SPE-9289-MS

8/18/2019 SPE-9289-MS

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SPE 9289

SP

ocietyo Petroleun EngIneers

of

'ME

EW METHOD TO CCOUNT FOR PRODUCING TIME

EFFECTS

WHE N

D R W D O W N

TYPE

CURVES RE

USED

TO N LYZE PRESSURE BUILDUP ND

OTHER

TEST D T

by

Ram

G

Agarwal

Amoco

Production Co

©Copyright 1980. American Institute of Mining. Metallurgical. and Petroleum Engineers. Inc.

ThiS paper was presented at the 55th Annual Fall Technical Conference and ExhiblliOn of the Society of Petroleum Engineers of AIME. held In Dallas. Texas. September 21 24. 1980.

The material IS sublect to correction by the author. Permission to copy S restricted to an abstract of not more than 3 words. Write: 6200 N Central Expwy Dallas. Texas 75206.

ABSTRACT

Currently,

type

curve analysis

methods

are

being

commonly

used

in conjunction

with

the

conven

t ional methods to

obtain

bet ter in terpreta t ion

of

well

t e s t

data. Although

the majority of

published

type

curves

are

based on pressure drawdown solu

t ions , they

are

often applied indiscriminately to

analyze both pressure drawdown and buildup data.

Moreover, the l imitat ions of drawdown

type

curves,

to analyze pressure

buildup data collected a f te r

short

producing

t imes, are not

well understood

by

the pract icing

engineers . This may often

resul t

in

an

erroneous in terpreta t ion

of

such buildup

t e s t s .

While

analyzing buildup data

by the

conventional

semi-log method, the Horner

method

takes into

account

the effect

of

producing

time. On the other

hand, for

type

curve analysis of the same se t of

buildup data, i t is customary to ignore

producing

t ime

effects

and ut i l i ze the exis t ing drawdown

type

curves. This causes discrepancies

in

resul ts

obtained by the Horner

method

and type

curve

methods. Although a few buildup

type

curves which

account

for

the effect

of

producing t imes have

appeared in the

petroleum

l i t e ra ture ,

they

are

ei ther l imited

in

scope

or somewhat

di f f icul t

to

use

In view of the

preceding,

a novel but simple

method has

been

developed which eliminates the

dependence on

producing

t ime

effects

and

allows

the

user to

ut i l ize

the exis t ing drawdown

type curves

for analyzing

pressure buildup data.

This method

may also be used to analyze two-rate, mult iple- rate

and

other

kinds of

tes ts by

type

curve

methods

as

well as the conventional methods. The method

appears to work for both

unfractured

and fractured

wells. Wellbore

effects

such as storage and/or

damage may be taken

into account except

in cer tain

cases.

The purpose

of

th i s paper

is

to present the new

method

and

demonstrate

i t s u t i l i t y and applicat ion

by means of example problems.

References

and i l lus t r a t ions a t end of paper.

INTRODUCTION

Type

curves

have

appeared

in the petroleum l i t -

erature since 1970 to analyze pressure

t ransient

(pressure

drawdown and pressure buildup) tes ts taken

on

both

un fractured and fractured wells. The

majori.ty of

type curves

l

-

g

which

have been developed

and

published to date were

generated using

data

obtained

from

pressure drawdown

solutions and

obvi

ously

are

most

su i ted

to analyze pressure drawdown

t es t s . These drawdown type

curves

are

also commonly

used

to analyze pressure buildup data. The applica

t ion of drawdown

type curves

in analyzing pressure

buildup data is not

as

bad as t may f i r s t appear.

As

long

as the

producing time,

t

prior

to

shut-in

is suff ic ient ly long

compared

toP

the

shut- in time,

[ that is t + ~ t / t

-

1], for

l iquid

systems,

i t is reasonablg to anRlyze pressure buildup

data using drawdown type curves.

However,

for cases

where

producing times prior

to pressure buildup

tes ts are of the same

magnitude

or

only s l igh t ly

larger than the

shut- in times [ that

i s , t

+ ~ t / t

» 1], the drawdown

type

curves may not bePused tg

analyze

data

from pressure buildup

tes ts .

The above

requirement on

the

duration of producing times

i s

the same

for

the conventional semi- log

analysis.

I f

pressure buildup data obtained af ter short

producing

time

are to be analyzed, the

Horner

method

lo

is

recommended

over the M H (Miller-Dyes-Hutchinson)

method.

9

The M H

method

is generally used

to

ana

lyze buildup data

collected a f te r

long producing

t imes, whereas the Horner

method

i s used

for

those

obtained af ter

re la t ively short producing

times.

Although

pressure buildup tes ts with

short producing

t imes may

occur often under

any

s i tuat ion,

they are

rather

more

common

in the case

of

dr i l l

stem

tes ts

and

pre-f ractur ing tes ts on low permeabil i ty gas

wells.

Thus, there i s a need for

generating

buildup

type

curves, which

account

for the

effects

of pro

ducing time. Some l imited

work

has been

done

in

th is regard. McKinleyll has published type curves

for

analyzing buildup data for a

radia l

flow system.

However, his buildup type curves were generated

on

the assumption of long

producing

t imes; and these

type

curves

are therefore very s imilar to drawdown

8/18/2019 SPE-9289-MS


2

A

NEW METHOD

TO

CCOUNT

FOR

PRODUCING

TIME EFFECTS

WHEN DR WDOWN TYPE

CURVES RE

USED TO

N LYZE PRESSURE BUILDUP

ND

OTHER

TEST D T

SPE 9289

type curves and are

obviously

unsuitable for

cases

where

producing times prior

to shut - in are

re la

t ively

short .

Crawford,

e t ar . ,12 pointed

out

the

above l imi ta t ions

for

McKinley

type

curves in ana

lyzing pressure

buildup

data from the

DST

tes ts .

They

also presented

buildup curves

for short pro

ducing times.

Since

the i r

curves

deal with speci f ic

values

of rea l producing

times prior

to shut- in ,

they are

l imited

in

scope and u t i l i t y .

Recently,

the effect of producing

time on

analysis of

pressure

buildup

data

using

drawdown

type curves has

been

discussed

by Raghavan.

13

His

study clear ly points

out the l imi tat ions

of

drawdown type curves

for

ana

lyzing buildup data collected af ter

small

producing

times.

A

family

of

buildup type

curves i s presented

both

for unfractured and fractured wells

with

pro

ducing

time

as a parameter.

Although these

type

curves

offer

a

defin i te

advantage over the exis t ing

drawdown

type

curves,

they

are di f f i cul t to use

because

of the

multip l ic i ty

of

type curves. In a

recent

paper,

Agarwal

14

also discussed

the

l imi ta

t ions

of

using drawdown

type

curves

for

analyzing

buildup data obtained af ter

small

producing times

but no de ta i l s

were

given.

These l imi ta t ions are

discussed

here

in

th is paper. Recently Gringarten,

e t a l .

15

,

presented

drawdown

type curves,

plot ted

in a s l igh t ly

different

form, and suggested some

gUidelines

regarding the port ions of buildup data

which may be

analyzed by

drawdown type curves.

Although these guidelines

may be

useful in cer ta in

cases,

the basic

problem

s t i l l remains.

To overcome the

above-mentioned

di f f i cul t i e s

and to e l iminate dependence on producing

t ime, a new

method

has been

developed.

This

method should pro

vide a

s ignif icant

improvement

over the current

methods because (1)

th is

permits

us to

account

for

the ef fects of producing

time,

and (2) data are

nor

malized

in

such a fashion tha t instead

of ut i l iz ing

a family

of

type curves with

producing

time as a

parameter, the

exis t ing

drawdown type

curves

may be

used. This concept

appears

to

work

for

both unfrac

tured

and

fractured

wells .

Wellbore

storage

effects

with

or

without damage may

also

be taken

into

account provided that producing time prior

to

shut-in i s long

enough to

be out

of

such wellbore

effects .

This

method

has

been

extended

to

include anal

ysis of data from two-rate tes ts

8

l 6 l 7

and mult ip le

ra te tes ts

8

,17 18

by

type curve

methods. Although

not

shown, t

appears

to have a potent ia l for

applying type curve

methods to

other kinds of

test ing.

This

new method,

although original ly conceived

for

type curve analysis

of

buildup data,

is

quite

sui table for the conventional semi- log analysis. I t

i s s imi lar

to

the Horner method because i t includes

the

ef fects

of

producing

time,

and

may

be used

to

determine

formation flow

capacity, skin

factor

and

the in i t i a l

reservoi r

pressure. However, t

has an

added

advantage.

I t

allows

the

plot t ing

of pressure

buildup

data,

with

and

without

producing time

e f fec t s , on

the

same t ime

sca le

as

the

graph paper.

This enables a be t te r

comparison of

data using the

MDH and Horner type

graphs.

Although

the

new method wil l

be developed using

the solutions

for

l iquid systems, i t s applicabi l i ty

to

gas

wells

wil l

also be indicated.

BASIS

OF DR WDOWN

ND BUILDUP

TYPE CURVES

A type curve i s a

graphical

representat ion of a

mathematical solution (obtained analyt ical ly or

numerically) for a speci f ic flow type. The solution

is

normally plot ted, in

terms

of

dimensionless

var i

ables, on

log-log

graph paper. The graph thus

pre

pared

becomes the type curve for the speci f ic flow

problem

with

given inner and outer boundary

condi

t ions. Depending on the type

of

solution drawdown

or

buildup),

drawdown and

buildup

type curves are

generated.

Drawdown Type Curves

s the name implies , these type

curves

are

based on

the

drawdown solutions.

The

pressure

draw

down solution for a well producing

a t

a constant

ra te as a function of flowing time, t may be writ ten

as

kh[Pi-Pwf(t)]

141.2 q B ~

where,

1)

(2)

Eq. (1) is a general solution

and

is not

meant

to be res tr ic ted

to any

part icular drainage shape or

well

location. The majority of the published

type

curves

l

-

8

for

both

unfractured and fractured

wells

are

based on

pressure

drawdown solutions

for l iquid

systems.

Examples of pressure

drawdown type

curves

for unfractured wells are those presented

by

Agarwal, et a l . , l

Earlougher

and Kersch

4

and

G r i n g a r t e ~ e t a l .

15

In

another publicat ion Grin

gar ten ,

e t a ~ , ~ p r e s e n t e d type curves

for ver t i

cal ly

fractured

wells with i nf in i t e flow

capacity

and uniform flux f ractures.

Type

curves

for

f ini te

flow

capacity frac tures were

provided

by Cinco

e t

a l . , 6 and Agarwal, e t

a l .

7

More regarding

the-Use of

above

type

curves

for

analyzing buildup

data wil l be said la ter .

Buildup

Type Curves

To

obtain

pressure buildup so lu t ions , superpo

s i t ion may be applied in

the normal

manner to pres

sure drawdown

solutions.

This provides buildup

pressures a t shut - in

t imes, af ter a producing

time,

t

Fig. 1

shows

a

schematic

of pressure

buildupPbehavior obtained following a

constant ra te

drawdown

for

a production period,

t Flowing pres

sures

p f ( t ) are shown as a

functioR

of flowing

time,

tWup to a production period, t , when a

buildup t es t is

ini t ia ted. Buildup

~ r e s s u r e s ,

p

( t + ~ t , are

shown as a function of

shut - in

time,

~ ~ ~ PInstead of taking a buildup

t e s t ,

i f the

well

was allowed

to

produce

beyond

t ime, t flowing pres

sures as shown by p fCt + ~ t would h a ~ e been

obtained.

Note

t h a ~ thg flowing

pressure a t

the

end

of

the production

period

which is denoted by p

f ( t

)

is same as the buildup pressure a t the instantWof p

8/18/2019 SPE-9289-MS


SPE 9289

RAM G. AGARWAL

3

shut- in which is shown as p

(6t=0).

Superposition

when applied to drawdown so ijtions provides the fol -

lowing.

kh[po-p ( t

+6t)]

1

ws p

141.2

qBlJ

(3)

The flowing

pressure,

p f ( t )

a t the end of

producing

period,

t

i s

g i v e ~ byP

p

kh[po-op f ( t )]

1 W

P

141. 2

qBlJ

(4)

Subtract ing Eq.

(3) from

Eq. (4)

and

subst i tu t ing

p (6t=O) for p f ( t ) ,

we

obtain

ws w p

kh[p ( t

+6t)-p

(6t=0)]

ws

p

ws

141. 2 qBlJ

Eq. (5)

provides

a basis for

buildup type

curves and

has

been ut i l ized

in

th is paper.

Before

discussing the

new method,

l e t

us review

the simplified

version

of

Eq. (5) which has been

commonly used in the past and has provided

the

basis

of ut i l iz ing

drawdown

type curves for analyzing

pressure

buildup

data. I f producing time, t ,

i s

s ignif icant ly

larger

than the shut- in time, P6t, t

i s reasonable to assume that

[ ( t +6t) / t ] 1.

Although approximate, this also £mpliesPthat

( t

+6t)

t ,

or

p

D[(t

+6t)D]

p

D[(t

)D]'

p p w p w P

Thus, Eq. (5)

can

be simplif ied as a pressure

buildup equation

as shown

below:

kh[p ( t

+6t)-p (6t=0)]

ws

p

ws

141.

2

qBlJ

A comparison of pressure buildup

equation

(6)

and the pressure drawdown

equation

(1) indicates

tha t they are similar a t leas t for

cases

where

pro

ducing

period,

t

i s s ignif icant ly

longer

than the

shut-in

time, I t also implies

tha t

(6p)drawdown

vs.

flowing time, t is equivalent to (Ap) v

L>

bOld

s .

shut

in time, 6t , where U1 up

(6p)drawdown (7)

(6p)b old = P ( t +6t) - p

(6t=0)

U1 up ws P

ws

(8)

Since Eq. (6) has been

derived

from Eq. (5)

based

on

the

assumption

of long producing period,

t

p

the difference

(9)

On Fig. I ,

the

above difference has been shown

as the

cross-hatched

area and may be defined as

(6p)

difference

p

f ( t )-p

f ( t

+6t)

w

p

w

p

(10)

or

(6p)dOff p (6t=0)-p f ( t +6t)

(11)

1 erence

ws w p

As producing

period

t

gets

smaller

or 6t

gets

larger , the difference

shoen

by

Eqs. (9)

through

(11)

can no

longer

be ignored and the use

of

draw

down type curves to

analyze pressure buildup data

becomes inval id . The impact of the

assumption

shown

by Eq. (9) wil l be

discussed

f i r s t in a generalized

fashion followed

by

i t s

impact on

type curves

for

specif ic

flow regimes. Final ly , the

new method

wil l

be discussed which accounts for producing

time

effects

for analyzing pressure buildup data.

Fig. 2 schematically shows pressure buildup

behavior obtained following

a

constant ra te

drawdown

but at the end

of

three s u e ~ s i v e l t incrEasing pro

ducing

periods,

t

such

that P3

>

P2

> Pl

The

cross-hatched areR shown at

the

end

or

each produc

t ion period denotes the

difference

between

(6p)d

d

and

(6p)b

Old represented

by Eq.

(10)

raw

own U1 up

or (11). Note that (6p)difference gets

smaller

as

the length of the

producing

period increases.

This can be

bet ter

shown by means of

Fig.

3

where (6p)drawdown vs.

flowing

time,

t

has

been com-

pared

with (6p)b

Old

vs. shut- in time, 6t with

U1 up

producing

period

t

as

a

parameter.

Although

sche

matic, Fig.

3 c l e ~ l y

indicates that there i s

a

s ignif icant difference between (6p)drawdown and

(6

p

)buildup for small

producing

periods.

However,

th is difference gets smaller as the length of

the

producing

period

increases.

Also note that for

a

given

producing

period, the

difference between

the

two

(6p)s is small at

early

shut-in

times

but

i t

gets bigger

as

shut- in time, 6 t

increases.

Fig. 3

clear ly indicates the l imi tat ions

of

using drawdown

type curves

for analyzing pressure buildup data

where

producing

period,

t ,

prior to shut- in is

re la t ively

small .

p

Next

we wil l

examine the

impact

of

this di f fer-

ence

on

type curve

analysis

for

the

specif ic

flow

regimes

(radia l

flow,

l inear

flow,

etc . )

and

discuss

the

new method

which accounts for

producing

time

effects .

UNFRACTURED

WELL

Inf ini te Radial System (s=O; Cn=O

Let

us

f i r s t consider

the

pressure

drawdown

solut ion for a well producing a t a constant ra te in

a radia l system.

8/18/2019 SPE-9289-MS


4

A NEW METHOD TO

ACCOUNT

FOR

PRODUCING

TIME EFFECTS WHEN DR WDOWN

TYPE

CURVES RE USED TO

ANALYZE

PRESSURE BUILDUP

ND

OTHER TEST DATA

SPE

9289

1

2 [ In( t

D

)

+ 0.80907]

(12)

Eq. (12)

is based on the assumption that well

bore

effects

(storage and skin)

are

negl igible

and

the dimensionless time,

tD 100 such

that the

log approximation

applies

to the

c. -solut ion.

Sub

s t i tu t ion of Eq. (12) in Eq. 1) p?ovides

kh[Pi-Pwf(t)]

141.2

2 [ In( t

D

) +

0.80907]

(13)

Eq. (13) is

a

pressure

drawdown solution

for

a

radia l system which also

forms the basis for

semi-log st raight l ine

on a

semi-log

graph

paper:

I f

Eq.

(12)

is subst i tu ted in Eq. (3) , the well

known

Horner pressure buildup

equation

is

obtained.

kh[p.-p ( t +At))

1

ws

p

141.2

(14)

In Eq.

(14)

the subscr ipt

D

may

be

dropped

i f

desired.

The above equation also takes in to account

producing time

effec t s .

Unfortunately Ap =

[p.-p

( t +At)) on the

l e f t

hand side of Eq.

(14)

r e q u i ~ ~ s

Rknowledge of

in i t ia l

reservoir pressure,

p. which is generally not known. Consequently,

Eq.

(14)

is not sui table for the purposes of type

curve

matching. However,

(Ap)b ld defined

by

Eq. (8) is generally known and

ng¥mally

used

for

type

curve

analysis . I f

Eq.

(12)

is subst i tu ted in

Eq.

(6),

the simplif ied pressure

buildup

equation is

obtained.

kh[p ( t +At)-p (At=O))

ws p ws

141.2

2

[In(At

D

) + 0.80907)

(15)

Eq. (15) i s the famil iar MDH (Miller-Dyes

Hutchinson) equation for

pressure buildup and

assumes that the producing

period

prior to

shut- in

is

suff i c i en t ly

long such tha t t ransients

during

the

flow period do not affect the subsequent

pressure

buildup data.

Thus, i t should be obvious that

Eq.

(15)

is

not

sui table

for

pressure

buildup

anal

ysis

(conventional

or type curve)

when

producing

times

prior

to shut-in

are

small.

However,

Eq. (5)

may be used,

as was

done by

Raghavan, to

generate

a

family

of pressure buildup

curves

with dimensionless

producing

time,

t

D

as a

parameter. Fig. 4

pre

sents such resultg

for

an in f in i te radia l system.

Dimensionless

pressure

change, PD

during buildup

has been

plotted

as a function o¥

3imensionless

shut-in

time, At

D

,

with dimensionless £roducing

time, t

D

as a

parameter.

Note that p D will be

defined

P

l a te r

as Eq.

(19) and

is

the

s ~ m ~ as

the

l e f t hand side of Eq. (5).

The

buildup

curve

with

t D oo

corresponds to the pressure

drawdown

solut ion

g ~ v e n

by

Eq.

(12). Since data are

plot ted

on

semi-log

graph

paper

(for tD 100),

the pres-

sure drawdown solut ion on F1g. 4 is a

st raight

l ine.

This f igure clear ly points out the

limitations

of

using pressure

drawdown

solut ion

in a

conventional

or type curve analysis mode to analyze pressure

buildup data

obtained

af te r short producing

t imes.

However,

buildup curves

may

be

ut i l ized

for

type

curve matching purposes as discussed

by Raghavan.

13

The obvious

disadvantage

is that this reqUires the

use

of a family of type curves.

To

overcome the

above diff icul ty, a new

method

has

been developed which

should allow

us

to analyze

pressure buildup

data by means of

pressure

drawdown

type

curves.

This new method may also be

used

to

perform the

conventional

analysis .

NEW

METHOD

Eq.

(5) , presented ear l ie r

as a

pressure

buildup

solution,

forms

the

basis

for

this

new

method.

Subst i tu t ion

of

Eq. (12)

in Eq.

(5)

kh[p ( t +At)-p (At=O)]

ws

p

ws

141.2

+

0.80907)

(16)

Eq. (16) thus becomes a

pressure

buildup solu

t ion for the in f in i te

radia l system. A

comparison

of pressure

drawdown,

Eq. (13) with Eq. (16),

indi

cates

tha t

pressure

drawdown

curves generated

by

Eq.

(13)

should

be

the same as the

pressure buildup

curves obtained

by

Eq. (16). To demonstrate

th i s ,

the

family of

buildup curves

shown in

Fig. 4 and

plot ted as a function of dimensionless shut- in

time,

At

D

, were replot ted as a function of the time group,

( t

D x

AtD)/(t

+ At)D Results

are

shown in

F i ~ 5. P

Note that i t is possible

to

normalize a

family

of

buildup

curves in to a single

curve.

Moreover,

this single curve is the same as

the pressure

draw

down curve. The

preceding

suggests that for ana

lyzing

pressure buildup

data by drawdown type

curves,

(Ap)b ld

data should

be

plot ted as a

Ul up

function of

( t x At) / ( t + At) rather

than jus t the

shut-in

t ime,

PAt. p

For the sake of s impl ici ty and

brevity

l e t us

define the

new

time

group

as

an

equivalent

drawdown

t ime, Atedt

or

fur ther

abbreviated as Ate

where

At

e

t x At

P

t

+

At

p

(17)

In

a

dimensionless form, Eq.

(17)

may be

expressed

as

8/18/2019 SPE-9289-MS


SPE 9289

R M G.

AGARWAL

5

t

pD

x Llt

D

)

tp +

Llt D

18)

The dimensionless pressure

change

during

buildup or a ra te change may be defined as

13

kh[p

t

+ Llt) - pws(Llt=O)]

ws p

141.2 qBfJ

19)

In

establ ishing

the

new

method, t

was

previ

ously assumed

that wellbore effects

such as storage

and skin)

are

negl igible . I t appears

that

skin

effect , s ,

may be considered in the development of

th is method.

Inf in i te Radial System

s10;

CD=O)

I f skin

effect ,

s ,

is introduced

in the

pres

sure drawdown

solut ion

given

by Eq.

12),

we obtain

1

2 [ In t

D

) + 0.80907] + s 20)

I f

we go through the same steps as we did

for

an inf ini te radial system s=O; CD=O) and instead

of

u t i l i z ing Eq. 12)

we

use Eq. 20), i t

wil l

be

obvious that

the new

method is equally

val id when

s10.

Pressure

drawdown

Eq. 13) and

the new pres

Sure

buildup Eq. 16) will respect ively become

Eqs. 21) and 22) as shown below:

kh[Pi-Pwf t)]

141.2

qBfJ

1

2 [ In t

D

) + 0.80907] + s

kh[p t +Llt)-p (Llt=O)]

ws

p

ws

141.2 qBfJ

(21 )

1

t x Ll t

D

[In pD + 0.80907] + s 22)

2 tp +

Llt D

The above

equations es t ab l i sh the

valid i ty

of

using

pressure

drawdown

type

curves

for pressure

buildup analysis even when skin is present .

Based on the

encouraging resul ts

obtained

thus

far ,

we

wanted

to

apply th is concept

of

the equiva

lent drawdown time, Llt

to

other wellbore

.effects

and also to other floweregimes. The attempt was

made

to establ ish

the valid i ty

of th is concept for

the

above

si tuat ions

by graphical

means rather than

the mathematical

solut ions .

Let us f i r s t consider

the inf ini te radial system with wellbore

storage

effects .

In f in i te Radial

System s=O; CD10)

To study the

effect

of s torage

on

buildup type

curves,

data

presented

by

Agarwal, e t a l . , 1

were

u t i l i zed .

Pressure

drawdown

data

PwD

vs.

tD

data

for s=O and CD=IOOO were taken from Table 3 of the

above paper. Eq. 5) was

used

to generate the pres

sure

buildup

data for

a

number of

producing times as

was done by

Raghavan.

13

Both

pressure

drawdown data

and pressure buildup data are

plot ted

on Fig. 6

semi-log graph paper)

as

a

function of

tn and

LltD

respect ively. Note

that a

family of

builaup curves

is

obtained with

producing

time, t 10

3

to 10

6

) as

a parameter.

These data are also

B20tted

on log-log

graph paper

as

shown by Fig.

7.

These

f igures

further

emphasize the

l imi tat ions

of

using

pressure

drawdown

curves

to

analyze

pressure

buildup

data

obtained

af ter short

producing

periods.

Fig. 7

shows that uni t slope l ines

for

buildup data

are

shi f ted to the r ight of the drawdown curves. If

dimensionless storage, CD

is

computed using the

buildup

data, the computed value of C

n

wil l be

erro

neously high.

Moreover,

i f

pressure

Duildup data

are

forced to match the

pressure

drawdown type

curve, the

computed

value of formation

flow

capacity

kh) will

be erroneously

optimistic.

The magnitude

of

error wil l

increase

with

decreasing producing

period.

Figs . 8 and 9 are the replots of pressure

buildup

solut ion on

semi-log

and log-log graph

papers u t i l i z ing the new

time

group.

Fig.

8

indi

cates that almost a l l pressure buildup

curves

are

normalized

except

two which correspond to dimension

l ess

producing

period, t D

equal

to 10

3

and 10

4

.

Although

these

curves doPnot seem to

appear

bad on

the semi-log paper,

they

look rather poor on the

log-log graph

in

Fig. 9. The

reason

for

th is

may be

obvious i f

we inspect the

following

equation

19

which

provides

the

time

for s torage ef fects to

become neg

l ig ib le .

For

s=O

23)

I f Eq. 23)

is

used for the subject

problem,

C

=1000,

the minimum producing time

required for

the

srorage ef fects

to

become

negl igible

wil l

be

equal

to 6 X 10

4

. Since the producing periods in the two

cases were only 10

3

and 10

4

respect ively, pressure

buildup data could not be normalized. Based on a

number

of

cases studied, t

appears

that i t is

pos

s ib le to normalize the

pressure

buildup

curves pro

vided

that the

producing

time, t D is a t least

equal

to or grea ter

than that giBen by Eq. 23).

Inf in i te

Radial

System

s10;

CD10)

Agarwal,

et

al . ,1 data

were

taken

for a

number

of

cases

for non-zero values of CD and

s .

Although

not shown

in

th is

paper,

resul ts

indicate

that

pressure

buildup curves are normalized when the

new method

is

used. The

lower

l imi t

of

the pro

ducing

time for s10

is

determined by

the following

equation.

20

8/18/2019 SPE-9289-MS


6

A

NEW

METHOD

TO ACCOUNT FOR PRODUCING TIME

EFFECTS

WHEN DR WDOWN TYPE

CURVES

ARE

USED TO ANALYZE PRESSURE BUILDUP AND OTHER TEST DATA

SPE 9289

tpD 60 + 3.5 s)C

D

24)

The preceding discussion establ ishes

the

val

idi ty of using

Agarwal,

et

a l s l

pressure

drawdown

type curves radia l flow

with storage and skin

effect)

for analyzing pressure buildup data provided

tha t

the

new

method i s used.

APPLICABILITY OF

NEW

METHOD TO OTHER TYPE CURVES

I t appears that t is possible to extend

th is

method to other

drawdown

type curves

which

have

appeared

in

the

petroleum

l i te ra ture . Two

se ts of

type

curves wil l

be

considered: 1) Earlougher and

K e r s c h , ~

and

2) Gringarten et al .

IS

Earlougher

and Kersch

Type

Curves

4

These type curves are based on pressure draw

down solution and are applicable to an

inf in i t e

radia l system

with wellbore storage and skin.

They

are

basically the same type curves

as that of

Agarwal

e t

al .

l

because both use

the

same

solu

t ion.

However,

they

are

dis t inc t ly

different in

appearance because

data are plotted

different ly . A

schematic

of

the i r drawdown type curves is shown in

Fig. 10, where p DC ) / t

D

has been plotted as a

fun:tion ofotD/COWwith cDe

l

as a p a r ~ e t e r . where

CD

lS

the dlmenslonless storage

coeff lc ient , and

is

defined as

25)

Although no proof i s demonstrated, their type

curves may be

converted for analyzing pressure

buildup data i f dimensionless time, t

D

, appearing

both

in

y-axis

and

x-axis is replaced

by 0 and

p D on

the

y-axis

is replaced

by P D as sh6wn in

FY

g

•

11. In

performing type c u r v e W a ~ a l y s i s ,

the

basic steps as

outl ined

by Earlougher and Kersch

4

remain the same except

for

some minor changes in

the

preparat ion

of

the data

plot ( ~ P b Old ) / ~ t vs.

U l up

e e

should be plotted

instead of

plot t ing ~ p / t vs t .

Limitations on the lower l imits

of

producing time,

t D given by Eqs. 23) and 24) should also apply

iR

th is

case.

Gringarten

et al . Type

Curves

I5

Recently, Gringarten

et a l .

IS

presented

another se t of

type curves in a form different

than

tha t

of

Agarwal et a l .

l

and

Earlougher

and

Kersch

4

• Pressure

drawdown

data

have

been plot ted

as

p D

vs.

tD/C

O

) with

cDe

l

as

a

parameter. Their

d r a w ~ o w n type curves are schematically shown

in

Fig.

12.

To convert their type curves for pressure

buildup data,

p D

i s

changed

to

p

and the

param

eter tD/C

O

on t ~ e abscissa s h o u l d w R ~

replaced

by

~ t e D / C O . In

using

the i r type curves

for

buildup,

~ P ) b

Old

vs. should

be

plot ted on the data

U l up

e

plot . Steps for type curve matching

remain

the

same.

The l imita t ions on

the

lower

l imit of pro

ducing time

as discussed ear l ier

should also apply

in th is case.

CONVENTIONAL ANALYSIS

USING

THE

NEW METHOD

Although the new method was

orig inal ly

con

ceived as to

be

used for type

curve

analysis pur

poses, t

also

appears

useful

for analyzing

pressure

buildup data by

the

conventional

semi-log

analysis

methods. This can be seen by rewri t ing Eq.

22)

in

the following

familiar

form:

or

[p ( t + ~ t ) - p

( ~ t = O ) ]

=

162.6

ws p ws

kh

( t x

~ t )

k

Ilog

P

+

log

( t

+

~ t )

2

P

¢ ~ c t r w

3.23 + 0.87 s]

log

k

r

2

t

w

-

3.23

+

0.87s]

26)

27)

where,

m

is the slope per log cycle,

s

is the usual

skin

effect and

~ t e

was

defined ear l ier

by Eq.

17).

m = 162.6 q ~ B

kh

28)

( ~ P ) b u i l d u p is the l e f t hand

side of Eq.

22)

and

was

defined ear l ier

by Eq.

8). The

form of

Eq. 26)

or Eq. 27) suggests

tha t a

graph

of

buildup

pressure,

p

or

( ~ P ) b Old

vs.

~ t e should

ws

U l up

be l inear on a semi-log

graph

paper. This will be

shown

l a te r by

means

of Fig.

16. The

slope of

the l ine

should provide the value of formation

flow

capacity, kh. Note that

the

graph ut i l iz ing is

similar to

the

Horner graph because t also t a k ~ s

into account the effect of producing time, t •

Moreover, th is graph

appears

more

general t h ~ n the

Horner graph because

the value of increases with

the increasing value

of

shut- in

time; as

opposed

to the Horner time group ( t + ~ t ) / ~ t where

i t

decreases

as

increases.

PThis

permits plot t ing of

buildup data

on

the

same time

scale using and

so

that

the

effect

of including

or

excluding the

e

producing time can be compared. Eq.

17)

also indi

cates

that

for

long producing times, when

( t +

~ t ) / t

1. Eq. 17) rever ts back as

p p

e

29)

Eq. 29) also provides the basis for making a

MDH plot for long

producing

times.

Eq.

26) may be

solved for the skin effect, s as

8/18/2019 SPE-9289-MS


SPE

9289

RAM

G.

AGARWAL

7

s =

P

1.151[ ws e

hr) - Pws

m

log

k + 3.23)

Pllc

r 2

t

w

0)

(30)

Note t ha t in Eq. (30) , p 1

hr )

should

be read

on the

semi - log s t r a i ~ ~ t l i ne or i t s exten

s ion .

The

i n i t i a l reservoir pressure, P i

or a fa lse

pressure, pi can be d i r ec t ly read from

the

s t ra igh t

l ine por t ion of the semi-log graph [pws

vs .

~ t e )

where is

equal

to producing

t ime, t

Inspec

t ion of e

Eq

.

(17)

ind ica tes t ha t t h i s cgrresponds

to

the Horner time ra t io , ( t + ~ t / ~ t equal to uni ty or

shu t - in t ime,

c lose

to

~ n f i n i t y . The

es t imat ion

o f i n i t i a l r ese rvo i r pressure by t h i s method wil l be

i l l u s t r a t e d l a t e r by means of a f i e ld example

and

wil l

be

shown

on

Fig.

16.

FIELD EXAMPLE

Pressure Buildup Analysis

Using

New Method

A f i e ld example taken from Gringarten

e t a l .

s15

paper wil l be

u t i l i zed

to i l l u s t r a t e

t h e ~ p p l i c a t i o n of the new method

to

analyze pres

sure bu i ldup

data

taken on

an ac idized

wel l . Both

conven t iona l

and type

curve methods

wil l

be used

to

analyze the data .

Results wi l l

be

compared with

those

of Gringarten e t a l .

15

To m ~ n t a i n

the

con

t inu i ty , par t of

the

information

appearing

in the i r

paper wil l be

reproduced

here.

Table

l i s t s the per t inent r ese rvo i r and well

data , along with

pressure- t ime

da ta bo th dur ing the

drawdown and bu i ldup per iods . Fig. 13 i s a graph

showing

well

pressures

bo th dur ing

the

constant

r a te

drawdown [vs . f lowing t ime, t ) and dur ing

the

subsequent

bu i ldup

[vs.

( t

+

~ t ) J . Buildup

data

were

r ep lo t t ed

using the ngw t ime group,

( t

x

~ t ) / t

+ ~ t ) or ~ t . On Fig.

13,

they

p p e

are plot ted

as

a func t ion of

[ t

+ ~ t

J.

Note t ha t

there

is a s ign i f i can t d i f f e r e n ~ e b ~ t w e e n the two

bu i ldup

curves.

Data

plot ted

simply as a func t ion

of shut- in t ime,

(shown

by open c i r c l e s ) appear

f l a t t e r

compared to the

second bu i ldup curve p lo t t ed

vs . and shown by

so l id

c i rc le s .

This

is to

be

e x p e c t ~ d because the

f i r s t curve does

not take in to

account the e f f ec t of producing t ime.

More wil l

be

sa id

about

t h i s l a t e r .

Although i n i t i a l r ese rvo i r pressure was no t

known

for

t h i s problem, t is

poss ible to

es t imate

t by

means

of

an expanded p lo t (no t

shown

here) o f

ear ly t ime

buildup and drawdown data and recognizing

tha t a graph of [ ~ P ) b ld vs . J is

equiva lent

U

up e

to

[ ~ P d r a w d o w n

vs .

f lowing t ime,

t ) .

Using the

known value of ~ P ) b old a t a given , the cor-

U up e

responding

~ P ) d r a w d o w n a t the same value of

flowing t ime,

t , was

es t imated. This

provided

p,

=

3251 ps i ,

which

was used

to

compute

~ p )

1VS.

drawdown

flowing t ime, t .

Fig. 14 shows a comparison of drawdown data

[p lo t t ed

as

~ P ) d

d vs. t and shown by t r i an -

gles)

with

the

p r ~ ~ ~ u ~ ~ n b u i l d u p data [ ~ P ) b old

vs .

)

(shown

by sol id c i rc le s ) . The c o m ~ ~ r 1 ~ g n

betweefi the two p lo t s i s exce l l en t . ~ p ) . d

data plot ted

as

a func t ion of c o n v e n t i o n a ~ u § k u ~ £ i n

t ime,

~ t , are also

shown

by a dot ted l ine

with open

c i rc le s .

Note t ha t between 200

and

250 minutes

there

is

a

depar ture

between

the convent ional

bu i ldup curve and the other two curves .

Gringarten

e t a l .

15

observed

a s im i la r depar ture

between

the

conven t iona l bu i ldup

curve

and

the drawdown

curve

a t

about 250

minutes and concluded tha t the bui ldup

data

beyond

th i s t ime

should

not be analyzed by

drawdown type curves . However, the

modif ied

bui ldup

plot

does not

suf fer from the

above

re s t r i c t ion .

I f

the

new

method i s used, the

major i ty

of data may

be

type

curve

matched.

Fig.

14 also sugges t s t ha t in

th i s case, the

r ea l shu t - in

t ime,

~ t ,

of 4281

minutes is only equal

to

(1347 x 4281)/(1347 + 4281)

= 1025 minutes in terms of equiva lent drawdown t ime,

.

e

Next

~ P ) b u i l d u

vs . ~ t e

data

were

type

curve

matched uS1ng Gr1nga¥ten e t a l s15 type curve as

shown

in Fig.

15.

Note t ha t

a very sa t i s f ac to r y

match

has been obtained. Computations for type

curve

analys is are shown in Table 2 and r e su l t s sum

marized in Table

3.

I t i s

a lso

poss ible to read the

i n i t i a l

pressure

d i r ec t ly from the log- log p lo t . To

accomplish t h i s , read

~ P ) b

ld a t

=

t This

provides

p, = p ~ t = O ) + ~ P ) b u ~ l d u P . In t h i s ~ a s e

p , = 3251 ~ s i , as shown

on ¥ i g . U ~ 5 .

1

To demonstrate app l i cab i l i t y of

the new

method

to

convent ional semi-log ana lys i s , bui ldup pres

sures , p were

p lo t t ed

on a semi-log

graph

paper

bo th

as

~ s f u n c t i o n

of conven t iona l shu t - in t ime,

(shown

by open c i r c l e s ) and the

equiva lent

t ime,

(shown

by so l id c i rc le s ) . This is shown in Fig. 16:

As expected, there is a s ign i f i can t dif ference

between

the

p lo t s .

In

a

way

t

is

s im i la r

to

com

par ing

a MDH plot with a Horner p lo t . However,

the

new

method i s be t te r

because

data

can

be compared on

an

equiva lent

time scale . I t

a lso

prov ides a

rea

sonable

s t r a igh t

l ine ,

whose s lope was

used to com

pute format ion

flow capac i ty ,

kh,

as

given by

Eq. (28). Eq.

(30)

was used

to

compute the skin

e f fec t ,

s. I t i s a lso poss ible

to

d i r ec t ly read

the

i n i t i a l pressure

from

the

semi-log

s t r a igh t

l ine or

i t s

extens ion where =

t

Resul t s

of both con

vent iona l

semi-log

andetypePcurve

analyses are

l i s t e d in

Table 3.

For comparison purposes , anal

ys i s r e su l t s ob ta ined by Gringar t en

e t a l .

15

are

a lso

shown in Table 3.

Note

t ha t

exce l l en t agreement has

been obtained

between

the

conven t iona l

semi-log and

type curve

methods

when the

new

method is

used.

Moreover,

resul t s also agree very wel l wi th t ha t of

Gringarten

e t

a l .

15

when t hey used the Horner method

for

the

semi-log analys is and the desuperposed data for type

curve matching purposes . Obviously,

the i r

M H type

r e su l t s shown in

Table

3 obtained by ignoring the

e f f ec t

of producing

t ime (us ing

semi-log or type

curve method) wil l be

wrong,

as

expected.

This was

a lso

pointed out by

Gringarten

e t a l .

15

In

regard

to the desuperpos i t ion p r i n c i p l ~ t

should

be

pointed out t ha t

t

i s not always

poss ible

to

desu

perpose the bu i ldup dat ; -because t

requi res

a

knowledge of pressure

vs.

time data

from

the

p reced ing flow per iod. I f the new method i s used,

8/18/2019 SPE-9289-MS


8

A NEW

METHOD

TO ACCOUNT FOR PRODUCING

TIME

EFFECTS WHEN DR WDOWN TYPE

CURVES RE USED TO

ANALYZE

PRESSURE BUILDUP ND OTHER TEST

DATA

SPE

9289

desuperposi t ion of data may not be

necessary.

Although not shown,

the

new

method

may be used to

desuperpose

pressure buildup data.

EXTENSION OF NEW

METHOD

TO OTHER KINDS OF TESTING

I t appears that the new method may be

extended

to analyze other types of

tes t ing

such

as

two-rate

and

multiple

rate

tes ts

in

an i nf in i t e

radial

system.

Once th is

method

is used,

data

may be ana

lyzed

by

both

the

conventional semi log method

and

type

curve matching techniques.

Two

Rate

Testing

8

, 1 6 1 7

A schematic of two rate tes t ing with rate

and

pressure history is shown

in

Fig. 17.

This type of

tes t ing consists of flowing

a well a t a

constant

ra te ,

q l

for

time, t

1

, when the rate is changed to

q2 during

the

incremental

time,

~ t . The flowing

pressure,

p f l ( t

1

a t

the

end of

the

f i r s t

flow

ra te , q l cl3 n be obtained from

Eq.

(21)

as

kh[Pi

-

Pwfl(t

1

)]

141.2 iJB

(31)

Note that the above equation assumes the

log

approximation.

By applying the superposi t ion pr in

ciple ,

we obtain the following equation

for

the

flowing pressure ,

Pwf2(t

l

+

during the

second

flow

ra te , q2

kh[Pi-

P

wf2(t

1

+ ~ t ) ]

141.2 iJB

Subtract ing

Eq.

(32)

from Eq.

31,

we

obtain

kh[pwf2(t

l

+ ~ t ) - Pwfl(t

1

)]

141.2(ql

- q2 iJB

+ 0.80907 + 2s]

(33)

Comparison of

the above equation

with

Eq.

(21)

indicates

that

a plot

of

[po - p f ( t ) ] vs. t

during

a pressure drawdown t es t i s l e q u i ~ a l e n t

to

plot t ing

of data as

obtained

during

the

second rate tes t ing .

Also note

that

Eq.

(33) reverts back to

the

pressure buildup

Eq.

(22) when q2 is se t equal

to zero.

This

is

to

be expected in view of

the

fact that a pressure

buildup t e s t

is

a

special case

of two rate tes ts

with q2 = O.

For a two rate tes t ing , l e t

us

define equiva

lent drawdown time, ~ t e 2 as

34)

Subst i tu t ing

Eq.

(34) in

Eq.

(33)

and

expressing

i t

for conventional semilog analysis , we

obtain

[Pwf2(t

l

+

~ t ) - Pwfl(t

l

)]

162.6

(q l . -

q2 BiJ

kh

[ 1 0 g ~ t e 2 )

+

log _::.k--c-

2

-

3.23 O.

87s]

PiJ

t

r

w

(35)

Eq.

(35) suggests that a graph of [pwf2(t

1

+

- P f l ( t

)]

or p f2( t

1

+ vs.

2 sfiou d

be

l i n ~ r with s l o p e ~

m. Formation

f l o ~ capacity

is

computed

as

kh

162.6(ql

-q2 BiJ

md-ft

m

and skin

effect , s , is

computed using

s

=

P = 1 hr) - Pwfl(t

1

1.I51[

wf2

e2

m

k

- log + 3.23]

PiJ

c

r

2

t w

36)

37)

An

example

will

be shown la ter for mult iple rate

tes t ing .

8/18/2019 SPE-9289-MS


SPE 9289

R M G. AGARWAL

9

Multiple

Rate Testing

8

, 1 7 , 1 8

A schematic of mult iple ra te tes t ing

is

shown

in Fig.

18.

This type of tes t ing consists of

flowing a well a t a constant ra te q1 for time, t

l

,

at

ra te

q2 for time

t1

to t2 and so on.

Say

the

f inal ra te

i s

q for time t 1

to

any

incremental

time,

~ t . Althgugh

not

s h o e ~ , pressures

are

denoted

as

Pwfl t

1

,

Pwf2 t

2

Pwfn-1 t

n- 1

a t

the

end

of f1rs t , secona ana t

n

_

2

t1me per10ds. P w f o ~ t )

are the pressures during the f inal

nth)

per10d.

I f

the steps

similar

to

those

shown

for

two ra te

tes t ing

are

followed

for

multiple ra te tes t ing, the

following equation

is

obtained.

k h [ p w f n ~ t ) -

Pwfn-1 t

n

-

1

)]

141.2 qn-l - q n ) B ~

+

0.80907

+

2s} 38)

where to

=

0;

qo = 0

and

2. Eq.

38) is very

general and should apply to any number of flow and

buildup

periods, provided that the

system

i s

behaving l ike an

inf in i t e

radia l

system

and log

approximation is valid . Eq. 38) also

suggests

tha t

multiple

rate t es t

data during

any

flow or buildup

period may be analyzed using drawdown type curves.

For

multiple

ra te tes t ing, the

equivalent

drawdown

time may

be defined

as

en

39)

For

conventional semi-log analysis , Eq.

38)

may

be wri t ten

as

[ P w f n ~ t ) - Pwfn-l t

n

-

1

) ]

162.6 qn_l -

q n ) B ~

kh [ l o g ~ t e n )

+

log

k 2 -

3.23

+

0.87s]

< p ~ c t r w

40)

Eq. 40) also suggests a l inear

relationship

on a

semi-log paper with slope, m where

41)

h

m

and

Eq. 40)

i s expressed for the skin effec t , s ,

as

follows:

P =1

hr)

-

Pwfn-l t

n

-

l

1 . 151 [wfn

en

s =

m

k

-

log

+ 3.23]

p ~ c r 2

t

w

42)

Next a computer

simulated

example will

be

con

sidered

to

i l l u s t r a t e

the application

to mUltiple

ra te

data.

SIMULATED

EXAMPLE

Multiple

Rate

Analysis Using New Method

To demonstrate the

application,

a computer gen

erated example will

be ut i l ized.

Table 4 l i s t s the

reservoir and well data for

a

gas well , where

product i s kept constant to el iminate

the

effectsgof

varia t ions

as

a

function of

pressure.

Rate

andgpressure his tor ies

are

shown on Fig. 19. Note

tha t the gas

well

was produced a t three different

flow

rates

15, 10 and 5 MMCF/D with intermediate

buildup

periods.

Each flow period was simulated to

be 1 2

day long

whereas

each

of the

f i r s t

two

buildup periods

was I day long.

The

th ird

and

the

f inal

buildup period

was 2

1 2

days

long. Pressure

vs. time data

for

each flow period are shown on

Fig. 19

by

means of

open

symbols, whereas

for

each

subsequent buildup period

by

means of corresponding

solid

symbols.

Multiple

ra te

t e s t Eq. 40) was

expressed in terms of

rea l

gas pseudo pressure

19

,

m p) and rearranged s l ight ly for the purpose. In

pract ical

gas

units , the following equation is

obtained:

m [ p w f n ~ t ) ] - m[pwfn-l t

n

-

1

) ]

qn-l - qn)

163 7T [log

M

+ log -;-;-_k___ ;;

kh en < P ~ c t ) i r w 2

- 3.23 + 0.87s]

43)

Eq.

43)

was used to

process

the

pressure vs.

time

data obtained for

each

t es t period. Results

are shown on Fig. 20

using

the same

symbols as

shown

on Fig. 19. The l e f t hand side of Eq. 43) denoted

by

~ p ) / ~ q l vs. the equivalent drawdown time,

have

been

plot ted

on

a

semi log

graph

paper

for

eacfin

t es t .

Note tha t

i t was

possible

to normalize

a l l

t es t data on the semi-log s t ra ight

l ine

obtained

using

single ra te drawdown dat3.

Although deta i ls

are

not

shown, the slope of

the semi-log s t ra ight

l ine provided the

value

of formation flow capacity

kh) which was consistent with the kh

value

entered

into the

program,

as expected. The

preceding

example

was used to demonstrate the appl icat ion and

establ ish

the val idi ty of the new method for

mul

t ip le

ra te

tes t ing data.

8/18/2019 SPE-9289-MS


10

A NEW METHOD TO CCOUNT FOR PRODUCING TIME EFFECTS WHEN DR WDOWN TYPE

CURVES

RE

USED

TO N LYZE

PRESSURE BUILDUP

ND

OTHER

TEST D T SPE 9289

Other Kinds of

Testing

Although

not shown

in

thi-s paper,

i t appears

that

the

new

method

may be applied to other kinds of

tes t ing methods such

as

in ter ference, constant pres

sure tes t ing,

e tc .

VERTIC LLY

FR CTURED

WELL

The new method was next applied to ver t ica l ly

fractured wells with

both

inf in i t e and f in i t e

flow

capacity f ractures.

Eq. (5)

again

fgrms

s ionless

time, DX

f

as

follows:

Results

are

discussed

below.

the basis

of

th i s

study.

Dimen

for a f rac tured well is defined

2.634 x 10 -

4

kt

2

C P ~ C t ) i

x

f

(44)

Where x

f

is

the

f racture hal f - length in

feet .

De ini t lons of real and dimensionless drawdown and

buildup

pressures

were

kept the

same.

Inf ini te

Flow

Capacity Fracture

Gringtrten

e t

al . s s pressure drawdown data

(p D vs. DX

f

for the inf in i t e reservoir

case

were

t a ~ e n

from

the i r Table 1.

Eq. (5)

was used

to

gen

erate a family

of

pressure buildup curves with

pro

ducing t imes, tpDx ,

as

a parameter. Results

similar to those of Raghavan are presented

in

Fig.

21. Since Raghavan

13

adequately discussed the

l imi tat ions

of using

pressure

drawdown

curves

for

analyzing pressure buildup data

for

frac tured

wells,

only certa in

key points

wil l be re-emphasized.

(i) Computed formation flow capacity wil l be

opt imist ic .

( i i ) Computed value of f racture length wil l be

pessimist ic .

( i i i )

The

character is t ic half

slope

l ine

may

not appear on

the

log-log paper.

Fig. 22 shows the replot

of

pressure buildup

data ut i l i z ing the new method.

Data

are plot ted

as

( t

\

( t

Pwds VS. ~ € D X f or pDx

f

x

Dxf l

pDx

f

DX

f

.

Nofe

tha t

eDx

f

i s the equivalent drawdown t ime,

expressed

in

the dimensionless form, for a ver t i -

cal ly

fractured

well. The majori ty of buildup

data

have

been normalized on the drawdown curve. I t was

rather a surpr ising

observation

in view of

the

fact

that

a t ime

group

developed for the radial system

should

also

be

applicable for

a

fractured

well

which

is

normally

associated

with l inear , e l l i p t i ca l and

radial flow regimes.

Although not included

in th is

paper, the pres

sure

drawdown

data

of

Gringarten,

e t a l . , s for

the

uniform

f lux

f racture case were also

considered.

Pressure

buildup

data

were generated

and

plot ted

using the new method. Once again i t was possible to

normalize

the

majori ty

of buildup data on the draw

down type curve. The plot was very similar

to

that

shown in

Fig. 22.

Finite

Flow Capacity Fracture

The new

method

was next applied to data for a

ver t ica l ly fractured with f in i t e

flow

capacity

f rac

ture . Constant rate pressure drawdown

data

of

Agarwal, et al . ,7 were used

to generate

a family

of buildup-type curves with producing time

as

a par

ameter. This

had to

be

done for

each

value of

dimensionless

fracture

flow

capacity.

These

resul ts

were replot ted using

the

new method. Once

again,

i t

was possible to normalize the majori ty of buildup

data on drawdown

type

curves. For the sake of

brevi ty , resul ts

are not presented here. However,

t should be suff ice to say that constant rate pres

sure drawdown type

curves

of Agarwal, et a l.

7

and

Cinco,

et

al .

6

, may be u t i l i zed to

a n a l y z ~ p r e s -

sure

buildup data. Requirement i s that ~ P ) b

. ld

data are plot ted as a

function

of rather ¥fian

up

the conventional shut- in time, ~ t . e

N LYSIS OF

G S WELL

BUILDUP D T

The development of

the

new

method, for

ana

lyzing

pressure buildup

data,

has been

discussed

mainly ut i l iz ing solutions for l iquid systems. How

ever,

i t

appears

that

the

method

may

be

extended to

include the analysis of data from

gas

wells, i f real

gas

pseudo-pressure m(p) of

AI-Hussainy, et

a l.

19

,

is used and var iat ions of vs. pressure-are

accounted

for. The l a t t e r ma§ be accomplished i f

real

times

in

the

new t ime group are replaced by

real gas pseudo-time, t (p) of Agarwal

l 4

. For

example, i f

pressure

bufldup

data

collected

af ter

short producing time

from

an MHF gas well are to be

analyzed

by

drawdown type curves,

the

following

procedure is recommended.

Graph {m[p

( t + ~ t ) l

-

m [ p w s ~ t = O ) l }

vs.

ws p

( tap

x ~ t a )

( t ~ t a ) on

ap

the data

plot ,

ut i l iz ing the appropriate

type curve.

Steps

outl ined

in Ref.

14

for

type curve

matching

remain

the

same. In

the above

t ime group, t and

represent

flowing

time, t , and

s h u t - i n a ~ i m e ,

~ t ~ P e x p r e s s e d

in

terms of real gas pseudo-time. I f

var iat ions

of

vs.

pressure

during the

t es t

period

appear

to gbe small , instead of

using pseudo

time, real times may be used.

CONCLUDING REM RKS

1.

A new

method

has been developed

to

analyze

pressure buildup data by pressure drawdown type

curves. I t provides a

s ignif icant

improvement

over the

current

methods

because

( i ) the

effects

of

producing time are

accounted

for;

( i i )

data

are normalized

in

such

a way

that

instead

of using a family of buildup

curves, the exist ing drawdown type curves

may be used;

( i i i )

wellbore storage and damage

effects

may

be

considered except under certa in condi

t ions.

8/18/2019 SPE-9289-MS


SPE 9289 RAM

G.

AGARWAL

2. This method can

also

be

used

to

perform

conventional semi-log

analysis

to est imate for

mation

flow

capacity,

kh, skin effec t , s ,

and

in i t i a l

pressure, Pi

I t appears similar to

the

Horner

method because

both methods

take

into account producing time effec ts . However,

this method i s

more

general and has the

advan

tage that

( i )

boL, the

M H

plot

and

the

p lo t

using

the

new method ut i l ize the common time scale ,

w.iich

permits

comparing the two

plots

and

determining the effec ts of including or

excluding the producing time;

( i i )

i t provides

a relationship

between

the

flowing

t ime,

t ,

during

a drawdown t es t

to an

equivalent t ime, , during a

buildup t es t .

e

3. For

long

producing

periods,

the

new

method

rever ts

back

to the M H

method.

4.

A

f ie ld

example

is included to demonstrate the

appl icat ion

of the new

method

and point

out the

ut i l i ty .

5.

This

method has been

extended

to

include the

analysis of two ra te and

multiple

rate

t e s t

data by both type curve and conventional

methods. This was shown

both

theoret ical ly and

also by means of example

problem.

6. Although

original ly developed for radia l

sys

tems,

this method appears to work well for

ver

t i ca l ly

fractured wells

with

inf in i te

and

f in i te

flow capacity

fractures.

7.

The method,

although

developed

using

liquid

solutions,

should be applicable to data from

gas

wells, as shown in the

paper.

8.

Final ly ,

i t

appears

tha t

the

new

method

may

be

applied to

a varie ty

of tes t ing

methods

such as

in ter ference

tes t ing

and constant pressure

tes t ing

to

name a

few.

NOMENCLATURE

B

c

g

C

h

k

log

In

formation volume factor, RB/STB

(res

m

3

/ s tock

tank

m

3

gas

compressibil i ty,

pSi-

1

(kPa-

1

tota l

system compressibil i ty,

pSi-

1

(kPa-

1

storage coeff ic ient , RB/psi (res.

m

3

/kPa)

dimensionless storage coeff ic ient

[see Eq.

(25)]

dimensionless f racture

flow

capacity (see

Ref. 7)

formation thickness, f t em)

formation permeabili ty,

md

logarithm to base 10

natural

logarithm

m

m(p)

)builduP

~ P ) d i f f e r e n c e

~ P ) d r a w d o w n

q

r

w

s

t

ta p)

t

ap

t

X

f

M

a

11

slope of the

semilog

s t ra ight l ine ,

ps i / log cycle (kPa/log cycle)

rea l

gas pseudo

pressure, psi

2

/cp

(kPa

2

/Pa s)

in i t i a l pressure, psi (kPa)

dimensionless pressure drop [see

Eq.

1)]

dimensionless

pressure

r ise or

change [see

Eq.

(19)]

wellbore

flowing

pressure, psi

(kPa)

pressure

at

the end

of t es t

period,

t n - I psi (kPa)

pressures during nth t es t period,

psi (kPa)

shut-in

pressure,

psi

(kPa)

pressure change during buildup,

psi

(kPa)

[see

Eq. (8)]

pressure dif ference,

psi (kPa)

[see

Eq. (10)

and (11(]

pressure

change

during

drawdown,

psi

(kPa) [see Eq. 7) ]

flow rate,

STB/D

or MCF/D

( standard

m

3

/D)

flow ra te

during the

f inal flow

period, STB/D ( standard m

3

/D)

wellbore radius,

f t m)

skin effect

flow time,

hours

to ta l

time up to (n- l) t es t

period,

hours

dimensionless time based on well

bore radius (see Eq. 2)

real gas pseudo

time

(see

Ref. 14)

real gas pseudo-producing time

dimensionless time

based

on half

f racture length

(see

Eq. 44)

producing

period,

hours

producing period during the

f i r s t

t es t , hours.

dimensionless

producing period

shut-in time

or incremental

time

during the f inal flow t es t period,

hours

real gas pseudo

shut-in time

8/18/2019 SPE-9289-MS


12

A

NEW

METHOD TO

ACCOUNT

FOR PRODUCING TIME EFFECTS WHEN

DR WDOWN

TYPE

CURVES ARE

USED

TO

ANALYZE

PRESSURE BUILDUP ND OTHER TEST DATA SPE 9289

t

ap

L1t

e

L1t

en

T

T

Subscripts

a

CD

D

e

eD

e2

en

f

g

i

j

n

p

t

w

REFERENCES

= real gas

pseudo

producing period

=

equivalent

drawdown time,

hours

(see

Eq. 17)

= dimensionless shut-in time based on

fracture half length

=

equivalent

drawdown time for two

rate test

hours (see Eq. 34)

= equivalent drawdown time for mul

t iple ra te

test

hours (see Eq. 39)

= reservoir temperature, oR

= fracture

half length, f t

(m)

=viscosi ty , cp (Pa s)

=

viscosity-compressibility

product

at in i t i a l condition, cp

= formation

porosity, fraction

=

constant,

3.14159

=

adjusted or

pseudo

=

dimensionless flow capacity

=

dimensionless

=

dimensionless, based on x

f

=

equivalent

=

equivalent,

dimensionless

=

equivalent

for

two

ra te

t es t

=

equivalent for multiple

rate

t es t

=

fracture

=

gas

=

in i t i a l

=

index

=

index

=

producing

=

total

=

wellbore

1.

Agarwal, Ram G., AI-Hussainy,

Rafi, and

Ramey,

H. J .

Jr . :

An

Investigation

of

Wellbore Sto

rage and Skin

Effect

in

Unsteady

Liquid Flow:

I

Analytical

Treatment,

Soc. Pet.

Eng. J.

(Sept. 1970) 279-290;

Trans., AIME,

249.

2. Wattenbarger, Robert A. and Ramey,

H.

J .

Jr . :

An Investigation of Wellbore

Storage

and Skin

Effect in

Unsteady

Liquid Flow: I I . Finite

Difference Treatment, Soc.

Pet.

Eng.

J. (Sept.

1970) 291-297; Trans.,

AIME,

249.

3.

Ramey, H. J .

Jr . : Short-Time

Well Test Data

Interpretation

in

the

Presence

of

Skin

Effect

and

Wellbore Storage,

J.

Pet.

Tech.

(Jan.

1970)

97-104; Trans.,

AIME, 249.

4. Earlougher, Robert C., Jr . and Kersch, Keith

M.: Analysis of

Short-Time Transient

Test

Data

by Type Curve

Matching, J. Pet. Tech.

(July 1974) 793-800; Trans., AIME, 257.

5. Gringarten, Alain C., Ramey, Henry J . J r .

and

Raghavan,

R.: Pressure Analysis

for

Fractured

Wells,

paper

SPE 4051 presented

a t

the SPE

AIME 47th Annual Fal l Meeting,

San Antonio,

Tex., Oct. 8-11,

1972.

6. Cinco-L,

Hever, Samaniego-V.,

F. and Dominguez

A.,

N.:

Transient Pressure Behavior for a

Well

With

a

Finite-Conductivity Vertical

Frac

ture,

Soc.

Pet. Eng.

J .

(Aug. 1978) 253-264.

7. Agarwal, R. G., Carter ,

R.

D.,

and

Pollock,

C.

B.:

Evaluation and Prediction of Perform

ance

of

Low Permeability Gas

Wells

Stimulated

by Massive Hydraulic

Fracturing,

J. Pet. Tech.

(March 1979) 362-372; Trans.,

AIME,

267.

8. Earlougher, Robert C.,

Jr . :

Advances in Well

Test Analysis, Monograph

Series,

Society of

Petroleum

Engineers,

Dallas (1977)

5.

9. Miller ,

C. C.,

Dyes, A.

B.,

and Hutchinson,

C.

A.,

Jr . :

The Estimations

of

Permeability

and Reservoir Pressure from

Bottom-hole

Pres

sure Build-Up Character is t ics , Trans., AIME

(1950) 189,

91-104.

10. Horner, D. R.: Pressure Build-Up in

Wells,

Proc.,

Third

World

Pet.

Cong., The Hague (1951)

I I 503-521.

11. McKinley, R. M.: Wellbore Transmiscibil i ty

From Afterflow-Dominated Pressure Buildup

Data,

J. Pet.

Tech.

(July

1971) 863-872;

Trans., AIME, 251.

12. Crawford, G.

E.,

Pierce,

A. E.,

and McKinley,

R. M.: Type Curves for McKinley Analysis of

Drill-Stem Test Data, paper SPE 6754, pre

sented

a t

the SPE 52nd Annual Fall Meeting,

Denver, Colo., Oct.

9-12,

1977.

13. Raghavan,

R.:

The

Effect

of

Producing

Time on

Type Curve

Analysis, J.

Pet. Tech.

(June,

1980) 1053-1064.

14.

Agarwal,

Ram G.:

Real

Gas Pseudo-Time - A New

Function

for Pressure Buildup

Analysis

of MHF

Gas Wells, paper

SPE

8270 presented a t SPE

54th

Annual Fall Meeting, Las

Vegas,

Nev.,

Sept. 23-26, 1979.

8/18/2019 SPE-9289-MS


SPE

9289

R M G. G RW L

13

15. Gringarten, A. C., Bourdet, D.

P. ,

Landel,

P. A., and

Kniazeff,

V. J . : A Comparison

Between

Different Skin

and

Wellbore

Storage

Type-Ruves to

Early-Time Transient Analysis

paper

SPE

8205 presented

at

SPE

54th

Annual

Fall Meeting,

Las

Vegas, Nev., Sept. 23-26,

1979.

16. Russell, D. G.:

Determination

of Formation

Characterist ics

From Two-Rate Flow

Tests,

J.

Pet.

Tech.

(Dec. 1963)

1347-1355; Trans.,

AlME, 228.

17. Matthews, C. S. and Russell, D. G.: Pressure

Buildup

and Flow Tests

in

Well, Monograph

Series,

Society of Petroleum

Engineers

of AlME,

Dallas (1967) 1 Chapter

5.

18.

Odeh,

A.

S. and

Jones,

L.

G.:

Pressure Draw

down

Analysis,

Variable-Rate Case, J . Pet.

Tech.

(Aug. 1965)

960-964;

Trans. , AlME, 234.

19. AI-Hussainy,

R., Ramey,

H. J .

Jr . and Craw

ford,

P. B.:

The Flow

or

Real Gases Through

Porous

Media, J. Pet.

Tech.

(May 1966)

624-636; Trans., AlME,

237.

8/18/2019 SPE-9289-MS


TABLE 1

RESERVOIR AND WELL DATA

(Field Example - Ref. 15)

Formation thickness h

Formation porosity ,

Wellbore

radius

r

Fluid viscos i ty

System

compressibi l i ty c

t

o ~ t i o n volume factor B

Production

rate q

Producing

period,

tp

30 f t

0.15 fraction PV

0.3

ft

1.0

cp

10 X

10-6

psi- l

1. 25 RB/STB

800 STB/D

1347

minutes

Drawdown

Buildul2

t

X

At)

At

(min)

45

85

192

297

417

654

983

1347

Pwf

3198

3180

3154

3141

3130

3116

3196

3097

At

Pws

Pws

t

~ t - p

At=O)

(min)

P l2sil s

3

3105 ·8

5

3108

11

9

3115

18

16 3125

28

30

3139

42

40

3146

49

66

3159

62

100

3171

74

138 3180

83

252 3195 98

334 3203

106

423 3208

111

574 3216 119

779 3222

125

1092 3228

131

1674 3234

137

2186 3238

141

2683 3242

145

3615 3246

149

4281 3246

149

TABLE

3

SUMMARY OF

RESULTS

(Field Example -

Ref.

15)

Analysis

Using This

Method

Type Curve

kh (md-ft)

s

2323

-5.15

2400

-5.31

0.184

3251

C Res

Bbl/psil

Pi

3253

Gringarten et al .

5

Analysis

Semi

loS

TY e

Curve

Horner

MDH D e s u E e q : ~ o s i t ion

kh

(md-ft)

2274

4279

2259

s

-5.0

-4.0 -5.1

C

Res Bbl/psi)

0.19

Pi

3253

3230

12

t

12

+ At)

2.99

4.48

8.64

15.61

29.74

38.6

62.6

92.9

125.0

210.5

269.

321.

396.

499.

612.

748.

842.

898.

962.

1036.

MDH

4095

-4.0

0.25

TYPE CURVE ANALYSIS

(Field Example -

Ref.

15)

Match

Point

i )

i i )

Eq.

[Ap M

10

psi

[PWD

t

[At)

10

min

[..l )

e M

CD

M

2s

[cDe )

1.0

M

Fora.tion

flow

caE:acit} :z

kh

kh

PD)

_-.:.M:....., ',- md-

f t

[Ap M

(141.2) qBjJ

kh

(0.17}(141.2}(800)(1.25)(1)

10

th 2400 md-ft

Skin e f f ec t z s

[At I

0.17

M

0.64

C

kh e M

(0.000295)

- - - / - J -

Res

Bbl/psi

jJ

tD

CD

M

(0.000295)(2400)(10/60)

Res Bbl/psi

(1)(0.64)

C

0.184

Res

Bbl/psi

(25)

is used

to

compute

C = (C)(5.615)

D 2n,hc r 2

t w

(0.184)(5.615)

2n(.15)(30)(IXIO-

5

)(.3)2

S

I

'2 .tn

-5.31

I 1

2

.tn 40821)

TABLE

4

RESERVOIR AND WELL DATA

(Simulated

Example)

Ini t ial reservoir pressure Pi

Reservoir temperature

T

500

psi

7

0

R

Formation permeability k

Formation thickness

h

40600

Hydrocarbon porosity •

Viscosity-compressibil ity product, jJc

t

)

Wellbore radius rw

'5 md

40 feet

5

3.93

X

10-

6

cp/psi

0.25 feet

8/18/2019 SPE-9289-MS


CONSTANT R TE

D R A W D O W N ~

TIME

FIGURE

1: SCHEllATIC OF PRESSURE BUILDUP BEHAVIOR FOLLOWING

A CONST NT R TE

OR WOOWN

FOR A PRODUCTION PERIOD, tp

TIME lor 61

FIGURE

3: SCHEllATIC SHOWING COMP RISON BETWEEN [ 6p)drawdoWn vs

t

ND [ 6P)buildup vs 6t)

8

J

6

:::l.

i -

II

CT5

;a

4

~

:fi

3

.

Q

2

o

TIME

FIGURE 2: SCHEllATIC

OF PRESSURE

BUILDUP BEHAVIOR

FOLLOWING CONST NT

R TE

DR WOOWN

OF SUCCESSIVELY INCREASING FLOW PERIODS, tp

LY

LY

DR WDOWN

SOLUTION

DI

ENSIONLESS

PRODUCING TIME

; D

00

l t

-----

-----

10

4

-

-» -

- 11 . . . . .

Jt '-- - --

/ - - 103

_...0

6

10

2

0 _ _

000

~ ~ ~

BUILDUP SOLUTION

1

lrf

DMENS IONLESS BU ILDUP TIME. 6 tD •

2.634 x 10 4 k6 t

0JlC

t

rw

2

FIGURE

4: BUILDUP TYPE CURVES FOR VARIOUS PRODUCING TIMES- SEMI-LOG GR PH

INFINITE RADIAL SYSTEM,

CD • 0 S • 0)

8/18/2019 SPE-9289-MS


.

.

<J

I

::I.

'i '

=

<I

N

+

:t::

:

~

:5

.

..

c

8

7

6

5

4

3

2

1

lit

BUILDUP

DATA

FOR VARIOUS

PRODUCING TIMES,

tPD

•

105

10

4

lrY

a 102

lrY 10

4

lcf,(tpOX6 t )

DIMENSIONI£SS EQUIVALENTTIME,

6 .0

•

tpO 0

FIGURE 5: N ~ I Z E O BUILDUP TYPE CURVE-SElII-LOG GRAPH

(INFINITE RADIAL

SYSTEM,

COoO;

•

0

0)

w r - - - - - - - r - - - - - - - - - - - - - - - - - - - - -

~

0.1

~

;

FIGURE 7:

LOG-LOG BUILDUP TYPE CURVES

FOR

VARIOUS PROOUCING

TIMES

(INFINITE RADIAL SYSTEM, C

O

l000;

SaO)

111

'

...

::>

VI

VI

~

::>

9

::>

VI

VI

VI

z

Q

...

C

6

::>

II>

5

I>

4

:>

::>

3

II>

II>

2

II>

C

0

6

5

4

DIMENSIONI£SS ~ / . . .

BUILDUP

PRODUC ING

TIME

1r/J/ SOLUTION

lD __o =O

..

/ / -

lIT'

10

4

//

__

...__ ... - - - - ...... ... -

----x--

/ / -

/ /

/;/ lrY

_-- _ D 6 l> --- ----6-- - 6 - - - - --

-06 - - -

111

DIMENSIONl£SS BUILDUP

TIME, 6 t

D

FIGURE 6:

BUILOUP TYPE

CURVES FOR

VARIDUS PRODUCING TlMES-SEMI-lOG GRAPH

(INFINITE RADIAL SYSTEM, C

O

o

l000; saO)

BUILDUP

DATA

FOR

VARIOUS

PRODUCING

TIMES,

tPD

•

10

6

0

105

10

4

"

lrY

la6

DIMENSIONI£SS EQUIVAI£NT TIME, 6t

eD

FIGURE 8: NORMALIZED

8UILDUP

TYPE

CURVE

(SEMI-LOG

GRAPH)

(INFINITE

RADIAL SYSTEM, C

O

loo0; S-o)

8/18/2019 SPE-9289-MS


...

10

~

i

::>

CI l

CI l

~

::>

9

::>

lID

CI l

;g 0.1

co

BUILDUP DATA

FOR VARIOUS

PRODUCING

TIMES

o let

) 10

4

t:

1 3

i

1tYO

2

1 14

10

10

10 8

10

6

10

4

10

2

10

10

0.01l - - - - . . . . . L . _ - - - . . . . . l . ; - - - - - l

J

~ J J

DIMENSIONLESS EQUIVAlENT TIME C:.t

eD

FIGURE 9: NORMALIZED BUILDUP TYPE

CURVES

LOG-LOG

GRAPH

INFINITE

RADIAL SYSTEM, C

D

l000

S-O)

[ 0 ~ 2 9 5

.

FIGURE 10: A GENERALIZED SotEMATIC OF EARLOUGHER

AND KERSCH

DRAWOOWN TYPE

CURYES

4

FIGURE 11: A SCHEMATIC OF EARlOUGHER AND KERSot

4

DRAWDOWII TYPE

CURVES

FOR A WELL WITH STORAGE

AND

SKIN

INFINITE

RADIAl

SYSTEM)-BY

PERMISSION

OF

MARATHON

OIL O M f ~ N Y

8/18/2019 SPE-9289-MS


~ ~

...

-

.

-;;;

Q

t ~

....

10

Q

~

--=-'

t

::>

VI

VI

...

i:

...

VI

VI

j

VI

10-

1

10-

1

Q

1 : ~

)..,<§»

~ O >

:-,. ;1 t

J f .

~

' > ~

~ <;

" ~ ~

. . : s ~ < ,

10

10

2

let

DIMENSIONlfSS

TlME,( tDor 6 V " 0.txXl295 ( ~ ) ere 6 t ,

CD

j

321Kl

DRAW-

BUILDUP

DOWN

.}

15

32<1>

/' .D'.- ..

.q

.... - - - - - - ~ - - - ~

,

...

-

~

vi

3200

f

...

Pwf

vs.

t

i:

;

"

:>

VI

t

• P

ws

vs.6t

e

Qi:

...

o

Pws

vs.6t

...

i:

0

31 l

D

TIME (MINUlB)

FIGURE 12: A SCHEMATIC

OF

GRINGARTEN Et Al

15

DRAWOOIIN

TYPE

CURVES FOR

A

WELL WITH STORAGE AND

SKIN

FIGURE

13:

WELL PRESSURES

Vs

TIME DURING PRESSURE ORAWllOIoIIi AND

BUILDUP PERIOD

(BY PERMISSION OF FLOPETROL) - (FIELD

EIWI'LE

- REF. 15)

200

. r r ~ r _ ~ ~

llKl

160

l I.l<1>

cl.

<l.I l

100

%

U

~ IKl

::>

VI

~

60

Qi:

...

<I>

l

6 t

- 4281 min

6 t -1025 min.

e.

_ ~ O ~ - - ~ - - - . c ; - - - - - - - ~ - - - - o - -

-

&7'

,r: '

" ( 6

p)

drawdown

vs.

t

• (6P)builduP

vs.

6te

o

(6p)buildUP

VS. 6 t

OL -____L -_ _ _ _L -____

--

__

--l'--__ - - J

5<JXl

o

1<JXl

lOO 3(XX)

400)

TIME

(MINUTES)

FIGURE 14: COMPARISON BETWEEN DRAWllOIoIIi

AND

BUILDUP

DATA USING

NEW

METHOD

(FIELD EXAMPLE - REF. 15)

EXTRAPOlATED

(6

p)

-

154

psi

AT

6 te - tp - 1347 min.

OR Pi -

3097

+

154

•

3251 psi

. /

200

}(

,

,

: _ :.P.

2:-

;

~ .

;-- ~ - . . . - _. ..

100

- I . . . . __ -I

: I _

,,

,.+ - -_

...

I

1:. . . ---------- ___

___

~ _ : : ; . r y ~

.,-

I

i .

,-I

I

r

-- - ----1

:

,/,

,

;

...

oC > , . .........

\)

<l

/"';''''/''''/

I I

I

-<

I

..-

....

1- " .".

/

• ( 6 P)b 'Id Vs 6 t .:

<:>

o [ //,,11/ ;

,

UI

up e_:

z

01

/'

... '" I

<

/ /'

'"

(

I I

%

a .

,

I I

U

10

' -,

...

:

~ ~ / f

:

I

Qi:

' I

::>

1 L . / . . I ' L ~ __ ._ _ . _ , ~ ~

___

.

_ _ :.._I_._.

' I

VI

0.1

~ ( ~ ~ r ( : ~ D )

"'-lO--

..

... -

- ~ - .

_'_I ~ . . . ' l O O

VI

...

i:

f -

..

1

1

10

100

1<JXl lOO

MAltH

POINT

( 6

pIMa

10 psi; (PWOIM " 0.17

6te (MINUTES)

( 6

telM

• 10 min; (ta'C

D

I

M

o.64

(CDe2S)

•

1.0

M

FIGURE 15: APPLICATION OF NEW METNOD USING

GRINGARTEN

Et A1 TYPE CURVES

(FIELD

EIWI'LE

- REF. 15)

8/18/2019 SPE-9289-MS


I L

'"

:>

V

V

.....

'"

..

0..

::>

9

::>

33aJ

3300

• P

ws

VS.6t

e

32 11

o P

ws

Vs.6t

EXTRAPOLATED PRESSURE

3Zf1O

US ING NEW METHOD

PI 3253

psl6'

32«1

kh

.. Z3Z3

md-It.

0

3220

5

..

-

5.31

SLDPE

3200

70 ps IIcycle

31S

3160

31«1

6te VI347 min.

3120

3}00

3011)

}

SHUT

IN

TIME, 6t or 6t /MINUTES'

e

FIGURE

16:

APPLICATION

OF NEW METHOD

TO

CONVENTIONAL SEMILOG-ANALYSIS

I

<

'"

9

...

(FIELD

EXAIf'LE

-

REF.

15)

1 t4

TIME, HOURS

FIGURE

18:

SCliEMTIC

OF

MULTIPLE

RATE TESTING

}O

I -

<0

q}

'-

I

qz

Vl

...

I

I

I

I

I t}

.1·

D.t_

FLDW

TIME, t, HR.

INITIAL

PRESSURE L

PRESSURE

DURING

_

_---

- SECOND

R TE

wfl

,/ Pwf

0.. -

,

i PRESSUREDURING:

FIRST

R TE

:

PRESSURE

TH T OOUlD

OCCUR

T FIRST R TE

.r-

D.t-

FLDW

TIME, HR.

FIGURE

17:

SCliEMATIC

RATE AND

PRESSURE HISTORY

FOR

A

TWO-RATE

TESTING.

q > ,

o

TIME

(DAYS'

FIGURE 19: RATE AND PRESSURE HISTORY FOR SIMULATED EXAMPLE

8/18/2019 SPE-9289-MS


'

'

;

VI

VI

'

..

0..

:;

Q

=

;

co

VI

VI

z

52

VI

z

Q

35

20

o FLOW

1

• BUILOUP 1

o

FLOW 2

• BUILDUP 2

FLOW

3

A BUILDUP 3

15

1 ~ 0 - ~ 3 ~ ~ ~ ~ ~ 1 ~ 0 - ~ 2 ~ ~ ~ ~ ~ 1 0 - 1 - ~ ~ ~ ~

TIME (DAYS)

FIGURE 20: NORMALIZED MULTIPLE RATE TEST DATA SIMULATED EXAMPLE

10

INFINITE RESERVOIR (xe'X

f

• 00

DRAWDOWN

SOLUTION

10-

2

10

10

2

DIMENS IONLESS

EQUIVALENTTIME.

6 t

eD

FIGURE 22: NORMALIZED

BUILDUP TYPE

CURVES lOG·lOG GRAPH

VERTICAllY FRACTURED

WEll

WITH INFINITE FLOW CAPACITY FRACTURE)

1 0 r ~ ~ ~ . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:g INFINITE RESERVOIR

X

f

•

00

FIGURE

21:

BUILDUP

TYPE

CURVES FOR

VARIOUS PRODUCING TIMES-lOG-lOG GRAPH

VERTICAllY FRACTURED WEll WITH

INFINITE

FLOW CAPACITY FRACTURE)

Documents

SPE-9289-MS