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Combined nalysis

o

Postfracturing

Performance

and

Pressure Buildup Data

for Evaluating

an

MHF as Well

James N. Bostic, SPE, Amoco Production Co.

Ram O. Agarwal,

SPE, Amoco Production Co.

Robert D. Carter,

SPE, Amoco Production Co.

Introduction

Low-permeability less than 0.1 md) gas wells must

be

stimulated with massive hydraulic fracture MHF)

treatments to be commercial. Once a well has been

stimulated, it

is

important to determine the length

and conductivity

of

the resulting fracture as ac

curately as possible to predict rates and reserves and

evaluate the effectiveness

of

the stimulation.

Two types of data normally are available to

perform the required analysis. The first type

is

routinely collected data such as gas flow rates and

flowing wellhead pressures. The second type

of

data

is

that from specially designed tests such as pressure

buildup tests. There are various methods available

analyzing both kinds

of

data,

but

all

of

the

methods have certain limitations.

Current analysis techniques are satisfactory for

finite-flow-capacity fractures when formation flow

capacity kh

is

known and sufficient production data

without shut-in periods

is

available to perform a

type-curve match as shown by Cinco

et al

1 and

Agarwal

et al

2

However, one disadvantage of

MH

treatments

is

that they often cause the stimulated

wells to produce large volumes

of

load water

returning fracture fluids) for several weeks or

months. Production data obtained during this

0149·2136/80/0010·8280 00.25

opyright 198 SO iety o Petroleum Engineers

cleanup period usually are not applicable for type

curve matching purposes with the currently available

type curves. Since type curves normally are plotted

on a log-log scale

and

exhibit the most character at

small dimensionless times, the loss of the early-time

production data disproportionately reduces the

length

of

the available data band and

is

critical to the

problem

of

obtaining a satisfactory match.

The purpose of this paper is to present a method of

combining the production

and

buildup test data so

that their concurrent analysis overcomes some

of

the

limitations

of

either individual

data

set. The

superposition principle is used in this combination,

and the analysis

of

the resulting

data

set should

provide a more accurate result than previously

possible. This combination has the potential to

generate field data curves

of

sufficient length that the

shape

of

the data curve may be sufficiently definitive

to determine formation flow capacity by type-curve

analysis. Additionally, this method makes it possible

to use production data that contain occasional shut

in periods

of

long duration.

The superposition principle, on which this paper

is

based, was described by van Everdingen and Hurst

3

and was used by Hutchinson and Sikora,

4

Mueller,

5

and Coats

et

al

6

to calculate pressure functions for

the analysis of aquifers associated with water-drive

oil reservoirs. Those pressure functions were called

This paper presents a method

o

combining

post

racturing performance data with

post racturing buildup data

or

analysis on the same constant-rate type curve. This

combination

o

buildup and production data is accomplished through the use

o

the

superposition principle. The method allows the use

o

production data that contain

occasional shut-in periods

o

long duration.

OCTOBER 198

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  resistance,,,4 response,

5

and influence

6

functions and are similar to the pressure function

P

F N

presented in this paper.

n

1965 Jargon and van Poollen

7

applied the

superposition principle to single-well drawdown test

data to generate a pressure function they called the

unit response function. They then used this unit

response function in the analysis of variable-rate

variable-pressure drawdown tests of oil wells.

n

1975

Ridl

ey

8

discussed a method of unified

analysis

of

well tests which used pressure data from

both drawdown and buildup periods. Ridley used a

statistical regression analysis method of parameter

estimation, but he assumed the fundamental form of

the pressure solution.

The method described herein extends these con

cepts to the combined analysis of buildup and

production data f ;'om MHF-stimulated gas wells.

t

also appears to offer certain advantages in both

accuracy and ease of use, in comparison with the

currently accepted analysis techniques.

Methods and Limitations

o

Postfracturing nalysis

A number

of

techniques have been presented in the

petroleum engineering literature for the analysis of

post fracturing pressure and/or rate data. Raghavan

9

presented a very comprehensive summary of pressure

behavior of fractured wells in 1977. Cinco and

Samaniego

10

and Lee and Holditch

11

also have

discussed the advantages and limitations

of the

various analysis methods. While some discussion of

the various available analysis techniques is included,

this paper is not intended to duplicate other com

petent and recent work by examining in detail all the

various analysis methods.

Basically, the various analysis techniques can be

divided into two major categories: conventional

analysis techniques and type-curve matching

techniques. The majority of the conventional

methods are applicable only to infinite-flow-capacity

fractures. (The modified Millheim-Cichowicz meth

od proposed by Lee et at 11

is

an exception to this

restriction, as

is

the analysis of the bilinear-flow

period data using the one-fourth root of time plot

suggested by Cinco et at 10 Both will be discussed in

more detail.) Of the various available type-curve

methods, some are applicable only to infinite-flow

capacity fractures, while others can be used with both

infinite- and finite-flow-capacity fractures.

1,2,10

Defining dimensionless fracture flow capacity

after Agarwal

et

at 2 as

kJ :

D

= , . . . . . . . . . . . . . . . . . . . . . . . . . .

1)

k f

a fracture performs as an infinite-flow-capacity

fracture only for high FCD values. Since MHF

treatments usually are associated with long finite

flow-capacity fractures, any technique used to

analyze MHF-stimulated wells should be applicable

to finite-flow-capacity fractures.

As mentioned previously, the only two con

ventional analysis techniques which address the

7 2

finite-flow-capacity concept are the modified

Millheim-Cichowicz method and Cinco's one-

fourth root analysis of the bilinear-flow-period

data. However, as Lee

et

at 11 and Cinco

et

at 10

point out, both of these techniques require that

formation permeability be known a priori. However,

if this condition

is

met, the type-curve matching

techniques also are simplified greatly and both the

reliability and uniqueness of their answers are im

proved, as discussed in Ref.

2.

Since type curves are fairly reliable where for

mation permeability

is

known and still can be used, if

somewhat less effectively, in the absence of a known

permeability,

we

prefer the use of type-curve

methods over other methods for the routine analysis

of MHF-stimulated wells. This especially applies

when time considerations will not permit the ap

plication of all available methods. Agarwal

et at

2

have shown that their finite-flow-capacity vertical

fracture type curves produce the same solutions as

the Cinco et at 1 type curves. For simplicity

throughout the remainder

of

this paper, all examples

with and references to finite-flow-capacity vertical

fracture type curves will use the Agarwal

et

at

type

curves.

Two sets of finite-flow-capacity vertical-fracture

type curves for

an

infinite reservoir were presented in

Ref. 2. One set of type curves was for a vertically

fractured well producing at a constant weI/bore rate

The other set of type curves was for a vertically

fractured well producing at a constant weI/bore

pressure

While the latter set of curves

is

much more

likely to approximate a realistic producing scheme

for an MHF-stimulated well, the former set also is

valuable as it can be used to analyze pressure buildup

data.

There are some limitations on the use of these type

curves. One

of

these limitations

is

that separate type

curves must be used for the analysis of buildup and

production data. Trying to match two sets of field

data on two separate type curves can be a difficult

and time-consuming procedure, especially in the

absence of a dependable estimate of formation flow

capacity kh

A second limitation applies to the Agarwal et at

2

technique of matching

1/

q (rate - 1) vs. time with the

constant wellbore pressure type curve. This

limitation

is

that the production must be continuous

from the time the well

is

placed on production. While

this limitation was not stated specifically in Ref. 2, it

is

obvious since the rate data

is

to be matched with a

type curve generated under the assumption

of

a

constant flowing bottomhole pressure. However,

experience has indicated that many wells producing

in low-permeability gas plays are subject to oc

casional shut-ins of extended duration due to

demand restrictions or other problems.

Additionally, there are some limitations that are

inherent in the use of either buildup test data or long

term production data. One limitation of pressure

buildup data

is

that it

is

normally impractical or

impossible to obtain long buildups. Since the buildup

tests require the well to be shut in, economic con-

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siderations may dictate that buildup time should not

be excessive. n addition, since the buildup data

is

to

be used for type-curve matching, the production time

immediately before the buildup must be much longer

than the buildup time.

12

Raghavan

13

has discussed

this limitation in detail and has provided guidelines

for estimating the extent

of

acceptable buildup data

based

on

prior production time.

While the buildup data suffers from

an

absence of

late-time data, the production

data

has the opposite

limitation. Early-time production data on MHF

stimulated wells

is

normally not analyzable due to the

production of large volumes of returning fracture

fluids during the first several weeks following

stimulation. Since type-curve matching normally is

done

on

a log-log basis and it is the early-time

data

which have the most characteristic curve shape, the

loss

of

this early-time data has an especially severe

effect

on

the quality

of

the analysis.

Superposition

The principle of superposition has been applied to

petroleum engineering problems for many years. Its

mathematical basis was described in the petroleum

literature by van Everdingen and Hurst in 1949.

3

Superposition can be used to apply known pressure

solutions for single-well constant-rate systems to

multiwell and/or multirate systems.

The general form of the superposition equation

describing the pressure history of

an

undamaged well

in an infinite system producing a slightly com

pressible fluid

is

n

E [ q jB j -q j - IB j - I ) [PD tn- t j - I )DJ] ,

j=1

(2)

where qo =0, to

=0,

and

p to)

=Pi . n SI metric

units, the numerical constant in the numerator

is

1.84.

For a fractured gas well, Eq. 2 becomes

1,424T

m[p to)] -m[p tn ) ] ~ m [ p t n ) ] =

n

E [ q j -q j - I ) [PD tn -

t

j-I)DXj1),

......

3)

j=1

where m p) is the real gas pseudopressure as

defined by AI-Hussainy

et

al.

14

and

tn

- t j - I )Dx

is

a dimensionless time difference based on the fracthre

half-length. n SI metric units, the numerical con

stant in the numerator

is

1.28 x

10

-

3 .

evelopment

o

FCN

Since

PD tn - t

j

_

1

DXf

= [m[p to)] -m[pc tn - t j - I )J }

kh)

1,424Tqc

where q

c

is a constant rate and Pc is the bottomhole

OCTOBER 1980

flowing pressure that would exist

if

the rate qc were

imposed on the well, Eq. 3 can be written as

1,424T

n ( kh )

~ m [ p t n ) ] = ~ j ~ ~ q j - q j - I ) ~ 1,424T

[m [p tn) )

m [ c

tn

- t j - I ) I ]

/qc] .. 4)

n

SI metric units, the constant 1,424

is

replaced by

1.28 X 10-

3

.

Simplifying Eq. 4 yields

n

~ m [p tn)]

=

E

qj -q j_ I ) [m [p to) 1

j=1

- m [Pc tn - t

j

_

l

) l}/qc .   5)

Define a new term as

PpCN t) = m [p to) 1

m

[Pc

t)

l}/qc-   (6)

Thus, PpCN t) can be thought of as the

m

[p to) 1 m [pc t) that would be generated

by a constant flow riite of 1

MscflD

(or 1 std m

3

/d in

SI metric units to avoid carrying a numerical con

stant in the following equations). Substituting Eq. 6

into Eq. 5 yields

n

~ m [p tn)] = E qj

-q j - I

)PPCN tn - t

j

_

l

)

j=1

n

=

q

1 - q 0 ) PPCN tn -

to)

+ E

j=2

qj -q j - I)PPCN tn

-

t

j

_

I

)·

(7a)

Since qo =0 and to =0,

n

~ m

[p tn)

1

=ql

PpCN

t ~ )

+

E

qj

- q j - I )

j=2

PPCN tn t

j

_

I

)· (7b)

Solving Eq. 7b recursively for

PpCN t

n

), we.

obtain

f o r n ~

P

PCN t

n

) = [

~ m [p tn

) ] -

q

-

q

- 1 )

PPCN tn

t

j

_

I

) ] /ql .   (8a)

For

the special case

of

n

=

1 Eq. 8a becomes

and PpCN is then the transformation to the

equivalent variable pressure history with a constant

rate. PPCN

is

similar to the influence function of

Coats et al.

6

or the unit response function

of

Jargon

et

al.

7

Example

The following example shows how PpCN is

calculated for a fractured gas well which has

produced continuously

at

a constant flowing bot-

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-- L

CTU L PRODUCTION

HISTORY

---1 APPROX

IM TE

PRODUCTION

HISTORY

~ ________________________ ______

o

TIME t

Fig -

Production history for

P

FCN

example.

tom hole pressure

Pwj.

For this case, the pressure

drawdown in terms

of

real gas pseudopressure is

Am [p

t

n ]

=

m [p t ) ] - m (p

wj

=

constant

=Am

(Pwj)

The rate history for this example is given in Fig. 1

where the solid line

is

the actual producing history

and the dashed lines represent a stair-step ap

proximation

of

this history where the size

of

the time

length of this approximation can be chosen as small

as necessary. I f this well had been produced at an

e ~ u i v l e n t constant rate

of qc

= 1

MscflD

or 1 std

m /d to avoid carrying a numerical constant), then

its pressure history would have been

n

PFCN(tn) =

[Am

(Pwj) - E (q j -q j - l )

j=2

PFCN(tn - t

j

-

1

)] Iq l ' ......... 9a)

for

n ~ a n d

PFCN

( t l )

=

Am

(Pwj) Iq l '

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

9b)

for

n=

1, where

PFCN(tn)

is defined by Eq. 6. Note

that Eqs. 9a and 9b differ from Eqs. 8a and 8b only

in the substitution

of

the constant

Am (p

wj) for the

function Am [p( t

n

  ] .

Use of P

FCN

The P

F N

vs. time data now generated using

drawdown data can be matched with the constant

rate type curves of Agarwal et al

2

or

o t h e r ~ .

Comprehensive instructions

on

the use

of

type curves

are given by Earlougher

15

and will not be repeated

here. Note that dimensionless pressure can be ex

pressed in terms

of

P FCN ( t

n)

as

PD =kh PFCN(t

n

)

/l,424T, ............. 10)

where

PFCN(tn)

is defined by Eq. 6. In SI metric

units, the numerical constant in the denominator is

1.28 x 10-

3

•

In the special case of constant drawdown

production, no advantage is obtained by using P

FCN

vs. time as opposed to 11q vs. time with the constant

wellbore-pressure type curve. The use

of

P

F N

vs.

time will be preferred when 1) a well is produced

with varying flowing bottomhole pressures, 2)

1714

available buildup data are combined with the

production data, and

lor

3) there are shut-in periods

in the production data.

Combination o Production

and uildup Data

The superposition transformation which results in

the calculation

of

P

F N

t) allows constant wellbore

pressure production data to be used with constant

well-rate type curves. This is valuable because

pressure buildup data also are analyzed with con

stant-well-rate type curves; thus, pressure buildup

data and production data can be evaluated on the

same type curve.

Since buildup tests in the absence of wellbore

effects) provide early-time data but may be too costly

in terms of lost production to run for long periods,

and production

data is

generally available for long

times but is unanalyzable at early times due to

fracture fluid cleanup

and/or

other production

problems, these two types of data tend to com

plement each other. Together, they can provide a

much longer data band for type-curve matching.

Another way to use the superposition principle is

to convert pressure buildup

data

to the equivalent

rate-time data. Equivalent rate data can be calculated

from Eq. 8 for any desired drawdown Am

[p(tn)]

if

P

F N

t) is known. I f the preceding producing

period is much longer than the buildup period,13

P

FCN

t) can be estimated from the buildup data

using

PFCN(At) =[m[p(At) ] -m[p(At=O)lJlqj . . 11)

where At is the buildup time and q j is the final

stabilized flow rate before buildup. Once obtained,

these calculated early-time rates are treated like any

other rate data and, thus, can be used in the

P

F N

calculations.

When this early-time data is combined with

production data for about a year, the resulting data

band will be significantly longer than the data bands

available from either buildup or production data

alone. This may help to minimize the problem of

non uniqueness of results which may exist with type

curve analysis techniques.

Use o the Real Gas Pseudotime

It is important to note that to use the buildup data

accurately with draw down-generated type curves, it

was necessary to apply the real gas pseudotime

ta

(p) concept

of

Agarwal

16

to the buildup data.

This

is

similar to the

real

gas pseudo pressure

m(p) of

AI-Hussainy

et

al

4

ta(p) compensates

for the variations

of

the effective Jl t product with

pressure, which appears in the dimensionless time

term. This quantity is given by

t , ( p )= r

p

~ )

Jpo Jl(p)c

t

(p)

dp ............ 12)

and replaces

Atl Jlc )

i in the dimensionless time

term.

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40 ------------------------------

38

0..

34

z

U

:

c.. g32

30

10

20 30 40 50 70

100

200 300 400

t DAYS)

Fig 2 - Analysis

of

simulated radial

flow

data

using P

FCN

technique.

Extension Over Shut In Periods

With Unavailable Bottomhole

Pressure Measurements

Eqs. 7 and 8 can be used to obtain the

PpCN

function

for wells that have shut-in periods extending over

several days or weeks. When the bottonhole pressure

is available during these periods, the procedure

described previously can be used in a straightforward

way for this purpose, treating the data as in a

standard variable-rate situation. However, when

bottomhole pressure information

is not available

during such shut-in periods, an extrapolation

technique is employed to synthesize the missing data .

An inspection of

any

of

the dimensionless pressure

vs

dimensionless time solutions for fractured wells

shows that beyond very early time the 10gPwD vs log

tDx curve is approximately linear over a short in

terval. This interval

is

usually

at

least twice the

t

Dx

value

of

the last available pressure point. This ob

servation, therefore, allows estimation

of PpCN

as a

function of time during shut-in periods, provided

that the shut-in time does not exceed the previous

producing time. Unacceptable extrapolations are

manifested in a computed P PCN curve that is

oscillatory.

This extension

of

the

PpCN

technique over long

shut-in periods is very significant. It serves to extend

the technique to a much larger group of candidate

wells and enlarges the amount of usable data on

many more wells.

Some Practical Considerations

Although the equations used to calculate P PCN t )

appear simple in concept, they can become extremely

tedious in application.

In

practice, a computer is

needed to perform the calculations. Preparation of a

program to perform these calculations is relatively

simple, and for convenience the program can include

routines to calculate m(p)14 and t

a

(p).16 f these

routines are included, the required data need only

include times, pressures, rates, and gas property

data

to generate PpCN t ) . As with any computer program

or analysis technique, the final results depend on the

OCTOBER 1980

TABLE NON·MH APPLICATION

- SIMULATED EXAMPLE

Input

Parameters

Formation permeability k md

Formation

thickness

h

ft

(m)

Reservoir temperature T 0 R

(K)

Analysis Results

Semilog s lope m, [ psi

2

/cp) McflD)]/cycle

[ MPa

2

/Pa·

s)/(m

3

/d)]/cycle

I

Formation

flow

capacity

kh,

md·ft

md·m)

0.004

50 15)

740 411)

6.05

X

10

6

10.1)

0.2 0.06)

quality

of

the input data. Inaccurately measured

rates or pressures will yield inaccurate values of

PpCN(t ) .

The calculation>

of

P PCN

t )

using Eq. 8 requires

repeated interpolation

of

some type to obtain the

terms

P p c N ~ t n

j

_

l

) for j= to n We agree with

Jargon et

al that

a logarithmic interpolation is pref

erable here, due to the nearly linear nature

of

the log

PwD

vs. log t

Dx

  relationship over the portion

of

the

curve normally subject to interpolation. Although no

support

is

presented in this paper, we also agree with

the observation

of

Jargon et al

7

that it is possible to

obtain erratic and oscillatory results

n

the

calculation of

PpCN

when drastic rate and/or

pressure chages are encountered.

N on MHF Applications

This technique should be applicable to any situation

in which the superposition principle can be used. In

particular, it should be possible to use this approach

to analyze gas and liquid flow data from wells that

have been given at most a small stimulation treat

ment.

Since PpCN corresponds to a constant rate of 1

MscflD or

1 std m /d this function plotted for

radial flow analysis on a semilog graph yields a

straight line

of

slope m from which formation flow

capacity can be calculated using

kh=

1 637T/m

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

13)

In SI metric units, the numerical constant in the

numerator is 1.47 x

10

- 3 .

Fig. 2 is a plot

of

P PCN vs. t on semilog paper

which illustrates this analysis method. The rate vs.

time data used in the calculation

of PpCN t)

for this

figure was obtained from a one-dimensional radial

flow reservoir simulator. 17 The important input

parameters and results of

the analysis are given in

Table 1.) This method can be a useful way to analyze

wells which, for various reasons, could not be shut in

for buildup tests or held at constant flow rates for

drawdown tests.

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T BLE

2 -

RESERVOIR PROPERTIES

-

SIMUL TED EX MPLE

Initial

reservoir pressure

Pi

psi a MPa)

5,500(38)

0.004

50

(15)

0.035

740(411)

Formation permeability k md

Formation

thickness

h ft m)

Hydrocarbon

porosity

¢HC

Reservoir temperature T oR K)

Initial

gas

viscosity

Ui,

cp

Pa·s)

Initial

gas

compressibility cgi

0.0284 (2.84 x 10-

5

)

psi-

1

Pa-

1

)

Half

fracture length

x

ft

m)

Fracture flow

capacity

k w, md-ft

md·m)

0.0001286 (1.86 x 10-

8

)

1,000 (305)

40 (12)

T BLE

3 -

TYPE·CURVE M TCHES

-

SIMUL TED EX MPLE

Match

k

X, k,w

No. • md)

ft)

m)

md-ft) md· m)

l

2

3

- -

0.004 995 303 39.8

0.0032 1325 403 42.4

0.0072 740 225 26.6

• Preferred match .

12.1

12.9

8.1

5 0 c 0 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

q

(MCFD)

2000

0

500

>-

=>

:r

Vl

200

0

40

240

t

DAYS)

Fig. 3 - Simulated example - production history.

10

~ l

w

oo

=>

V>

V>

1

V>

V>

Z

V>

Z

w-

e

P

FCN

( J l S i ~ C P

\

I MCFD 7

oMENS IONl£SS

FRACTURE

CA PAC

ITY FC D

I O ~

: I I

I

I

I

-----,

MATCH

POINT

:

I I

I I I

I : I

_ _____ _____ L ____

I. 0 10 100 1000

, (DAYS) .k w

FCD kif

DIM

ENS IONl£SS TIME, DX,

Fig. 4 - Simulated example - type·curve match.

1716

320

Note

that

due

to

the length

of

the time interval

covered by the production

data

to be analyzed, in

terference or boundary effects frequently will be

observed in the analysis

of

normal-permeability

wells.

Type Curve Matching

Although type-curve matching

is

now an accepted

pressure transient analysis technique, the problem

of

sometimes locating more than one possible match

remains. Agarwal et

l

2

suggested that if

kh

is

known from a pre fracturing buildup test, then a

y-

axis value

of

the type curve can be calculated for a

given y-axis value

of

the tracing paper overlay (field

data). With this relationship fixed, the tracing paper

overlay is moved in the x direction only until the best

match is obtained. This simplifies the type-curve

matching

and

usually results in only one acceptable

match.

When no acceptable match is obtained, it may be

because

of

inaccuracies in the prefracturing testing

or

because the prefracturing test was representative

of

a

much smaller portion of the reservoir than that

examined by the post fracturing buildup and

production data. When this occurs, a

two

dimensional type-curve match must be attempted.

One method

of

limiting the range

of

possible type

curve matches is to calculate a value of kh from a

plot of P

F N

vs. log t as discussed in the section on

non-MHF

applications. For most MHF-stimulated

wells, the calculated kh from this type

of

analysis will

be somewhat optimistic since the well probably will

have not reached pseudoradial flow yet. This

calculated value, therefore, should be the maximum

possible value

of kh for the well, and a PwD value

calculated from it could then serve as a maximum

PwD value. This should minimize at least partially the

freedom of movement during the matching process

and perhaps provide better answers.

Once one or more potential solutions have been

obtained from type-curve analysis, the acceptablity

of

the solutions can be checked using a two

dimensional reservoir simulator such as the one

described by Agarwal et

al.

2

This is done by making

a comparison

of

actual vs. simulated buildup and

production

data

to determine which

of

several

possible solutions is preferable. Lee et

al.

have

suggested

that

a reservoir simulator could be used in

place

of

other techniques to determine

k xf

and

kfw, but experience indicates that such trial-and

error history matching techniques can be more time

consuming and expensive than those suggested here.

Of

course, systems with multiple layers, areal

heterogeneities, variable fracture flow capacities

or

other nonuniform conditions may be analyzable,

at

present, only through trial-and-error simulation.

Examples

Simulated Example

To

test the

P

F N

analysis technique, the two

dimensional reservoir simulator mentioned pre

viously was run with the reservoir properties given in

Table 2. A well was produced with a constant flowing

JOURNAL OF PETROLEUM

TECHNOLOGY

8/9/2019 SPE-8280-PA

7/9

bottomhole pressure of 1,238 psia (8.538 MPa) for

150 days, shut in for 30 days, and then produced at

1,238 psi a (8.538 MPa) for another 130 days. The

rate vs. time results

of

this simulation are shown in

Fig. 3.

Simulated rate-time

data

and buildup pressure

time

data

obtained at the end of the final production

period along with the appropriate gas properties then

were processed through a computer program

to

compute PpCN

I).

Next, a data plot of PpCN vs.

t

was made on the same scale as the log-log type curve.

The resulting type-curve match is shown in Fig. 4.

Match point data are

PpCN) M = 10

6

(psi2/cp)/(McflD)

[1.66

x 10

3

(MPa

2

/Pa·s)/(m

3

/d)],

I)

M =

100 days

=

2,400 hours,

tDXf) M = 2

x

10 -

2

,

PwD) M

= 0.19, and

FCD = 10.

Fracture flow capacity, permeability, and fracture

length then were calculated from Eqs. 1, 10, and 14,

where Eq.

14 is

where

t

is in hours. In SI metric units, the numerical

constant in the numerator is 3.6 x

10 9 .

The results

are given in Table 3 as Match 1.

Although these results were in excellent agreement

with the simulator input data, it should be noted that

two other type-curve matches with the FCD =

10 and

F

CD

=

5 curves were possible. This is the

non-

uniqueness

of

match problem

that

frequently has

hampered type-curve matching methods. Analysis

of

these other matches would have yielded the results

indicated as Matches 2 and 3 in Table 3.

These three sets of results illustrate the typical

range

of

answers that can be obtained from type

curve matching when estimates

of

kh from

prefracturing buildups are not available. f a

pre fracturing buildup estimate

of

kh had been

available, only the first (correct) type-curve match

would have been possible.

Field Example

An actual field example

of

the use

of

the

PpCN

analysis technique also is given. Fig. 5 shows the

reported 3-year producing history

of

a low

permeability gas well. Both prefracturing

and

post fracturing buildup tests were available on this

well. The reservoir properties estimated for this well

are given in Table 4, and the post fracturing buildup

data are contained in Table 5.

The available rate-time and pressure-time data

were used

to

calculate

PPCN t).

A type-curve match,

which honored the

kh

calculated from the

prefracturing buildup test, then was made,

and

the

results are shown in Fig. 6. (Note that the buildup

portion of the data is indicated

on

Fig. 6.) The match

point obtained was

OCTOBER 1980

~ r ~

\.

q

(MCFDI

\ ----

000

500f-

1975 1976

YEARS

1977

Fig. 5 - Field example -

production

history.

TABLE

4 - RESERVOIR PROPERTIES - FIELD EXAMPLE

I

nitial

reservoir pressure Pi

psia MPa)

Flowing

bottom

hole pressure Pwf,

psia MPa) -

5,100(35)

1,273 (8.777)

0.0027

56 (17)

0.035

725 (403)

Formation permeability

k

md

Formation

thickness h, ft m)

Hydrocarbon

porosity

cPH

Reservoir temperature T, 0 R K)

Initial

gas

viscosity

Ui,

cp Pa

·s)

Initial gas

compressibility

e

g

 

0.0264 (2.64 x 10 -

5)

psi - 1 Pa - 1 )

I

0.0001465 (2.12 x 10 - 8)

TABLE 5 - POSTFRACTURING BUILDUP DATA

- FIELD EXAMPLE

Time Pressure

hours) psia) MPa)

0.0 1,273 8.78

0.1 1,415 9.76

0.4 1,567

10.80

1.0 1,692 11.67

2.0 1,803 12.43

4.0 1,947 13.42

8.0 2,040 14.07

12.0 2,193 15.12

16.0 2,292 15.80

20.0 2,358 16.26

24.0 2,415 16.65

32.0 2,507 17.29

40.0 2,583 17.81

48.0 2,647 18.25

56.0 2,703 18.64

63.0 2,746 18.93

100.0 2,929 20.19

118.0 2,996 20.66

130.0 3,037 20.94

134.3 3,051 21.04

Final rate before shut·in: 1,600 McflD 45 760 m

3

/d .

1717

8/9/2019 SPE-8280-PA

8/9

10

DIMENSIONLESS

FRACTURE CAPAC ITY FCD

7

1

T

-- ----

I

1

I

l[ t

N ~ ~ I VI

I

6

VI

l O I - . ~ -  

- . J :

Q

10-

1

I

I 3

,

,

I

I

...

I

l h 0 _

10-

I I I

_.1 . ______

1. ______

J

10 100 1000

t

IDAYS)

F •

kfW

CD

kx

f

10 -3 ------::_--- --;-_--- ---::-_---- ---::-_--- --::-_--- ------

10 '

10-4

10-

3

10-

2

10-

1

DIMENSIONLESS

TIME, to

x

f

Fig. 6 - Field example - tYPEl·curve match.

P

FCN

  M =

10

6

(psi

2

/cp)/(McflD)

[1.66x 10

3

(MPa

2

/Pa·s)/(m

3

/d)],

t)

M

=

10 days

=

240 hours,

tDX/)M

= 2.2x

10-

4

,

PwD)

M

=

0.146,

and

FCD = 50.

The results, calculated as before using Eqs. 1, 10,

and

14, were

k

=

0.0027 md,

x f = 2,400 ft (730 m), and

kfw = 324 md-ft (99 md ·m).

The two-dimensional reservoir simulator then was

run using these parameters and the production

schedule of this well. The results of this simulation

are shown in Fig. 7. To predict reserves

and

future

rates for this well, the reservoir simulator

then

was

run

in a predictive

mode

for

the

expected life

of

the

well with an appropriate drainage area assigned.

onclusions

Based

on

the

work

presented in this

paper, the

following statements appear valid.

1.

Superposition can be used to generate

P

FCN

t ) , which

is

the transformation to the

equivalent variable pressure history with a constant

rate

of

a well/reservoir system

that

has

produced

a

variable rate with a known pressure history.

2. Production

and buildup

data can be combined

into a single P

FCN

vs. time curve which can be used

for type-curve matching.

3.

The combination

of

production and buildup

data offers a significant advantage because it

produces a much longer field data curve, which

greatly enhances type-curve matching by curve

shape.

4. The P

FCN

technique can be used to extend type

curve matching

to

wells with intermittent

production

periods.

5. The PFCN technique also should be applicable

to

data

from non-MHF

wells.

6. The r e ~ u l t s of a type-curve match can be

checked and long-term rates

and

reserves can be

1718

2000

q

MCFD)

1000

500

CASE CUM. PROD.

\

SIMULATED

-

8 8

MMCF

'lr ACTUAL

- 809

MMCF

100 L..L..J....L..L-:-:19=75

.........

..L..L.L...L..

.........

':-'::'--'-J....L..I...L.J....L...L..'-:1L, 97='7 ...L...L.1...LLJ....l...J...J

Fig. 7 - Field example - simulated/actual production

comparison.

predicted with a two-dimensional reservoir

simulator.

Nomenclature

=

formation

volume factor, RB/STB

(m

3

/m

3

)

gas compressibility at initial reservoir

pressure, psi - 1 (Pa - 1)

total compressibility at initial reservoir

conditions, psi -1 (Pa -

1 )

F CD

dimensionless fracture flow capacity

h formation thickness, ft (m)

k = formation permeability, md

k

f

=

fracture permeability, md

m = slope on a semi log plot, [(psi2/cp)/

(McflD)]/cycle {[(MPa

2

/Pa·s)/

m p)

t::.m p)

(m

3

/d»)/cycle

J

real gas

~ s e u d o p r e s s u r e

psi2/cp

(MPa /Pa·s)

difference in real gas pseudopressure,

psi2/cp

(MPa

2

/Pa

·s)

P =

pressure, psia

(MPa)

Pi

= initial reservoir pressure, psia

(MPa)

PwD =

dimensionless wellbore pressure

Pwf

=

flowing bottomhole pressure, psia (MPa)

P

FCN

= pressure function, (psi2/cp)/(McflD)

[(MPa

2

/Pa·s)/(m

3

/d»)

flow rate, MscflD (std m

3

/d)

producing time, days

real gas pseudotime, hr cp . psi - 1

(hr/Pa.s. Pa -1

dimensionless time

dimensionless time based

on

half

fracture length of a vertical fracture

t:: t = buildup time, hours

T = reservoir temperature , 0 R (K)

w = fracture, width, ft (m)

xf

= half fracture length, ft (m)

p- = viscosity, cp (Pa .s

P-i =

viscosity

at

initial reservoir pressure, cp

(Pa·s)

total porosity, fraction

hydrocarbon porosity, fraction

JOURNAL OF

PETROLEUM TECHNOLOGY

8/9/2019 SPE-8280-PA

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References

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

Transient Pressure Behavior for a Well With a Finite

Conductivity Vertical

Fracture,

Soc. Pet. Eng. J (Aug.

1978) 253-264.

2. Agarwal, R.G., Carter, R.D., and Pollock, C.B.:

Evaluation and Performance Prediction

of

Low

Permeability Gas Wells Stimulated by Massive Hydraulic

Fracturing, J

Pet. Tech

(March 1979) 362-372; Type

Curves for Evaluation and Performance Prediction of Low

Permeapility Gas Wells Stimulated by Massive Hydraulic

Fracturing,

J

Pet. Tech.

(May 1979) 651-654;

Trans.

AIME,267.

3. van Everdingen, A.F. and Hurst, W.:

The

Application of the

Laplace Transformation to Flow Problems in Reservoirs,

Trans. AIME (1949) 186, 305-324.

4. Hutchinson, T.S. and Sikora. V.J.: A Generalized Water

Drive Analysis, Trans. AIME (1959) 216,169-177.

5.

Mueller, T.D. : Transie nt Response of Nonhomogeneous

Aquifers, Soc Pet. Eng. J (March 1962) 33-43; Trans.

AIME (1962) 225,33-43.

6. Coats, K.H., Rapoport, L.A., McCord,

J.R.,

and Drews,

W.P.:

Determination of Aquifer Influence Functions From

Field Data, J Pet. Tech. (Dec. 1964) 1417-1424; Trans.

AIME,231.

7.

Jargon, 1.R. and van Poollen, H.K.:

Unit

Response Func

tion From Varying Rate Data, J Pet.

Tech

(Aug. 1965 965-

969; Trans. AIME, 234.

8. Ridley,

T.P.: The

Unified Analysis

of

Well Tests, paper

SPE 5587 presented at SPE 50th Annual Technical Con

ferenceandExhibition, Sept. 28-0ct.l, 1975.

9.

Raghavan, R.: Pressure Behavior

of

Wells Intercepting

Fractures,

Proc.

Invitational Well Testing Symposium,

Berkeley, CA (1977).

10. Cinco L., H. and Samaniego V., F.: Transien t Pressure

Analysis for Fractured Wells, paper SPE 7490 presented

at

SPE 53rd Annual Technical Conference and Exhibition,

Houst on, Oct. 1-3, 1978.

OCTOBER 1980

11. Lee,

W.J.

and Holditch, S.A.:

Fracture

Evaluation with

Pressure Transient Testing in Low Permeability Gas Reser

voirs. Part I: Theoretical Backgro und, paper SPE 7929

presented at SPE Symposium on Low Permeability Gas

Reservoirs, Denver, May 20-22,1979.

12. Agarwal, R.G. , AI-Hussainy, R., and Ramey, H.1. Jr. : An

Investigation of Well bore Storage and Skin Effect in Unsteady

Liquid Flow:

1.

Analytical

Treatment,

Soc Pet. Eng. J

(Sept. 1970) 279-290; Trans. AIME, 249.

13. Raghavan, R.:

The

Effect of Producing Time on Type Curve

Analysis, J Pet. Tech. (June 1980) 1053-1064.

14.

Al-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: The

Flow

of

Real Gas Through Porous

Media,

J

Pet. Tech

(May 1966) 624-626; Trans. AIME, 237.

15. Earlougher, Robert C. Jr.: Advances in Well Test Analysis

Monograph Series Society

of

Petroleum Engineers, Dallas

(1977) 5,24-27.

16. Agarwal, R.G.: 'Real Gas Pseudo-Time' - A New Function

for Pressure Buildup Analysis of

MHF

Wells, paper SPE

8279 presented

at

SPE 54th Annual Technical Conference and

Exhibition, Las Vegas, Sept. 23-26,1979.

17.

Carter, R.D.: Solutions

of

Unsteady-State Radial Gas

Flow,

J

Pet.

Tech

(May 1962) 549-554; Trans. AIME, 225.

SI Metric onversion Factors

ep

X 1.0*

ell

ft

x

2.831 685

psi x

6.894 757

·Conversion factor

is

exact.

E-03

E-02

E+OO

Pa·s

m

3

kPa

JPT

Original

manuscript

received in Society

of

Petroleum Engineers

office

July

20

1979. Paper accepted for publication April

18

1980. Revised manuscript

received Aug. 11, 1980. Paper SPE 8280) first presented at the SPE 54th Annual

Technical

Conference and

Exhibition

held in Las Vegas. Sept. 23-26. 1979.

1719