11. SPE-18172-MS OK
of 16
Transcript of 11. SPE-18172-MS OK
-
8/9/2019 11. SPE-18172-MS OK
1/16
SP
Society of PetroIeun
Engineers
SPE 8 72
Pressure
Transient
nalysis of Wells With Finite
Conductivity
Vertical
Fractures
in
Double Porosity
Reservoirs
by
H
Cinco-Ley,· PEMEX and University of Mexico and H.-Z. Meng,· Dowell Schlumberger
MemberSPE
Copyright 1988. Society of Petroleum Engineers
This paper was prepared for presentation at the 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in
Houston. TX. October 2-5. 1988.
This paper was selected for presentation by an SPE Program Committee following review of information contained in
an
abstract submitted by the
author s). Contents of the paper. as presented. have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the
author s). The material. as presented. does not necessarily reflect any position of the Society of Petroleum Engineers. its officers. or members. Papers
presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is
restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of
where and by whom the paper is presented. Write Publications Manager. SPE. P.O. Box 833836. Richardson. TX 75083·3836. Telex. 730989 SPEDAL.
ABSTRACT
This paper
presents
the
results
obtained in
the
study
of the
transient behavior of a well intersected by a finite conductivity
vertical fracture in a double porosity reservoir. Two models are
considered to
take
into account the fluid
transfer
between matrix
blocks and fractures: the pseudo-steady-state matrix flow model
and the transient
matrix
flow model.
A general semianalytical model and simplified fully
analyt
ical models are presented. t is demonstrated that these systems
exhibit the basic behavior of a well with a finite conductivity
fracture: that is bilinear flow, pseudolinear flow and pseudoradial
flow
in addition
to
the transition
flow periods.
Each of
thes e flow
periods is
under
the influence of the different states of
the
fluid
transfer between matrix and fractures;
that
is
fracture dominated
period, transition period
and total
system dominated period.
t
is shown that correlating
parameters are the
dimension
less
fracture
conductivity kfbf
)v,
the
fracture
storativity coeffi
cient wand the interporosity flow parameter Af
or
the dimension
less matrix hydraulic diffusivity 1)maD).
t
was found, for the
transient
matrix flow model,
that the
pressure behavior exhibits 1/8 slope in a log-log graph during the
bilinear flow dominated by the trans ition period of the fluid trans
fer. Hence a graph of pressure versus t
l
/
S
yields a straight line
passing
through the
origin.
During
the pseudolinear flow, and if the fluid transfer is in
the
transition period, a log-log graph of
the
prerssure versus
time
exhibits 1/4 slope straight line. This means that a graph of p
versus t
l
/
4
yields a
straight
line. Hence it is concluded that bilin
ear flow is not the only type of flow that exhibits the one quarter
slope type of behavior.
References and illustrations at
end
of
paper.
645
Type
curves
are
presented
to
analyze data falling in
the
bi
linear - pseudolinea r flow regions. The effect of wellbore storage
are also included. The general semianalytical models yields simul
taneous the
constant
flow rate and
the
constant pressure solutions
as well as the pressure derivative function for the constant rate
case.
INTRODUCTION
In recent years interest has been growing for the evalua
tion
of hydraulically
fractured
wells producing in double porosity
reservoirs.
Although
considerable efforts have been dedicated
to
study
either fractured
wells in homogeneous reservoirs
or
wells
producing in double porosity
reservoirs,there is not
a complete
study that includes fractures in double porosity reservoirs.
t has been shown in the
pastl
that well intersected by
fi-
nite conductivity vertical fracture in a homogeneous reservoir can
exhibit several flow periods: bilinear, pseudolinear and pseudo
radial
in
addition to
the
transition
between
them.
For each of
these flow periods there is a specific graph of interpretation that
produces a
straight
line
portion
for
the
pressure
data. The
pres
sure data
within the
bilinear flow, the pseudolinear flow and
the
pseudoradial flow exhibit a straight line in a graph of pressure
versus tl/4,
t
l
/
2
and log t respectively. The complete behavior
of the system
can
be
correlated by a
parameter
called dimen
sionless
fracture
conductivity kfbf)v. In order to identify the
different flow regimes type curves have been presented in
terms
of
pressure
l
,2
or
pressure derivative
3
•
4
•
On the other hand, the studies on the behavior of wells
in double
porosity
reservoirs have considered two kinds of models
for the matrix-fracture fluid transfe r: the pseudo- steady-s tate flow
model Warren and Root
5
) and the
transient
matrix flow model
de Swaan
6
and
KazemF). t
has been established that both
models exhibit
three
well-defined
flow
periods: at early time,
the
behavior is dominated by the storativity o f the natural fractures,
at
intermediate time
the fluid
transfer
from matrix to fractures
-
8/9/2019 11. SPE-18172-MS OK
2/16
2
PRESSURE
TRANSIENT
ANALYSIS OF ,WELLS WITH FINITE
CONDUCTIVITY
VERTICAL
FRACTURES IN DOUBLE POROSITY RESERVOIRS SPE 18172
becomes more
important and
the pressure in the network of natu
ral fractures
tends
to stabilize originating the so called transition
period
and
at long time the system is dominated by the storage
capacity of the total system. The behavior of a double
porosity
system can be correlated by
the
fracture storage capacity param
eter w and the
interporosity
flow parameter
or
dimensionless
matrix hydraulic diffusivity l1maD).
For the case of an
unfractured
well in a double porosity
reservoir a semilog
graph
shows two parallel
straight
lines repre
senting the
natural
fracture dominated
flow
period
and
the
total
system dominated
flow period. In between these parallel
straight
lines there is a transition period sometimes called stabilization
period; here the pressure curve is represented by a rather flat line
for the
pseudo-steady-state
matrix flow model and by a straight
line whose slope is
half
the slope of the parallel straight lines for
the transient
matrix
flow model.
Several type curves have been presented to interpret tran
sient pressure data of naturally fractured reservoirs
8
9 .
These
type curves are expressed in terms of pressure or pressure deriva
tive. Recently Houze et al
10
developed a model to study the
behavior of wells intersected by infinite conductivity vertical frac
tures in
double porosity reservoirs.
They
presented
type
curves
that can be used to estimate w, ~ f x J and the formation flow
capacity kh if a large span of pressure data is available. Their
models are semianalytical and consider
the
pseudo-steady-state
type
of
matrix-fracture fluid
transfer
and
both
the uniform flux
fracture and the infinite conductivity fracture.
Later Lancaster
and
Gatens
11 presented a methodology for
analyzing pressure data for fractured wells in double porosity
reservoirs.
Their
work is
based on the
use
of the type
curves
developed by Houze et al. Ben-Naceur and Economides
12
utilized
a finite difference model to simulate the behavior of a naturally
fractured
reservoir
intercepted by
a vertical fracture
of
finite con
ductivity. Type curves for pressure draw down and cumulative
production are
provided
and the
effect
of formation
anisotropy
is
discussed.
The objective
of
this work is to provide
an
analytical solu
tion for the behavior of a well
intercepted
by a finite conductivity
vertical
fracture
in a double porosity reservoir.
Two
models for the
matrix-fracture fluid
transfer
are considered, the pseudo-steady
state matrix flow model and the
transient
matrix flow model.
In
this paper
we
first present simplified fully
analytical
solutions for
both
the bilinear
and
the pseudolinear flow periods. We then
present a general semianalytical model for the behavior of
the
system including the effect
of
wellbore storage. Finally we discuss
the
application
of the specific
graph
of analysis for pressure
data
in this type of systems.
SIMPLIFIED
ANALYTICAL
MODELS
Basic
Flow Equations
Let us consider a well intersected by a finite conductivity
vertical fracture producing in a double porosity reservoir, as in
dicated
in
Figure 1. The fracture has
a half length xJ, a
width
b a permeability kJ'
and
penetrates the formation vertically.
The reservoir is a homogeneous double porosity system of bulk
permeability kJb' thickness h and of infinite lateral
extent.
The
reservoir is composed of a fracture network
and
matrix blocks; the
fracture
network possesses a
bulk fracture
porosity 0
Initial Condition
P/ lv(xv,
tv = 0) =
0
4)
where the dimensionless variables are defined as follows:
Dimensionless wellbore pressure drop
PaV = kJbh(Pi -_Pa(t)] 5)
o.qBJL
for hydraulic
fracture
a =
fl,
for fracture network a = f2,
Dimensionless time
tv , = f k
Jb
t
(6)
,
-
8/9/2019 11. SPE-18172-MS OK
3/16
SPE
18172
H. CINCO-LEY AND H.-Z. MENG
Dimensionless
fracture
conductivity
7)
Dimensionless coordinates
8)
9)
Reservoir Linear
Flow Model
As mentioned before, the reservoir is represented
by
a frac
ture network and by
matrix
blocks. I t is assumed
that
the char
acteristics
of both
the fracture network
and matrix
blocks do not
vary throughout the reservoir. The flow from the reservoir into
the hydraulic fracture occurs through the fracture network only,
as generally considered in the past for double
porosity
reservoirs.
At early
time
the flow in the neighborhood
of
the
fracture
is essential linear as indicated in Figure 4. The transient flow in
the
double porosity reservoir
can
be described by:
Transient
Matrix
Flow
for -1 ::; ZD ::; 1
and
tD 'f > 0
Boundary Conditions
Initial Condition
Pseudo-Steady-State
Matrix
Flow
and
1 -
w OPmaD = -A/PmaD
- Pj2D)
otD ' f
Initial conditions
for 00
<
ZD
<
00
Pj2D(ZD, YD, tD 'f =
0)
=
0
PmaD(tD 'f = 0) = 0
and 00 < YD < 00
10)
11)
12)
13)
14)
15)
16)
17)
647
Boundary conditions
for 1
::; ZD
::;
1
and
tD 'f
> 0
lim Pf2D(ZD,
YD,
tD 'f)
= 0
z Do rY D O O
for
tD 'f
> 0
where the dimensionless
parameters are
defined as:
Dimensionless wellbore pressure drop
for fracture network a = 12,
for rock
matrix a
= rna
Dimensionless
fracture
storativity
w
=
¢>jb
C
t j2
_ ¢>jb
C
t j2
¢>/b
C
tf2 ¢>maCtma - ¢>ct}t
Dimensionless
fracture
network
area
A
AjbhmaVb
j = = Ajmahma
ma
Dimensionless
matrix
hydraulic diffusivity
Interporosity
flow coefficient
Dimensionless fracture flux
18)
19)
20)
21)
22)
23)
24)
25)
The use of these dimensionless variables allows
to
present
the solution for the flow equations in a simple general form. A/D
and l1maD are used only in the transient
matrix
flow model
and ..\t
is a parameter of the pseudo-steady-state matrix
flow
model.
The pressure behavior of the
system
is
obtained
by solv
ing simultaneously the partial differential equations 1 and 10 for
transient
matrix
flow
model and equations 1, 14 and 15 for pseudo
steady-state
matrix flow model
with
initial and
boundary
condi
tions as described previously. The Laplace Transform is used
to
derive
the
solution.
The
wellbore pressure solution for
both the
transient
matrix
flow model and the pseudo-steady-state matrix
flow
model
can be
expressed
in
Laplace space as
26)
where
I s
is given by:
Transient matrix flow model
27)
-
8/9/2019 11. SPE-18172-MS OK
4/16
4
PRESSURE TRANSIENT ANALYSIS OF WELLS
WITH FINITE
CONDUCTIVITY
VERTICAL
FRACTURES
IN DOUBLE
POROSITY
RESERVOIRS
SPE
18172
Pseudo-steady-state
matrix flow
model
f(
)
(1 - 1 1 )Af
II = 1 1 --'--"--'-
l -W) I I+Af
(28)
Equation 26
has
the
same form as
the
solution presented by
Cinco
and
Samaniego
13
for
fractured wells
in
homogeneous reser
voirs;
in
that
case
sf(s)
=
s.
The
solution
in the
real space can be calculated by using
Stehfest
14
Numerical Laplace inverter. Now
let's
examine
the
early
and
long time behavior
of the
solution.
Fracture Storage
Dominated
Flow Period
At
small values
of
time,
the
fluid
production
is due
to the
expansion
of the
fracture network;
that
means
that the
contribu
tion
of
the
matrix
is negligible. This
flow
period in double porosity
system has been called
fracture
storage dominated
flow
period .
For this case:
(29)
Hence:
(30)
By comparing this equation
to the equation
presented by
Cinco
and
Samaniego
we
find
that
a
fractured
well in a double
porosity reservoir behaves
at
early time as a fractured well
in
a
homogeneous reservoir with the fracture network properties. As
a consequence,
the
pressure behavior might exhibit the bilinear
and the
pseudolinear
flow
periods.
Early
Time
For the bilinear
flow
equation 30 becomes:
(31)
whose inversion is:
(32)
Intermediate Time
For
the
pseudolinear flow equation
30
becomes
11 11
P
D
-
w - 2s3/2w1/2 3(k/b/
DS
(33)
which yields:
(34)
Transition Flow
Period
After the fracture network dominated period,
and if 1 1
is
very small, there is a period when
flow in the matrix
is essentially
linear
and
the well production is caused by the expansion in the
matrix. Under these conditions:
648
For the transient matrix flow model, we have
sf(s)
= A/Dv'TfmaDS
Hence, equation
26
gives:
For bi linear flow,
11
PwD
= r = = = = = = ~ ~ ~ ~ ~
J2(k/b/
D A ~ ~ T f ~ : D S 9 / 8
and
inversion
of
this equation yields
11 1/8
P D
-
t
w -
r( ) l2 k
b )1/2At/4 1/8 D 'I
S V.to / / D
/DTfmaD
For pseudolinear flow,
11 11
PwD
= 2(A/Dv'TfmaD)1/2
S
6/4 3(k/b/)DS
this gives
1/4
11 t
D
, 11
PwD=
I
r ~ ) 2 A 1 / 2 , . , 1 / 4 3(k/b/)D
4 /D maD
For
the pseudo-steady-state matrix flow
model
S f S ) ~ A /
and
the
wellbore pressure
PwD
is:
Inversion
of
this equation yields:
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
We can observe that according to equation 43 the wellbore pres
sure is independent
of
time. This equation represents the limiting
value of pressure during the transition between fracture network
dominated period
and
the period dominated by the
total
system
(matrix
fracture).
Total System Compressibility Dominated Flow Period
At large values
of time the
compressibility
of
the
total
sys
tem
dominates the pressure behavior
of
the reservoir, under these
conditions
s f s )=s (44)
and
equation
26
becomes
(45)
-
8/9/2019 11. SPE-18172-MS OK
5/16
SPE
18172
H. CINCO-LEY AND H.-Z.
MENG
As mentioned before for
the
fracture network dominated period,
this equation includes both the bilinear
and
the pseudolinear flow
periods.
For
the
bilinear
flow
behavior
46)
which yields:
47)
For
the
pseudolinear
flow
behavior
48)
which gives
49)
Note
that
these equations
are
applicable for
both
the
transient
matrix flow
model
and
pseudo-steady-state
matrix flow
model.
GENER L SEMI N LYTIC L
MODEL
Let us consider a well intersected by a finite conductivity
vertical fracture
in
a homogeneous, isotropic reservoir. According
to
Cinco-Ley et all the fracture pressure behavior for constant
rate
conditions
in
dimensionless form is given by:
50)
where q D is the dimensionless flow rate per unit of fracture length
going from the formation into the fracture.
The
pressure drop between any point within
the
fracture
and the
wellbore can
be
expressed as:
51)
This equation is derived by assuming incompressible
flow
within
the
fracture Cinco-Ley
et all
have shown
that
this as
sumption is valid for practical values of dimensionless time). A
combination of equations 50
and
5 yields:
52)
6 9
Gringarten et
l have shown
that the
solution for double
porosity reservoirs can be obtained in Laplace space by simply
replacing
the
variable
s
by 8/(S). f s) is defined
in
equations 27
and 28
for
transient matrix flow and
pseudo-steady-state
matrix
flow
respectively. Application
of the
Laplace transformation to
equation
52
and
replacement
of s
by
s/(s)
gives
111, ,
r:;;-:;. ,
PWD S)
- - qfD(Z
,S)KO ZD
-
Z
)ys/(s)dz
4 1
53)
The
fracture is symmetric
and
homogeneous; hence
qj
D
ZD,
s
= qjD(-ZD,S)
and
equation 53 becomes:
1
, , r:;;-:;.
PWD S) - - qfD(Z ,S)[KO ZD - Z )ys/(s)
2 0
,
+KO ZD
+z h/s/(s)]dz +
(kj:j)D f D f qjD(z ,s)dz dz'
54
This equation can be solved
if we
discretize
the
fracture
half
length)
in n
segments
of
uniform flux.
The
first
integral of
the left
hand
side of equation 54 can be approximated as: .
Zi
and
Zi+1
are the
beginning
and the
end of
the
ith
segment.
The
second integral
of equation
54 can
be
expressed as:
56)
This equation assumes segments
of
equal length
and
con
siders ZDj
to be
located
at
the
j th
segment.
Substitution of
equa
tions 55
and
56
into
54 gives:
I f we
write this
equation
for every fracture segment
we
ob
tain
a system
of
n equations
with
n + 1) unknowns
(qjDi
s), i =
1, n, and
PwD
s)). One additional equation results if V:e recall
that the
flow
entering the fracture is equal
to
the well flow rate;
that
is
58)
-
8/9/2019 11. SPE-18172-MS OK
6/16
6
PRESSURE
TRANSIENT ANALYSIS
OF
WELLS
WITH FINITE
CONDUCTIVITY
VERTICAL
FRACTURES IN DOUBLE POROSITY RESERVOIRS
SPE 18172
The
unknowns are found by solving
the
system
of
equations
59)
The formulation of this model has several advantages over
the semianalytical
model
used in the past
1
.
First, it avoids lengthy
calculations because it takes into account superposition of time ef-
fects in Laplace space; second,
the
constant pressure solution can
be calculated directly from the
constant
rate solution and third,
wellbore storage effects can be added into the constant
rate
soul
tion with minimum of computation. A more detailed description
and
the application
of
this
general
purpose
semianalytical
model
will be presented
in
a subsequent paperl5.
GENERAL DISCUSSION ON SOLUTIONS
After analyzing
the
equations derived in the previous sec
tions we
can
study the pressure behavior of a
fractured
well in a
double porosity reservoir according to Table 2. Here, it is indi
cated that the behavior of this system is the result of a combina
tion
of the
behavior
of
a fractured well in a homogeneous reservoir
and the behavior of a double porosity reservoir.
It has been established that a fractured well in a homo
geneous reservoir might exhibit the following flow periods:
a)
bilinear flow, b) transition, c) pseud olinear flow, d) transition
and e) pseudoradial flow. On
the
other hand, a double porosity
reservoir exhibits three flow periods: a) fracture dominated flow,
b) transition
and
c) flow dominated by the storativity
of
the
total system
fracture
+ matrix).
Table 2 indicates that a fractured well in a double porosity
reservoir
can
exhibit fifteen different flow periods.
In
the follow
ing we discuss the first nine flow periods; that is, those falling in
the
bilinear, transition
and
pseudolinear flow regions.
Cinco
and
Samanieg0
2
showed
that
a
graph
ofPwD(kfbf)D
versus tD"" (k b f)h yields a single curve for the bilinear and pseu
dolinear flows as shown in Figure 5. They also showed
that
the
transition between bilinear and pseudolinear flows l asts about one
log cycle.
At early
time
the pressure behavior of the
system
is dom
inated by the storage capacity of the
natural
fracture network;
that
is, the expansion in the system is provided by the fractures
only. The pressure
at
the wellbore during this flow period is given
by equation
30
for both the transient
matrix
flow
and
the pseudo
steady-state matrix flow.
This
equation is similar to the equation
representing the pressure behavior of a fractured well in a homo
geneous system. Here
the
dimensionless
time
is affected by
the
parameter
W; that
is;
[PwD(tD",,/W,
(kfb
f
)Dlf.actu.e netwo.k dominated
= [PwD(tD"", (k
f
b
f
)Dlhomogeneou6 (60)
Figure 6 presents a log-log graph of
PWD(kfbf)D
versus
tD ,,(kfbf)h
for different values of
w.
We see
that
the curves
for different values
of
ware displaced towards smaller values of
dimensionless time. The case for w = 1 represents a fractured well
in a homogeneous reservoir see equation 45). During the fracture
network dominated flow period, the wellbore pressure behavior
follows one
of
the curves according
to
the value
of w and
during
the period dominated by the total system the wellbore pressure
follows the curve for w
= 1.
65
Three well defined regions form the total behavior of the
system: bilinear flow region, transition flow region
and
pseudo
linear flow region. They are separated by horizontal lines, as
indicated
in
Figure
6. The lower region is
dominated
by bilinear
flow; the intermediate region corresponds
to
the transition flow
and the
upper region represents the pseudolinear flow.
The
transition flow period for the double porosity behavior
has
an asymptotic
behavior represented by
the
case for
w
= 0
for
any value
of
the interporosity flow parameter (1JmaD for the tran
sient
matrix
flow model and \for the pseudo-steady-state matrix
flow model).
Figure 7 represents
the
asymptotic behavior
of
a fractured
well duri ng the transitional regions
fracture
- fracture
+
matrix)
for the transient
matrix
flow model. Here, the correlating param
eter is
1JmaD
/ (k b f )h according to Appendix A. Figure 8 presents
the behavior during the transition period for the pseudo-steady
state matrix flow model; here correlating parameter is V(kfb
f
)h.
The pressure behavior solutions for
both
matrix flow mod
els are
bound
by
the
curves
in
Figures 6,7
and
8.
That
is, the
wellbore pressure solution for
the transient matrix
flow model is
given by a combination
of Figures 6 and 7 as shown in Figure 9.
On the other hand, the solutions for the pseudo-steady-state ma
trix flow is given by a combination of Figures 6 and 8 as presented
in
Figure
10.
As pointed out before, Figures 9 and 10 represent asymp
totic behavior of the system. This becomes evident in Figure
11 where data points
are
graphed for 71maD/(k
f
b
f
)h
=
10 -
4
and
w = 10 -
3
,10-
2
,10-
1
for the transient matrix flow model. A simi
lar graph
is presented
in
Figure
12
for the pseudo-steady-state ma
trix flow model for V(kfbf)h = 10 -
2
and w = 10 -
3
,10-
2
,10-
1
•
Figures 9 and
10
can be used as type curves to analyze a
pressure
test
whenever a large
span of
pressure
data
is available.
These type curves can be simplified if we assunle
that
the pres
sure behavior is dominated by only one flow region bilinear or
pseudolinear ).
For bilinear flow
and
transient
matrix
flow,
it
is shown
in Appendix A that, a graph of PwD(kfb
f
) ~ 2 7 1 : : D and tD"" •
,
(k
b
)1/2 1/4 ·
f
· ·
f
PwD f f D 7 maD versus 7 maD D"" gives
anu
les 0 curves pres-
sure
and
pressure-derivative ratio) for different values of
w
as
shown in
Figure
13. This
type
of
graph
provides a single line
for the stabilization period reducing the uniqueness problem in
type curve matching. For
the
case of pseudo-steady-state
matrix
flow a similar graph can be obtained if we graph PWD(kfb
f
) ~ 2
AV
and
tD""
. ~ D k f b f ) ~ 2
A 1/4 versus ).,tD"" as a function
of w
Fig
ure 14).
f
he
data are within the pseudolinear flow region the unique
ness problem
in
type curve matching can be
reduced,
for transient
matrix
flow model, if curves are presented in a graph of PwD1J: ;D
, 1/2
and tD"" .
PwD1J
ma
D
versus 1JmaDtD",,/W Appendix A). Two fam-
ilies of curves appear in this graph, one representing different val-
ues ofw and the other different values of the group 71: ;D/(k
f
b
f
)D
Figure 15). For
the
pseudo-steady-state
matrix
flow the corre
sponding graph for the pseudolinear flow is in terms of PwDA
I
/
2
d
1/2 f
an tD"" ·PWDAf versus AID"" /w and the correlating parameters
for the families of curves are w and ) . ~ / 2 / k f b f ) D Figure 16).
-
8/9/2019 11. SPE-18172-MS OK
7/16
SPE 18172
H.
CINCO-LEY
AND H.-Z. MENG
METHOD OF
ANALYSIS
Speciflc Graph
of
Analysis
In
the preceding sections,
we
see
that
the behavior
of
some
flow periods
can
be represented by rather simple equations. Such
are
the cases for flow periods falling
in
both the bilinear and the
pseudolinear flow regions. These flow periods
are
the cases num
ber
1, 2, 3, 7, 8
and
9
in
Table 2. Following we discuss
the
application
of the specific graph of analysis for pressure
transient
test
data in this type
of
systems.
Bilinear Flow Analysis
According to equations 32
and
47 the wellbore pressure
varies with
the
fourth root
of
time
at
early and long times under
constant
rate conditions for both matrix flow models.
At
early
time
the wellbore pressure pseudo-pressure) is given by:
Oil:
61)
Gas:
62)
Oil:
63)
Gas:
Two straight lines can be shown in a bilinear flow graph
( P vs. t
1
4
;
the first one represents
the
period
dominated
by
the natural
fracture
network and the second one represents the
period dominated by the
total
storage capacity of the reservoir.
Both straight lines intersect the origin as indicated in Figure 17.
The conductivity ofthe hydraulic fracture is calculated from
the slope of
either
straight line as shown by Cinco and Samaniego
2
for a homogeneous reservoir. The parameter
W
can be estimated
if the two
straight
lines
are
present; w is given by:
65)
t must
be kept in mind that this analysis is valid when
data fall on the bilinear flow region.
Trilinear Flow Analysis
This flow period takes place in
the
bilinear flow region when
most
of the expansion is provided by the matrix
under
linear flow
conditions. That is, in this type
of
flow matrix linear flow is su
perimposed to
the
bilinear flow in
the
hydraulic fracture. We call
651
this Trilinear Flow period. The solution in terms of dimen
sionless variables is given by
equation
38. For real variables
Oil:
66)
Gas:
During this
trilinear flow period , the
wellbore pres
sure exhibits 1/8 slope in a log-log graph as shown in
Figure
13.
Hence a graph of pressure versus t
1
8
yields a straight line pass
ing through the origin as shown in Figure 18. The slope of the
straight
line allows the
computation
of
f b ~
as follows:
Oil:
68)
Gas:
A
r ;
= btlfgqT 4
fbV 'rna / 1/4 /
mtl lh(kfb,) l
2k
'b
«CdtJl)l 8
69)
This parameter appears to be important in imbibition
rate
calculations.
Equations
66 and 67 assume that
w
is very small;
that is
Ct}t Cdrna.
In order
to
apply equations 68 and 69
both
the fracture
conductivity and the
formation
bulk permeability must be known.
If both
the
trilinear
flow and the final bilinear flow
are
present in
a test we can
compute f b ~
from:
A
r ;
_ btlfo· mbf
fbV ' rna-c
Ublo . mtlf
Pseudolinear Flow Analysis
70)
As explained before,
there are
two flow periods
with
pres
sure being a function of the square
root
of
time
within the pseudo
linear flow region.
During
the period dominated by the storativity
of
the
natural
fracture
network
the
pressure drop for
constant
rate
flow is given by:
Oil:
Gas:
bplfoqBJt r. b
A.Pwl = -h- k (-1.-)
1/2
v t + plIo
:r
f
IbJlW pCt
t
)
- bpllgqT i b
A.m Pwl - h
(k
(-I.)
1/2
t + plfg
:r I IbJlW pCt t
71)
72)
where bpll
o
, bpll
g
are unit
conversion factors see Table 1) and bpl
f
is defined as:
Oil:
73)
-
8/9/2019 11. SPE-18172-MS OK
8/16
8
PRESSURE TRANSIENT
ANALYSIS
OF
WELLS
WITH FINITE CONDUCTIVITY
VERTICAL FRACTURES
IN DOUBLE
POROSITY
RESERVOIRS
SPE
18172
Gas:
qT xf
bplfg
=
cp l fghk fb f
(74)
where
cplfo
and
Cplfg are unit
conversion constants (see Table 1).
At
large values of
time
the compressibility of
the
total
system
af
fects
the
flow, hence:
Oil:
(75)
Gas:
(76)
According
to these equations
a
graph of :1p
(or :1m(p)) ver
sus
t
1
/
2
gives two
straight
lines of slope mplfl>
m
p
lf2 having
the
same intercept to
the origin as
shown
in
Figure 19.
Both the hydraulic fracture conductivity and
length
can
be
estimated from the
slope and intercept
as
discussed
by
Cinco-Ley
el ap3,.
The
can
be
estimated.from:
=
(rn,plf2)2
mplf1
(77)
t
should
be
pointed
out
that
this type of
analysis apply to
both matrix flow models.
Formation Bilinear
Flow Analysis
This
flow
period
occurs
in the
pseudo linear flow region
when
most of the expansion is provided by the
matrix under
linear
flow
condition. Here
the
matrix linear flow is superimposed to
the
linear flow
in
the reservoir. We call this Formation
Bilinear
Flow period.
According
to
equation 40
the
wellbore pressure is
given
in terms of
real
variables by:
Oil:
:1p
- 5
fbfo
qB
V t +
b
8)
wf - h
A 1/2k1/2k1/4(A.
)1/4 plfo
7
x
f fb fb rna
'l'JLCt
Gas:
A )
5 bfgqT
t ;
~ Pwf = 1/2 1[2 1/4 V
t
+
bplfg
hZfAfb
k
fb
krna(4)JLCd
1
/
4
(79)
where
5fbfo, 5
f
bJg
are unit
conversion constants (see Table 1).
Equations
78
and
79
indicate
that a
graph of
pressure
data
versus
the fourth root of time during the formation bilinear
flow period
gives a
straight
line and intercept as shown
in
Fig
ure 20. Also a log-log graph of the pressure versus time should
exhibit 1/4 slope
straight
line
as
shown
in Figure
15.
Note
that
this
type
of
pressure behavior
occurs during
the
pseudoline ar flow
dominated by
the
transition
period of the fluid transferror
10 -
2
as
shown
in Figure
15 for
the transient
matrix flow model.
Note that the
product f b ~
can
be estimated
from the
slope and the conductivity of the fracture can
be
calculated from
the intercept.
65
Pseudoradial
Flow Analysis
At
large values of time, both matrix flow models exhihit
the
so called pseudoradial flow after a
transition
period.
The
pressure varies following
the radial
flow equations based on
the
effective wellbore
radius
2
• The semianalytical
model discussed
in
previous section
can be
used
to compute the
pressure
behavior
for
the
pseudoradial flow period and
the transition
flow period.
Nothing more
will
be
said
about the
pseudoradial
flow since
this
has been
covered extensively
in the literature.
Flow Regime Identification and Type Curve Matching
Although the application of any specific
graph
of analysis to
pressure data
from
a
fractured
well
in
a double
porosity
reservoir
is straight forward, a careful flow regime identification process
must be the
first
step
in
interpretation.
The
use
of
log-log
graph
of pressure
drop and
pressure
derivative function
t :1p'
is recommended for flow regime diag
nosis. Figures 13 through 16 can be used for type curve matching
process for
the
bilinear and pseudolinear flow regions if a large
span of
data
is available.
t
should
be pointed out
that
the
parallel
straight
line fea
ture
that characterizes double porosity reservoirs in a semilog
graph
for
radial
flow is also present in a log-log
graph
for both
bilinear and pseudolinear flow.
The
log-log
graph
also shows a
straight
line
during the
sta
bilization
period
for
the transient matrix
flow model;
the
slope
of this line is one half of
the
parallel
straight
lines (trilinear and
formation
bilinear flows).
For bilinear flow, a log-log
graph
shows parallel
straight
lines
of one quarter
slope for both
the
pressure
drop :1p and the
pressure derivative function
t :1p'
as shown
in
Figure
21. For
the
pseudolinear flow,
although the t :1p'
curve shows one half slope
parallel
straight
lines as shown
in
Figure 22,
the :1p
curve does
not necessarily shows this feature unless
the
well is intersected by
a highly conductive hydraulic fracture.
Recent studies
16
,17 have shown
that
the
use of
the
function
:1p/t :1p'
simplifies
the type
curve matching technique when ana
lyzing pressure data for homogeneous reservoirs. t
appears
that
this function is dimensionless and fixes
the
vertical matching in
such a way
that
it is only necessary to move
the
tracing
paper
with data
in
the horizontal direction (time
axis).
Figure
23
shows a new
set of
pressure-derivative
ratio
type
curves for bilinear flow
behavior of
a
fractured
well
in
a dou
ble porosity reservoir including
the
wellbore storage effect. Here
a log-log
graph
of PwV/(tV:r:f • versus
TlrnaVtV:r:f
for =
10-1,
10 -
2
,
10 -
3
,
10 -
4
and CVfTliaV/ J kfb
f
)v
=
10 -
7
,10-
6
,
10-
6
, 10-4, 10 -
3
, 10 -
2
, 10-
1
is presented for the transient ma
trix flow model. Two families of curves define
the
behavior
of the
system. The wellbore storage effects
are
characterized by curves
with the parameter CVfTl iav/
J kfb
f
)v.
The unit
slope log-log
straight
line is represented
in
this
graph
by a horizontal line of
P w V / t V : r : f P ~ V ) equal to 1.
-
8/9/2019 11. SPE-18172-MS OK
9/16
SPE
18172 H. CINCO-LEY AND H.-Z. MENG
For the purpose
of
demonstration,
a set of pressure data
points
are graphed
for
w = 10-3
and CDfT/,tad/ J(kfb
f
)D
=
10 -
6
. for the system under consideration. At the early time the pres
sure beh avior follows one
of
the wellbore storage curves as shown
in
Figure
23.
At
the
intermediate time the
pressure behavior fol
lows one of the fracture-matrix transition curves indicated by w.
A trilinear flow" period exits for this case
and is
indicated by a
horizontal line with a PwD/(tDI1J
.
~ D ) value of 8. At large time
the pressure data follows the total
system
curve for the bilinear
flow
period
as indicated by a horizontal line at a value of 4.
It is evident that the type-curve presented in
Figure
23 is
superior
to
the type-curves shown in Figures 9
and
13 in type
curve matching.
When
field
test data are plotted
in
terms
of
Ap/ At .
Ap )
on
a log-log paper,
both
y-axes (field data and
type-curve) coincide
and
only horizontal movement is required in
type-curve
matching.
Therefore the uniqueness
problem can be
reduced significantly.
For the case of
pseudo-steady-state
matrix flow a similar
graph
can
be obtained if
we
graph PwD/(tDI1J
.
~ D ) versus A,tDI1J
for various values of w and C D f \ ~ / J(kfb
f
)D as shown in Figure
24. Here
the
bilinear
flow
period
is
indicated
by a
horizontal
line
with a PwD/(tDI1J . ~ D ) value of
4
Figure 25 shows the pressure-derivative ratio type-curves
for the pseudolinear flow behavior of a
fractured
well in a dou
ble porosity reservoir with the wellbore storage effect. A log-log
graph of PwD/(tDI1J . ~ D ) versus T/maDtD", for various values of
w and D f ~ D is presented for transient
matrix
flow model.
Here a horizontal line
at
a value
of
4 indicates the formation
bilinear flow"
period
and the line at a value of 2 represents the
pseudolinear flow period for this type of system.
For the pseudo-steady-state
matrix flow
the corresponding
graph for the pseudolinear flow as shown in
Figure
26 is in
terms
of
PwD
/
t
DI1J
•
~ D )
versus
Al
DI1J
and the
correlating
parameters
for the families of curves are
w
and CD1'1/'>,. In
this
case a horizon
tal line at a value of 2 is indicative
of
the ~ e u d o l i n e r flow period.
n
the absence of wellbore storage effects an unique
type
curve
can be
presented for both bilinear and pseudolinear flow
regions and for a given matrix flow model.
This type
curve is a
graph o f p w D / ( t D I 1 J p ~ D n ) versus
T/maDtDI1J
(or AtDI1J )' where n
I f
is the slope of the log-log curve of
Ap
or
tAp.
Figure
27 presents the type curve for double porosity be
havior
within
the bilinear and pseudolinear flow regions for the
transient
matrix flow model. This type curve is applied to esti
mate
n,
wand T/ma/h;"a; hence the matrix block size h
ma
can be
estimated if
a value for
T/ma
is available
and the nature of
flow
geometry
can
be identified from n (i.e. n = 4 bilinear, n = 2
pseudolinear)
Finally Figure
28 shows the type curve for the pseudo
steady-state
matrix flow model to
be
applied
within
the bilin
ear
flow and pseudolinear flow regions; here a log-log graph of
P w D / ( t D I 1 J p ~ D n ) versus
\tDI1J
is presented. Estimation of
w
f
and n
is possible through
type
curve matching.
65
ON LUSIONS
From the results of
this
work the following conclusions are
warrented:
1. A general semianalyt ical model and simplified fully analyti
cal models have been developed to
study
the basic behavior
of a well
intersected
by a finite conductivity vertical
fracture
in
a double porosity reservoir. I t is shown
that
the behavior
of
this system
is
the
result
of
a combination
of
the behavior
of a fractured well in a homogeneous reservoir and the be
havior of a double porosity reservoir. This type of
system
can exhibit fifteen different flow periods.
2.
Two
models
are
considered
to take into
account the fluid
transfer between
matrix
blocks
and
fracture network, namely
the transient matrix flow and pseudo-steady-state matrix
flow. t is shown
that
correlation parameters are the dimen
sionless
fracture
conductivity (kfbf)D,
the fracture
stora
tivity coefficient wand the
interporosity
flow parameter At
for pseuso-steady-state matrix flow model,
the
dimension
less
matrix
hydraulic diffusivity
T/maD
for transient matrix
flow model.
3. Two new flow periods are defined for the transient
matrix
flow models: the trilinear flow and the formation bilinear
flow. To our knowledge these types of flow behavior have
not been described before.
4. The trilinear flow" is present for
the
bilinear flow regime
during the stabilization
period
for small values of
w
(::;
10 -
2
. n
this
case the
matrix
linear
flow
is superimposed
to
the bilinear flow in the hydraulic fracture. This flow
period
can
be identified from a log-log graph of the pressure versus
time with a straight line of a
1/8
slope. Hence a graph of
pressure versus t
l S
yields a
straight
line and
the
slope is
inversely
proportional
to
the fracture
conductivity.
5. The
formation
bilinea.r flow" occurs during the pseudolin
ear flow when most of the fluid expansion is provided by
the matrix
under
linear flow condition for small values of
w
(::; 10 -
2
. Here a graph of pressure versus tl/4 yields a
straight
line
with
an
intercept. The
slope of this
straight
line is inversely
proportional to the
fracture area hz and
the intercept is inversely proportional to k b
J
Z
f.
A log-log
graph of pressure versus time also exhibits
1/4
slope straight
line. Hence it is concluded that the bilinear flow
in the
hy
draulic
fracture
is not the only type of
flow
that exhibits the
one quarter slope on a
log
versus log
t
plot.
6. New pressure-deriva tive ratio type-curves including
the
well
bore storage effect are presented for
both
the transient
and
pseudo-steady-state
matrix flow models
under the
bilinear
flow
and
pseudolinear flow periods. A log-log
graph of
PwD/(tDI1J
.
~ D ) versus T/maDtDI1J
is
used
to
analyze pres
sure
transient test data
for
the transient
matrix flow system.
Shnilarily, a graph of log PwD/(tDI1J ·
~ D )
versus log
AtDI1J
should be used for the pseudo-steady-state matrix flov sys
tem.
7. A new formulation is presen ted for the selnianalyticalmodel
to allow the consideration of double porosity and ,wellbore
storage, as well as the
computation
of the
constant
pressure
solution.
This
formulation saves
computation time
consid
erably, since the superposition in
time
is simplified in the
Laplace space.
9
-
8/9/2019 11. SPE-18172-MS OK
10/16
10
PRESSURE TRANSIENT ANALYSIS OF WELLS
WITH
FINITE CONDUCTIVITY
VERTICAL
FRACTURES IN DOUBLE POROSITY RESERVOIRS
SPE 18172
8. A single type curve is also
presented
to identify and evaluate
double porosity behavior for
both
bilinear
and
pseudolinear
flows.
CKNOWLEDGEMENTS
The authors wish to thank the management of
Dowell Schlumberger for permission
to publish
this
paper.
NOMENCL TURE
tf
t
Ctma
h
m
Fracture area per unit of
matrix
volume
Fracture
area per
unit of bulk volume
Dimensionless
fracture
area
Formation volume factor
Fracture width
Dimensionless wellbore storage
constant
Fluid compressibility
Fracture system
total compressibility
Total system compressibility
matrix system
total
compressibility
Formation
thickness
Matrix height
Hydraulic
fracture
permeability
Reservoir bulk permeability
Matrix permeability
Slope of the
straight
line
on
the specific
graph
of
analysis
6.m pwf) Gas pseudo-pressure change
Pwf Wellbore pressure
Pi Initial reservoir pressure
6.pwf Wellbore pressure change
PWD
Dimensionless wellbore pressure drop
PW Dimensionless pressure derivative
q
Well flow
rate
qf
Flow
rate
going
into the fracture per unit of length
q: a Fluid transfer
rate
per unit of rock volunle
rw
Wellbore radius
Laplace transform variable
Producing time
Dimensionless
time
Volume
Fracture
half-length
a, f unit conversion
constant
< J Reservoir porosity
< Jfb Fracture
porosity
< Jma Matrix
porosity
p. Reservoir fluid viscosity
At Interporosity flow coefficient
.
Hydraulic diffusivity
w
Dimensionless
fracture
storativity
SUBSCRIPTS
a rock matrix
b bulk
b bilinear flow
dimensionless
11 hydraulic
fracture
12 fracture network
Ib
fracture
referred to
bulk
volume
65
IbI
formation
bilinear
flow
initial,
ith matrix
block
ma
matrix
pi pseudolinear flow
t total
tl
trilinear flow
REFERENCES
1. Cinco-Ley H., Samaniego-V.,
F.
and Dominguez, N.:
Tran-
sient
Pressure
Behavior for a Well with a Finite-Conductivity
Vertical Fracture , Soc. Pet. Eng.
J.
(Aug. 1978) 253-264.
2. Cinco-Ley, H. and Samaniego-V., F.:
Transient
Pressure
Analysis for Fractured Wells ,
J. Pet.
Tech. (Sept. 1981)
1749-66.
3. Wong, D.W., Harrington, A.G. and Cinco-Ley, H.: Appli
cation
of the
Pressure
Derivative Function in the Pressure
Transient Testing of Fractured Wells ,
SPEFE
(Oct. 1986)
470-480.
4
Alagoa, A., Bourdet, D. and Ayoub, J.A.: How to Simplify
the Analysis of Fractured Well Tests , World oil (Oct. 1985)
97-102.
5.
Warren,
J.E. and Root,
P.J.:
The Behavior of
Naturally
Fractured
Reservoirs ,
S o £ ~ _ P e t
Eng. l : (Sept. 1963) 245-
255; Trans.
AIME,
Vol. 228.
6. de Swaan, O.A.:
Analytic
Solutions for Determin ing Natu
rally
Fractured Reservoir
Properties
by Well Testing , Soc.
Pet. Eng.
J.
(June 1976) 117-122; Trans. AIME, 261.
7 Kazemi, H:
Pressure
Transient Analysis of
Naturally
Frac
tured Reservoirs with Uniform Fracture Distributiou , Soc.
Pet. Eug. J. (Dec. 1969) 451-458.
8.
Bourdet, D.,
Alagoa, A. Ayoub,
J.A. aud Pirard,
Y.M.:
New Type-Curves for Tests of
Fissured Formations ,
World
Oil (April 1984).
9. Bourdet, D. and
Gringarten,
A.C.: Detennination of Fis
sure Volume
and
Block Size in Fractured Reservoirs by Type
Curve
Analysis , paper SPE 9293 presented at
the
1980
SPE Annual Technical Conference
and
Exhibition, Dallas,
September 21-24.
10. Houze, O.P., Horne, R. and Ramey,
H.J.
Jr.: Infinite Con
ductivity
Vertical Fracture in a Reservoir
with
Double Poros
ity Behavior , paper SPE 12778 presented at the 1984 SPE
California Regional Meeting, Long Beach, April 11-13.
11.
Lancaster, D.E.
and
Gatens, J.M.
III: Practical Well Test
Analysis
Methods
for Hydraulically Fractured Wells in Double
Porosity Reservoirs , paper SPE 15924 presented
at
the
SPE Eastern Regional Meeting, Columbus, Ohio, Nov. 12-
14,1986.
12. Ben-Naceur, K. and EconOlnides, M.J.:
Production
from
Naturally
Fissured Reservoirs
Intercepted
by a Vertical Hy
draulic Fracture , paper SPE 17425 presented
at
the SPE
California Regional Meeting, Long Beach, Ca. , March 23-25,
1988.
13. Cinco-Ley, H., Samaniego-V., F. and Rorlriguez, F.: Ap
plication
of the Pseudolinear Flow Model
to
the Pressure
-
8/9/2019 11. SPE-18172-MS OK
11/16
SPE 18172 H. CINCO-LEY AND H.-Z.
MENG
Transient Analysis of Fractured Wells , paper SPE 13059
presented at
the
1984 SPE Annual Conference
and
Exhibi
tion, Houston,
Texas,
September
16-19, 1984.
14. Stehfest, H.: Numerical Inversion of Laplace
Transforms ,
COllununications
of
the ACM
(January
1970), 13, No.1 ,
47-49.
15.
A General
Purpose
Semianalytical Model for Vertically
Fractured
Wells
to be submitted to SPE.
16. Mustafa, 0., Yeh, N.-S.
and
Reynolds, A.
Jr.:
New Deriva
tive
Type
Curves for Well Test Analysis ,
paper
SPE 16810
presented at
the
1987 SPE Annual Technical Conference
Exhibition,
Dallas,
September
27-30.
17.
Doung,
A.N.:
A
New Set
of Type
Curves for Well Test
Interpretation
Using
the
Pressure-Derivative
Ratio , paper
SPE 16812 presented at
the
62nd Annual Technical Con
ference and Exhibition of the SPE of AIME, Dallas, Texas,
Sept. 27-30, 1987.
APPENDIX
A
Correlation
Parameters
for
the Type
Curves
for
Bilinear
and
Pseudo
inear Flow Periods
According to equations 26 and 27
the
pressure
behavior in
Laplace space for a
fractured
well in a double porosity reservoir,
during the bilinear and pseudollnear flow periods, considering the
transient
matrix
flow
model
is given by:
_ 1 - S 1]maD
PwD = - k3 fV - k2 - ( k2 )
ffD ffD ffD
(A
-1)
Hence:
This
equation
indicates that a
graph
of
PwD(kfbf)D
has two in
dependent
group
of
variables;
tD",f(kfbf)"b and 1]maD/(k
f
b
f
)h.
f
we use the pseudo-steady-state matrix flow model (equations 26
and
28), we find that a
graph of PwD(kfbf)D
will have two inde
pendent
group
of
variables;
tD""(kfbf)h and
V(kfbf)h.
Now let us
assume
that
the
flow process is
dominated by the
bilinear flow; that means
that
in equation 26 we can
approximate
(A -
3)
Hence:
(A-4)
By defining the group
S/1]maD
we can write
From
here:
(A-6)
655
This equation indicates that
a
graph of PWD(kfb
f
)if21/; ;:D can be
presented in terms of
two
group of
variables:
wand 1]maDtDa:r
A similar conclusion can
be
reached for
the
case of the
pseudo-steady-state matrix flow model; that is, during
the
bi
linear flow
the
pressure behavior at the wellbore can be expressed
in terms
of
PWD(kfbf )if2>..V
4
as a function
of wand
1
D
:z:r
f we assume that
the
pseudollnear flow
dominates the
pres
sure behavior
of the
system; that is ,
the
coth function
in
equation
26
can be approximated as
follows:
Equation
26 becomes:
1 a
coth a) - + -
a
3
11 11
PwD
= 2s(f(W,1]maD,s)jl/2 +
3s(k
f
b
f
)D
or:
11
11
(A -7
(A -
8)
PwD
= 3/2 + 3 (k b
21]maD (S/1]maD) (f(w,S/1]maD)jl/2 S f f D
or:
(A
-
9)
1/2
1/2
11 £-1( 1 11 1]maD
PwD1]maD = 2 [f( / » 1/2 +
3(k
b
w, S 1]maD f f D
(A -
10)
This
equation
indicates
that
the group PwD1]; ;:D versus 1]maDtD
will show two fantilles of curves; one representing different values
of wand the other one
representing values of 1]: ;D/(k
f
b
f
)D.
For
the
case of pseudo-steady-state matrix flow, an equiv
alent
graph
will
be
in terms
of
PwD>..V
2
versus
> JD ',
and
the
fanti
lies of curves correspond s to different values of wand >..F2
/(kfbf
)D.
-
8/9/2019 11. SPE-18172-MS OK
12/16
Table 1. SI Preferred
Units,
Customary
Units,
and
Unit
Conversion Constants Used in
These
System
Parameter
SI
Preferred
Customary Units
k
I m
2
md
h
m
It
qo
m
3
/D
STB/D
qg
m
3
/D
MSCF/D
I
Pa ·s
cp
B
m
3
/m
3
RB/STB
¢> fraction fraction
t
Pa psi
p kPa
psi
m(p) kPa
2
/Pa.
s
pse/cp
t
hours hours
Cl
o
1842 141.2
Cl
g
1293 1424
f
3.6 X
10 -
9
2.637 X 10 -
4
Ob/o
34.97
44.1
Ob/g
24.57 444.75
O/b/o
24.73 31.18
O/b/g
17.36 314.47
Opl/o
0.196 4.064
Opl/g
0.1375 40.99
tpl/o
1928.94 147.86
tpl/g
1354.1 1491.2
Otl/o
382.4
118.9
Otl/g
268.43 1199.1
C
m
3
/Pa
It
3
/psi
Fig.
1 • Vertically Fractured Well
in
a Double Porosity
Reservoir
656
\
S ~ 18172
Table 2.• Pressure Behavior for a Vertically
Fractured Well
in a Double
Porosity
Reservoir
Fractured Well
Behavior
Bilinear
Transit ion
Pseudolinear
Transition
Pseudoradial
Double Porosity Reservoir
Behavior
Fracture
Network 1
Transition
Fracture + Matrix
Fracture Network
Transition
Fracture
+ Matrix
Fracture Network
Transition
Fracture
+ Matrix
Fracture Network
Transition
Fracture
+
Matrix
Fracture Network
Transition
Fracture
+ Matrix
2
3
4
5
6
7
8
9
10
11
12
13
14
15
:.
' - ~ - ~ . '
• '
-. :
:
.....
:::
. .. ,
• , , .... ·0
,',
... , .
:'. :
,'..
' ,; .
. ',
'
M
a t r i ~ ' . ' . :.:. '
.
< k ~ a t p m ~ . h'I
. .....
:
- ~ . ' .: .
.
.
.'.
...... /
..•.....
.• ..:
..... . . . . • .
",
.
" . "
. .
, .'
. .•..: ::..
, . . . . . .
" .
. ' . . . .' \ • '. ' , ' ' . ....
: ~ . . .
..
-: f
,
-'
. ~ . . . .
••••
10
•
.
.. > : < .. "
• I•• •· . . . . . ; •. .
':''''
.... ',
. : . , '
:.
Fra.cture
,
.. .
..
'
. .
\
.
•••
to
:. , ,, ,, .:' .
. .
.
..
.
..
,,: ..:
. .
. ....
' . ; ;
. :' . . .
Hydraulic Fracture
Fig. 2 • Slab Matrix Blocks in a Double Porosity Reservoi
-
8/9/2019 11. SPE-18172-MS OK
13/16
ell
Fig. 3 - Hydraulic Fracture Model
V . =0
t t \ \ t \61
j
t \ t I
-
..
XI
Hydraulic Fracture
Fig. 4 -
System Under
Reservoir Linear Flow Conditions
.....
.....
II:
--
Q
Q
10
3
1
2
Pseudolinear Flow
10
1
10
0
10 -
1
10 -
2
- - - ~ - - ~ - - ~ - - - - - - - - ~ - - ~ ~ - - ~ - -
10-
6
10-
5
10-
4
10-
3
10-
2
10-
1
10
0
10
1
10
2
10
3
t xf
krbr)i,
Fig. 5 - Pressure Behavior for a
Vertically
Fractured
Well
10
3
10
2
1
1
.....
II:
--
Q
10
0
Q
10 -
1
Bilinear
Flow
10-
2
10-
6
10-
5
10 -
4
10-
3
10-
2
10-
1
10
0
10
1
10
2
10
3
t xf krbr)i,
Fig. 6 -
Pressure Behavior
for the
Fracture
Dominated
Period
657
.....
10
1
Q
10
0
Q
10 -
2
10 -
6
10-
5
10-
4
10-
3
10-
2
10 -
1
10
0
t xf krbr)i,
Fig. 7 - Asymptotic
Behavior
for a
Fractured
Well in a
Double Porosity Reservoir During Transition
Period Transient Matrix Flow)
10
3
10
2
Arf(krbr)i,
10-
3
10-
2
.....
10
1
10-
1
.....
1
II:
- -
10
0
.... \..0
Q
u -
Q
10 -
1
10-
2
10 -
6
10-
5
10-
4
10 -
3
10-
2
10 -
1
1
0
10
1
1
2
1
3
t xf krbr)i,
Fig. 8 -
Asymptotic
Behavior
for
a Fractured Well in a
Double Porosity Reservoir During Transition
Period Pseudo-Steady-State Matrix
Flow)
1
3
1
2
1
1
.....
II:
- -
1
0
Q
Q
10-
1
10-
2
10-
6
10 -
5
10 -
4
10 -
3
10-
2
10 -
1
10
0
1
1
10
2
10
3
t xf
krbr)i,
Fig. 9 -
Type
Curves
for Pressure Behavior of
a
Fractured
Well
in
a Double Porosity Reservoir
Transient
Matrix Flow)
-
8/9/2019 11. SPE-18172-MS OK
14/16
U3
3
Arf
krbr)b
10
2
10-
3
10-
2
10
1
10-
1
..0
1
Q
10
0
Q.
10-
1
10 -
2
10-
6
10-
5
10-
4
10-
3
10 -
2
10 -
1
10
0
10
1
10
2
10
3
tOxl krbr)b
Fig. 10 -Type
Curves
for Pressure Behavior of a Fractured
Well
in a
Double Porosity
Reservoir
Pseudo
Steady-State
Matrix Flow)
10
3
Q
10
2
..0
10
1
---
Q
10
0
.
10 -
1
10 -
4
10-
3
10 -
2
o Data
Points
for
=
10-
1
1 0
2
10
3
,
17rnaO/(kr
b
r)2 = 10-
4
10-
1
10
0
10
1
10
2
10
3
10
4
tOxl krbr)b
Fig. 11 -Pressure Behavior of a Fractured
Well
in a Double
Porosity
Reservoir
Transient
Matrix Flow)
10
3
10
2
..0
101
---
Q
Q.
10
0
10 -
1
10-
3
10 -
2
o Data
Points
for
= 10-
1
1 0
2
10
3
,
Arf krbr)2
= 10-
2
10 -
1
10
0
10
1
tOxl
krbr)b
Q
10
2
I'
e
>=
INQ
10
1
;;:;
1.0
..0
C.
Q 10
0
_ it
Q.
10-
1
H
Q
10-
2
CI
:::
a I
10-
3
10-
2
10-
1
Q
10-
3
it
10-
9
10 -
8
10 -
7
10 -
6
10-
5
10-
4
10-
3
10-
2
10 -
1
10
0
17rnaO
t
Oxr
Fig.
13
-Bilinear Flow Pressure and Pressure Derivative
Behavior
of a Fractured
Well
in a Double
Porosity
Reservoir Transient
Matrix Flow)
10
2
10
1
10-
3
10-
2
10-
1
1.0
10
0
10 -
1
10-
2
10 -
3
pressure derivative
10 -
4
10-
7
10 -
6
10 -
5
10-
4
10 -
3
10 -
2
10-
1
10
0
10
1
ArtOxr
Fig.
14
-
Bilinear
Flow
Pressure and Pressure Derivative
Behavior of
a Fractured
Well
in a Double
Porosity
Reservoir
Pseudo-Steady-State Matrix Flow)
10 -
2
10 -
3
10 -
4
~ ~ ~ - - - ~ - - - ~ - - - - - ~ - - - ~ - - - - - ~
10-810-710-610-510-410-310-210-1100 10
1
10
2
10
3
17rnaO
t
Oxr/w
Fig.
12
-Pressure
Behavior
of a Fractured
Well
in a Double Fig. 15 -Pseudolinear Flow Pressure and Pressure Deri-
Porosity
Reservoir
Pseudo-Steady-State
Matrix vative
Behavior
a
Fractured Well in
a Double
Flow) Porosity Reservior Transient
Matrix Flow)
658
-
8/9/2019 11. SPE-18172-MS OK
15/16
..IN. .
10 -
3
pressure
derivative
10 -
4
- - - - - - - - - - - - - - - - - - - - - r - - - r - - - r - - -
10 -
5
10 -
4
10 -
3
10 -
2
10 -
1
10
0
10
1
10
2
10
3
10
4
10
5
)"rtDxr w
Fig. 16
-Pseudolinear Flow
Pressure and
Pressure Deri
vative
Behavior
of a Fractured
Well
in a Double
Porosity
Reservoir
Pseudo-Steady-State Matrix
Flow)
..
o
Q..
-
8/9/2019 11. SPE-18172-MS OK
16/16
113-
9
113-
8
113-
7
113-
6
113-
5
113-
4
113-
3
113-
2
113-
1
3
13
3
1
I1maDtDxr
Fig. 23 -Type Curves for Bilinear Flow Behavior of a
Fractured Well in a Double Porosity
Reservoir
With
the Wellbore
Storage Effect (Transient
Matrix Flow)
3
4
3
3
c:l
-
3
2
I
c:l
3
1
:l
4.0
3
3
113-
1
113-
6
113-
5
113-
4
113-
3
113-
2
H3
1
Hl
13
3
1
3
2
.\rtDxr
Fig. 24
-Type
Curves
for Bilinear
Flow
Behavior
of
a
Fractured
Well
in a Double Porosity Reservoir
With
the Wellbore Storage Effect (Pseudo
Steady-State M atrix Flow)
11
1
aD/(k
r
b
r
D
=
10-
4
113-
9
113-
8
113-
7
113-
6
113-
5
113-
4
113-
3
113-
2
113-
1
3
3
HI1
I1maDtDxr
Fig.
25
-Type Curves for Pseudolillear Flow Behavior of
a Fractured Well in a Double Porosity Reservoir
With the Wellbore Storage Effect (Transient
Matrix Flow)
SfE
8 72
113-
1
CDf0\ = 10-
4
10-
3
10-
2
10-
1
1.0
113-
6
113-
5
113-
4
113-
3
113-
2
113-
1
3
3
3
1
3
2
.\rtDxr
Fig.
26 -Type
Curves for
Pseudo inear
Flow
Behavior of
a Fl:actured
Well
in a
Double Porosity
Reservoir
With the Wellbore Storage Effect (Pseudo
Steady-State M atrix
Flow)
113-
9 113-8
113-
7 113-6
113-
5 113-4 113-3
113-
2
113-
1
3
3
3
1
I1maDtDxf
Fig. 27 - Type Curves for Double Porosity B ehavior
Under the Bilinear and Pseudolinear Flows
(Transient Matrix
Flow
Model)
113-
6
113-
5
113-
4
113-
3
113-
2
113-
1
3
3
.\rtDxf
Fig.
28 -
Type Curves for
Double
Porosity
Behavior
Under the Bilinear and Pseudolinear Flows
(Pseudo-Steady-State
Matrix Flow Model)
