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Determining Fracture Orientation

From Pulse Testing

DJebbar

Tlab

SPE, and Edwin O. Aboblse, SPE, U. of Oklahoma

Summary.

This paper presents a procedure for determining a complex orientation

of

a vertical fracture and the formation permeability

from pulse testing. Generalized correlations relating the quotient of dimensionless response amplitude and dimensionless cycle period

(D.PD/D.tcycD)

to dimensionless time lag are presented. The correlations can be used for analyzing pUlse-test pressure response at an un

fractured observation well resulting from pulsing a vertically fractured active well. Detailed procedures for the design and analysis of

pulse testing

of

a vertically fractured well are presented.

Introduction

The pressure behavior of fractured wells has been investigated ex

tensively, partly because of the large number of wells that are

hydraulically fractured.

As

a result

of

these investigations, several

reservoir and fracture characteristics can be determined from

pressure-transient testing. One valuable well-testing method that

has received limited attention is pulse testing. 1-5 This paper dis

cusses a method whereby pulse-test response data

of

a vertically

fractured well can be used to determine the formation permeabil

ity and fracture orientation Fig. I).

Uraiet

et al.

6 applied the uniform-flux solution to the analysis

of an interference test of a vertically fractured well. They showed

how the solution may be used to analyze pressure data in adjacent

observation wells to determine fracture orientation in a manner

analogous to standard interference tests.

Pierce

et at.

7 described a method to use pulse tests to determine

both the compass orientation and the length of hydraulic fractures.

The method, however, requires both post- and prefracture pulse

test parameters and thus cannot be applied to wells that already have

intersecting fractures.

Another method to analyze pulse tests in fractured wells was

presented by Ekie

et

at. 8

They used the uniform-flux solutions

of

Gringarten

et al. 9

to generate correlations that can be used to de

termine the orientation of a vertical fracture, provided that the for

mation permeability and/or fracture length are known. The objective

of this study is to develop generalized pulse-test correlations and

to outline a detailed procedure for the design and analysis of pulse

tests

of

vertically fractured wells. The method proposed

in

this study

uses correlations

of

the ratio

of

dimensionless response amplitude

to dimensionless cycle period and the correlations of Ref. 8 to de

termine the compass orientation of the vertical fracture and the for

mation permeability.

Pulse·Test Pressure

Response

Several fluid-flow models have been proposed in the literature to

describe the pressure behavior of hydraulically fractured reservoirs.

The vertical-fracture model developed by Gringarten

et al.

9 is

used

in this study.

In pulse testing, the flow disturbance is generated by changing

the flow rate periodically. The pressure response at an observation

well located at a distance D.r from the pulsing well at any time t

during the general m period can be obtained by superimposing the

responses resulting from flow-rate changes from the beginning of

the test to the m period. Consider the sequence

of

flow and shut-in

periods shown in Fig. 2.

I f

t is assumed that the pulse periods are

equal, the shut-in periods are equal, and all pulse rates are equal,

then their pressure response at the observation well resulting from

the fractured pulsing well is given by

141.2JLB

[

m- I

p x,y,t)=Pi-

qIPD xD,yD,tD)+

E qi+l_

ql

)

kh i=1

XP xD>YD [,v-

 ;

[, 1+R)

+

- IFH 1 -

R)

lJ)1

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

I)

Now

at Nigerian Natl. Petroleum Corp.

Copyright 1989 Society of Petroleum Engineers

SPE Fonnation Evaluation. September 1989

where

D.tl =D.t3=D.t5=D.t

pulse periods),

D.t2=D.t4=D.t6=D.t

shut-in periods), and R=the ratio of the pulsing period to the shut-in

period.

Consider three consecutive periods:

m, (m

+

I),

and

(m

+2). Let

tA,

tB,

and te denote the times at the points of tangency

of

the

pressure-response plots

of

the three periods and let

tem,O, tem.

1

and

tem,2,

respectively, denote the time lags for each

of

the periods

Fig.

2).

The pressure response can be expressed as

141.2BJL [ m

PA

=P i - qIPD XD,yD,tDA)+

E (qi+1 -qi)PD

kh

i=1

.

x [XD,yD,(tDA

Di

) ] }

2)

141.2BJL[ m+1

PB=Pi - qIPD XD,yD,tDA)+ E

qi+l-q;)PD

kh

i=1

X[XD,yD, tDB-t

Di

) ] }

3)

l41.2BJL [

m+2

and

Pe=P;- qIPD(XD,YD,tDC)+ E (qi+1

-qi)PD

kh i=1

X[XD,YD, tDe-t

Di

) ] }

4)

where

tDi= D.tD li 1+R)

+

E

- I ) i+I(1-R)] . . . . . . . . .

5)

2 L j=1

and PD can be evaluated with the Gringarten and RameylO and

Uraiet

et

at.

6 solutions. By definition

of

the time lag, the tangent

to the pressure response at tA is also the tangent at te and is parallel

to the tangent at

tB

Furthermore, the slope of the straight line con

necting the two points

tA

and

te

on the pressure-response curve

is equal to the tangents at

tA

and

te.

The various slopes may be

obtained by differentiating Eqs. 2 through 4. Because

1 [ YD2) l l - XD

)

exp erf

4--./

/rtD 4tD ~

+err ] }

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

6)

~

the tangents can be easily obtained. Also, the slopes can be repre

sented as

I f we equate the slopes obtained by the differentiation of the pres

sure equation to that obtained from Eq. 7 for all three points, we

obtain three equations with three unknowns:

tem,O,

tem,

1

and tem.2

The computation for the time lags that satisfy the three resultant

equations simultaneously can be simplified

if the three time lags

are assumed equal. Ekie

et

al.

8

tested this assumption for the pulse

ratios ranging from 0.3 to 0.7 and found the approximation to be

459

8/9/2019 28. SPE-11027-PA

2/8

0

0

r / pulsing

well

V f r a c t u r e o

, Responding

Wells

t ~

0

y

(x,y)

lh : Uniform flux

e I fracture

-x f

(0,0)

x

f

X

Fig.

1 Pulse

testing

of

uniform-flux fracture.

valid. Hence the assumption that tfm O

=tfm,l

=tfm.2 =tfm was used

in this study. Consequently, only one equation is necessary to solve

for the root of the resultant equation. For consistency, the deriva

tive at

PB

is chosen for the required equations and is of the form

g(tfm)=(141.2Bp.lkh>tqIPD'(XD'YD,tDB)

+ EI

(qi+l-qj)PD [XD,yD(tDB-to;)]}+ PC-PA .

  (8)

j=l te- fA

iii

c.

a

~ < l

CD

w

n n

0-0

w(/)

w

«a:

a:

w

ll.t

:r:a:

0::>

. . J /)

U en

w

a:

c

...

0

Developing

ulse·Test

orrelations

In Eq. 8, time lag

tfm

is the independent variable, because tA, fB,

and te can be expressed in terms

of

time lag, tt, cycle period,

t:.t

cyc

 

and pulse ratio, R , where

R'=t:.tll::.t

cyc

=

l / ( l+R).

',

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

9)

Dimensionless quantities are used in all the computations to reduce

the number of correlations to be generated. To accomplish this,

the following definitions are used:

tfD=ttlt:.t

cyc

, ,

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

10)

t:.tcycD

=O.OOO264kt:.t

cyc

lc/ C

r

p.t:.Tl,

................... 11)

t:.rD=t:.rlx

r

.Jx

2

+y2

Ix ,

......................... (12)

and 9=tan-

l

(xly) .

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

13)

These two dimensionless quantities are related implicitly in Eq. 8.

Consequently, the solution procedure involves choosing a dimen

sionless cycle period and then finding the corresponding dimen

sionless time lag for a selected pulse ratio. The response time can

then be used to calculate the response amplitude from

ilp=PB -PA

+(Pc

-PA)/(tc -tA)(tB - tA) (14)

The relation to the dimensionless response amplitude is given as

t:.PD=(kh/141.2qp.B)ilp

........................... 15)

I f

Eq.

15

is divided by Eq. 11, we obtain

t:.PDI :.tcycD

=

(26.

8264>crht:.r2IqBt:.tcyc)t:.p,

16)

which can be calculated without knowing the permeability.

Three different correlations can be obtained from the preceding

formulations: I)

t:.t

cyc

D(tfD,9,t:.rD,R'),

(2)

ilpD(tfD,9,t:.rD,R'),

and

3) il p

D

It:.t

cy

cD(tfD,9,t:.rD,R'). Correlations 1 and 2 have been

presented by Ekie

et al.

8 Correlation 3 is generated in this study

with the results from Eqs. 8 and 12.

Results

A large number

of

figures and tables giVIng the values of

t:.PDIt:.tcycD

for the first four pulses were generated for a

dimensionless-radial-distance range of

0.2

:S

t:.rD

:S 1.4 and pulse

ratio varying from 0.3 to 0.7.

11

The dimensionless cycle period

investigated depends on dimensionless radial distance, t:.rD, and

the fracture orientation, 9, and varies from 0.01 to 7.04. This dimen-

TIME, t,

min

Fig,

2 Pulse test

terminology,

460 SPE Formation Evaluation, September 1989

8/9/2019 28. SPE-11027-PA

3/8

lD

Orientat on

of

Fracture, e

15

30

C

45

u

60

>

75

u

-

0.1

90

8/9/2019 28. SPE-11027-PA

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TABLE 1-RESERVOIR PROPERTIES

Reservoir and Well Data

Viscosity,

p

cp

Porosity, C/> fraction of bulk volume

Thickness, h, ft

Flow rate, q STB/D

FVF,

B RB/STB

System compressibility,

c

t

, psi-

1

Distance between Fractured Well A

and Responding Well

B,

M B ft

Distance between Fractured Well A

and Responding Well

C,

i lr

c,

ft

Fracture half-length, x

I,

ft

Approximate permeability, k md

Pulse-Test Data

Pulse period, ilt minutes

Shut-in period,

RM,

minutes

Pulse ratio,

R

Time lag, t

f

  Well B, minutes

Time lag,

t

f

  Well

C,

minutes

Response amplitude, t..p,

Well B, psi

Response amplitude, t..p,

Well C, psi

0.6

0.2

15

900

1.0

1.24x10-e

280

200

200

8

84

36

0.7

12.5

8.75

30

60

response amplitude. The reservoir properties are the formation per

meability, porosity, and thickness; the fluid viscosity; the total com

pressibility of the fracture half-length; and the vertical fracture

orientation. The following procedure provides the guidelines for

the proper design and analysis

of

pulse tests of vertically fractured

wells with the correlations presented in Ref. 8, in conjunction with

the t:J.PDlt:J.tcycD correlations presented

in

this study.

De.lgn

Procedure

Designing a pulse test requires the determination of two criteria:

the pulse time and the expected pressure response. The proper pulse

time must be determined so that the test falls around the midpoint

of the range of effectiveness,

2

and the expected pressure response

must be calculated so that we can predetermine the pressure-gauge

sensitivity required. The following procedure is recommended.

TABLE 2-PRESSURE DATA

t..p

t

(psi)

(minutes) Well B

WeliC

---

10

0.005 6.442

20

12.687 29.567

30 27.090 56.134

40

42.663

82.502

50

58.384

107.16

60 73.826 130.14

70

88.798

151.50

80 103.22 171.39

85 110.22 180.84

90

116.75 188.76

100

122.60 187.96

105 122.77

183.45

110

121.99 178.12

120

118.92 166.90

130 117.06 162.58

135

119.39 168.07

150 133.30 193.69

180 167.90 245.88

200

190.28

275.98

220

203.48 283.32

230

200.07

269.43

240 194.36 254.49

260

193.90 256.83

270 201.75 271.73

300 230.50 316.23

462

I

/:

Possible

Orientation

/ I

Using Only Well C \

/ . ; o o ~

;

.

./

30

0

Well C

/

/

/

Fracture Plane

\ Y k I l ~

Possible Orientation I

Using Only Well B

Fig. 7-0rlentatlon

of

fracture plane In Example 1.

I. Select a pulse ratio,

R .

A pulse ratio near 0.3

is

recommended

if even pulses will be used in the analysis, while a pulse ratio close

to 0.7 will be more suitable for analysis with odd pulses because

the response amplitude is greatest at those pulse ratios.

9

2 Choose a dimensionless time lag that ensures that the test falls

within the range of effectiveness. A dimensionless time lag of 0.14

or 0.17 is recommended, depending on whether the odd or even

pulses, respectively, will be used to analyze the results of the test.

3

Calculate the dimensionless radial distance with the distance

between wells and fracture half-length.

4. Determine the dimensionless cycle period from the

t:J.tcycD(tev)

vs. tev correlations presented in Ref. 8 Estimates of

fracture orientation from other sources may be used for selecting

the correct set of correlations; otherwise a fracture orientation of

45° should be assumed.

5. Determine the dimensionless response amplitude from the

t:J.PD(tm)2

vs.

tev

correlations reported in Ref. 8.

6. Estimate the cycle period with

t:J tcyc =cP/A.ctt:J.r2tcycDIO.OO0264k. .

................... 17)

An approximate value

of

the formation permeability

is

needed for

this calculation.

7. Estimate the response amplitude by

t:J.p=(141.2q/A.Blkh)t:J.PD

........................... 18)

Hence, the proper gauge sensitivity can be determined.

350

300

250

a:

.0

0

200

. .

:

c

w

150

..

a:

/

3OP8I

_____

  -

W

::>

100

III

..:

III

900

a:

w

:

50

600

0..

0

300

..J

u..

0

0

50 100 150 200

250

300

TIME MINUTES

Fig. 8-Pulse·test response

for

Well B (Example 1).

SPE Formation Evaluation, September 1989

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350

ii

300

0.

Ii

250

0

.c

a:

200

UJ

. .

a:

150

u.i

:::l

f-

  n

c(

n

100

a:

J

900

a:

l.

50

600

0

300

..J

LL

0

0

50

100 150

200 250

300

TIME MINUTES

Fig. 9-Pulse-test response

for

Well C Example 1).

Analysis Procedure

After a pulse test

is

run in a vertically fractured well, the follow

ing procedure can be used to determine the fracture orientation and

the formation permeability. The tangent method

1

is

used

in

the fol

lowing procedure.

1. Plot pressure change vs. time on a Cartesian graph.

2. Determine the response amplitude and time hig from Fig. 2.

3. Calculate

the

cycle period and pulse ratio with R =lJ.tl(lJ.t

ws

+

lJ.t).

4. Calculate dimensionless time lag and dimensionless radial

distance.

5. Calculate lJ.pvllJ.tcycv with Eq.

16.

6. Determine the fracture orientation from the /1pvllJ.tcycv corre

lations for a chosen pulse, pulse ratio, and dimensionless radial dis

tance with the dimensionless time lag and the /1pvllJ.tcycv

calculated

in

Step 5.

7. Determine the dimensionless cycle period from the correspond-

ing set of lJ.tcycv tev) correlations.

8

8. Calculate the average formation permeability with

k=4>p.c

tlJ.r2tcycVIOJXX)264tcyc (19)

9. Determine the dimensionless response amplitude from the cor-

responding set

of lJ.pv tev)2

correlations.

8

10. Calculate the value of formation permeability from Eq.

18

and compare

it

with the result obtained in Step

8.

xample

Example 1 illustrates how the proposed procedure may be applied

in

the design and analysis of pulse tests of vertically fractured wells.

1.0

First Odd Pulse

4ro =1.4,

R

= 0.7

e

I

15

3D

45

1.5

x 10 - w . . . . u - _ - - I . _ ~ ~

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

. i . . . . _ - - I . _ . . . L . - ~ - - . J . . . . L . L . . J

0.01

0.1

1.0

Fig.

10-Compass

orientation

of

vertical fracture with respect

to Wells A and B Example 1).

SPE Fonnation Evaluation, September 1989

TABLE 3-PUlSE-TEST PARAMETERS

Time lag,

t ,

minutes

Response amplitude,

Ap,

psi

B

12.5

30.0

Well

C

8.75

60.0

The reservoir has the properties shown in Table 1. Fig. 7 shows

the relative locations of the wells involved in the test and Table

2 shows pressure data.

Design of the Pulse Test

1. Select the pulse ratio. Assume that we wish to analyze test

results with the first odd pulse; then a pulse ratio

of

0.7 will give

the optimum response.

2. Determine dimensionless time lags. Because an odd pulse will

be analyzed, a dimensionless time lag, tev

=0.14, is

desirable.

3. Determine dimensionless distance. If the test is designed with

Well C properties, then lJ.rv =200/200= 1.0.

4. Determine the dimensionless cycle period. When the lJ.tcycv

tev)

vs.

tev

correlations

of

Ref. 8 are used for the first odd pulse

attev=0.14,

e=45°, lJ.rv

=1.0,

and R =0.7, then tcycD=0.0889.

5. Calculate the dimensionless response amplitude. When the

lJ.pV tev)2

vs.

tev

correlations in Ref. 8 are used for the first odd

pulse at the same values (Step 4)

of

tev, e, lJ.rv, and R , then

lJ.pv tm)2 =

-0.00011.

6. Calculate the response cycle. From Step 4,

lJ.tcycv

=0.0889/0.14=0.653.

With Eq.

17

the response cycle

is

lJ.t

cyc

=

(0.2)(1.24

x

10 -6) 0.6) 200)2 0.635)/0.000264 8) = 1.8

2.0

hours. Thus, from Eq. 9, pulse

period=R x 120=84

minutes and

shut-in period =

120-84=36

minutes.

7. Calculate the response amplitude. From Step 5, /1pv=

-0.000111(0.14)2 = -0.0056. With Eq.

18

/1p= 141.2 900) 0.6)

1.0) -0.0056)/ 8) 15)= -3.56 psi [-24.55 kPa]. Hence,

pressure-gauge sensitivity can be specified.

Analyzing the Pulse Test.

1.

For pressure-response plots, Figs. 8 and 9 show pressure

responses of Wells

Band

C, respectively.

2. Determine pulse-test parameters with the first odd pulse from

Figs. 8 and 9 (see Table 3).

3. Calculate the cycle period and pulse ratio. Cycle period,

lJ.tcyc=lJ.t+lJ.tws=84+36=120 minutes. Pulse ratio,

R =

lJ.tllJ.tcyc =84/120=0.7.

4.

Calculate dimensionless time lag,

tm,

and dimensionless dis

tance, lJ.rv.

tm

= 12.5/120=0.104 (Well B),

tm

=8.751120=

o

u

>-

u

0..

0

8/9/2019 28. SPE-11027-PA

6/8

c

1.0 r - - - - - - - - - - - - - - - - - 0.1

Ar : 1.4,6

:3 0

0

0.058,

~

I

I

I

I

I

I

I

10

'

0.075

ArO:1.0 ,6 :45

0

:

0.01

:i ~ ~

I

I

I

I

0.01

0.1

1.0

Fig.

12-Correlatlon of

the dimensionless cycle period (Ex

ample 1).

0.073 (Well C),

ArD =280/200=

1.4 (Well B), and

ArD =2001200

= 1.0 (Well C).

5. Determine the ratio of dimensionless response amplitude to

dimensionless cycle period with Eq. 16.

For

Well B,

ApD/AtcycD =26.826(0.2)(1.24 x 10 -

6

)(15)(280)2(30)/(900)(1.0)

(2.0)=0.13039. For Well C, Ap

D

IAt

cyc

D=26.826(0.2)(1.24x

10 -

6

)(15)(200)2(60)/(900)(1.0)(2.0) =0.13306.

6. Obtain the compass orientation of the vertical fracture. For

Well B (Fig. 10), teo =0.104 and Ap

D

/At

cyc

D=0.13039, 0=30°.

For

Well C (Fig. 11), teo=0.073 and Ap

D

/AtcycD=0.13306,

0=45°.

The orientation of the fracture that results in one consistent direc

tion from the analysis

of

Wells Band C pressure data (Fig.

7)

is

the true compass orientation. Hence, the compass direction of the

vertical fracture is established as 0° north (north/south direction,

Fig. 7).

7. Determine the dimensionless cycle period with Ekie et al. 's

AtcycD(teo)

vs.

teo

correlations, and Fig. 12. For Well B, for

teo=O.104

and 0=30°, AtCYCD(teo) =0.0579. Thus, AtcycD

=0.0579/0.104=0.5567.

Thus, for Well C, for

teo=0.073

and

0=45°,

AtcycD(teo) =0.0749 and AtcycD =0.0749/0.073 = 1.026.

8. Calculate formation permeability. With Eq. 19 and the results

of Step 7, for Well B, k=(0.2)(0.6)(1.24x 10-

6

)(280)2(0.5567)

North

Well D

\

, \

Fractured

-  \

Pulsing Well E

Well C

Possible Location

of

Fracture Plane Using

Data from Well C

/1'

/

v

/

/

/

,157 j2

\[

I,

\ Possible Location of

\ Fracture Plane Using

Data From Well D

Fig. 14-0rlentatlon of fracture plane In Example 2.

464

a.

C

8/9/2019 28. SPE-11027-PA

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TABLE 5 EXAMPLE 2 ANALYSIS

RESULTS

psi

tf,

minutes

tiD

ilro

R

ilPolMc't 'o

o from Fig. 15

• After second fracture values.

8

Well C

0.711

1.0

0.025

1.2

0.5

0.6388

o

Well D

0.085

11.5

0.2875

1.2

0.5

0.1135

18

The possible fracture orientations are indicated in Fig. 14. As

in

Ref. 8, two possible fracture orientations were obtained from

the analysis. Because only two wells had sufficient data to deter

mine fracture orientation by the method

of

this study, the disparity

in

fracture orientation could not

be

investigated further. The different

fracture orientations may

be

a result

of

heterogeneities in forma

tion storage. The storage calculated from the two different wells

Wells C and D) vary by a factor of 1.6. The extent of possible

heterogeneities is further reflected in the significant variation of

hydraulic diffusivity for the different well pairs.

For

the purpose of comparison, the fracture orientation obtained

with the method of this study and the results reported in Refs. 7

and 8 are shown in Table 6.

Another possible interpretation of the results obtained

is

that the

pulsing well may be intersected by more than one vertical fracture.

This conclusion is supported by the second fracture obtained by

Ekie et

al.

8 and this study. It is unlikely that the various pulse

testing analysis techniques can be used to establish the direction

ofmUltiple fracture because of the underlying assumptionsof a single

vertical fracture in the theoretical developments. This interpreta

tion is even more likely because the pulsing well was subjected to

two massive fracture treatments.

On the basis of the results presented, it seems that another frac

ture with a general trend in a direction perpendicular to the frac

ture intersecting Well C may exist.

It

will be useful to confirm the

existence of multiple fractures

by

other methods.

Discussion

Two previous studies

7,8

presented techniques that could be used

to determine fracture orientation and/or fracture length of a verti

cally fractured well from pulse testing. These methods have cer

tain limitations and this study has overcome some of them.

For instance, the Pierce et

at

7

method requires pre- and post

fracture pulse-test data. This requirement

is

not necessary to apply

the method presented in this study. Also, the Pierce

et

at method

lacks generality and relies on a simulator to generate the necessary

correlations. The generalized correlations presented

in

this inves

tigation are easy to use.

Ekie

et

al. 8 developed excellent correlations that could be used

in determining the orientation of the fracture and/or fracture half

length. This method, however, cannot be used in determining the

formation permeability. This limitation is overcome by use

of

the

ilpDIilteyeD correlations.

Conclusions

1 The pulse test of a vertically fractured well may be analyzed

to determine the compass orientation

of

the fracture and the aver

age formation permeability

of

the reservoir zone influenced

by

the

test.

2. In the design of pulse tests of vertically fractured wells, a

dimensionless-cycle-period/pulse-ratio combination that will result

in

a dimensionless time lag of 0.14 for odd pulses and 0.17 for

even pulses ensures that the test will fall into the effective range.

3. f the angle between the fracture plane and the line connect

ing the responding well and pulsing well is < 60°, then the

ilpDlilteyeD correlations will provide more accurate values for

compass orientation of the fracture.

4. For a given dimensionless distance, pulse, and pulse ratio,

if the calculated value of ilpDliltcycD falls above the 0° fracture

orientation values,

it could mean that the observation well is con

siderably influenced by the fracture.

SPE Fonnation Evaluation, September 1989

1.0 .--------------------

o

o

>.

o

8/9/2019 28. SPE-11027-PA

8/8

t = time, minutes

tD = dimensionless time based on fracture half

length

t£ =

time lag, minutes

t£D = dimensionless time lag

t£m,i = ith time lag after m period, minutes

tlt = pulse period

tl t

cyc

= cycle period

tltcycD

= dimensionless cycle period

tltws

=

shut-in period

x = distance along x axis, ft [m]

XD = dimensionless distance based on fracture

half-length

x = fracture half-length in x direction, ft [m]

y

=

distance along y axis, ft

[m]

y = dimensionless distance based on fracture

half-length

j = compass orientation of fracture plane

hydraulic diffusivity, md-ft/cp

[md' m/Pa' s]

p.

= viscosity, cp [Pa' s]

> = porosity, fraction

Subscripts

A,B,C

=

points

of

tangency or wells

f = fracture

i = initial or index

j = index

m = pulsing or shut-in period

References

I.

Johnson,

C.R.,

Greenkorn, R.A., and Woods, E.G.: Pulse-Testing:

A New Method for Describing Reservoir Flow Properties Between

Wells, JPT Dec. 1966) 1599-1604; Trans., AIME, 237.

2. Brigham, W.D.: Planning and Analysis of Pulse-Tests,

JPT May

1970) 618-24;

Trans.,

AIME, 249.

466

3. Kamal, M. and Brigham, W.E.: Pulse-Testing Response for Unequal

Pulse and Shut-In Periods,

SPEl

(Oct. 1975) 403-09;

Trans.,

AIME,

259.

4. Kamal, M. and Brigham, W.D.: Design and Analysis of Pulse Tests

With Unequal Pulse and Shut-In Periods,

JPT Feb.

1976) 205-12;

Trans.,

AIME, 261.

5. McKinley, R.M., Vela, S., and Carlton, L.A.: A Field Application

of

Pulse-Testing for Detailed Reservoir Description,

JPT

(March 1%8)

618-24; Trans.,

AIME, 243.

6. Uraiet, A., Raghavan,

R.,

and Thomas, G.W.: Determination of the

Orientation

of

a Vertical Fracture by Interference Tests,

PT

(Jail.

1977) 73-80;

Trans.,

AIME, 263.

7. Pierce, A.E., Vela, S., and Koonce, K.T.: Determination

of

the Com

pass Orientation and Length

of

Hydraulic Fractures by Pulse Testing,

JPT Dec. 1975) 1433-38; Trans., AIME, 259.

8

EIde,

S.,

Hadinoto, N., and Raghavan, R.: Pulse-Testing of Verti

cally Fractured

Wells,

paper SPE 6751 presented at the 1977 SPE

Annual Technical Conference and Exhibition, Denver, Oct. 9-12.

9. Gringarten,

A.C.,

Ramey, H.J.

Jr.,

and Raghavan, R.: Unsteady

State Pressure Distributions Created by a Well With a Single Infinite

Conductivity Vertical Fracture, SPEl (Aug. 1974) 347-60; Trans.,

AIME,257.

10

Gringarten, A.C. and Ramey, H.J. Jr.:

The

Use

of

Point Source So

lution and Green's Functions for Solving Flow Problems in Reservoirs,

SPEl

(Oct. 1973) 285-96;

Trans.,

AIME, 255.

11. Abobise, E.O.: Analysis of Pulse Tests of Vertically Fractured Wells ,

MS thesis, U. of Oklahoma, Norman, OK (Dec. 1981).

SI

Metric

onversion

actors

bbl

x

1.589 873

E Ol

m

3

cp

x

1.0*

E 03

Pa's

ft

x

3.048*

E Ol

m

md

x

9.869233

E 04

p

2

psi

x

.6.894757

E OO

kPa

psi-I

x

1.450377

E Ol

kPa-

1

Conversion factor is exact.

SPEFE

Original SPE manuscript received for review Aug. 27, 1982. Paper accepted for publica·

tion Oct.

5,

1987. Revised manuscript received Sept. 15, 1988. Paper (SPE 11027) first

presented at the 1982 SPE Annual Technical Conference and Exhibition held in New

Orleans, Sept. 26-29.

SPE Formation Evaluation, September 1989