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SPE 6959
Pressure Transient Analysis
o
Dually Fractured Reservoirs
Abdullah AI-Ghamdi, SPE and Iraj Ershaghi, SPE, University of Southern California
Copyright 1996 Society of Petroleum Engineers Inc
li f.,la., SPE. P.O.
a
133136, Rtc
4••••
TX 75013-3836. U.S.A.,
f 01-214-952·905.
Abstract
Dual fracture models are examined as a more realistic al
ternative to dual porosity models for the representation of
naturally fractured reservoirs. A
major
component of
the
fracture system is
the
network
of
microfracture which by
virtue of their lower permeability respond somewhat later
than the
macrofractures. A delineation of micro racture re
sponse versus matrix response is made using the proposed
conceptual models. It is
demonstrated
that the microfrac
tures response
may
at times be mistakenly attributed to
matrix.
Introduction
Studies published on diagnostic plots
of
pressure transient
test data indicate strong similarities amoung certain cases
of conceptua l reservoir models. In particular, diagnostic
plots expected for
naturally
fractured reservoirs are often
times
not
developed because
of
either inadequate test du
ration or well bore controlled conditions. A major question
in
the
testing
of
naturally fractured reservoirs is explana
tion for causes of non-development
of
transition period
l
This transition
was predicted by Warren
and
Root
2
in
their dual porosity conceptualization of naturally fractured
reservoirs. Other researchers have also predicted the tran
sition periods for layered type response
3
as well as systems
of
triple porosity.4
In the dual porosity conceptualization, an assumption
is made as to
the
nature
of flow
units with interporosity
propert ies. Specifically, two types of flow units are con
sidered. First is a system
of
tight matrix with substantial
storativity for fluid.
The
second unit is the network of
fractures with high fluid conductivity. In this study, the
above model is extended to a more realistic one where the
effects
of
microfractures are also included.
The
objective is
to predict response duration for these subsets and develop
guidelines for interpretations
of
pressure transient test data
misinterpreted because
of the
selection
of
an inappropriate
model.
SPE Journal, March 1996
Models
for
Naturally Fractured Reservoirs
Over
the past
several years, numerous models for char
acterization
of naturally
fractured reservoirs NFR) from
pressure transient tests have been suggested. One com
monly used model is the double porosity model pro
posed by Barenblat
and
Zeltov
5
and introduced into the
petroleum literature by Warren and Root.
The
idealized
model introduced by Warren and Root Fig.I) consists
of
a set
of
orthogonal fracture planes dividing
the
matrix into
equal blocks.
Production
at
the
well bore is essentially con
trolled by
the
fractures. The fracture system contains a
small fraction of indigenous oil, yet with hydraulic con
ductivities superior to that of
the matrix,
act as primary
conduits for flow in
the
reservoir. Matrix rock, however,
contains the bulk of fluid in place and provides pressure
support to the fracture system. While this model has been
the backbone of various analysis techniques and simulation
applications, certain modifications are necessary to bring it
closer to realistic representat ion of NFR.Among the mod
ifications suggested is
the
work
of
Abdassah and Ershaghi
4
who introduced
the
Triple Porosity Model. In this model,
two distinct matrix systems of different flow and storage
capacities are recognized in
addition
to
the
fracture sys
tem. Another modification was introduced by Bourdet and
Johnston
3
where matrix blocks also contribute to produc
tion
at
the well bore.
In this paper new conceptual models are proposed
to
dif
ferentiate between
the
microfractures and the macrofrac
tures. Dual fracture systems consisting
of
macrofractures
and microfractures Fig. 2 are introduced as
the
basis of
the
reservoir architecture.
The
theoretical basis
of the
pro
posed models are developed and
the
anticipated pressure
transient response on
the
pressure derivative plot are then
compared to those
of the
existing models.
Both
the
double porosity and
the
triple porosity models
predict transition periods reflecting matrix support to
the
fracture system. In actual field tests, indications are, at
times, and for certain tests, these transition periods may be
observed. However, there are cases where
the
response of
natural ly fractured reservoirs have lacked a clear definition
indicating matrix
support.
One purpose
of
this paper is
to ascertain the similarities and differences between the
support from tight
matrix
and that
of
the more permeable
microfractures.
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PRESSURE TRANSIENT ANALYSIS OF DUALLY FRACTURED RESERVOIRS
SPE 6959
Triple Porosity Model
The triple porosity model consists of two matrix systems
with different properties (Fig. 3a). One basic assumption
of the
triple porosity is that
the
two matrix system are
not in communication with each other. The model uses
two
(A S) and
two (w s) relating each matrix system to the
fracture system.
The matrix
with larger permeability will
respond first, followed by the response
of
the tighter ma
trix
at a later stage. The general response of this model
on the pressure derivative plot (Fig. 3b) shows 3 horizon
tal
line segments separated by two troughs representing
a transition period when each matrix type provides pres
sure
support
to the system. The three line segments cor
respond to
the
fracture response, the fracture and matrix
1 response, and the response of the total system, respec
tively.
This
model also assumes
an
unsteady state (gra
dient flow) between the fracture and each matrix system.
This last assumption will only influence the transition pe
riod by limiting the depth of depression to a value
of
0.25
which is
half
the value corresponding to the infinite acting
response. The concept of triple porosity can be further
extended to represent the proposed Dual Fracture sys
tem with one matrix
type
in addition to
the
two fracture
systems.
The
triple porosity model has been tested with
(A) values between 10-
5
-10-
9
) representing matrix inter
porosity flow. However, in the new proposed model of dual
fractures,
the
interporosity
flow
between
the
two systems
of
fractures has values between 10-
1
- 10-
4
) reflecting
expected higher permeability for
the
microfractures.
Proposed Conceptual Models
Two of the conceptual models that can be employed to rep
resent dual fracture systems are discussed here. The first
model is similar to
that
of the triple porosity but with the
microfracture system replacing one of the matrix systems
(Fig. 4).
This
model assumes no interporosity flow be
tween the microfracture and the matrix systems, yet both
support the
macrofracture system.
The second model assumes pressure support from the
matrix
to
the microfractures which in turn support the
macrofractures. The macrofractures
and the
microfrac
tures both contribute
to
the overall hydraulic conductivity
and
to
the
production at the test well. (Fig. 5 a,b).
Dual
Fracture
System
Fractures and fissures occur in many sedimentary rocks.
f the
fracture size distribution can be delineated into
two broad categories representing macrofractures
and
mi
crofractures,
then the
macrofracture system will
dictate
the very early time response of pressure transient tests.
The
response of
the
microfracture system will only be dis
tinguished if
the ratio
of the microfracture permeability
to
that of
the
macrofracture is small; i.e.
AI::;
0.001 .
Otherwise the two fracture systems will respond practi
cally at the
same
time.
This
results in
the
pressure
tran
sient response to be similar to that
of the
double porosity
94
model with longer and a more steady extension of the first
straight
line representing the combined response
of the
two
fracture systems.
The first model
to
be considered is similar to that
of
the
triple porosity model with the microfracture system
interacting with
the
macrofracture system but not with
the matrix. This model also assumes
that
production
at
the
wellbore is primarily from the macrofracture system.
In
the
second model, the microfractures play
an
ad
ditional role by receiving support from the matrix and
transmitting support to the macrofracture system.
The
matrix system will only provide pressure support to the
microfractures
and cannot
transmit fluid directly
to
the
macrofractures. This model assumes pseudo steady-state
flow between the two fracture systems and between the
matrix
and the
microfracture systems. Model 2 can be
divided into two submodels. Model 2a assumes
that
only
the
macrofrature system produces at
the
well bore, while in
model 2b, the assumption is
made
that production is con
tributed from both macro and microfracture systems. The
contribution of each fracture system will be proportional
to
its
permeability
ratio K).
The proposed models are equivalent to double poros
ity, triple porosity,
or
double permeability models under
certain limiting conditions. For example
the
assumption
that
the storativity
of the microfracture system is zero will
change the above models
to
the double porosity model. As-
suming
that
the storativity of the
matrix
system is zero and
changing the range of
the
A
and
w for
the
microfracture
to
that
of another matrix system will change the above
models
to the
double porosity model for model 1
and
2a.
Model
2b
will be equivalent
t
the double permeability.
Changing the range of the values
of
A and w for the mi
crofracture
to
that
of
a new
matrix
system will produce
the
triple porosity model for all of
the
three new models.
Mathematical Representation
Model 1. The first model as mentioned is similar to
that
of the Triple Porosity model. The solution of this
model was derived by Abdassah
and
Ershaghi
3
. This so
lution was not intended for A values outside
the
range
10-
5
- 10-
9
) which is representative of
the
two matrix
systems. In
the dual
fracture system,
the
microfracture
system replaces one of the matrix systems with A values
in
the
range of 10-
1
- 10-
4
. The dimensionless pressure
solution (including the effect of wellbore storage and skin)
in
the
Laplace space is:
X
= [ S + ~ J t , w 7 - 1 ) s t a n h J t , w 7 - 1 ) S
J -
1)
anhJ -
1) s
1 (2)
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PRESSURE TRANSIENT ANALYSIS OF DUALLY FRACTURED RESERVOIRS
SPE 26959
and
have
the
potential
to
further explore
the
complex na
ture of naturally fractured reservoirs.
Discussion
Conceptual models representing naturally fractured reser
voirs by the formulations discussed here, predict several
cycles of time data before tight matrix can be recognized.
Examination
of
a
number of actual
pressure buildup
and
drawdown tests Table 1), that exhibit a double porosity
behavior for naturally fractured reservoirs, indicate pres
sure
support
of
the
matrix develops in relatively
short time
1
- 3 cycles) since the first point recording. Considering
the large expected contrast between the matrix permeabil
ity and that
of
the fracture resulting in an interporosity
flow
parameter A
in the range of 10-
5
-
10-
9
, one can
predict that the matrix support will actually require more
time to develop. The exhibition of an early transition pe
riod can
be
attributed to the presence of the microfracture
system with considerably larger permeability than that of
the matrix. In the dual fracture model, the response of
the
so called microfractures is characterized by a transi
tional period similar to
that
of the matrix. On the pres
sure derivative plot, a trough is developed at a much earlier
time,
td
10
1
- 10
4
. On the other hand the response of
the
matrix is manifested by a second trough
that
comes at
a later stage.
The
dual
fracture model provides
an
explanation for
many field tests where reservoirs known to be naturally
fractured are responding in a way similar
to that of
homo
geneous formation. In wells with high skin
and
wellbore
storage,
the
first trough is very likely to be masked. f
the permeability
of
the reservoir rock is very low tight
matrix) with
m
representing the matrix in
the
range
of
10-
5
-10-
9
,
the
second trough
may
require days
or
weeks
and
may never
be
detected within realistic
test
durations.
Conclusion
The delineation of fractures into two broad categories, a
macro and a microfracture, is a
step
forward toward a more
realistic representation of naturally fractured reservoirs.
The pressure support
of
the microfractures is similar to
that of
the
matrix. On the pressure derivative plot,
the
presence of microfractures can lead
to the
formation of
transition zones at substantially earlier time. These zones
may be mistakenly interpreted as matrix support. The
proposed models provides
an
explanation for
the
observa
tion
of
early pressure
support emanating
from a network
of
microfracture
and
often attributed to
the tight
matrix
rocks. The models also provide a general explanation for
the observation or lack of observation of single or double
transition periods on the
test
data from
naturally
fractured
reservoirs. The concept
of
dually fracture reservoir can
lead to better
estimation
of reservoir
parameters
including
the partition coefficients corresponding to the volumetric
contribution of macro
and
microfractures in addition to
the matrix.
96
The proposed models suggest that the pressure response
of
the tight
matrix rocks require extended
test duration to
be observed. A more realistic design
of
pressure test dura
tion can be implemented for improved characterization of
naturally
fractured reservoirs.
Finally, with downhole recording and by minimizing the
effect
of
well bore storage, the influence of the microfrac
tures
support
can be best exhibited.
Acknowledgement
First author wishes to thank the management ofthe North
ern Area Production Engineering at Saudi Aramco for
their continual support of his graduate study at the Uni
versity of Southern California. This study is supported by
the Center for Study of Fractured Reservoirs at USC.
References
1.
Odeh,
A. S.: Unsteady-State Behavior of naturally frac
tured
reservoirs, JPT,
March
1965, 60-66.
2.
Warren,
J.E
and
Root,
P.J.: The behavior
of
naturally
fractured reservoirs. SPE. J.
Sept.
1963, 245-255; Trans.
AIME.
3. Bourdet, D. and Johnston: Pressure
behavior
of layered
reservoir
with
crossfiow.
Paper SPE
13628,presented
at
the SPE California regional meeting, Bakersfield, CA,
March
27-29, 1985.
4.
Abdassah,
D.
and Ershaghi,
I.: Triple porosity models for
representing
naturally fractured reservoir. PHD disserta
tion, USC, JULY 1984.
5.
Barenblatt, G.
I., Zeltov, Ju.
P.
and
Kocina, I.
N.: Basic
concepts
in the theory of seepage of homogeneous liquids
in fissured
rocks.
Soviet
J.
App. Math. and Mech., 1960,
XXIV,
no5, 1286-1303.
6. Stehfest, H.: Algorithm 386, Numerical inversion of
Laplace transforms, communication
of
the ACM,
Jan.
1970, 13, no. 1, 47 - 49.
7.
Streltsova,
T.
D.: Well
Pressure
Behavior
of
a
naturally
fractured reservoir SPEJ, Oct. 1983) 769-780. .
8. Crawford, G.
E.,
Hagedorn, A. R., PIerce, A. E.: AnalYSIS
of
Pressure
Buildup
Tests
in a
naturally
fractured reser
voir, JPT, Nov. 1976, 1295-1300.
9.
Strobel, C.
J.,
Gulati,
M. S.,
Ramey, Jr.,
H.
J.:
Reser
voir
limit Tests
in naturally fractured reservoir-A field case
study using type curve, JPT,
Sept.
1976, 1097-1106.
Appendix A
The dimensionless equations describing the flow in the
macro-fracture, microfracture,
and the
matrix system re
spectively are:
[8
PFd
18PFd]
8PFd'F
=
W F
rd 8rd
Otd
Af Pfd
- PFd)
A.l)
[8
PJd 1
8PJd]
8Pfd
'J
=
w
rd 8rd
Otd
AJ(Pfd -
PFd)
m
Pmd
-
PJd)
A.2)
8P
m
d
-Am Pmd
- P
J
d)(A.3)
w8t ; ;
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SPE
26959
ABDULLAH AL-GHAMDI AND IRAJ ERSHAGHI
The boundary conditions for the above system of equations Solutions are possible when:
are:
At t =0) :
PF r)
=
PJ r)
=
Pm r)
=Pi
At r
=
w) : PF t)
=
PJ t)
=
Pw t) A.4)
lim
PF
=
lim
PJ
=
lim
Pm
=
Pi
..
.. ..
B
-21rrw
[k
h 8PF k h 8P
J
]
q =
F
+ J
P 8r 8r
....
A.5)
A.6)
Transforming the above system of equations to the Laplace
domain, then substituting equation A-3 in equation A-2
obtains:
Where:
Pd
td
rd
I;kh
I; J
c
t
h
KF
KJ
WF
wJ
WFSP
Fd
- AJ P
Jd
- P
Fd
) A.7)
= [WJS
+
Am
- A ] P
Jd
wms+ m
=
=
=
=
=
=
+AJ P
Jd
- P
Fd
) A.8)
I;kh)
141.2
qpB P
-
P
w
)
0.000264 I;kh)t
[I;4Jct h p r ~
r
rw
kJhJ
+
kFhF
J
C
t
h
)F
+ J
c
t
h
)J
+ J
c
t
h
)m
kFhF
I;kh
1
KF
J
C
t
h
)F
I;4J
c
t
h
c
t
h
)J
I;4J
c
t
h
A.9)
A.10)
A.11)
A.12)
A.13)
A.14)
A.15)
A.16)
A.17)
A.26)
or
A.27)
Solutions
to
equation A-27 are:
A.28)
A.29)
A
A.30)
Substituting
and
into equation A-24 and A-25 one
obtains:
where:
P
Fd
=
a1BIKo 0-1rd) + a2B2Ko 0-2rd) A.31)
P
Jd
=
B1Ko 0-1rd) +
B2KO 0-2rd)
A.32)
A.33)
A.34)
Applying the boundary condition from A-5:
B
=
1
-
a2)K
o
0-2
r
d) B
1 aI)Ko O-lrd) 2
A.35)
Wm
1-WF-WJ
A.18) Therefore:
A,
=
kJh
J
2
aJ
I;kh
rw
Am
=
kmh
m
2
am I;kh
rw
A2
Let:
X = wJs +
Am
_
m
A
W
m
8 + m
The solutions to equations A-7 and A-8 are:
P
Fd
=
AFKo o-rd)
P
Jd
=
AJKo o-rd)
A.19)
A.20)
A.21)
A.22)
A.23)
Substituting equation A-22 and A-23 into A-7 and A-8:
[KF0-2 -
WF8
-
AJ]
AF
+
AJA, = 0
AJAF
+ [ 1 KF)0-2
- X -
AJ] A, =
0
SPE Journal, March 1996
A.24)
A.25)
Bl =
b
=
8 1 - aI) KFa2 + 1 -
KF)
0-2KO 0-I)Kl 0-2)
8 1 - a2) KFal +
KF)
0-1Ko 0-2)K
1
o-d
A.36)
A.37)
A.38)
Therefore, the dimensionless pressure solution in the
Laplace domain is
A.39)
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PRESSURE TRANSIENT ANALYSIS OF DUALLY FRACTURED RESERVOIRS
SPE 26959
where
r
a2 -
al
al - 1) ICpa2 + ICp)
al - a2
(A.40)
(A.41)
Incorporating
the
effects
of
skin
and
wellbore storage, the
inner boundary conditions change into
P
w
=
qB
=
+
Cd
8Pp)
Pp - Srw
r r at well bore
..
-27rrw
k
h 8Pp k h 8P
I
)
p
p--
I I--
I
8r 8r
C 8P
w
d7Jt
O.S936C
[ ¢ch)p
+
¢ch)1
+
¢ c h ) m ] r ~
(A.42)
(A.43)
(A.44)
The
dimensionless pressure solution in Laplace space be
comes
r/J
=
I < ~ c r )
Ngmen_clatyre
PPd, Pld, Pmd =
Pi
=
P
w
t
=
td
rd
=
rw
=
h
=
ICp, I I
Wp, WI, Wm
=
AI,
Am
98
1
S
CdS +
ICpa2 + 1 - I p
a2
- 1
Kf cr2) + S
ICpal + 1 -
I p
al - 1 Kf crt) +
S
KP cr2) + S
a2 - 1 Kf cr2) +
S
Kf crt} +
S
al - 1) Kf crt) +
S
Ko cr)
crKt{cr)
(A.45)
(A.46)
(A.47)
(A.4S)
Dimensionless pressure in the
laplace domain for macrofracture,
microfracture and
the matrix
systems respectively.
Initial pressure (psi).
Pressure at the production well
(psi).
Time
(hr).
Dimensionless time.
Dimensionless radial distance.
Well bore radius
ft).
Thickness (ft).
Ratios of permeability thickness
defined by equations A.14 and
A.15 respectively.
Storativity ratio for the macro
fracture, microfracture and
matrix system respectively.
Interporosity flow parameters
for the microfracture and the
matrix
system respectively.
S =
Skin factor.
Cd
=
Dimentionless well bore storage.
C
=
Well bore storage constant
(bbl/psi).
S
=
Laplace variable.
¢
=
Porosity.
C
t
=
Total
Reservoir Compressibility
psi-
l
.
kh
Permeability thickness (md-ft).
q
=
Flow rate
STB/day).
I
=
Viscosity (cp).
B
=
Formation volume factor.
RB/STB)
aI,
am
=
Shape factors for the micro-
fracture
and the matrix
system respectively.
K
o
, Kl
=
Modified Bessel functions
of the
second kind oforders zero and
one respectively.
I<J
=
Ratio of the
modefied Bessel
functions.
cr
=
Argument of Bessel functions.
X
=
Defined by equation 2 in
the
first model, and by equation
A-21 in
the
second model.
r/J,
,,{,
Ap,
AI,
aI,
a2, B
l
, B2, b, cr, crl, cr2, r, D. = Quanti
ties defined by equations in the second model.
Table-1
No.
of Time Cycles
to
the End of the Transition Period
since the First Data Point
Source
Test
ofcycles comments
Warren
and
Root
2
B/U
2 < n < 3
Fig.1
Warren
and
Root
2
B/U
< n < 2
Fig.-D2
Bourdet
3
B/U
2 < n < 3
double
perm
Streltsova
7
DID
3 < n < 4
Crawford
8
B/U < n < 2
Test
A
Crawford
8
B/U
< n < 2
TestB
Crawford
8
B/U
< n < 2
Test C
Crawford
8
B/U
< n < 2
Test
D
Crawford
8
B/U < n < 2
Test
E
Strobel
9
DID
2 < n < 3
Well 2
Strobel
9
B/U
3 < n < 4
WellS
uthors
Abdullah AI-Ghamdi is a doctoral candidate research assistant
at
the University of Southern California. He holds a B.S. and
an
M.S. degree
in
Petroleum Engineering from USC. He worked
as a
production
engineer with
the Northern Area
Production
Engineering of Saudi Aramco. Iraj Ershaghi is Omar B. Milli
gan Professor of Petroleum Engineering
and
the director
of
the
Petroleum Engineering Program at the University
of
Southern
California. He holds a
Ph.D.
degree in Petroleum Engineering
from the University of Southern California.
SPE Journal. March 1996
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SPE
26959
ABDULLAH AL-GHAMDI AND IRAJ ERSHAGHI
....
Fig. 1 Double poros ity model .
Fig. 2 Dual fracture model .
....
e ~
Matrix 1
...
Macrofracture
~ t
t
atrix 2
Fig
3 a Triple porosity model
....
01L-
1 00
4
1F1r)
1 ~ 1 2
td
I
ig.
3b Pressure res'ponse
of triple
pOrosity mOdel'.'
SPE Journal, March 1996
e ~
Matrix
acrofracture
e-t
t
crofracture
Fig -4
Modell
~
Matrix
icrofracture
.-
+
acrofracture
I
Fig 5a Model2a
.
e ~
Matrix
icrofracture
~ ~
+
acrofracture
Fig 5b Model2b .
1. l e r ~
10
2 Ie = 10
2
3, hfc
10)
4 Ar= 10
5. Ie r 10
5
n - . ~ 10 ,1
00 = 1.6
x10
OOm= 5.1 X10'2
.
A . f ~ 1 0
o1 - _ ~ ~ ____ ___1... ...l
_ _ _ ___
101
1[·00 1[ 02 1[41[6 1[81[10 1[12
td
Fig.6 The general response of model 1 .
99
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PRESSURE TRANSIENT ANALYSIS OF DUALLY FRACTURED RESERVOIRS
SPE 26959
10.00
01.00
t-'
'U
0
c..
TI 0.10
0.01
1.
2.
3.
4.
5.
/ - ,-
10 '
A t
1
Af-
10
A f ~
10';;
A f ~
10(
100
1E4
1E6
td
l "ig .7 TI1e general response of model 2-a .
IC
..
0.99
J P1=
10
{t)F-
10'
~ r =
10.
2
1EB
1E10
10 00 r------------------
1.
A t
10
2.
-
1
0 '
3.
1,1
:
10'
4.
A, - 10;
5.
t -
10
0.01
100 1E4
1E6
td
[ Fig.R The general response
of
model 2-b .
l
10.000
s 10 \
1.000
C
u
- 100 / \
',---
0
t:0
0.100
0
c..
A f ~
TI
A'rn=
10·
0.010
b lF=
10'
rot=
10
0.001
100 1E4 1E6
td
1C
Am-
OlF=
t1 f =
1EB
1EB
L
ig .9· y p i ~ a l dual f r a c t u r ~ response(modeCZ-a).
~ ~ -
100
0.7
10'
10'
10
1E10
1E10
-J
10.000
s
~ 1 0
.- \
C D ~
100
I \
1.000
/
\
/
0
-'
'U
0.100
IC
=
0.99
0
10
c..
/ f
A.f=
::0
Arn=
10.
8
0.010
OOF= 10
J
t}f=
10
0.001
1
100 1E4
1E6
1E8 1E10
td
[ i ig.IO
The combined response of the two tractures .
l
10000
s
~ 1 0
~ \
C D ~
100
i
1.000
/
\
N
'-....
1 1
0
0.100
/
IC = 0.99
0
- t ~
10'
..
::0
A m=
10
0.010
C ~ f '
10'
~ ) r =
10.
2
0.001
1
100 1E4 1E6 1E8 1E10
td
l i g ~ i I
. h
e
early trough representing microfTacture response J
10.000
s =10
/ r -
Go= 100 (
1,000
/
0
t-'
'U
0.100
I
0
f
10·
..
- t ~
TI
Am""'"
10
8
0.010
t ) F=
10
3
Of=
10']
0.001
100
1E4
1E6
1EB
1E10
td
Fig .12 Delay of the microfracture response.
SPE Journal. March 1996
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