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The

Influence

o

Vertical

ractures

Intercepting

ctive and Observation Wells

on Interference Tests

Naelah

A

Mousli, *

SPE.

U

of Tulsa

Rajagopal Raghavan, SPE, U. of Tulsa

Heber Cinco-Ley,

SPE. Petroleos Mexicanos and U. Natl. de Mexico

Fernando Samaniego-V., SPE. Ins . Mexicano del Petroleo

bstract

This paper reviews pressure behavior at an observation

well intercepted by a vertical fracture. The active well

was assumed either unfractured or intercepted by a frac

ture parallel to the fracture at the observation well. We

show that a vertical fracture at the observation well has a

significant influence on the pressure response at that

well, and therefore wellbore conditions at the observa

tion well must be considered. New type curves presented

can be used to determine the compass orientation of the

fracture plane at the observation well.

Conditions are delineated under which the fracture at

the observation well may influence an interference test.

This information should be useful in designing and

analyzing tests. The pressure response curve

at

the

observation well has

no

characteristic features that will

reveal the existence of a fracture. The existence of the

fracture would have to be known a priori or from in

dependent measurements such

as

single-well tests.

Introduction

In this work, we examine interference test data for the in

fluence of

a vertical fracture located at the observation

well. All studies on the subject of interference testing

have been directed toward understanding the effects of

reservoir heterogeneity or wellbore conditions at the ac

tive (flowing) well. Several correspondents suggested

our study because many field tests are conducted when

the observation well is fractured. They also indicated

that it

is

not uncommon for both wells (active and obser

vation)

to

be fractured. To the best of our knowledge,

this is the first study

to

examine the influence of a ver

tical fracture at the observation well on interference test

data.

Two conditions at the active well are examined: an ac

tive well that

is

unfractured (plane radial flow) and an ac-

• Now with Arabian American Oil Co.

0197· 7520/82/0012·9346 00.25

Copyright 1982 Society of Petroleum Engineers of

I

ME

DECEMBER 1982

tive well that intercepts a vertical fracture parallel to the

fracture at the observation well. The parameters of in

terest include effects of the distance between the two

wells, compass orientation of the fracture plane with

respect to the line joining the two wellbores, and the

ratio of the fracture lengths at the active and observation

wells if both wells are fractured.

The results given here should enable the analyst

1) to

interpret the pressure response at the fractured observa

tion well, (2)

to

interpret the pressure response when

both the active and the observation wells are fractured

(3)

to

design tests to account for the existence of a frac

ture at one

or

both wells, and (4)

to

determine quan

titatively the orientation and/or length

of

the fracture at

an observation well. We also show that one should not

assume a priori that the effect

of

a fracture on the obser

vation well response will be similar to that of a concen

tric skin region around the

wellbore-i.e.

idealizations

to

incorporate the existence of the fracture, such as the

effective wellbore radius concept, may not be

applicable.

Mathematical Model and ssumptions

In this study, we consider the flow

of

a slightly com

pressible fluid of constant viscosity in a uniform and

homogeneous porous medium of infinite extent. Fluid is

produced at a constant surface rate at the active well.

Well bore storage effects are assumed negligible because

the main objective

of

our work

is

to demonstrate the

fluence of the fractures. However, note that wellbore

storage effects may mask the early-time response at the

observation well. Refs. 1 and 2 discuss the influence of

wellbore storage on interference test data. We obtained

the solutions to the problems considered here by the

method of sources and sinks. 3

The fracture at the observation well was assumed to be

a plane source of infinite conductivity.

4

The condition

of uniform pressure over the fracture surface was

933

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Y

I

xm ------l

LINE SOURCE

WELL

OR

UNIFORM-FLUX

FRACTURE

A

=

ACTIVE

WELL

0= OBSERVATION

WELL

INFINITE-CONDUCTIVITY

VERTICAL FRACTURE

Fig. 1-Schematic of system.

satisfied by dividing the plane source into a number

of

equal segments and then assigning a flux to each seg

ment such that the pressure drops at the midpoints of the

segments were identical. Details

of

this approach are

given in Ref. 4. Modifications to this approach, incor

porated primarily to reduce computing costs and com

puter storage, are discussed

in

Appendix A.

The active well was modeled by a line source to

simulate plane radial flow and by a plane source

uniform-flux

5

)

to simulate a fracture. The resulting

pressure response at the observation well was determined

by the principle

of

superposition. The procedure used to

obtain the pressure response at the observation well

in

this study is discussed in more detail in Appendix A. We

did not consider an infinite-conductivity fracture at the

active well nor a uniform-flux fracture at the observation

well

in

detail mainly because

of

computer time limita

tions. All results given here are applicable as long as the

fractures can be represented by the uniform-flux

or

the

infinite-conductivity ideal izations.

When both wells are fractured,

we

assume that the

fractures are parallel. However, the model used in this

study can determine the pressure response for any value

of

the angle between the fracture planes. A review

of

the

literature indicates that in a given reservoir the fractures

usually are oriented along a specific azimuth) direc

tion.

6

Thus, the assumption that the fractures are

parallel

is

realistic. MousIi

7

considers the pressure

response when the fractures are not parallel.

Before proceeding, we offer some remarks on the

assumption used to model the fractures. Many feel that

any solution based on the infinite-conductivity idealiza.

tion is unrealistic for application to field data. Unfor

tunately, they fail to recognize that the pressure behavior

at a fractured well is governed by a· combination

of

parameters and depends on the dimensionless fracture

conductivity, F

cD

defined by

k w

F cD = x (1)

f

where

k is

permeability,

w is

fracture width, xf

is

frac

ture half-length, and

subscript

denotes the fracture.

It is

934

o

1

&oJ

II

>

(J)

(J)

&oJ

f

(J)

J)

&oJ

J

-,

Z 10

o

iii

z

'

i

o

1

1

1 1 l t

RATIO OF DIMENSIONL ES5 TIME TO DIMENSIONLESS DISTANCE SQUARED I

OL

f r ~

Fig. 2-Pressure response at the observation well when the

observation well is fractured

r

D = 0.4).

well-documented that if F

cD

500, the behavior

of

a

finite-conductivity fracture cannot be distinguished from

that

of an

infinite-conductivity fracture. In most fluid in

jection projects, the value of F

cD

for propped hydraulic

fractures is large

~ 5 0 0 )

since

x

is small. Also, field

experience suggests that many acid-fractured or in

advertently fractured due to high injection pressures)

wells respond as if they are uniform-flux fractures.

5

Thus, the assumptions regarding fracture conductivity

used

in

this study appear to be more than adequate for

potential applications.

All results given in this study are presented in terms

of

dimensionless variables for convenience.

The dimensionless pressure drop

is

given by

. (2)

where p

is

pressure at location

x

,y) at time

t,

h is

thickness,

q

is surface flow rate, B

is

formation volume

factor, and

l is

viscosity.

The dimensionless time

is

defined as

3.6

X 10

-6kt

¢ctJlL/

3)

where

¢ is

porosity

of

the porous medium,

C

t

is

system

compressibility, and L

f

is

total

fracture length at the

observation well.

The dimensionless distances are defined

by

the follow

ing equations.

X

XD=- (4)

L

f

Y

YD=-

(5)

L

f

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

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10

0

0.

0:

0

a:

0

'

:

::>

'

'

a:

'-

'

'

...J

Z

0

iii

z

0

~ ~

-1

10

A

- ~ - L - L L L ~ ~ ~ ~ ~ ~ ~

__ L L U i l L

__

~ L U ~

1° 0-

2

1

1

1 1

2

RATIO Of DIMENSIONLESS TIME

TO

DIMENSIONLESS DISTANCE SQUARED,

OLf

Fig, 3-Pressure response at the observation well when the

observation well is fractured r 0 =

1).

m

xmD= - 6)

L

f

d

d

D

=-

7)

L

f

and

r D = . J X ~ D + d D 2

.

 

8)

X m and d are the distance between the two well bores

along the X and Yaxes respectively (see Fig. 1). We

now discuss the results obtained in our study. As already

mentioned, the procedure used to calculate the pressure

response is given in Appendix A.

Results

For each boundary condition at the active well, we ex

amine eight values of r D in the range 0.2 ::; r D ::; 2.

f

r D

was greater than two, the fracture(s) had a negligible in

fluence on the pressure response for most of the cases ex

amined here. For each value of

r

D we examined several

values

of

the angle of orientation, a. In this paper, the

angle of orientation is defined as the angle between the

fracture plane and the line joining the two wellbores

(Fig. 1). Values chosen for

a

were based primarily on

the sensitivity of the solution to this parameter. We did

not examine conditions in which the fracture at the

observation well intercepted either the active well or the

fracture at the active well because

of

the lack

of

potential

applications.

Plane Radial Flow at the Active Well

Fig. 2 presents the dimensionless pressure response at

rD=O.4

and is typical of the results obtained

in

this

study. The parameter of interest is the angle of orienta

tion,

a.

The dashed line is the expected response

if

there

is no fracture at the observation well; it is the line source

solution.

8

Upon examining the shape of the curves in

Fig. 2, it is clear that there is no distinctive characteristic

or imprint to indicate to the analyst that a fracture exists

DECEMBER 1982

0.0

a:

o

lS

1

w

a:

iil

U

'

::

'

U

U

'

..J .,

Z 10

o

iii

z

'

o

ORlfNTATION, Q

0

90

RATIO

OF

DIMENSIONLESS TIME

TO

01 MENSIONLESS DISTANCE SQUARED, I

Olf g

Fig.

4-Pressure

response at the observation well when the

observation well is fractured r

0

=2).

at the observation well. Equally obvious is the drastic in

fluence of a vertical fracture at the observation well on

the pressure response.

f

the fracture is neglected,

serious errors will result in the estimate of the formation

parameters and erroneous conclusions regarding reser

voir heterogeneity will be made.

It is also clear from Fig. 2 that the pressure response at

an observation well that intercepts a fracture cannot be

modeled by assuming the fracture to be an annular skin

region concentric with the well bore. An annular skin

region around the wellbore would influence the pressure

response only

if

storage effects are significant. 2

The curves for larger values of

r

D indicate that the in

fluence of the fracture begins to diminish as

r

D ap

proaches one (Fig.

3),

and that the existence of the ver

tical fracture can be neglected if rD is greater than two

Fig.

4).

Uraiet et ai.

9

and

Cinco-Ley

and

Samaniego-V. 10 have found that the influence of a ver

tical fracture at the active well on the observation well

response diminishes as r y approaches one, where r y is

defined in terms of the fracture length at the active well.

The curves in Fig. 2 also indicate that at a fixed value

of

r

D the dimensionless pressure response becomes less

sensitive to the orientation angle if the orientation angle

is greater than 45 0. Consequently,

if

the compass orien

tation of a vertical fracture is to be determined by an in

terference test, care should be taken in analyzing the data

if

it appears that a is greater than 45 0. It is also evident

from Figs. 2 and 3 that the curves for a=90° should be

used to design interference tests if the observation well is

fractured. We make this recommendation because the

magnitude of the pressure change is the smallest when

a =90°. Note that the influence of a

is

small for values

of

tDL/rD

2> 10.

When we compared the results obtained here with the

line-source solution, we found that the line-source solu

tion falls between the curves for 15 °

::; a::;

90° if

0.4::;rD::;2. For small values of rD the trace

of

the

line-source solution intersects the responses for several

values of a. As the value of

rD

increases, the line-source

solution shifts toward the curve for a = 90 0 and almost

overlies this curve when

r

D = 2. When

r

D is less than

935

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cL

0

a:

0

w

a:

::J

U)

U)

w

a:

'-

U)

U)

Z

e

U)

z

w

URAIET ql ,

SOLUTION

THIS

STUDY

I

ORIENTATION.

a

.,

10

RATIO

OF

DIMENSIONLESS TIME TO

DIMENSIONLESS

OISTANCE SQUARED, t D L r ~

Fig. 5-Effect of location of fracture on the observation well

response.

0.4, the response

of

the line-source solution falls bet

ween the culVes for ex equal to 15 ° and 45 ° provided that

tDL/rD 2 is less than 0.6. For larger values

of

tDL/rD 2,

it falls between the

cUlVes

for

ex

equal to 0° and 15°.

Fig. 5 compares the effect of fracture location on the

obselVation well response for

rD

=0.4. The solid lines

depict the response when the fracture

is

located at the

obselVation well, and the dashed lines represent the

response when the fracture

is

located at the active well.

The results for the latter case are taken from Ref. 9. Note

that Uraiet

et al

9

considered the pressure response

caused by a uniform-flux fracture at the active well,

whereas our work assumes that the conductivity is in

finite. The results shown in Fig. 5 establish two points. *

First, the location

of

the fracture does not appear to af

fect the obselVation well response. Second, the fracture

type (infinite-conductivity or uniform-flux) appears to

have a negligible influence on the well response. These

results are not self-evident for a number of reasons. For

example, the flux distribution around an infinite-

*This result is valid only

if

r

0

0.4.

DIMENSIONLESS RADIAL

DISTANCE, ro 0 4

FRACTURE RATIO

1 0

o INFINITE - CONDuCTIVITY

A : UNIFORM - FLUX

0.

0

a:

0

w

a:

::J

U)

U)

w

a:

0.

U)

U)

W

..J

Z

0

iii

z

w

1

10

ORIENTATION.

a

l ~

IJ 'x

2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ L - ~ ~ ~ ~ ~

1°10 2 10-' 1

10

1 1:

RATIO Of DIMENSIONLESS TIME

TO

DIMENSIONLESS DISTANCE

SQUARED. Ol f

Fig. 7-Pressure response at the observation well when both

wells are fractured F

L,

=1, r

0

=0.4).

936

cL

0

a:

0

w

cr

::J

U)

U)

w

a:

'-

if

U

W

..J

1 j l

Z

e

U)

z

w

DIMENSIONLESS

TIME,

f Dl

f

Fig.

6-Effect

of the dimensionless distance,

r

0 on the

observation well response.

10

conductivity fracture

is

significantly different from that

surrounding a uniform-flux fracture.

5

Similar results

were obtained for other values of

rD

Fig. 5 justifies the

earlier obselVation that the results given here are ap

plicable as long

as

the fractures can be represented by the

uniform-flux or the infinite-conductivity idealizations.

Fig. 6 shows the effect

of

a change in

rD

for specific

values

of ex As

expected, at any given time the dimen

sionless pressure drop decreases as

rD

increases for all

values of ex This graph re-emphasizes the need to ac

count for the vertical fracture at the obselVation well if

rD

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0.0

0.

o

s 1

V>

V>

V>

UJ

..J .,

Z 10

D

in

z

UJ

o

o

= INFlf I

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should be negligible. However,

if

both wells are frac

tured, then from Fig. 10 it is clear that there can be cir

cumstances when the influence

of

the fractures can be

significant for much larger values

of rD

Our computa

tions indicate that the influence of the fractures becomes

negligible at much larger values

of

r

D if F L <

I

The effect

of

the change in the radial distance,

r

D for

a given value

of 0:

are similar to those

in

the previous

case.

For

a given value

of F L ,

the magnitude

of

the

response is a function of 0: and rD. Since the pressure

drop is not directly proportional to the dimensionless

radial distance, care should be taken in designing a test

if

the wells are fractured.

Figs. and

12

show the effect of

F L

on the dimen

sionless pressure response at two radia{ locations, 0.4

and

1.5,

respectively. Also shown

is

the dimensionless

pressure response for the case of

F

L

j

= ex> no fracture at

the active well).

In

Fig. 11 we note that as

F L

j

increases, the

magnitude

of

the dimensionless pressure drop increases

for both values

of 0:

considered here. The same holds

true for rD

=

1.5 and 0:=90° Fig. 12). However, the

dimensionless pressure drops decrease as

FL

j

increases

if 0:= 15°.

After examining all the solutions for this case, we

found that if

rD

: :;

0.8,

the dimensionless pressure drops

will increase as

F L

increases for all values

of 0:.

However,

if

r > 0.8 the dimensionless pressure drops

will decrease as

F L

increases

if 0: is

in the range

0°:-:; 0::-:; 30° (curve/ shift to the right), whereas the

dimensionless pressure drops will increase as

F

L

j

in

crease when 0:

is

in the range

45 °

: :; 0: : :;

90 °

curves shift

to the left).

Comparing the pressure responses for F L

j

=2 and

FL = ex>

we find that the change in the pressure

response

is

small if

F

L

j

becomes greater than 2. We also

found that

if

rD 2: 1.5

and

F L 2: 2,

the response

is

not

sensitive to

FLJ.

Therefore, for all practical purposes the

solutions for

F

L

j

=2 can be used for all values

of

F

  j

>2.

Discussion and Conclusions

The first objective

of

this work was to demonstrate that a

vertical fracture at an observation well can have a signifi

cant impact on the pressure response at the observation

well. Many assume that the skin effect at the observation

well can be neglected

if

there are no storage effects. As

shown here, this assumption is good only

if

the skin

region can be considered an annular region-i.e., in

finitesimally

thin.

Our second objective was to

describe qualitatively the effect of a fracture at the obser

vation and active wells on the observation well response.

Third, we delineated conditions under which the in

fluence

of

the fracture at the active and/or the observa

tion wells can be neglected. This is an important finding

useful in designing and analyzing interference tests. The

fourth objective

of

this work was to enable the analyst to

obtain quantitative information regarding the effect

of

vertical fractures on interference test data. For example,

the results given here can be used to determine the com

pass orientation

of

a fracture. Appendix B discusses an

example application for determining fracture orientation.

This example represents only one

of

the many possible

938

uses

of

the solutions obtained in this study. All dimen

sionless pressures obtained in this study are documented

in Ref. 7.

The results

of

this study highlight the need to incor

porate the wellbore conditions at both the active and the

observation wells. The characteristics of the porous

medium can be determined only after the wellbore condi

tions are incorporated properly.

The conclusions

of

this study are

as

follows.

1.

The existence

of

a vertical fracture at the observa

tion well has a significant influence on its response.

f

the active well

is

unfractured and

if

r

D <

2, the existence

of the fracture should be included in the analysis of

pressure data. f the active well

is

fractured, the ex

istence

of

the fractures can be ignored

if

rD

2:

2 and

FL 2: 1. f F L < 1

the existence

of

the fracture can be

n e ~ l e c t e d onl/if rD > >2.

2. The pressure responses at the observation well do

not possess any special features

or

characteristics to in

dicate to the analyst that a fracture exists at the observa

tion well and/or the active well. Therefore, the existence

of

these fractures would have t be determined in

dependently. Single-well pressure transient tests can be

used to identify the fractures. This requires running an

additional test at the observation well and measuring the

pressures at the active well during the interference test.

3. At a fixed value of rD the dimensionless pressure

drop becomes less sensitive to 0:

if

0:

is

greater than 45 ° .

Consequently,

if

the orientation of a vertical fracture is

to be determined, then care should be taken in analyzing

the data if 0: appears greater than 45

°

4.

The value

of

the pressure response

is

dependent on

the value

of rD

and 0:. Thus, care should be taken in

designing a test since the dimensionless pressure drop is

not directly proportional to the magnitude of rD Our

results show that the curves for 0:

= 90°

should be used to

design interference tests

if

the obervation well is frac

tured.

5.

f

the active and the observation wells are fractured

and FL

>

2, the curves for FL =2 can be used to

analyze{est data to obtain quantitative information.

6. f

F L 2:

1 and r

D >

2, the effect

of

the fractures on

the o s e r v ~ t i o n well response will be negligible.

Nomenclature

=

formation volume factor,

res m

3

/ stock-tank m 3 RB/STB)

C

=

total system compressibility, kPa - 1 psi - 1 )

d = perpendicular distance from the active well

to the fracture plane, m ft)

d =

dimensionless normal distance from the

active well to the fracture plane based

on LI

FeD

=

dimensionless fracture flow conductivity

FL =

fracture length ratio

h

formation thickness, m ft)

k

=

formation permeability,

/-tm

2

md)

k1

=

fracture permeability, /-tm

2

md)

LI = infinite-conductivity fracture length, m ft)

M

=

number

of

time intervals in which the

dimensionless time

is

divided

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N = number of equal length segments in which

the infinite-conductivity vertical fracture

is divided

p

reservoir pressure, kPa (psi)

q flow rate at the active well, stock-tank

m

3

/d STBID)

qD

= dimensionless fracture flow rate

qf = fracture flow rate per unit length,

d

m

3/

s

/

m

(cu ft/sec/ft)

r

=

radial distance between the center

of

the

active well and the center

of

the

observation well, m (ft)

rD dimensionless radial distance between the

center of

the active well and the center

of

the observation well based on L

f

at the

observation well

rD = dimensionless radial distance between the

center of the active well and the center of

the observation well based

on

L

f

at the

active well

r

w = wellbore radius, m (ft)

t = flowing time, hours

t

DL

f

=

dimensionless time based on

L

f

w fracture width, m (ft)

x -

distance along

x

axis, m (ft)

xD dimensionless distance along x axis based

on

L

f

x m =

horizontal distance between the active well

and the observation well, m (ft)

x mD

=

dimensionless horizontal distance between

the active well and the observation well

based on L

f

Y distance along Y axis, m (ft)

Y

D

dimensionless distance along

y

axis based

on L

f

a

=

angle between the

x

axis and

r

(compass

orientation), degrees

YJ = hydraulic diffusivity, Jlm

2 /[(Pa'

s)/kPa

I]

[md/(cp/psi

-I ]

Jl fluid viscosity, Pa' s (cp)

¢

porosity, fraction

Subscripts

cD

= dimensionless flow conductivity

D

=

dimensionless

f

= related to the fracture

i = initial; ith fracture segment

j

=

segment index

f

=

time index

w

=

wellbore

cknowledgments

Portions

of

this work were completed by N.A. Mousli in

partial fulfillment

of

the requirements for the MS degree.

Mousli acknowledges the support

of

the Arabian

American Oil Co. (Aramco). Computer time was pro

vided by Aramco (Shell Oil Co. Fellowship) and the U.

DECEMBER 1982

of

Tulsa's

Dept. of Petroleum Engineering.

We

are

grateful for this assistance.

References

I. Chu,

W.C.,

Garcia-R.,

J.,

and Raghavan, R.: Analysi s of In

terference Data Influenced

by

Wellbore Storage and Skin at the

Flowing Well, J Pet. Tech. (Jan. 1980) 171-78.

2. Tongpenyai, Y. and Raghavan, R.: The Effect of Wellbore

Storage Effects on Interference Tests, J Pet. Tech. (Jan. 1981)

151-60.

3. Cars aw , H.S. and Jaeger, J.e.: Conduction of Heat in Solids,

second edition, Oxford at Clarendon Press (1959).

4. Cinco-L., H., Samaniego-V., F., and Dominguez, A .N.:

Unsteady-State Flow Behavior for a Well Near a Natural Frac

ture, paper SPE 6019 presented at the SPE 1976 Annual Fall

Technical Conference and Exhibition, New Orleans, Oct. 3-6.

5

Gringarten,

A.e.,

Ramey, H.J. Jr., and Raghavan, R.:

Unsteady-State Pressure Distribution Created by a Well with a

Single Infinite-Conductivity Vertical Fracture, Soc. Pet. Eng.

J

(Aug. 1974) 347-60; Trans., AIME, 257.

6. Donohue, D.A.T., Hansford, J.T., and Barton, R.A.:

The

Ef

fect of Induced Vertically-Oriented Fractures on Five-Spot Sweep

Efficiency, Soc. Pet. Eng. J (Sept. 1968) 260-67; Trans.,

AIME,24O.

7. Mousli, N.A.:

The

Influence of Vertical Fractures Intercepting

Active and Observation Wells on Interference Tests,

MS

thesis,

U

of

Tulsa (July 1979).

8

Theis,

e.V.: The

Relation Between the Lowering of the

Piezometric Surface and the Rate and Duration of Discharge of a

Well Using Ground-Water Storage, Trans., AGU (1935)

519-24.

9. Uraiet, A., Raghavan, R., and Thomas, G.W.: Determination of

the Orientation of a Vertical Fracture

by

Interference Tests, J

Pet.

Tech

(Jan. 1977) 73-80.

10 Cinco-L., H., and Samaniego-V., F.: Determination

of

the

Orientation of a Finite Conductivity Vertical Fracture

by

Transient

Pressure Analysis, paper SPE 6750 presented at the SPE 1977

Annual Technical Conference and Exhibition, Denver, Oct. 9-11.

II. Richardson, L.F.: The Deferred Approach to the Limit,

Philos. Trans., Roy. Soc. London, Ser.

A

(1927) 226,299-361.

PPENDIX

A

Determination o the Pressure Response

As shown by Cinco

et al.,

4

the pressure drop caused by

an infinite-conductivity vertical fracture at the observa

tion well and plane radial flow at the active well can be

written as

. [ (X D

2

+YD

2

 

] I

rID rXmD+O.5

= - V2El -   - J J

4tD

20 05

XmD .

q

D(X D,

T

D)e - {[ x D

-

x D 2 +(y D

-d

D)2]/[4(t D 7 D ]}

tD

-TD)

·d .x D dTD (A-I)

The condition of uniform pressure

over

the fracture sur

face can be satisfied, as indicated by Cinco et al., 4 by

dividing the plane source (fracture) into a number of

segments and assigning a flux to each segment in such a

way that the pressures at the midpoints

of

each segment

are identical. I f we now replace the time integral in Eq.

A-I by M discrete time steps and divide the fracture into

939

8/9/2019 27. SPE-9346-PA

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N segments, we can write the following system of

equations.

N

PjDi =PDi +

qDiAij,

A-2)

j=l

for 1:5

i :5 N.

Here

P Di

represents the pressures at the

midpoints

of

each segment

of

the fracture and is given by

N

Ie,

M

l hE i [ XD

i

+d

D

  ]

+

4tD f I

j=l

where

_erf[X_Di_-X_mD+_o.5_-i_IN

]

2.JtD

-tD(/)

94

E l [(XD

i

-X

m

D+O.5- j N)2] ]

. A-3)

4[tD

-tD(f)]

and

E I (x) = - Ei( - x). . A-4)

Here

M

represents the time step under consideration,

qDi

represents the fluxes in each segment

i td

is identical to

tDL

in the text, and

qD (XD,

tD)=qD(X,

t )L/q.

The

product

qDiAij is

given by

The expression PDi

is

given by

N

I

j=l

In the system

of

equations given by Eq. A-2, all terms

on the right side are known except the dimensionless

flow rates, q

Db

at time level M. These rates can be ob

tained if we note that

P D(x

Di,d D,tDM) =PjD(xDi+

I,d

D,tDM)

and that

N

qDi,M=O.

i l

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Using these two relations, we may rewrite Eq. A-2

as

follows.

A

-A2J

A2]

A3d

A 12

-A22

A 22

A

32

)

A

IN

  A

2N

)

A

2N

  A

3N

)

Solving the matrix system Eq. A-5 will yield the

dimensionless flow rates for all fracture segments at any

time level

M.

Consequently, the pressure at the observa

tion well can be calculated by substituting these rates

in

anyone

of

the system

of

equations given by Eq. A-2. As

mentioned

in

Ref. 4, to ensure

an

accurate solution a

minimum number

of

fracture segments

is

required. A

discussion of the determination of the minimum number

of

segments

is

considered next and represents

an

impor

tant part

of

this work.

The dimensionless pressure drop

at

the observation

well was first calculated by arbitrarily selecting

rD =0 6

and cx=O. Several values for the total number of

segments,

N N=20,

30, etc. ), were considered.

We determined the optimal number

of

segments by first

determining the two consecutive N values that would

yield pressures within 1

of

each other

at

any instant

in

time. The smaller

of

these two values would be con

sidered tlie optimal number

of

segments. We found the

following.

o

(A-5)

1 The percentage difference between the calculated

dimensionless pressure drops at any given time de

creased as

N

increased. The percent change

in

the dimen

sionless pressure drop was less than one when N was in

creased from 90 to 100.

2. The percent differences between the calculated

dimensionless pressure drops for two consecutive N

values decreased as the dimensionless time in

creased-i.e.,

the solution became less sensitive to the

total number

of

segments at late times.

Similar results were obtained for rD

=0.2

and

cx=90°

It is

apparent from this analysis that 90 segments (or

more) are needed to simulate the infinite-conductivity

vertical fracture to obtain accurate solutions. Conse

quently, we first decided to use 90 segments

in

this

study. However, because of computer size (memory)

and time limitations,

it

was not possible to use 90

segments for long time intervals. The problem was over

come by applying

Richardson s

extrapolation

technique.

TABLE A 1 VALIDITY OF APPLYING RICHARDSON S

EXTRAPOLATION TECHNIQUE

Dimensionless Pressure Drop at the Observation

Dimensionless

Well for Case 1:

fo

=0.6 and a=OO

Time

Extrapolated

tOL,

tOLJ

f

02

N=30 N=60

Pressure<

N=90

0.001

0.2778 E -02 0.1530 E-03 0.2009 E-03 0.2169 E-03 0.2183

E-03

0.002 0.5556 E-02 0.1942 E-02 0.2268 E-02

0.2377 E-02

0.2377

E-02

0.003

0.8333

E-02

0.5323

E-02

0.5951 E - 02 0.6160

E-02

0.6159

E-02

0.004

0.1111 E-01 0.9668 E-02 0.1058 E-01 0.1088 E-01 0.1087 E-01

0.005 0.1389 E-01 0.1457

E-01

0.1573

E-01

0.1612 E-01

0.1611 E-01

0.006

0.1667

E-01

0.1980

E-01

0.2118

E-01

0.2164

E-01

0.2163

E-01

0.007 0.1944 E-01 0.2522 E -01 0.2679 E -01 0.2731 E-01 0.2731 E -01

0.008 0.2222

E-01

0.3073

E-01

0.3248

E-01

0.3306 E

-01

0.3306 E

-01

0.009 0.2500

E-01

0.3628 E

-01

0.3820

E-01

0.3884 E-01 0.3882 E

-01

0.01

0.2778 E-01 0.4184 E-01 0.4390 E -

  1

0.4459 E-01 0.4457 E-01

0.02 0.5556

E-01

0.9568 E -01 0.9871 E -

  1

0.9972 E -01

0.9969 E -01

0.03

0.8333

E-01

0.1426 E+01 0.1461 E+OO 0.1473 E+OO 0.1473 E+OO

0.04 0.1111 E+OO 0.1842 E+OO 0.1881 E+OO 0.1894 E+OO 0.1894

E+01

0.05 0.1389 E+OO 0.2219 E+OO 0.2260 E + 00 0.2274 E+OO 0.2274 E + 00

0.06

0.1667 E+OO 0.2563 E+OO 0.2606 E+OO 0.2620 E+OO 0.2620 E+00

0.07 0.1944 E+OO 0.2881 E+OO 0.2926 E+OO 0.2941 E+OO 0.2940 E

+00

0.08

0.2222 E+OO 0.3177 E+OO 0.3223 E+OO 0.3238 E+OO 0.3237 E+OO

0.09 0.2500 E+OO 0.3454 E + 00 0.3500 E+OO 0.3515 E+OO 0.3515 E+OO

Extrapolated pressure. P 0 = t P O N.60

-

t P 0) N.30

DECEMBER

1982

941

8/9/2019 27. SPE-9346-PA

10/12

0

Q.

(/)

/)

w

a:

Q.

. \

I \

\_----INFINITE

-CONDUCTIVITY

\ FRACTURE

A ~ 5 2 m _ ~

\

\

\

. \ \ 8

Fig. B-1-Relative pOSition of Observation Well C with

respect

to

the Active Wells A and B.

10

2

FLOWING TI

ME,

t,

hours

Fig.

B-2-

Type-curve match of Observation Well C,

In

terference Test 1 (A-C).

•

I TWO

POSSIBLE ORIENTATIONS

~ I N DATA

FROM

wELL

PAIR

A-C

...............

15°

A 0 - - - 5 2 m ~ - . . . . . . . _

--

 -.......

FRACTURE

\\.,

ACTUAL ORIENTATION USING

DATA FROM WELL

PAIRS

A-C

AND

B C

Fig. B-3-Actual orientation of the fracture plane.

942

To check the applicability of this technique, the

dimensionless pressure drops were calculated by using

30,60,

and 90 fracture segments for

rD

=0.6 and a=O

The extrapolated pressure drops then were calculated

by

Richardson s method using the pressure drops for 30 and

60 segments (see Table A-I). From Table A-I it

is

evi

dent that the agreement between the pressure drops ob

tained by using N=9 and the pressure drops obtained

by

Richardson s extrapolation technique

is

excellent.

Similar results were obtained for other dimensionless

radial distances and orientation angles.

This analysis proves that Richardson s extrapolation

technique

is

valid in this case. Hence, it has been used

throughout this study to calculate the dimensionless

pressure drop at the observation well using the pressures

calculated for 30 and 60 segments, respectively.

The procedure outlined also can be used if the active

well intercepts a fracture. To simulate the fracture at the

active well, the exponential integral on the right side of

Eq. A-I should be replaced by the following expression.

e +erf

[

rf

I -

x Du

) I

+

u

]

d r u

2.JrDu 2.JrDu

J rDu ·

Here, t Du XDu, and Y u are given by

t u =4 t DL

f

xF

L

f

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

XDu =2XD

XF

Lf

  (A-7)

and

YDu

=2YD

XF

Lf

 

(A-8)

where F L

f

is the fracture length ratio.

APPENDIX B

Example pplication

In this section we illustrate the determination of the com

pass orientation of the fracture at the observation well by

using the results obtained in this study. The example is a

computer-generated case since no field data are available

to

us at the present time. Fig.

B 1

shows the relative

position of the observation well, C, with respect to the

two active wells, A and B. (The need for two active

wells will be explained later.) The pertinent reservoir

data and the pressure data from Interference Test I (A-C)

and Interference Test 2 (B-C) are given in Table B-1.

The observation well was tested before the in

terference test (single-well test). It was found that this

well was intersected by

an

infinite-conductivity vertical

fracture. Estimates

of

the formation permeability,

k

and

fracture length, L

f

 

obtained from the single-well test are

given in Table B-1. Pressures also were recorded at the

active wells during the interference tests. These data in

dicated that the active wells were unfractured. The for

mation permeability at each active well was calculated

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by the conventional semilog method. These estimates

were approximately the same as the permeability at the

observation well. This result indicated that the porous

medium was relatively homogeneous

in

the area sur

rounding the three wells.

Using the value of

L

f

 

the dimensionless radial

distances between Wells A and C and between Wells

B

and C can be calculated as follows.

rA 52

rDA=-=-=O.4

. . . . . . . . . . . . . . . . . . . .

(B-l)

L

f

131

and

rB 79

rDB = - = - =0.6. .

. . . . . . . . . . . . . . . . . . .

B-2)

L

f

131

Interference Test 1 was analyzed

by

matching results

with the appropriate type curve. The curve for a= 15°

matched the data the best Fig. B-2). The formation

permeability was determined from the pressure match

point as follows.

PD=7.3XlO-

2

That is,

k=9x 10-

4

/Lm2.

The fracture length, L

f

 

is calculated from the time

match point as follows:

tDL

f

=7.36 x

10-

3

= [3.6 x

10

-6(9 x

10

-4 Lm

2

) 10 hours)]

-;-

[0.15 2.32 X

10

-6 kPa -1

(7x

10-

4

Pa·s) L

f  

m)2].

That is,

L

f

= 134 m. These values are in very good agree

ment with the values obtained from the single-well tests.

As shown in Fig. B-3, there are two possible positions

for the fracture plane

if

this result is used. Thus, even

though the orientation

of

the fracture plane with respect

to Line AC is known a= 15°), a second interference

test is needed. The results

of

the second interference test

were matched with the type curve corresponding to

rD=0.6. We found that the curve for a=30° matched

the data the best. From this match we calculated

k

and

L

f

and found them to be in agreement with the first in

terference test. Thus, the compass orientation now can

be determined as shown in Fig. B-3.

At this stage, it should be clear that the interference

data need not be matched with the type curves corre

sponding to

rD

=0.4 and 0.6. Since values of k and L

f

are known from the single-well tests, the pressure vs.

time data may be converted to dimensionless form and

TABLE B-1-RESERVOIR

AND PRESSURE

DATA

DECEMBER

1982

Reservoir and Well Data

Porosity

1>

fraction of bulk volume

Formation thickness h m

System compressibility c

t

kPa -

1

Viscosity

/l,

Pa·

s

Formation volume factor B res m 3/stock-tank m 3

Flow rate q, stock-tank dm 3/s

Distance between Active Well A and Observation Well C,

rA

m

Distance between Active Well B and Observation Well C, rB m

Permeability k / lm2

Fracture length L

t

, m

Pressure Data

0.15

18

2.32 x 10-

6

7.0x10-

4

1

0.2

52

79

8.9x10-

4

3

Interference Test 1

(A-C) ,

Interference Test 2

(B-C),

Flowing Time

t

(hours)

o

4

5

8

10

16

20

24

30

37

40

50

60

80

100

t:.p

(kPa)

o

26

40.5

78

110.6

180.8

222.5

258.7

307.4

364.4

386.7

t:.p

(kPa)

o

13.8

21.3

35.0

53.7

61.2

89.0

115.4

172.4

222.5

'Determined from an additional test at the observation well and pressure

measurements at the two active wells.

* * Determined from an additional test at the observation well.

943

8/9/2019 27. SPE-9346-PA

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placed on the type curves. The angle x then can be read

directly. But note that matching the data with the type

curves increases the level of confidence

in

the analysis

since it provides an opportunity for confirming the

results from the single-well tests.

In this example, values of r were calculated from the

pressure data obtained from the single-well test at the

observation well. However, if these data were

unavailable, r would be a parameter. All type curves

obtained in this study then should be used to match the

measured pressures.

Finally, we emphasize that the graphical matching

procedure suggested here can be replaced by a computer

approach if the dimensionless pressure vs. dimensionless

time data generated in this study are digitized. A

nonlinear least squares routine that minimizes the dif

ferences between the observed pressures and the

calculated pressures should be used. The differences

may be minimized by varying transmissivity, fracture

length, and fracture orientation.

944

Regardless of the approach used, it is imperative that

the pressure responses be measured at the observation

and active wells. In addition, single-well tests

may

be

needed to determine wellbore conditions at the active

wells.

SI Metric onversion Factors

bbl

x

1.589 873 E=OI

= m

3

cp

x

1.0

E-03

Pa s

cu ft

x

2.831 685

E-02

= m

3

ft

x

3.048*

E-Ol

m

mL

x

1.0*

E OO = dm

3

psi

x

6.894 757 E OO kPa

psi

-1

x

1.450 377

E-Ol

= kPa-

1

Conversion factor is

exact

SPEJ

Original manuscript received

in

Society of Petroleum Engineers office July 18. 1980.

Paper accepted for publication Dec.

10, 1981.

Revised manuscript received Aug.

9,

1982. Paper SPE 9346 first presented t the 1980 SPE Annual Technical Con-

ference and Exhibition held in Dallas Sept. 21-24.

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