8/10/2019 17. SPE-10043-MS

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SP

SPE

1 43

Society of

Petroleum

Engineers

Evaluation Of Hydraulic Fracturing By Transient Pressure

nalysis

Methods

by Heber Cinco, University

of

Mexico nd P M X

Member SPEAIME

Copyright 1982, Society of Petroleum Engineers

T h i ~

paperwas

presented at the International Petroleum Exhibition and Technical Symposium of the Society

of

Petroleum Engineers held in

Bejlng, China, 1826 March, 1982. The matenal

IS

subject

to

correction by the author. Permission

to

copy is restricted

to

an

abstract

of not

more than

300

words. Write

SPE

6200 North Central Expressway, Dallas, Texas, 75206 USA. Telex 730989

Product ion

of e i t he r wells

completed

in

low

permeabi l i ty

reservoirs or damaged wells

has been

poss ib le

because

of hydraul ic f rac

tur ing .

The

es t ima t ion of both the geometrIc

and f low c h a r a c t e r i s t i c s

of

a f rac ture rep re

sent a useful information for the ca l ib ra t ion

of

f rac ture

des ign methods

and permits

wel l flow behavior .

Transient pressure

wel l

t e s t ana lys i s

has

been

used with

success to es t imate

well

condi t ions and

r e se rvo i r s

parameters .

vent iona l methods of

in te rp re ta t ion

are

on r ad ia l

flow .

This i s

a

l imi t a t i on

when app l ied to f rac tured

wells

because they

e xh ib i t other

type

of

flow

a t

d i f f e ren t

t imes

in a t e s t .

Several

a l thors have presented

d i f f e ren t

to

c a l c u l a t e

both

r e se rvo i r

and

These methods include

lip

vs

l . r- t ,

the

lip vs ,

the

semilog

lip

vs t

and

type

curve matching.

Among

these techniques , th e

type

curve method

deserves

spec ia l a t t en t ion because it al lows

both the ana lys i s of pressure da ta and the

de tec t ion of

d i f f e ren t

flow

regimes.

Transient pressure

techniques

have

been proved

to be an

e xc e l l e n t formation

eva lua t ion

tool .

In te rp re ta t ion of wel lbore

Ids values of forma

and to de tec t

some he te rogene i t ie s in

the

r e se rvo i r .

These

were deve i n i t i a l l y ,

fo r

low

cond i t ions

and l a t e r

modified

to

take

in to

considera t ion d i f fe ren t types of

flow

geometry.

At

the

same

t ime,

s t imula t ion techniques

were deve

to

increase

the

of both

damaged

wells or wells

producing

low permeab i l i ty

r ~ s e r v o i r s

Hydraul ic

f rac

639

tur ing

s tands as on of the most

e f f e c t i v e

s t imula t ion

methods

because

i t s

a pp l i c a t i on

allows production of

wel l s

to be

I t

was

recognized

e a r ly

t ha t wells

in te rcepted by a

f r a c tu re

have d i f fe ren t

f low

behavior than unf rac tured wel ls , consequent ly,

appl ica t ion of ana lys i s methods

based on

theory

to

these

cases

can y ie ld erroneous r e s u l t s .

Many s tud i e s

l

45

have

been

pub l i shed

to

examine

d i f fe ren t

flow s i tua t ions

for

tured

wel ls . Table

1 pre sen t s a summarv

of these

pub l i c a t i ons .

I n i t l a l l y , most

works

1

-

10

dea l t with steady s t a t e flow toward

f rac tured wel l s ;

both

hor izonta l

and v e r t i c a l

f r a c tu re s

were

cons ide red

and

the

main

ob jec

t ive was to determine the e f fec t

of a

f r a c tu r

on

wel l produc t iv i ty .

The f i r s t

the

uns teady-s ta te

f low behavior wells was present

ed

Dyes

e t a l

They inves t iga ted

the -

e f fec t

a a

ver t i ca l

f r a c tu re

on

the semi log

l ine and concluded t h a t

the

l i n e i s affec ted

when

a

ture extends over

f i f t een percent of

the

dra inage radius .

La te r , Pra t s

l3

showed t h a t

a

wel l

in te r s ec ted

by

an i n f i n i t e conduct iv i ty ver

t i c a l

f r a c tu re

exh ib i t s an

ef fec t ive wel lbore

radius

equal to one ha l f of the f rac ture

length;

t h i s conclus ion was reached before by

Muskat 1

Russe l l

and T ru i t t

l

s tudied the

t r a n

of

r e se rvo i r

the

f rac ture

wai::

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EVALUATION OF HYDR ULIC FRACTURING BY TRANSIENT PRESSURE ANALYSIS METHODS

SPE 1004

a

c or re c t i on fa c to r

for the kh

values

ed from

semilog

graph fo r

cases

when

the

t u re pe ne t ra t i on

i s

high. This method was

extended by

Raghavan e t a1

Lee 1 6 used

a

numer ical s imula to r to

the

e f f e c t of both

v e r t i c a l and horizon

t a l conduct iv i ty

f rac tures

and

pre se n t

ed cor re la t ions to es t imate rese rvoi r and

f r a c tu re parameters .

The uns teady-s ta te l inea r f low

to f rac tured wells

by

and Millheim and Cichowicz

1 8

They

po in ted

out

t h a t

a

graph

of

wellbore

p

versus the square

root

of t ime

wf

s t r a i g h t l i n e whose slope

i s

por t i ona l to the f r a c tu re

i s

c a l l e d

l i nea r flow

Wattenbarger and Ramey

1

9

s tudied the

t r a n s i e n t

flow behavior towards a gas wel l

by an

i n f i n i t e conduct iv i ty

f r ~

included

non-Darcy

flow in the

format ion

and

concluded

tha t

t h i s

e f f e c t

i nc r ea se s the of the l inea r f low

s t r a i g h t

l ine .

a lso indica ted

the

non-Darcy flow causes an

ext ra

pressure

drop which i s flow

r a t e dependent .

A

numerica l s imula to r to study produc

t ion o f wells in te rcepted by

a

f i n i t e conduc

t i v i t y f r a c tu re

was by

Sawyer

e t

a l

They

showed

assump

t ion

of i n f i n i t e

f rac ture conduct iv i ty

can

l ead to se r ious er ro r s

when

wel l

performance.

Gringar ten , Ramey and 2S reexam

ined

the solu t ions

for

t r ans ien t

flow

fo r

f r ac tu r ed

wel l s

and

study

three

models:

i n f i n i t e

conduct iv i ty

v e r t i c a l

f rac ture

uniform

f lux

v e r t i c a l and uniform

f lux hor izonta l f rac ture I t was demon

s t r a t e d t h a t these

cases

e xh ib i t

three flow

;

i n i t i a l l y

there

i s

a

l i ne a r

flow

and

a f t e r

a

t r a n s i t i o n

flow the system

reaches a flow. They in t roduced

the type curve as

a

tool

and a method to

both

formation and

f r a c tu re

parameters .

This work was

extended

by Cinco

e t a l

7 , Raghavan e t l ~ ~

and

and

Hadinoto

3

to

the e f f e c t

i nc l i ne d f ra c tu re s , p a r t i a l

pene t r a t ion and

constant

pressure outer

boundary, respec t ive ly .

Later , it was demonstra ted by Cinco e t

a1

3 3

tha t the

i n f i n i t e

f rac ture conduct iv i ty

assumption i s not va l id

when

pressure

the f rac ture i s

considerable ,

tha t

i s ,

when

the dimensionless f r a c tu re

less tha t

300. A

f i n i t e f rac ture

model

was

fo r

these cases

and

indica ted tha t th i s type of

does

not exhibi t the l inea r f low , and

as

a

consequence, the l inea r flow graph

ana lys i s

i s

not re l i ab l e . Simi

conclu

s ions

were

by

Ramey e t

a l

and

Agarwall e t

64

The ef fec t of c losed exte rna l boundaries

on the

behavior

of f i n i t e

conduc t iv i ty f r ac

tUre

was

s tudied

by Barker and Ramey34. They

showed

tha t

a t

values of producing

t ime

t h i s type systems reaches pseudo-s teady

s t a t e

flow condi t ions

and demonst ra ted

t h a t

the use of

type

curves

3 3

,

ava i lable a t t h a t

t ime can lead , in

some

cases ,

to

a

problem.

Sco t t

35

t ime power

su re

data

for smal l

the

o f t ime.

He

showed

pre ssu re behav

io r

of a well

by

a f i n i t e conduc

v e r t i c a l f r a c tu re

can be approximated

of a t ime power being the

dependent on the

f r a c tu re

conduct iv

Exper ience

has shown

t h a t

wells a low

permeabi l i ty

r e se rvo i r ,

bottomhole pressure .

have beeen

Agarwall

e t

a l

an

i n f i n i t e conduct iv i ty

f r a c tu re

and the l a t e r

the f i n i t e f r a c tu re conduc t iv i ty

case . Type curves are given

in these works

to es t imate both format ion and

f r a c tu re

c h a r a c t e r i s t i c s from ana lys i s

of flow

r a te

da ta .

The

on f r ac tu r ed

by Ramey and

Cinco

and

now

ava i lable

wellbore

has

Raghavan

2

Type curves are

when wel lbore

s torage ends in a t e s t .

Flow r e s t r i c t i o n s with in or around

a

f ra

ture

can

af fec t

dramat ica l ly

the

e f f e c t i v e

ness of a

f r a c tu re . This

s

s tud ied

by

severa l au thors

5

,

43,44,45 I t

was

shown t h a t f r a c tu re

causes an ext ra

pressure

drop

of a well .

Type

curves for

da ta t h i s case has

Raghavan

2

, Cinco-Ley and

t i s

well

known tha t

Darcy ' s Law i s not

va l id

for

high ve loc i ty

flow ra t e s ;

t h i s

ca

occur

when f l u ids

flow a f r ac tu re .

Guppy

e t

a l

4 1

showed

t h a t

wells

a f fe c t e d by

non-Darcy

f low

wi th in

the f rac

t u re

e xh ib i t

an apparent which

i s flow

r a te

They

concluded

t h a t

es t ima t ion

o f t rue f ra c tu re

conduc t iv i ty

a t l e a s t two

t e s t s with d i f fe ren t

In

1981,

Cinco and If presented

a

genera l

theory for

the

t r ans i en t flow

towards

a

ver t i ca l ly f rac tured wel l .

found tha t in addi t ion to the l i nea r and

pseudorad ia l

f low ; a f r ac tu re with

in te rmedia te or

low

exh ib i t s

th

b i l i n e a r

flow

periOd and pre ssu re da ta

whe

versus

the four th

roo t o f

t ime

ld

l i ne

whose s lope i s

to the square roo t o f f r ac tu re

con

8/10/2019 17. SPE-10043-MS

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SPE 10043

HE ER

CINCO-LEY

3

duc t iv i ty .

New type

curves

were to

overcome

the

problem in

data

ana

l y s i s .

In the

next

sec t ion, a descrp t ion of

flow

models used today fo r t r ans ien t

pressure

i s

in

f rac tured wel ls i s

Modern methods

of

models

fo r

f rac tu red wel ls :

cons ider four

a) I n f i n i t e

conduct iv i ty

ver t i ca l f rac ture

b) Uniform

f lux

v e r t i c a l f rac ture ,

c)

Uniform

f lux

hor izon ta l f rac ture and

d) Fin i te

conduct iv i ty ver t i ca l

f r ac tu r e .

In

some cases , th ree d i f fe ren t o f

outer

boundary condi t ions

are used; i s ,

i n f i n i t e

reservoi r ,

closed or cons tan t

p r ~

sure condi t ions .

A

v e r t i c a l

f rac ture

i s

considered

to

an

i n f i n i t e conduct ivi ty pressure

along the f rac ture i s neg l ig ib le

and

l eng th

x

f

.

The

f rac ture

the formation and produces from a

square reservoi r whose

s ide i s 2xe as

shown

in Figure. 1 . This model assumes

t h a t

flow

in to the wellbore i s only

through

the

f r ac tu re . As mentioned before,

t h i s

system

exh ib i t s a l inear flow

per iod

and a

r a d i a l flow per iod . The

f lux

d i

the f rac ture var ies with

time;

ly a

uniform f lux takes

place, as t ime

increases , the f lux

changes

and becomes con

s t an t when

the

pseudo- rad ia l

flow

i s es tab

l i shed .

This model

i s

s imi la r

to the i n f i n i t e

conduct iv i ty v e r t i c a l f rac ture

in

severa l

Fig.

1).

The

d i f f e r en ce

between

systems

occurs a t

the

boundary o n d ~

t i on

a t the

f rac ture . The

uniform f lux f rac

t u r e has a cons tan t f lux and a var iable pres

the

f rac ture ; it

a l so

exh ib i t s

pseudoradial flow

The

of th i s model i s shown in

Figure a hor izon ta l c i r c u l a r f rac

tu re of in an

i n f i n i t e

s lab r ese r

voi r i s The f lu id ex t r ac t ion

from

the

reservo i r occurs a t the f rac ture

face through a uniform

f lux

d is t r ibu t ion .

This

system

a l so

exhibi t s

l inear and

pseudor

d i a l flow

periods .

Figure 3 shows a ver t i ca l

f r ac tu r e

in

an i n f i n i t e

s lab

reservoi r .

The

f rac ture

has a permeabi l i ty

l

a

width

and a ha l f

length

x

f

. The formation i s

t o t a l l y In a v e r t i c a l

di r ec t ion

64

t u r e

and it i s

l imi t a t ed

by a

lower

and an

upper impermeable boundar ies . The t r a n s i e n t

behavior for t h i s

system

can inc lude

flow

per iods as

indica ted in

Fig.

4; in

, there

us

a

l i n e a r flow with in the

-

f rac ture , t h a t can be fol lowed by

the

b i l i n

ear flow

then

a l i n e a r

flow

in the format ion

may

be presen t and even tua l ly

the

pseudoradia l

f low per iod i s reached.

Modified

vers ions o f the

these

models

have

been

presen ted

too,

are the cases of

damaged f rac tures

29,

32, 4 , 5 Fig. 5 and 6

and

heterogeneous

f ractures

4o

,43.

may correspond

There are

of

ana lys is for

each flow regime; ,

flow

data

must

be

analyzed wi th a graph of versus

~ for b i l i n e a r flow

data

the 6p versus

4ft-graph must be used and

the

semilog graph

6p

versus

log t must

be appl ied

to data

on

the pseudo- rad ia l flow per iod .

The general

so lu t ion

fo r the pressure

behavior

in

a

r e se r vo i r

i s expressed

in

terms

o f dimensionless var iab les . For f rac tu red

wel ls the

va r i ab le s

are used:

Dimensionless Pressure Drop.

Oil

wel l :

kh 6 P

141.

2 qSlJ

Gas

wel l :

Dimensionless Time

4

2.637

x 10

k t

lJ

C

t

X

f

;?

2

rhllctr

I

,t- W

4

2.637

x 10

k t

lJc

t

r

f

2

Dimensionless

Frac tu re Conduct ivi ty

1)

2)

3)

4)

5)

6)

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4

EVALUATION

OF

HYDRAULIC

FRACTURING

BY TRANSIENT PRESSURE ANALYSIS

METHODS

SPE

1004

Fracture

Skin Factor

Flu id

Loss

Damage:

li

s

~ _

1)

k

s

Choked Frac tu re :

Trx

k

s

7

8)

Next

a

discuss ion

i s presen ted on

the

bas i s and

appl ica t ion o f each

method of

ana lys is .

(6p versus \It

This

method

was developed

for f in i t e

conduct ivi ty v e r t i c a l f rac tures of small

s to rage capaci ty and i s

based

on the b i l i n e a r

flow

theory .

This

behav io r

i s

a

r e s u l t

the

superposi t ion

o f two l inear f lows; one

flow

i s incompress ible and occurs

wi th in the

f rac ture and the

other i s

a compressible

flow

in

the

format ion. A b i l inea r

flow

e x i s t s

when

the

flow

in to the

wellbore

i s due to the

expansion of

the

system

in the formation and

f rac ture

e f f e c t s

have not yet a f fec ted

the

well

The

dimensionless wel lbore

pressure

change for a t e s t can be as:

2.45

P

wD

1 4

9

or:

Oil

wel l :

6p

44.1

(10)

h

f

(kfb

f

Gas

well :

6m p)

4 4 4 7 5 q T t

4

hf k fb

f

1/2(pc

t

k) 1 / 4

(11)

These equat ions

indica te

tha t the

pre

sure change i s both

to

h f k f b f ) ~ 2

and d i r ec t ly propor t iona l

to

fourEh

roo t of t ime.

According to

Eqs.

10

and

11.

a graph

o f 6p

(or 6m(p)} versus

4/1:

gives

a s t r a i g h t l i n e passing

through

the or igin of slope rn

bf

as ind ica ted in F i g . ~

a l so

i nd ica t e s tha t

af t e r

the l ine

por t ion

the curve could

be

concave

o r downwards depending upon

the

dimensionless f rac ture

conduct iv i ty .

64

From

t h i s

graph

the product hf k fb

f

V

can be es t imated

as

fo l lows:

Oil

well :

44.1 qBp

Gas wel l :

The b i l i n e a r

flow

ends

a t :

t 0.1

Debf

(k b

fo r D

>

3

f f D

0.0205

for 1.6

f )

f )

w

c

Q

f )

/ )

W

.-J

Z

0

f )

Z

ltJ

0

10

10

3

5 7 1 1 9 1 II .. 5 1 5 7 8 9 1 3

51171191

I )-110

Sf s

0 R SrS Ch

2

10

Fig.

2 4 r ~

versus Sts or

(StS)Ch

for damaged fractures

DIMENsrONLESS TIME, TO

Fig. 25 -Type curve for an infinite conductivity vertical fracture with wellbore storage

664

8/10/2019 17. SPE-10043-MS

27/27

('f )

Cl

4 1

Cl

U

..0

4 11

2 \

II

-

Cl

;;

UNIFORM-FLUX

WELL IN AN INFINITE RESERVOIR, Xe/Xf

QO

DIMENSIONLESS

STORAGE CONSTANT, C

OXf

O ~

5X 10-

3

10-

2 - - - - : r - - . . - : i

DIMENSIONLESS TIME. tOXf

Fig. 26 -Type curve for a uniform flux vertical fracture with

well bore storage

nd

o f Wellbore

Storage

Effec t s

10

1

~ ~ L

10-

1

10 10

2

10

3

10

4

kfb

f

D 2/3

F

2

t

Oxf

= Y toxf

C

Of

)

3