Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture

8/10/2019 Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture

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SPE 6014

TransientPressure ehaviorfora WellWithaFinite Conductivity

VerticalFracture

HEBER ClN(X) L.

F. SAMANIE(30,V.

N. DWINQUEZ A.

MEMBERS WE-AIME

ABSTRACT

A mathematical model was developed to study tbe

transient behavior o/a well with a finite-conductivity

vertical /racture in an iniinite slab reservoir. For

values 0/ dimensionless time 0/ interest, tD 210-3,

the dimensionless wellbore pressure, PwlD~ can be

correlated by the dimensionless group,

wkl/xf k ,

where w, kf, and x

i

are tbe width, permeability, and

half length o/ t e /racture, respectively, and k

represents the {ormation perrneabi ity.

Results when plotted as a [unction of pwfD us

10g tD give, for urge t~, a 1. 151+iope straight

line; hence,

semilogaritbmic pressure analysis

methods can be app ied. When p[ottcd in terms of

log pwjD vs log tD, a family o/ Curves 0/ cbaraCter.

istic shape result, A type-curve matching procedure

can be used to analyze early time transient presszdre

data to obtain the formation

and jracture

characteristics.

INTRODUCTION

Hydraulic fracturing is an effective technique for

increasing the productivity of damaged wells or

wells producing from low permeability fortnations.

Much research has been conducted to determine the

effect of hydraulic fractures on well performance

and transient pressure behavior. The results have

been used to improve

the design of hydraulic

fractures. Many methods 1-14 have been proposed to

determine formation properties and fracture charac-

teristics from transient pressure and flow rate data.

These methods have been based on either analytical

or numerical solutions of the transient flow of fluids

toward fractured wells. Recently, Gringarten

et al.14

made an important contribution to the ‘analysis

of

transient pressure data of fractured wells, They

presented a type-curve analysis and tttree basic

solutions: the infinite-fracture conductivity solution

(zero pressure drop along a verticai fracture), the

uniform flux solutiori for vertical fractures, and the

Original manu; cript received AnSociety of Petro:eum Ertglneem

office Aug. 15, 1976. Paper mcepted for

publication Feb. 17, 197?.

Revised nranuscrlpt received April 28, 1978. paper (SpE 6014)

first preaentcd at the SPE-AIME 5 Itt Annual Fmll Technical

Conference and Exhibition, held in New Orleans, Oct. 3-6, 1976.

0037-9999/78/0008-60 14 00.2S

@ 197s Society of Petroleum Englneerc of

AIME

INSTITUTO MEX\CANO DEL P ROLEO

MEXICO CITY, MEXICO

uniform flux solution for horizontal fractures.

Although the assumption of an infinite fracture

conductivity is adequate for some cases, we must

consider a finite conductivity for large or very low

flow capacity fractures. Sawyer and Locke15 studied

the transient pressure behavior of finite-conductivity

vertical fractures in gas wells. Their solutions

cannot be used to analyze transient pressure data

because only specific cases were presented.

In this study, we wanted to prepare general

solutions for the transient pressure behavior of a

well intersected by a /inite.conductivity vertical

fracture. The solutions sought should be useful for

short-time or type-curve analysis. We also wanted

to show whether conventional methods could be

applied to analyze transient pressure data for these

conditions. A combination of both methods, as

pointed out by Gringarten

et aft,

14

should permit an

extraordinary y

confidence level concerning the

analysis of field data.

STATEMENT OF THE PROBLEM AND

DEVELOPMENT OF FLOW MODELS

The transient pressure behavior for a fractured

well can be studied by nnalyzing the solution of the

di~ferential equations that describe this phenomenon

with proper initial and boundary conditions.

To

simplify the derivation of flow models, the following

assumptions are made.

1. An isotropic, homogeneous, horizontal, infinite,

slab reservoir is bounded by an upper and a lower

impermeable strata.

The reservoir has uniform

thickness,

h, permeability,

k,

and

porosity, +,

which are independent of pressure.

2. The reservoir contttins a slightly compressible

fluid of compressibility, c, and viscosity, p, and

both properties are constant.

3. Fluid i~ produced through a

vertically

fractured

well intersected by a /ully

penetrating, /initer

conductivity /racture of half length, Xp width, W?

permeability, kf, and porosity, @ . These fracture

{

haracteristics are constant. F uid entering the

wellbore comes only through the fracmre.

A system with these assumptions is shown in

Fig. 1. In addition, we assume that gravity effects

are negligible and also that laminar flow occurs in

the system, Under these conditions, the flow

AUGUST,1~

2s3



+4

;> { ,,;:

phenomenon may be described by the diffusivity

equation in two dimensions.

16 TO

facilitate the

solution of this equation, two flow regions will be

considered — (1)

the reservoir and (2) the fracture.

F RAC TU R E F LOW M OD E L

‘l”he fracture is considered as a homogeneous,

finite, slab, prous medium of height, h, half length,

xl, and width, w. Fluid enters the fracture at a rate

q (x, t)

per unit of fracture length, and flow across

the edge of this

porous

medium is ne~ligible

because the fracture width is very mall compared

with the fracture length. lhe Iaat assumption allows

us to consider a linear flow in the fracture and

permits simulation of well production by a uniform

flux plane source of b and w, located at the wellbore

axis (Fig. 2).

Unsteady-state

flow in the fracture may be

described by the equation,

a2Pf ~

qffx.t) ~f~cft apf

——

‘+kf ~= kf at

ax2

O<%<%=........(l)

&

subject to the, following conditions.

Initial conditia,

pf(x, t=o) =

Ppo~x~xf”

WELLBORE

lAtPERM\ABLE

BOUND~RIES

i

~ ~

I

[

[ I

I FRACTURI

[ I

I “

I i

I

[ I

I

t I

I

I I

I

--

.

FIG.

1 —

F I NI TE -C O ND U C TI VI TY VE R TI C AL

F RAC TU RE I N AN I NF INI TE S LAB R ES ER VOI R.

F I G .

2-

F RAC TU R E F LOW M OD E L,

W

4

Boundary conditions,

aPf q

zE-

~o”-ziqi

and

m

E-

O” ”””””’ ““”

(2)

. X=X

f

/

In Eq. 1, q x, t) is a source term that represents the

fluid flow from the reservoir to the fracture.

The solution of Eq. 1 with initial and boundary

conditions Siven by Eq. 2 is expressed in dimen=

sionless form by this equation:

-/

2n+l

qfD(x’8T)

2n-1

z-

where

pf ~9 ‘D) =

‘D =

t.

D

and

[

(xD-x’

)2

-

4(kf@ct/k$fc f ~)

1e

x’ d~ *

3)

Wpf w ]

141.2 qwB~

‘f

0.000264 kt

$W pf2

2qf(x’ ,T)

qfD(x’90 =

*

4)

% ‘f””

Eq. 3 gives the dimensionless pressure &op in

the fracture at location XD and dimensionless time

tD. ‘his equation was obtained by applying Green

and source functions and the Newman product method

extensively discussed by Grin8arten and Ramey.22

SOCIETY OF PSTaOLWM CNG1NES8S

JOURNAL



R ES ER VOI R F LOW M OD EL

The transient pressure behavior in the reservoir

may be

studied by considering the fracture as a

plane source of height, b, length, 2X1, and flux

density q@ t) (Fi8. 3). The dimensionless pressure

&op at any point in the reservoir may be obtained

from the following equation:

o

-1

-[

Iyx’ *+yD2

4 (tD-T)

1

qD(%’,T)e -

(tD-T)

dx’ d’c . w (5)

where

P ~9Y~9tJ =

qD(x’ ,T) =

and

Wpi-p(x,y,t)]

141.2 ~BIJ

[email protected]~f

qw

L. . . . . . . .

‘D = X=

, . (6)

Eq. 5 also was derived using Green and source

functions.

To solve Eqs. 3 and 5 simultaneously, continuity

between the two flow regions must be established,

The dimensionless pressure drop pff) (x~, t~) and

flux density

qlD XD, tD)

in the fracture model must

eqUd the dimensionless pressure drop pD (xf)J yD$

tD) and flux density qD (xD, tD) on the plane

source of the reservoir model, respectively. That

is,

pfDtxD$tD)

= pD(xD}YD=O, tJ)) 0 . 0 (7)

%(x,t)

PLANESOURCE(FRACTURE

FIG. 3

-

R ES E RVOI R F LOW M OD E L.

A1’GUW’. 1978

and

qfD(x ~)

- qD ~, tD) o . . . .(8)

for

A combination of Eqs. 3, 5, 7, and 8 and use of

Poisson’s summation formula yields

1

2

— tD+—

*1

1

~ cos (n7rxJ

cfDf

7r2rlfD n=l n

I

-q ~Dn2w2tD

l-e

I

.

o

-1

-nf

Dn2T2 (tD-~)

-x’) e

n . 1

dx t d~

., ..., . . . . . . .,, ,,. ,.

(9)

‘k’ ft

where ~

fDf =

Txf @ct

and

k c ~

%D

—.. . . . . . . . .

= Wfc f t

(lo)

Eq. 10 is a Fredholm integral equation where the

Mknown is

qfj (%D ~f$. CfDf

is the dimensionless

fractw: storage capacity, and qfD is the dimension-

less hy&aulic diffusivity of the fracture.

METHOD OF SOLUTION

Eq. 9 can be solved by discretization in time and

space so thar the fracture is divided into 2N equal

segments (Fig. 4) and time is divided into K different

intervals. It is assumed that fr cture flux has a

stepwise distribution in time r.:~d space. In other

words, the flux density qD~,e of a fracture interval

w



is constant for a given segment

i

and time inteival 1.

For a fracture segment j, Eq. 9 becomes

-

112Ti2

DAtK, k-l

-e

1

where

I

12H

SEGMENTS

/.

I

)

, , ,

1

, ,

I

1 1 1

1

1 1

1

“ ,-

123

.

F I G . 4-

F RAC TU RE D IVI DE D I NTO N E QU AL

SEGMENTS.

z%

- 1-1/2

‘Di N

‘%,2-I= ‘DL - ‘D&l

+ e r f

and

- ~i,j%

n

[ II

- 6i,jEi

9

The arguments of the erf and Ei functions are

defined

a

j-i+l/2

~i,j 2N

B

~ j -j-i-l/2

9

2N

= j+;y2

‘id

and

6

~j=H L

9

By writing Eq. 11 for all fracture segments, a

system of equations is obtained where the unknowns

are the g~i,t ‘s. Solution of such a system for each

time interval produces values for the fracture flux

distribution.

The dimensionless pressure drop at

any point of the system can be calculated by using

the discretized forms of Eqs. 3 and 5. Although the

theory presented here does not consider formation

damage near the fracture caused by fracturing fluid

SOCIETY OF PETROLEUM ENGINEERS JOURNAL



loss, the equations may be modified to include a

variable iikin damage along the fracture.

DISCUSSION OF RESULTS

A computer program was writtsn to determine the

flux distribution and dimensionless pressure drop

along the

fracture.

A sensitivity analyais was

conducted to obtain accurate results. We found

that soluticma do not change appreciably when more

than 20 segments are taken per fracture half length,

x . Results also indicated that the solutions were

/

accurate enough for practical purposes if at least

10 intervals were considered in each log cycle of

logarithm of dimensionless time. Therefore, in all

cases studied, the fracture half len@h was divided

into 20 equal segments and 10 time intervals wete

taken in each log cycle of dimensionless time.

Cases were simulated for values of CfDf ranging

from 2 x 104 to 10-3 and values of q p from 10

to 108. ‘Iheae ranges were based on published

fracture characteristics data. Analysis of the

results showed that as soon as most of the fluid

produced at the wellbore comes from the formntion

(i.e., the expansion of the fracture system is

negligible), solutions can be correlated by one

parameter that depends on CfDi and ql~ constants.

Fortunately, this holds for times of interest. This

correlating parameter was found to be

kf

W

(12)

cfDf ‘fD =

. . ...*

~kxf” “,”

An important feature of this variabie is that it does

not depend on the porosity and total compressibility

of the formation and fracture. It is essentially the

dimensionless fracture flow conductivity,

With regard to the symbol for this correlating

parameter, Ramey 23 suggested using a product of

two dimensionless variables, such as

‘f

‘fD = ~

and

The first is the relative fracture permeability and

the second represents the dimensionless fracture

width. Large values for the product (kfD tu~D) rep-

resent highly conductive fractures; conversely,

small values represent fracturea of low conductivity.

Small values of the product may be caused either by

low fracture permeability or large /racture

length.

For much of the following discussion, we refer to

the condition of “low fracture conductivity” —

remember that we mean a dimensionless conductivity.

EitAer low

fracture permeability or long fracture

lcn@h, or both, may be the physical phe ‘mena

involved.

Solutions for the atcady-stato flow case were

co~related by Pratsls using the “relative fracture

capacity, ”

which may be expressed as

a ‘T&’

The use of the term “capacity” is a misnomer. The

correct term is

“conductivity.”

In the following,

the dimensionless fracture conductivity will be

considered as (k ~ .

{

tu(D). Although

there

was a

constant n in

t e

original correlating parameter

(see

Eq, 12), and (tr/2) in Prats’ expression, wc

&oppcd the constants

for the

sake of simplicity.

The solutions obtained in this study were

compared, where applicable, with solutions published

in the literature. Results for a highly conductive

fracture (CfDf = 10 ‘3, q ID = 107, and

kfp wjD =

I@/m) ~;how excellent a8rcemcnt with the infinite-

conductivity

solution

of Gringarten

et al.lg

Differcnccs between rhc

two

solutions arc less than

1% for small values of dimensionless time, and

less than 0.025% for other times of interest.

Fig. 5 shows the dimensionless flux along the

fracture at different values of

D. For small values

of [D, the fhtx

density is uniform along the fracture.

Also, for small times, the flow rate from the

formaticm into the fracture, is less than the total

WCII f14Dwrate.

This results from the storage

capacity of the fracture. For intermediate and

large vtdues of t~, the well flow rate is generated

by the expansion of rhc system outside the fracture;

under these conditions, the total area under each

flux density curve in Fig. 5 is equal to unity. And

finally, for large values of

tD,

the flux density

becomes stabilized as discussed by Gringarten

et al. for an infinite-conductivity fracture. Fig. 5

‘ 7fD ’+ .= lo’

It

@kf .,04

rack

3iJ

~>s

tB=2 -1

t*rl~ lo-4

Ialo’+

—

Ttlls STUDY .

6RlM6ARTEN,~.

(Iatlnll. contucllsltp solution)

1

1

I

I

.2

,4

.s

.8

XD?*

F II G . 5

- FLU X D ISTRI B UTION AT VARIOU S

TI ME S ALQNG A H IG H LY C OND U C TI VE

VE R TI C AL F R AC TU R E .

AllOUST, 1970

m



also shows the stabilized flux distribution presented

by Gringartcn

et al. Good

a8reement was found

between both solutions.

It is of interest to know the effect of

fracture

conductivity on rite

stabilized

flux density along

the fracture,

Fig. 6 shows that for a highly

conductive ftacture (i , kfD w fD 2 300), t he flux

density is high at the portions of the fracture away

horn the wellbore. As

fracture

conductivity

decreases, the flux density changes so that flow

entering the. portion of the

fracture close to the

wellbore becomes steadily more important. For

instance, in a low conductivity fracture (kp I fl) =

0.63), about 70% of the flow comes from the nearest

half of the fracture. However,

approximately two-

thirds of the total flow comes

from the farthest

half

in a highly conductive fracture (k@ w\D 2 300).

‘Ibis emphasizes the importance of creating high

conductivity fractures

to

overcome the flow

restrictions created by the wellbore damage zone.

These findings agree with results presented by van

Poollen.zO

Fig. 7 shows a graph of stabilized dimensionless

pressure &op vs dimensionless distance along the

fracture

fOr W/cd

values of

k/D wffj.

Thk

pres sure

&op is the difference between the pressure at any

point on the fracture and the pressure at the tip of

the fracture. The curves on this figure show that,

for

highly conductive fractures, the pressure drop

along

the fracture ‘,is small and sometimes

negligible. As the fracture conductivity decreases,

the pressure &op becomes increasingly greater,

and as the fracture permeability approaches the

formation permeability, the pressure drop distribution

(not shown here) corresponds to that for radial flow.

Fig. 7 also presents the results published by Prats18

for steady-state flow. Excellent agreement was

z

“-

-“ s

z -

e

W

k

=(

t

I 1 1

.2 .4 .* ,0

+

os ,

FIG . 6 —

S TAB ILI ZE D F LU X D IS TRI BU TI ON F OR

D I FF E RE N T F R AC TU R E C ON J XI CTI VI TI E S.

zm

.

found,

The wellbore pressure drop reduction caused by

a

fracture

is usually handled as a pseudo-skin

factor,

St, which is defined as the difference

between the dimensionless pressure drop for a

fractured well and that

for an

unfractured well.

Although s is a function of tD, it becomes a

{

unction o the geometry of the system only for

]ar8e values of tD.

l%e pseudo-skin factor for a fractured well in an

infinite reservoir may be applied to fractured wells

in a finite, circular reservoir whenever the radius

of influence, ~f, of the fracture is smaller tnan the

external radius of the reservoir. The radius of

influence of a fracture is defined as the radius

beyond which the pressure distribution created by

the fracture is similar, for practical purposes, to

that

for

radial flow. The radius

of

influence for an

infinite-conductivity fracture is about 4 x . This

means that values of s obtained for an infinite

/

system may be used for a finite reservoir when

re/x ~ 4, These results also are valid for a finite-

d

on uctivity

fracture because rfi

for this case is

less than r~i for an infinite-conductivity fracture.

Fig. 8 presents SI as a ~nction of dimensionless

frac~re conductivity, ~/D k D, for a system where

re/rw

= 2,000 and ~e/x, =

f

O. A particular case is

presented in this figure to compare solutions from

this study with those published in the literature.

Fig. 8 shows, that s is negative, indicating an

increaae in well pro uctivity. It also shows that

there is a strong variation of s for small values of

/

@

klD,

and as the value of racture conductivity

increases, Sf approaches a stabilized value. Data

published by McGuire and Sikora21 and Pratsls

appear tc agree well with the results of this study.

A general correlation for the fracture skin factor,

St, may be obtained if S 1 is expressed

as a

function

-

0

-“

I

—

THIS

STUDY

k

PRATS(Stoo4y-Stoto)

I

o

F IG . 7 - D I ME NS IONLE S S P RE S S U RE D ROP

D I S TR I B U TI O N AL O NG A F I N I TE -C O ND U C TI VI TY

F RAC TU RE (t D > 5).

SOC IE IWOF

ETROLEUM

E N G I N E E R S

OURNAL



.

Of WjD kiD

$ x//’w’

l%is can be shown by

cmnbining the line source solution and the definition

for fracture skin factor. Fig. 9 shows that a graph

of (s/ + k %//rw) V$ (w/L @) ttiil give a $ingle

curve chat may be used to estimate Sf if values

for

‘“ ‘w’ and ‘~Dk{D ace ‘govided’

he .dlmcnsion ess wellbore pressure drop vs the

logarithm .of dimensionless time for ,various values

of k D wff) is shown in Fi8. 9 and presented in

{ab e 1. Analysis of these results shows that for

times

of itjterest$ tD ~ l@3, solutions can be

correlated using only one

parameter. If practical

values

of q D and CID1 are considered, a unique

{olution wi ‘ be obtained for a constant value of

‘fD ‘/D’

Fig. 9 indicates that, as the fracture

conductivity increases, the dimensionless wellbore

pressure drop for a fixed time decreases, and for

wfDkFd

greater than

300,

the solution is essentially

equal to the infinite-conductivity solution of

Gringarten et al. All the curves on this figure do

follow, for large times, a straight line of slope

1.151, characteristic of the semilo8arithmic methods

of pressure analysis. Also indicated by a dashed

line is the approximate start of the semilogarithmic

I

I

I

A

— THIS SIUOY

-3

A MCGUIRC@ndSIKORA

4

FIG . 8 —

P S EU DO= S KI N F AC TOR F OR A F INITE -

C ON DU C TI VI TY VE RTI CAL F RAC TU R E.

F I G . 9- P S E U D Ck SK IN F AC TOR F OR A WE LL WI TX-1A

F I NI TE -C ON D U CTI VI TY VE R TI C AL F R AC TU R E ,

straight line for different values of w DkfD. This

{

ime varies between

tD

equal to 2.5

or very

low

fracture conductivities, and tD equal to S for high

fracture conductivities. This is in agreement with

the findings of Gringarten et al,

for

infinite-

TABLE 1

— DIMENSIDNLES8 PRESSURE FOR A WELL WITH

A FULLY PENETRATING, FINITE=C43NDUCTIVITY

VERTICAL FRACTURE

~~(/+ - Pwf)

Pwm =

“ —

1412 Qt3/A

t~ =

0.W0254

M

+/Actx,z

Pwfr)

kfwl kxf

= kfDwfD

tf)

o.211 lr 2f7

—. . .

Iolr

2olr

loon

1 x 10” 3 0.5449 0.2443 0. 1733 6m6 Em E65i i6

2

3

4

6

6

7

8

9

1 x

10-2

2

3

4

6

6

7

8

Q

1 x 10-1

2

3

4

5

6

7

8

Q

1

2

0.6330

0,7024

0.7520

0,7926

0,6273

0.6576

0A846

0.2090

0,W13

1.0s37

1.1661

1.2838

1.326e

1,2602

1.4266

1.4676

1.504Q

1.5357

1 . 772s

1.9263

2.0414

2.1340

2.211s

2.2164

2.3372

2.3607

2.4371

2.757$

0 2681

0.31s0

0.3432

0.3633

0.28W

0,3s 3

o.4cm

0.4224

0.4341

0.6181

0.5788

0,6272

0.6682

0,7040

0.7361

0.7653

0,7s21

0,8170

1.W1O

1.1269

12282

1.3CW2

1.37s4

1.4iot

1.41m

1,5424

1.5865

1.6m

tfj

-—

3

4

5

6

7

8

9

1 x 101

2

3

4

6

6

7

8

0

1 x 102

2

3

4

5

6

7

8

Ix @

O,a?

G O

3.0914

3.2W?

3.28SS

3.3684

3.4312

3.48s3

3.5414

3.6883

4.0670

4.2304

4.3417

4.4327

4.5m7

4.5763

4.6351

4.8578

5sm41

6.2387

5.3505

S.4220

5.5532

mm

5.7270

5.768S

5.s353

17

= 3

2.2144

2.3212

2.40s2

2.463 3

2.54s0

2.W5

2.6561

2.s205

3.2~

3.3430

3.4540

3.8458

3.6228

3.68s2

3.74s0

3m05

4.1486

4,34s2

4.4s2s

4.6045

4.8056

4.7736

4.68S4

4.6SS3

4.Q5m

0.2056

0.2280

0.2475

0.2632

0.2770

0,2ss3

o.3m

0.3110

0.3207

0.3630

0.4460

0.4= 0

0.5297

0.8030

0,5229

o.62tB

0.6456

0.6s91

0.s453

0. M65

1.ms

1.1442

1.2121

1.2715

1.3243

1.3719

1.4153

1.7156

kfwl kxf

2fl

1.S016

2.0371

2.1435

2.2311

2.3056

2.3705

2.4270

2.47S4

2$620s

3.0214

3.1643

32753

3@3s81

3.4430

3.50s5

3.8683

3.6208

3 668

4.16%

4.3131

4,4247

4.5156

4.5s28

4.65W

4.7185

4.7711

O.llfxl

0.1277

0.1424

0.1553

0.1666

0.1773

0,1871

0.1W2

0,2047

0.2720

0.3221

0,3833

0.3W8

0.4304

0,456s

0.4640

o.5mo

0,5316

0,7015

0.82CB

0.9143

0.SS18

1.0562

1.1163

1.1682

1.21m

1.2577

1.5546

lolr

1.73s4

1,8730

1.07ss

2.0671

2.1414

2.2ml

2.2684

2.3147

2.6553

2.8561

2.swB

3.lfns

3.2007

3.2775

3.3440

3.4a27

3.4553

3.6013

4.0038

4.1475

42ss0

4.3502

4.4272

4.4s39

4.5528

4.6055

0.0346

0.1120

0.1265

0.1392

O,lom

0.1610

0.1708

0.1796

0.1s81

0.2540

0.3047

0,34%

0.3502

0.4122

0.4408

0.4665

0,4905

0.5129

0.6620

o.8cm8

0;6S40

0.0712

1.0374

1.0355

1.1472

1.1s30

1.236s

1.5320

207r

1.7175

1.85$~

1.9577

2.0450

2.1193

2.1830

2.2412

2.232q

2.6331

2.6338

2.0768

3.0576

3.17s4

3.2551

3.3217

3.ss04

3.4330

3.776 3

3.s815

4.1252

4.2367

4.3278

4.4049

4.4716

4.83m

4.5631

0.W14

0.0SS6

0.1130

0.1258

0,1360

0.1472

Oclm

0.1658

0.1742

0.2407

o.2scr3

0.3310

0.3662

0,3074

0.4255

0.4514

0.4753

0.4s75

0.5651

0.7545

0.6774

0.9544

1.=05

1.0754

1.1300

1.1768

1,21=

1.5152

1oo17

—,

1.8W8

1.6~ 0

1.9308

2.0270

2.1013

2.1659

22231

2.2745

2.6150

2.6157

2.9= 5

3.0805

3.1602

3.2370

3.3035

3.3623

3.4148

3.76m

3.s633

4.1070

4.2166

4.30s7

4.3657

4,4535

4.5123

4.5650

Alx dm . 1970

2s



,

.

81

I 1

I

1 1/

A

p=-unlfonll nux WLullon

r

IllFllllTCWilOSCTIVITVOLUTION

CnlnoABTcn,u

o

1

1

I

,0-

108

IOs

,, ,+y ‘o

F IG , 10 —

km Vs t D F OR A WE LL WITH A FINITE -

C ON D U C TI VI TY VE R TI C AL F R AC TU R E .

con du ct ivit y fr act ur es . Th en t pr ovid ed that sufficient

data on the straight-line portion of the pressure

curve

are

available before boundary effects influence

a test, the formation flow capacity may be obtained

in the usual way by conventional semilog analytical

methods. Also plotted in Fig. 9 is the uniform flux

solution for vertical fractures presented by

Gringarten et

al.

This

solution

follows

the infinite-

conductivity solution at small values of times. For

intermediate times, this solution behaves as a

variable conductivity fracture solution, eventually

following a finite-conductivity fracture solution of

‘jD~fD

about equal to 4.4.

Fig. 11 shows the results of Fig. 10 plotted as a

function of the logarithm of pWD vs the logarithm of

ID. At small values of dimensionless time, the

curves have a distinct form for different values of

lo ~ .

.

.

9

8

*

a

II

‘fD ‘ID “

This feature of the solutions can be used

to analyze field data by a type-curve matching

technique. From this kind of analysis we can

determine the formation permeability, k, the half-

fracturc length, x,, and the fracture cr?nductivity,

k,w.

We assume that estimates for formation

porosity,

+, fluid viscosity, p, and total

compressibility, Ct, are available.

Log-1og

type-curve

matching is a technique

commonly used in well test analysis. As pointed

out by Gringarten et

al,, a combination

of this

technique with conventional semi log analytical

methods permits a hi8hly confident analysis of

field data.

Solutions presented here do not include wellbore

storage effects. However, the /racture

storage

capacity in a highly conductive fracture creates an

effect on the transient wellbore pressure behavior

similar to that caused by

wellbore

storage capacity.

Recently, Ramey and Gringarten12 presented finite-

difference solutions for the transient behavior of a

well crossed by a high-volume, infinite conductivity

vertical fracture. They defined a dimensionless

storage coefficient that appears equal to C D~ Fig.

[2 presents the dimensionless pressure so ution for

~,D/ = 0.1, and also shows data presented by,Ramey

and Gringarten for the same case. A good agreement

exists between the two solutions; differences are

less than 2.5%.

Although results

for the transient pressure

behavior of a fractured well in a finite drainage

system are not presented in this study, they may

be generated by means of a desuperposition

technique.19 AIso, wellbore storage effects can be

9919.’

II***,

,, ,8 *..,*

. . . . . .

m-,

.:1”:;

.,

““”-’-T.

.

.,,

. .. ...i.

r

I i :

---

--+-- --:- .- { . 4 . .

:..

..-

;_,

+; .; :.:,

.,

.

,“

,

,

tm’FR

I 1

,

1 1

, , ,

..:, j

I+tlll t I

~

--l

‘

1.-L.LL L 1..1ill .. :

..1..:.1---- L.-.1

F SEMILO~~ ~ ,;

.- .-,

, ,

1 I

J

-....-.,

.-. . .

.

r-l-

----

I

I

I [ [1

r

,..

I i 11 1 11 ill I

1

I 1

,

1

:-I

d .

.

, ,

—.

I

.—

I I t

1 I

,,,

1

t I 1

, , ,

k

.,, ,,, .

t

. . ———

io-s ‘ ‘ “ ‘* ’*i’&i ‘ ‘ “ *“”7

‘“’”40 ~ “

* ’.. IO*

lo~

:.00; 264kt

t,8—

+ P Q *

FI G . 11

—

f fD Vs ~D FOR A

F I NI TE -C O ND U C TI VI TY VE R TI C AL F R AC TU R E .

m

SOCIETY OF PETROLEUM ENGINEERS JOlkNAL



incorporated into the solution by using the principle

of superposition.

We expect that in certain field

cases, the pressure behavior for a fractured well

may not follow the solutions presented here, These

deviations may be related to several causes, such

as a high dependence of fracture conductivity on

pressure, wellbore storage, and partial penetration,

to mention some. Sometimes, the nature of the

deviation can be inferred from a careful examination

of field data, For instance, a weH crossed by a

fracture with a conductivity highly dependent on

pressure will exhibit a different behavior in both

bui Idup and drawdown tesrs, in a drawdown test,

the fracture conductivity will decrease continuously,

while in a buildup test, fracture conductivity will

exhibit an increase. In Mb cases, pressure

data

will cut across the curves presented in Fig, 11.

EXAMPLES OF APPLICATION

The examples presented here are synthetic

pressure drawdown tests

for a well

crossed by a

finite-conductivity fracture. The pressure data were

analyzed using a type-curve matching procerdun.,

This technique consists of plotting the presw re

&op, Ap, on the ordinate vs flowing time, t, on the

abscissa of a log-log paper of the sane size as

Fig. 11. Normally, a tracing paper is placed over

the type. curve, and the major grid lines are traced

for reference. The grid of Fig. 11 is used to plot

actual data on the tracing paper. Next, the data

plot is moved vertically and horizontally

over

Fig.

11, keeping the grids of the type-curve and those

of the data plot parallel to each other until the

best match is obtained with a curve of Fig. 11.

From this figure, the ‘value of klD UfDcorresponding

to the curve that fits the pressure data fs read. A

convenient match point is pieced and the values of

(Ap)M

and (At)M are read from the data plot. The

corresponding values lying directly under this point

on Fig. 11 are (pW@M ~d

(~D)M.

The

formation permeability, h, and the half-fracture

length, xi, may be obtained by substitution of the

match point data into the expressions for

1

I

I

c of~ *:1 0.1

,.~

10+

10”1 10-1 I

10

t,

F IG . 12 — Wt D VS t D F OR A H IG H -VG Li.J ME , I NE I NI TE -

C O ND U C TI VI TY F R AC TU R E .

TAf3LE 2-

PRES8URE DRAWDOWNDATA FOR A WELL

CRC%8EDBY A FINITE=CONDLJCTIVITY FRACTURE

(EXAMPLE 1)

+ = 0.3

h=30ft

Ct

= 20 x 10+ pal-*

p = 0. 85 C p

B = 1.6s bbl/S TB

%

= 260 S Tf 3/ )

rw

= 0.26 ft

D r a w d ow n D st a

Pi - Plvf

Jh&/

JE 9 L

0,25

0.00

1.

2.s

s.

10

20

30

40

00

80

70

80

90

lW

160

57

68

79

108

134

188

210

238

201

200

228

311

321

334

343

384

dimensionless pressure drop, pu D, and the

himension less time, tD, respectively. e definitions

of pw~ and @ are given by Eq. 4. Once these two

parameters are known, the fracture conductivity,

klw,

can

be

estimated from

kfD wtD

data match. The

definition of

kf~w@

is given by Eq. 13.

E XAMP LE 1

This represents the results of a pressure drawdown

test on an oil well. Pertinent drawdown and

reservoir properry data are given in Table 2. Fig.

13 shows the application of the type-curve matching

technique for this case. A good and unique match

is obtained for

kfD wfD =

2rr and it appsars that the

test was not run long enough to reach the semilog

straight line. Thus, type-curve matching is the

best method to analyze the data of this test.

The formation permeability may be estimated from

the pressure match, Ap =

100 psi, pw~D= 0.47, and

data from Table 2, as follows:

‘pwfD)M

= 0.47 =

(k md) (30 ft) (100 psi)

141. 2(250, STB/D) (0.85

,Cp)

(1.65 ,bbl/STB]

I

’

I

M ATCH POI HT

L

Niloo Dll,t o.4?

10

Allloohr

,lC I.C

I0

I

t:s1J

1

101

Id

1

1

t

Io? IOJ

,0.1

I 10

,08 ,~

tlhr81

FIG , 13 — AP P LIC ATION OF TH E TYP E -C URVE

MATCHING

TE CH NI QU E (E XAMP L E 1).

A U GI Sr . 1 97 s

261



This yields

k- 7.76m~.

The half-fracture length then is calculated from

the time match, t=

100hours and tD = 1.6; that is,

(tD)M = 1.6 =

0.000264(7.76

m d 1 0 0

hrs)

0. 3(0.85 Cp)

(20x10-6 P& (xf2 ft2)

Hence,

‘f

=

158.4 ft.

An

estimation of the

f racture conductivi ty

jw,

may

be obtained from the kfD jD match and the

valuesofk andxf already computed; thus:

(kfw red-f t)

‘fDwfD = 2

= (158.4 ft)(7.76 rnd)

and

kfw = 7.72 x 103 md-f~.

Now, the pseudo-skin factor is calculated from

Fig. 9 by using

kfDwjD

and %f/ru, values. Since

kpwp = 2rrandxflrw =633.60,

‘f

= -5.65”

Although the dimensionless fracture conductivity

has an intermediate value, the fracture is large

enough

to yield a good improvement in well

productivity.

EXAMPLE 2

Table 3 presents the data fora drawdown test in

a fractured well. Figs. 14 and 15 show the applica-

TABLE 3-

PRESSURE DRAWDCWNTEST DATA FORA

FRACTURED WELL (EXAMPLE 2)

4 = 0.18

h=55ft

C, = 1 8x 10 -1 3w 1

p =

l,8cp

B =

1.4 bbl/STB

%.

= 195sTB/D

t

~ =o,25ft .

Drawlown Dsta

—.

/+ -

Pwf

Pi - Pwf

I )

-i @ - J__ _@Q-

houm)

1

81

24

2s3

2

103 30 307

3

128 40

333

4 144 50 358

.5

157

60

378

6 17C 70 3m

7

182

80

411

8 192 90

424

9 201 100

439

10

207

120

45s

12

223

150

4s4

14 2s

200

522

16

247

250

54s

20

267 3W

571

tion of the type-curve and the semilogarithmic

techniques, rcapertively. Fig. 14 indicates that a

good data match is obtained with the curve for ,

k ~9@~D

1

= 10 rr.

Furthermore, some data points of

t is test fall in the semilog straight line.

The pressure match may be taken as I$p = 100

PSi 2UK /)w /D = 0.4. Similarly, the

tirric

match may

be chosen as

At =

100 hours and tn = 3.3. These

match points,

in conjunction

data given in Table 3 yield

k _ 5.05 md

Xf = 83,23 ft

%

with- the additional

wkf = 13.2

X

10= md-ft

and

‘f

= -5.06

From Fig.

15, the slope of the semi logarithmic

straight line is

m = -285

psi/cyck and P1 h~~r =

2,732 psia; then,

162.6

qB1.i ~ -

k=. ~

162. 6x195x1. 4x1.8

-285x55

k= 5.1 md

*

g

MATCH POIMT

II

1

1

;—

4

I I

AP~loo?Di,,o*o 4?

J

~t,look~, .1. .9.3

II

I

1

. .

1

d’

10-2

10 ’

lt

10

I

;02

lG’

o

10 At (h~)ld

103

F I G .

14 —

AP P L I CATI ON OF TH E TYP E -C U RVE

M ATC H IN G TE C HN IQU E (E XAM P LE 2).

o

00

2400 -

\

00 ”

Pwf

o

( psia)

.

,m~

m=-aC~C &

2000 -

L

10-’ 1

10

102 103

r (

hm)

FI G . 15 — S E MILOG G RAP H FOR E XAMP LE 2.

SWIETY OF PETROLEUM ENGINEERS JOURNAL

62



I

‘1 hour-pi

‘f

= 1.1513

rn

-W3

[oM;r:]3.22751

2732-2600

‘f

= 1*1513

-285

-

Log

[

5.1

.18X1.8X18X10-1X (.25)2

1

+ 3.2275

‘f

=

-5.04”

Results

from both procedures are in excellent

agreement; however, additional data concerning the

fracture geometry may be ~ound from the rype-cutve

analysis. Comparison of Fig. 14 with Fig. 15 is

interesting. Only four data points lie to the right of

the dashed line in Fig. 14, while 10 or 11 points

seem ro lie on the semilog straight line in Fig, 15.

This would include all points to the right of the

arrow in Fig. 14. This results because the analytic

solutions approach the semilog straight line

asymptotically. This practical application of the

criteria for the start of the semilo~ straight line

indicates the rules may be stretched with acceptable

results.

If pressure data are not available for early times,

a uniqueness problem

arises when type-curve

matching is applied. This means that data will

match any of the curves in Fig. 11 because they

are similar for large

values of time. As a

consequence, fracture geometry parameters cannot

be estimated, and the only applicable technique

will be the semilogarithmic method.

CONCLUSIONS

The main purpose of this study was to provide a

solution that could be applied to analyze transient

pressure data for wells with a finite-conductivity

vertical fracture. From the results of this

investigation, the following conclusions can be

reached.

1. Solutions for the transient pressure behavior

for a well with ‘ a finite-conductivity vertical

fracture can be correlated by two dirnensiotdess

parameters, CIDf= w~fc~l /nxf et, and ~/D = k f+c t /

+c j t “

2. For practical values of time, solutions can be

correlated as a function

of one parameter,

$L) W/D = kfw/kx/. This parameter.is the dimension-

less fracture conductivity. A decrease in the

dimensionless fracture conductivity may be caused

by a decrease in fracture permeability,

an increase

in fracture length, or botb. This appears

to be an

important reason why type-curve matching with

the

original

fracture

type curves sometimes resulted in

small apparent fracture lengths for large fracture

jobs.

3,. For’ k~D wfD values equal to or greater than

300, t he finit e-ccmduct ivit y solut ions a re for a ll

practical purposes

identical to the infinite-

conductivity vertical fracture solution of Gringarten

et a l .

40 The uniform-flux, vertical fracture solution of

Gringarten

et al. behaves like the infinite-

conductivity solution at small values of time; at

intermediate times, it follows a variable fracture

conductivity solution, For large values of time, it

follows a finite fracture conductivity solution of

“? ‘~e=u;~~” when plotted as

a function of

dimensionless wellbore pressure drop vs the

logarithm of dimensionless time

do

follow (for

large values of time) straight lines of slope equal

to 1.151. Thus, commonly used semilogarithmic

methods of analysis can be used.

6. The approximate start of the .w:milogarithrnic

straight line is a function of

k~D w~D.

The dimension-

less time for this point ranges from 2.5 to 5 for

practical values of fracture conductivity, but this

range

may be stretched to lower times when

necessary.

7. Results when plotted as a fimction of log

~w~D vs log

tf)

show, at small values of

t~,

a set

of curves

of distinct shapes for the different values

Of

kjD wjD .

Hence, a type-curve matching procedure

can be used to obtain both the formation and

fracture characteristics.

8. Pressure data for a well with a low or an

intermediate conductivity fracture (k~D WjD < 300)

does not exhibit a one-half slope, straight line in a

log-log

graph.

6’ =

f3=

c=

c, =

k =

k=

p=

9/ =

9W =

t=

w.

x,y =

NOMENCLATURE

relative capacity parameter defined by

Prats,ls dimensionless

formation volume factor, bbl/STB

compressibility, psi-l

fracture storage capacity, ft3/psi

formation thickness, ft

permeability

pressure

fracture flux density, STB/D -

ft

well flow

rate, STB/D

time, hours

fracture. width, ft

space coordinates, ft

Almlsr, 197s

‘w



.

T

= hydraulic diffusivity, md-psi/cp

P

= viscosity, cp

@ = porosity, fraction

SUBSCRIPTS

D

= dimensionless

/ = fracture

i

= initial

t = total

w= wellbore

SPE C I A L FUNCTIONS

1.

2.

3.

4.

5.

6.

7.

8.

9.

d% r-

references

P ra t s, M., Hazebroek, P., and Strickler, W. R.:

‘Jl ffectf Vertical Fracturea on Reservoir

Behavior

- CompreaaiMe.Fluid Case,” Sot.

Pet. 8mg. J.

(June 1962 ) 87-9 4; Tram s . , ASME, Vol . 225.

Scott, J. o.:

~c~e Effect of Vertical Fracturea on

Tranai~nt Pressure Behavior of Wells, ” J.

Pet . Tech.

(Dec. 1963) 1365-1369; Trans., AIME, Vol. 22S.

Ruascll, D. G. nd Truitt, N. E.: ’Transient Pres-

sure Behavior in Vertically Fractured Reservoirs,’”

J. Pet. Tech. (Oct. 1%4) 11s9.1 170; Truss., AIME,

Vol . 231.

Lee, W.

J ., J r .:

“Analysis of Hydraulical ly Fractured

Wells Wit h P r es su r e Buildup Teats ,” paper SPE

1820 presented at the SPE-AIME 42nd Annual

Fall

Meeting, Houston, Oct. 1-4, 1967.

Millheim, K.

Kt

and Cichowicz, L.: “Testing and

Analyzing Low-Permeability Fractured Gaa

Wells,”

J. Pet. Tech (Feb. 1968) 193-19S; Trans.,

AIME,

Vol. 243 .

Clark, K. K.:

‘~Transient Pressure

Testing of Frac-

tured Water Snjection Wells,” J. Pet. Tech (June

1%S) 639-643;

Trans.,

AI ME , Vol. 243.

Wsttenbarger, R. A. and Remey, H. J.,

Jr.: “well

Test Interpretation of Vertically Fractured Gas

Wella, t$ ), Pet, ?’ecb. (May 1969) 625-632; Trans..

AIME, Vol. 246.

van Everdingen, .% F. and Meyer, L. J.: ‘~Analyais

of Buildup Curves Obtained After Well Treatment,”

/. Pet. Tech (April 1971) S13-S24; Tram., AIME,

vol.

2s1.

Evans, J. G.:

~~~e Use

of pressure

Buildup

Infor-

mation

to Analyze Non-Respondent Vertically

10.

11.

12.,.

13.

14.

15.

16.

17.

1s.

19.

20.

21.

22.

23.

Fractured Oil Wells,” paper SPE 334S

presented et

the SPE.AIME Rocky Mountain Regional Meeting,

Billings, MT, June 2-4, 1971.

Rsgtmvsn, R., Cady, G. V,, and Ramey, EL J., jr.:

~~Well Test Analysis for Vertically Fractured

Wells, t? ,J, Pet, Tech. (Aug. 1972) 1014-1020;

Trams,, AIME, Vol. 253,

Schrider,

L.

A.

nd Locke, C. D.: ‘$Effectiveness of

Different Hydraulic Fracturing Trcatmenta in “ ow

Permeability

Reservoirs,”

pt per

SPE

453u presented

t the SPE-AIME 4Sth Annual Fail Meeting, Las

Vegas, Sept. 30-Ott. 3 1973.

Ramey, H. J., Jr.,

nd Gringsrten, A. C.: ‘iEffect of

High Volume Vertical Fractures on Geothermal

Steam Well Behavior,)$ Proc., Second United Nationa

Symposium on the Use

nd

Development of (3eothermai

Enc?gy,

San Francisco,

May

20=29, 1975.

Locke, C. D. and Sawyer, W. K.: ‘gConstant Preasuro

Injection Test

in II Fractured Reservoir-History

Match Using Numerical Simulation

nd Type Curve

Analyais,C~ paper SPE S594 presented

the SPE=

AIME 50th Annual Fall Technical Conference and

Exhibition, Dallas, Sept. 2S=Oct. 1, 1975.

Gringmten, A. C., Ramey, H. J., Jr., and f?aghavan,

R.: ~6Applied Pressure Analysis

for Fractured Wells, ’

J. Pet. Tech.

(July 197S) 887-S92 .

Sswyer, W.K., Locke, C. D.,

nd Overbey, W.K., Jr.:

tcsimulation of Finite@apacity Vertical Fracture

in a

Gas Reservoir,’$ paper SPE 4593

Presented at

the SPE-AIME 4Sth Annual Fall Meeting, Las Vegas,

Sept. 3&Oct. 3, 19730

Matthews, C. S. and Russell, D, G.: Ptessurs BMift

and Flow Teats im

Wells,

?donwrrwahseries, society

of

Petroleum Engineers, Dallaa (1967).

Csrslfiw, H. S. and Jaeger, J. C.: Conduction of

Heat in Solids, 2nd cd., Oxford at the Clarendon

Press (1959) 275.

Prats, M,: Effect of ve~ica~ Fractures on Rese~oir

Behavior - Incompressible Case,” Sot. Pet. Eng. ~.

(June 1961) 105-1 1S; Tnrtisd, AIME, VO1. 222.

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

R.:

t Unateady-statePreaswe Distributions Created

by a

Well With a Single infinite.Conductivity Vertical

Fra~ture,~~ Sot.

Pef, Eng. J* (Aug. 1974) 347-360~

Tram,, AIME, Vol. 257.

van Poollen, H. K,: c~ProductivWy vs Permeability

Damage in Hydraulically Produced Fracture,”

Drill. and

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Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture