
observed by Pedrosa 6 in homogeneous reservoirs. the reservoir is stress sensitive.
Figure 3 shows a semi-log graph of dimensionless buildup pressure Taking into consideration stress sensitivity, the solution does not
versus shut-in times, where the two upper curves correspond to CD = affect the end of the storage effect nor the transition period duration.
2000 and the two lower curves to CD = 200 for the same values of the
permeability modulus. From the analysis made, it is inferred that the variation of fracture
properties with pressure can drastically increase the estimate of parame-
Wellbore storage affects the interpretation of buildup tests for
ters that are commonly used in the characterization of naturally fractured
a reservoir where rock properties change with pressure: as wellbore
reservoirs, and also parameters such as wellbore storage.
storage decreases, it is more difficult to identify the effect of effective
stresses on pressure (permeability modulus), even during short periods If stress sensitivity is not taken into account, erroneous conclusions
of time. can be reached about the reservoir characteristics. As seen in Figs. 6
It is known that the parameters w and " determine the beginning and 7, a type-curve mathing for diagnostic pouposes and specially using
and the end of the transition zone. This feature is observed for an real data, closed outer boundaries effects may be interpreted as stress
undamaged well and CD = 0 when varying "YD. The trend of curves sensitive effects.
is towards the dimensionless pressure increase, this fact is graphically
observed in Figure 4 (conventional semi-log Homer plot).
Figure 5 shows the effect of stress sensitivity on pressure buildup,
for a naturallly fractured reservoir. For long shut-in times, the plot
suggests that conventional analysis can be performed for determining
the initial permeability of the fractured reservoir. The same performance
was observed by Kikani and Pedrosa 7 for a homogeneous reservoir. It
is clear that when the permeability modulus is zero, relations (7) and (9)
cannot be used. This case corresponds to a non-stress sensitive reservoir
and the zero order solution. It is exactly the solution for a naturally
fractured reservoir. The model also provides the selection of type of
flow for the matrix, having different effects for the solution. When
matrix flow is considered unsteady-state type, the transition period is

128 SPE Advanced Technology Series, Vol. 2, No.1

 NOMENCLATURE 4. Cinco-L., H. and Samaniego-V., F.:" Pressure Transient Analysis
for Naturally Fractured Reservoirs ", paper SPE 11026, presented
= fractured area, f t 2 at the 1982 SPE Annual Technical Conference and Exhibition,
= bulk fracture area, /t 2 New Orleans, Sept 26-29.
= dimensionless fracture area 5. Olarewaju, J. S. and Lee, W. J.:"A New Pressure Transient Anal-
= formation volume factor, RBI ST B ysis Model for Dual-Porosity Reservoirs", paper SPE 15634, pre-
CI = compressibility, I/psi sented at the 1986 SPE Annual Technical Conference and Exhibi-
C, = fracture compressibility, l/psi tion, New Orleans, Oct 5-8.
Ct = total compressibility, lips; 6. Pedrosa, O. A., Jr.:"Pressure Transient Response in Stress-
CD = dimensionless wellbore storage Sensitive Formations ", paper SPE 15115, presented at the 1986
F('1V,tV) = function describing flow in the matrix for different SPE Regional Meeting, Oakland, April 2-4.
geometries 7. Kikani, J. and Pedrosa, O. A., Jr.: "Perturbation Analysis of
= function describing flow in the matrix for different Stress-Sensitive Reservoirs", SPEFE (Sept. 1991) 379-386.
geometries 8. Stehfest, H.:"Numerical Inversion of Laplace Transform ", Comm.
= net pay thickness, f t of The ACM, (January 1970) 47-49.
= matrix block thickness, f t
= fracture permeability, md
= fracture initial permeability, md
APENDIXA
= matrix permeability, md MATHEMATICAL MODEL
= dimensionless pressure
= fracture pressure, psia The diffusivity equation that describes the flow in the fracture
= initial reservoir pressure, psia
system for an oil reservoir is given by:
= matrix pressure, psia 2
= flow rate, BID 8 Pj +! 8P, + CI (8P,)2 +
= radius, /t
&r2 r &r &r
(A.i)
= dimensionless radius 1- &k, (&P,)2 = £tP,(CI +c,)&P,
re = drainage radius, f t k, &P, &r k, &t
reD = dimensionless drainage radius On the other hand, the permeability modulus, which is the param-
rw = wellbore radius, f t eter that measures the dependence of hydraulic permeability on pore
s = laplace parameter pressure is defined as:
S = skin factor
= time, hr
(A.2)
= dimensionless time = O.0002637kt!4>ctJlr!
= producing time, hr The substitution of (A.2) in (A.1) gives the following equation:
= shut-in time, hr
= bulk volume, ft 3
&2pj l&P,+( + )(&P,)2
= matrix volume, ft 3 &r2 +;- &r 'Y CI Tr
= dimensionless matrix diffusivity (A.3)
Jl &P,
= porosity, fraction k, tP,(CI + c')Tt
= initial porosity, fraction
= permeability modulus In stress sensitive formations pressure gradients close to the well
= viscosity, cp
are high, thus the square term is not neglected. Finally, in the previous
equation the term q* must be included, since it represents the fluid
= density, gr I cc
transfer from the matrix to the fracture, leaving the diffusivity equation
= dimensionless storavity ratio
in the fracture network as:

ACKNOWLEDGEMENTS
(A.4)

The authors wish to express their appreciation to Petroleos de

Venezuela, S. A. and the managements of lNTEVEP, S.A. for granting where:
permision to publish the present work.
q.= J t
&P,(T)
---qum(t
&T
- )
T dT (A.5)
o
REFERENCES
(A.6)
1. Warren, J. E. and Root, P. S.:" The Behavior of Naturally Frac- Where V APum is the pressure gradient within the matrix per
tured Reservoirs ", SPEJ (Sept 1963) 245-255; Trans., AIME, unit of pressure drop in the fracture. In equation (AA) porosity and
228. permeability of fracture are assumed to be dependent on pressure ,
6
2. Kazemi, H.:" Pressure Transient Analysis of Naturally Fractured expressing said dependency as:
Reservoirs with Uniform Fracture Distribution", SPEJ (December
1969) 451-462 , Trans., AIME, 246. kj = kif (e-"Y(P'-p/») (A.7)
3. de Swan-O., A.:" Analytic solution for Determining Naturally
Fractured Reservoirs Properties by Well Testing ", SPEJ (June tPf = tPi,(e-c/(P,-p/») (A.S)
1976) 117-122, Trans., AIME, 261..
SPE Advanced Technology Series, Vol. 2, No. I 129
 By substituting previous expressions in equation (A.4), the equa- Initial and boundary conditions in dimensionless form are:
tion that rules the behavior of fluid flow in a dual-porosity and stress Initial Condition:
sensitive reservoir is obtained. taking into account the effects of storage
and damage. It is also possible to consider the reservoir as finite or
infinite extension.
PD(rD,O) =0 for rD > 0 (A.I6)

Boundary Conditions:
a) External
Finite reservoir:
(A.9)
(A.I7)

Initial Condition: Infinite reservoir:

P,(T,O) = Pi for all T > 0 (A.IO) lim PD(TD,tD)=O (A.IS)

rD-oo

Boundary Conditions: b)Internal:

a) External:
Finite reservoir:
8PwD
CD - - -
(-.,DPD
e aPD)1
- - 1
- rD=l-
8tD 8rD
(A.ll) (A.19)

Infinite reservoir:
Making transform, as propossed by Pedrosa6 , i.e.
(A.I2)
1
PD = --In(I
'YD
- 'YDU) (A.20)
b) internal:

< 'YDU < 1 equation (A.14) is

-q = Cs a;t Ir=r",-
with 0 transformed into:

kifhTw .,(Pi-P/) aPf I

/JOt e aT r=r", (A.l3)

Pwf = PJ(T, t)lr=r", - STwe-.,(Pi- P/) a~f

The following dimensionless variables and parameters are defined:

r Transforming now initial and boundary conditions we have:

TD=-
rw
Initial Condition:
P
D
= kifh(Pi - Pf(T, t))
OtqB/J

km(~Ct)tT~
"1D = 7-'-77=-'~~
kiJ(~Ct)mh~ Boundary Conditions:
a) External:
Finite reservoir:

The equation (A.9) is obtained in dimensionless form: (A.22)

Infinite reservoir:
2.~ (TD aPD) _ 'YD (aPD)2 _
TD arD aTD aTD- lim U(TD,tD)=O (A.23)
rD-OO

e"D D
P [ 8PD
w 8tD + (1 - W)AJD
jtD 8PD(T)
0
1
8T F("1D, tD - T)dT b)Intemal:

(A.14)
Where pressure gradient at the matrix '\7 !:l.Pum in dimensionless
form is given by the function:
(A.24)
00

F("1D,tD) = 4"1D Le-'1D(2n+l)2,,2 tD (A. IS)

n=o

130 SPE Advanced Technology Series, Vol. 2, No.1

 On the other hand, we assume that U can be expanded in series The zero order problem (Uo) corresponds to the solution of a
of power in 'YD as follows: naturally fractured reservoir that is not stress sensitive.

U = Uo + 'YDUI + 'YD 2 U2 + ... (A.25)

Likewise, we expand I--Y~U obtaining:
-1 -a- ( rD--
auo) = ( 1 auo
) w--+
rD arD arD 1 - 'YDU atD
1
1- "YD U
= 1 + "YD U + 'YD 2U
2
+ "YD 3U 3 + ... (A.26) (A.33)

By substituting (A.25) and (A.26) in equation (A.21) and grouping

in powers of"YD it follows that:
Initial condition:

(A.34)

Boundary conditions:
+ "'D{...L..L..(r ~)-w(~+u,~)
I rD BrD D 8rD 81D 0 B'D

-(l-w)AJD lo'D (8UJ;r) +Uo~ )F(7)D,tD-T)dT }+ (A.27) a) external:

+ ",2 {I 8 ( ~) (U2~
ID ~e.:o rD 8rD -w
8(u, U) +~)
8'D +8iD 08'D 0 1
Finite reservoir without external flow:

-(l-w)AJD lo'D (UJ~+ :r (UOUl)+~)

F(r/D,tD - r)dr} + .. =0 (A.35)

In a similar way for boundary conditions:

Infinite reservoir:

lim Uo
rD-oo
=0 (A.36)

b) Internal:

(A.28)
(A.37)
,1 {CD [(U;~+ 8:D (UoUr)+~)
-S..L..(~)]
8CD 8rD rD=l
-(~)
8rD rD=l
}+-o••• -
auo)
UOw = UOlrD=1 - S ( a;:-
Finite reservoir without external flow: D rD=1

Pedrosa6 showed that in homogeneous reservoirs (non fractured),

(A.29) the first order solution (UI ) is one order of magnitude lower than zero
order. thus UI is of the order of 10-3 • He also stated that when
taking dimensionless production times large enough UI is negligible.
Finite reservoir with external flow:
Then, in our case we calculate Uo and the ratio (A.32) that expresses
the dimensionless pressure in terms of the zero order solution, and it
(Uo + UI + U2 + ........... )(reD, tD) = 0 (A.30) provides us with the required information. For our purpose it is sufficient
Infinite reservoir: to consider:

lim (UO
rD-oo
+ UI + U2 + ...........) = 0 (A.31)
(A.38)
From the expansion of series of U and from relation (A.20) it
follows:

PD = --In(l
1
"YD
2
- 'YDU 'YD U) O - I - •. (A.32)

SPE Advanced Technology Series, Vol. 2, No. I 131

 THE ZERO ORDER PROBLEM External boundaIy:
• Infinite Reservoir
Finite reservoir without external flow:
Equation (A.33) in Laplace space is:

2d2 (Jo d(Jo 2 )"

TD - d
TD
2 + TD-
dTD
- aTDI(a vo = 0 (A.39)
a(JO
- a (TeD, tD)
TD
=0
Where function f(s) determines the type of flow in the matrix that
can take the cases of unsteady-state flow.

I(a) = w + (1 - w)AIDF(71D, a) (AAO) The solution of equation (A.39) for this case is:

Where:

- Va/TID
F(71D, a) = Vc-T:
71D/a tanh --2- (AAI)
(JO(TD, a) =
pseudosteady-state flow: f3K1(f3TeD )lo(f3TD) + 1311 (f3TeD )Ko(f3TD) (AA9)
aLl
I(a) = w(I - w)a + A (AA2)
(I-w)a+A
Initial Condition:
Where:
(JO(TD'O) = 0 (AA3)
Boundary Conditions:
Ll = f3K,(f3TeD)aCD1o(f3) - aCDSf31,(f3)-
a) external: 1311 (f3)! + 13/1 (f3TeD )aCDKo(f3)-
aCDSf3K1(f3) + f3K 1(f3)
lim (Jo
rD-oo
=0 (AA4)

b) internal:
Thus the solution at the well is given by:
- d(Jo 1
aCDUOw - -(1, a) = -
dTD a (AA5)
- - d(Jo
Uow = Uo(I, a) - S-d (1, a)
TD
The solution of equation (A.39) is:
Uow =
K1(f3TeD)lo(f3) + /1 (f3TeD)Ko(f3)+ (A.50)
(JO(TD, a) = Alo(TDval(a))+ Sf3[K1(f3 TeD )11 (13) - 11 (f3 TeD )K1 (13)]
(AA6)
BKo(TDVal(a)) a61

Where A, B are constants and K ,I are the modified Bessel func-

tions of zero order or second and first class, respectively. Then, pressure
in a infinite naturally fractured reservoir, including storage and damage with Lll = 1
effect is given by:
The zero order solutions, both for the finite or infinite reservoir
correspond exactly to the case of the naturally fractured reservoir with-
(JO = KO(TDf3) out stress sensitivity, with effects of damage and storage, besides being
(AA7)
a{f3K1(f3) + aCDKo(f3) + Sf3K1(f3)} able to consider two different types of flow at the matrix. These solu-
tions obtained in Laplace space are numerically reversed to real space
where: 13 = Val(a) . through Stehfest" reversal algorithm. On the other hand, the combined
By substituting (A.47) in (A.45) the dimensionless pressure of the effects of storage, damage and stress sensitivity of the well cannot be
well is obtained: dissociated since these mechanisms considerably affect pressure at short
and intermediate times. All these effects are studied by the formula:
(JOw =
Ko(f3) + Sf3K1 (13) (AA8)
a{f3K1(13) + sCDKo(f3) + Sf3K1 (f3)}
(A.5I)

• Finite Reservoir
Equation (A.39) is solved for the finite case with the initial condi-
tion, the internal boundaIy conditions and f(s) exactly defined as in the where L -1 is Laplace reverse transform, that in our case is nu-
case of the infinite reservoir varying only the external boundaIy. merically treated8 •

132 SPE Advanced Technology Series, Vol. 2, No.1

 SI Metric Conversion Factor.
bbl x 1.589 873 E-Ol = m3
psi x 6.894 757 E 00 = kPa
cp x 1.000 000 E-03 = Pa.s
Cll ft x 2.831 685 E-02 = m
3

ft x 3.048 000 E-Ol = m

in x 2.540 000 E+Ol = mm
sq ft x 9.290 304 E-02 = m'1.

Author.

Valmor. Cell. is a research mathematician at INTEVEP, S.A.

He works at the Geophysics Section. He received a BS degree
in mathematics from the "Universidad Central de Venezuela"
Caracas, Venezuela"
Ramon Silva is head of reservoir simulation section at Intevep,
SA Before moved to Intevep, he worked at Maraven SA as
senior reservoir simulation engineer. He had an assignment
in KoninklijkelSheil Exploratie en Produktie Laboratorium in
Rijswijk,The Netherlands in the Recovery Processes Department.
Silva holds a BS degree in Petroleum Engineering from the
"Universidad Central de Venezuela" Caracas, Venezuela"
Miguel Ramon •• is a senior research engineer at INTEVEP, S.A
were he works at the reservoir simulation section. He received
a Petroleum Engineering degree from the "Universidad Central
de Venezuela" Caracas, Venezuela"
Jaime Guerra is a research associate at the Universidad
Central de Venezuela" , Opt. of Applied mathematics, Science
Faculty. He earned MS degree in Warsaw University and
PHD degree in Uppsala University in Scientific Computing
Department.
Giovanni Da Prat is a Division Reservoir Engineer of
Schlumberger-Argentina. He holds a MS degree in Geo-
physics and a PHD in Petroleum Engineering, both from Stanford
University. He is the author of the book "Well Test Analysis for
Fractured Reservoirs Evaluation"(ELSEVIER) 1990.

(SPE 23668)
SPE Advanced Technology Series, Vol. 2, No. I 133
 ·0' New model 150
Pedrosa

'20

W.O 001
.0 A. s 10·r

..
o >d'
B? Co - 2000
5 • 0
60

30

00
4T.
.,' .0'

Fig. 1 - Comparison of analytical solution with Pedrosa analytical Fig. 2 - Effect of wellbore storage and stress sensitivity on pressure-
solution. buildup behavior for a naturally fractured reservoir.

I~ 0
2' 0 Yo ~ 007
Yo. 0.1
W' 0.001
120 W·O.OOI A • 10-1
200
A • 10-1
s· 0 )"D:OO~
'0
160
<>-0
rE' ro ~OO3
60
)[,:001
'20
Yo- 0

)0
80

00
10' 10' 10' .0' 10' '0
10 0 \0' 10
10
D. To
To

Fig. 3 - Type-curves for well bore storage and stress sensitivity for a Fig. 4 - Buildup pressure behavior for stress-sensitive formation for
naturally fractured reservoir. naturally fractured reservoir.

134 SPE Advanced Technology Series, Vol. 2, No. I

 Fig. 5 - Effect of stress sensitivity on pressure buildup for Fig. 6 - Effects of permeability modulus on pressure behavior of closed
naturully fractured reservoir. naturally fractured reservoirs.

Il

Yo' 0 t
w -001
A '10"

Fig. 7 - Effects of finite reservoir size on pressure behavior of (SPE 23668)

stress-sensitive naturally fractured reservoir.

SPE Advanced Technology Series, VoL 2, No.1 135