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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.

93

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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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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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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


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)

97

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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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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'.'


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

8/18/2019 SPE-26959-PA



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

Documents

SPE-26959-PA