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Transient Response of Nonhomogeneous Aquifers

ABSTRACT

T.D.MUELLER

MEMBER IME

Many

investigators have used

the response of

the

dimensionless aquifer to a unit

pressure

drop or

a

unit fluid-withdrawal volume

to

calculate the

performance of an

aquifer in

supplying

water

influx

to an

oil

reservoir. n the past, these response

functions have been calculated with the

aid of

the

Laplace

transform With

the

advent

of

ultra high-

speed

digital computers, it becomes practical to

solve for

the response functions with finite-differ

ence techniques. The computer method also permits

extension

of the dimensionless-aquifer concept

to

include

the nonhomogeneous aquifer wherein

the permeability

and

other properties vary

as a

function

of

the

space

co-ordinates. This

paper

gives

results of

calculating the

response

functions

for a series

of nonhomogeneous aquifers. Response

functions

are presented for both

linear

and radial

aquifers whose

thickness,

permeability-viscosity

ratio and porosity-compressibility vary

These

functions are new and should prove

useful

to the

petroleum

engineer

in

analyzing the behavior

of

nonhomogeneous aquifers.

Results are presented

in the form of

charts

that can be easily used by

the field engineer.

INTRODUCTION

Aquifers which

surround

many

oil and

gas

reser

voirs

have the

ability to

supply

water

influx

to

such

reservoirs as

oil and

gas

are withdrawn. This

water

influx, called

natural

water drive, provides

one of

the

most effective driving mechanisms for

the

production of oil

and

gas. In producing a reser

voir,

therefore,

it behooves one to make the maximum

use

of

natural water

drive.

To achieve the

maximum

use,

the

reservoir engineer must be able to predict

the performance

of

an aquifer under a

variety of

production

schemes that may

be

proposed for

the

reservoir.

Unfortunately,

the

physical

properties

which

dictate

aquifer

behavior

often are known

only

within

limits.

Seldom

do

wells penetrate

the

porous

strata

of the aquifer. Even when they do,

Original manuscript received in Society of Petroleum Engi

neers

office

Oct. 20, 1960. Revised manuscript received Feb.

7,

1962. Paper presented at 31st Annual California Regional

Meeting of SPE, Oct. 20-21, 1960, in Pasadena, Calif.

MARCH 1962

SPE 1604-G

ST ND RD OIL

CO

OF C LIFORNI

SAN FRANCISCO CALIF.

quantitative information regarding porosity,

perme

ability and water compressibility is seldom avail

able. t

is known

however, that the

water efflux

from most aquifer systems is governed by a single,

relatively simple,

linear,

partial

differential equation.

Also, the general physical location of the aquifer

boundaries

often

are known. A

technique originally

proposed by Hurst

and

van

Everdingen

and Hursr

has been

found

useful in analyzing reservoirs in

this situation. The idea

was

later expanded by

van Everdingen,

Timmerman

and

McMahan

3

to

inc ude

the

mathematical

technique

of least-squares

fitting. This latter method will

be

referred to as

the VTM

method.

The basic assumptions of the VTM method include

the

following.

1.

The location of

the physical

boundaries of the

aquifer are known.

2. The flow conditions

at

these physical bound

daries are

also known.

3. The aquifer is homogeneous;

e.g., thickness,

permeability, water compressibility and

porosity

are constant

throughout.

In the VTM method a material balance is made

on

the fluids entering

and leaving

the

reservoir.

In

the

balance equations,

the

water-influx term is

represented as the product of the water influx from

an arbitrarily-selected, dimensionless aquifer system

times an unknown conversion number.

This

balance

can be formed as many times as there

are

data

points in

the

history

of

the reservoir. Each time,

the conversion

number

can be evaluated. I f the

reservoir engineer has picked the

correct

dimension

less

aquifer

to

represent

the real

aquifer, the con

version number remains constant for all

balances

that

have

been

made

over the

history

period.

If

such

a situation occurs,

the

reservoir engineer can

then

predict the performance

of

the reservoir

for

any

type

of production scheme by using

the

function

associated

with

that particular dimensionless

system and the derived conversion

number.

These

functions will be referred

to

as

the

response func

tion of

the aquifer.

If the aquifer is nonhomogeneous (e.g., if the

porosity, permeability,

thickness,

porosity,

or

lReferences

given

at end of

paper.

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compressibility

vary

within

the system),

it may not

be

possible

to

find

an adequate dimensionless

aquifer system to represent the true aquifer

system.

The purpose of the

work

reported here is to give

solutions or response

functions

for an equivalent

dimensionless aquifer whose properties

are

truly

nonhomogeneous.

These

results

were derived

with

finite-difference techniques using

a

high-speed

digital computer. The results

are

presented as

charts

and are

suggested

for

direct use

in the

VTM method.

AQUIFER

EQUATION

I t can be shown that the behavior of

a

slightly

compressible fluid, one-dimensional system

IS

governed by the following equations.

Linear System:

~ (kh

ap)

= hc ap

. . (1)

ax

/l ax at

. 1

a (rkh

ap) ap

2)

RadIal

System: - - =

hc

.

.

r

r

/l r at

where

k

= permeability coefficient

of

porous media

in

perms

(cu ft-cp/ft-day-psi or

.15801

darcy,

/l =

viscosity

of

flowing

fluid,

cp,

p '

fluid pressure, psi,

h

= formation thickness,

ft,

=

porosity,

dimensionless,

c

= fluid compressibility

vol/vol/psi,

x

= linear

dimension, ft,

r

= radial

dimension,

ft, and

t = time, days.

In the

VTM method,

the aquifer is assumed to be

homogeneous. For such systems the thickness h

can be

cancelled from

the equations

and

all co

efficients can be

brought

to the right-hand side.

Further, to

convert these

equations to

dimension

less systems, the

following change of

variables is

made.

Distance

Time

Pressure

r

kt

- C1-P

Radial System:

r

t =

p

R1

c/lRr

C

2

3)

x

1=

kt

C1-P

Linear

System:

X=

cf1

L2

P

L

C

2

4)

where

R1

= inner boundary

of

radial

aquifer,

ft,

L

= length

of

lirtear aquifer,

ft,

r = dimensionless

radius,

x dimensionless

length,

t dimensionless

time,

P dimensionless pressure, and

C1 and C 2 = constant depending on boundary condi

tions.

If

the dimensionless variables

are

substituted 1fl

34

the aquifer equation, it becomes

a p

Linear System:

2

ap

at

Radial

System:

.i i

ap

T

aT aT

. . . . . . (5)

ap

aT

6)

These

equations represent the dimensionless,

homogeneous aquifer system solved by

varIOUS

investigators

for

use

in

the

VTM

method.

PREVIOUS

SOLUTIONS OF

AQUIFER

EQUATION

To discern the basic relationships

between

aquifer

systems, a classification

scheme

is

proposed

here

for

the many solutions that have been obtained

for

the homogeneous aquifer equation.

These

solu

tions

represent

the

work of several investigators.

This

classification is given

in Fig.

1.

The

first major division is made on the basis

of

the

two possible one-dimensional

geometries

-

radial or linear. Each solution of the gross geo

metries can be

put in

one of

two

classes.

1.

Aquifer infinite

in

extent

-

The

outer boundary

is

at infinity. Such systems obviously do not

occur

in nature.

However,

Muskat

4

states

that an aquifer

system

can be considered infinite if the pore volume

of

the

aquifer

is

1,000

times the

pore volume

of the

oil reservoir.

In

such systems, the pressure dis

turbance caused by the withdrawal

of

oil

from

the

reservoir is never fel t

at

the

true

outer

boundary

of

the aquifer, so that its exact position

is im

material.

Therefore,

it can be

considered

an

infinite

system.

2 Limited in extent -

The

outer boundary of

the

aquifer

is known to

terminate

at some specific

place, and

the pressure

disturbance is

known

to

be

fel t

there.

From the Muskat

definition,

a

limited aquifer

can be

defined

as any

aquifer

that

is

not

infinite.

A

limited aquifer is defined

as

one

in

which the pore volume is less than 1/1,000 of

the pore volume of the

oil reservoir. Even in

these

relatively small systems, the history

of

the oil

reservoir

may

be of

so

short

a duration

and

the

pre

ssure

drop

in the oil reservoir

so

small that the

BOUND RYCOtlDITIOIlS

FIG. CLASSIFICATION

OF

SOLUTIONS TO HOMO

GENEOUS-AQUIFER

EQUATION.

~ O t I E T Y 0 .

PETR OLEI IM

} ; : > G I : > E E R ~ JOI:R: '>AL

8/10/2019 SPE-1604-G

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pressure disturbance

may not

have reached the

outer boundary. In such situations, the problem

solutions of

the

infinite

aquifer

are

directly appli

cable. The

sol ution of the infinite and

finite aquifers

must correspond during these early time periods.

The next

subdivision of solutions to the aquifer

equation

is

made on

the

basis of the

boundary

condition applied at the inner

boundary.

The inner

boundary

condition reported

in petroleum literature

is designed

to

solve

two

problems.

1

Knowing

the pressure

history

at the

inner

boundary, what is the quantity of water influx

into

the

reservoir?

2. Knowing

the

history

of

water

influx into

the reservoir,

what is

the

pressure

history which

results?

van Everdingen and Hurst

2

have

designed

solu

tions of

the aquifer

equation to

answer

either of

these

problems. They have named

the

separate

solutions, corresponding to Problems

1 and 2, as

follows:

1) the

constant-terminal-pressure

solution,

and (2) the

constant-terminal-rate

solution. Using

the

superposition

theorem, they propose constructing

a solution

to

anyone-dimensional aquifer problem

from

these

two

solutions.

The

next division in the proposed classification

of solutions of the aquifer

equation

is made on

the

basis of

the

type of

boundary

condition

applied

at the outer

boundary. This

boundary condition is

only material,

of

course,

in

the

finite

reservoir.

Two boundary conditions occur most frequently

in

natural

systems.

1. The

outer boundary

is closed

The

outer

boundary is

sealed

by

some impermeable

formation.

I t

may be that the aquifer

sand

thins out due

to

normal sedimentary processes or

that

a

fault

com

posed

of

impermeable rock has

been

thrust opposite

the

outer boundary. In

any

event,

the

mathematical

condition

for

the

closed reservoir

is

to

have the

space

derivative

of

pressure at the outer boundary

be

equal to

zero.

PRESSURE

CHANGE AT

INNER6 UMOARY

t

t, - 0

DIMENSIONLESS

EFFLUX

Q

DIMENSIONLESS TIM

FIG. 2 - EFFLUX FROM DIMENSIONLESS

AQUIFER FOR ONE PRESSURE CHANGE.

MARCH

1962

2 The

pressure

at

the outer

boundary

remains

constant

This condition

can

occur

where the

aqui

fer

outcrops

into a

constant source of water such

as

a

lake, river,

or ocean. In

this situation, the

pressure at the outer boundary remains

constant

for

all

time,

regardless

of the amount of

water that

is withdrawn

from

the aquifer at the inner boundary.

USE

OF

DIMENSIONLESS

SOLUTION

OF

AQUIFER EQUATION

Solutions

of

the

aquifer

equations are

usually

presented as tables

or

charts. The functions are

referred to

as

response functions. The use

of

these

response

functions for water

efflux

for

the

homoge

neous aquifer will be demonstrated, and the tech

nique later will

be shown to

apply

to the

non

homogeneous

system.

Define

Radial

System:

Linear

System:

where

Dimensionless Dimensionless

Production Time

Qc

kt

Q=

t =

7TW ch Rl P

I

o -PI.T

o

)

c/1 Rl

Q = Q

c

kt

t =

c/1

L2

ch L

W PI,

0 -PI,to

)

Qc = cumulative water efflux over

time

period, cu ft,

Pi,O

= pressure

at

initial

time

at

lOner

boundary, psi,

PI.to = pressure

imposed

at inner boundary at

time zero, psi,

W

= fraction

of

radial system occupied by

aquifer, dimensionless,

and

W

=

width

in

linear

aquifer,

ft.

The

method of

solving

for

the

amount of water

flowing from a

system is demonstrated

with

the

radial system. The inner

boundary of a

radial

system,

initially at a

constant pressure

through

out,

is given

a step decrease in

pressure.

This

pressure

jump

or change can be designated

as

PI,O - PI,T ).

The pressure

PI.to

is

held constant

at

the i n n ~ boundary

for an indefinite period. A

diagram

of this

is

shown in

Fig.

2.

With

the constant pressure

drop

imposed the

inner boundary, the cumulative water efflux increases

as a

function

of

dimensionless

time. The amount

of

efflux is obtained as before,

from

the response

function curve. These are the curves that have

been

supplied

by

Hurst and

others. It

can be shown

that

these solution-function

curves depend

on

the

aquifer

size and boundary condition imposed

at

the

outer

boundary. The dimensionless cumulative water

efflux

can

be

ot-tained from

the response-function

curve

for

any desired dimensionless

time.

In the

example given in

Fig.

2, the amount of

water at

time

Tn

is required. To convert the dimensionless

Qat this time into

a

dimensioned quantity of water,

the dimension

conversion factor

is

used. Define

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the cubic

feet of

water that has flowed out

of

the

aquifer as Q1

Q

1

= Q(tn-O) 7TW ch

T lJ,o

-PI,t

o

)

where

Q t

-0

= cumulative dimensionless water

flowing

of aquifer

during time-period

zero to tn.

To extend the

development,

a further change in

pressure is

imposed after

a

dimensionless

time [1

This pressure change is

also continued

through time

This is

diagrammed in

Fig.

3.

A solution

to

the problem of the

total

quantity

of

water flowing from the aquifer during

the

time

period to [n is obtained

in

two steps.

The

flow

calculated

for the

first pressure change

is

assumed

to have occurred independently of the

second

pressure change.

The

amount of

flow

caused

by

the second pressure change

is

calculated by

assuming

i i

to

be

zero. A solution for the amount

of water flowing during

this

second

time period

is

obtained as though the starting

time

were zero

and the flow had occurred for the time ([n -

t1)

The

amount

of water

given up

as

efflux can be obtained

from the efflux response curve by entering the time

axis at

-

it)

and deriving

the

Q

this

period.

This

amount of water

is

defined

as

Q[t,;

- t1 )

To

compute the dimensioned water, the usual con

version

formula is used. Define this quantity as Q2

Q

2

= Q tn-tt) 27TW ch

y

(PI, to -PI,t t) .

The

amount of water derived independently for

the second

pressure

drop Can be

added to

the

amount

that was obtained for the

first pressure

drop,

to

obtain the amount of water that the

aquifer

produced as efflux for the two time periods.

The

reason that these two independently

derived solu

tions can

be

added

and

still have a valid solution

is

that the basic differential

equation

is

linear.

Any

two

solutions of a

linear

differential equation

can

6

PRESSURE

CH NGE T

INNER

BOUMO RY

--.:--

p,I,-r

tl l

l -_

_________ --t-

P

I,Q

- Pr, t

o

)

~ L ~ __________

II tn

1:

Q(in-o)

Q tn l

l

- - - - - -

D I ~ N S I O M l E S S

EFFLUX

.1

Q

...

; (t,

-

t,

- - - - - ' ' ' ~ I

OIM IISIOIL SS TIM

t,

FIG.

3 -

EFFLUX FROM DIMENSIONLESS

AQUIFER FOR TWO PRESSURE

CHANGES.

be added,

and

their

sum must

also

be a

solution.

This principle has

been

referred to as

the super

position

theorem

by

van Everdingen and Hurst.

2

This reasoning can

be

extended to a large number

of pressure changes.

Define

this amount

of

water

as Qc

+ - . - . + Q(t -t

PI - PI ]

n m

m l

m

NEW

SOLUTIONS

FOR

AQUIFER

EQUATION

REPRESENTING

HOMOGENEOUS SYSTEM

A

review

of

Fig.

1

shows that

some

of

the aquifer

systems of practical interest for use in the VTM

method

have

not

been solved.

Although

it is not

the

primary objective

of this

paper,

these cases

were solved

with

a

finite-difference technique

using a

digital

computer.

These results complete the possible solutions

for the homogeneous aquifer; they

correspond

to

the constant-terminal-pressure case for limited

linear and limited-radial aquifer systems where

the

outer

boundary

pressure is held constant.

In

addition,

the

constant-terminal-rate case was

solved

for

the

limited linear aquifer where the

outer boundary is closed.

The

inner boundary

condition

for

the

former

homogeneous aquifers is

a constant

terminal pressure.

With a

constant

pressure at the

outer

boundary

during the

early

dimensionless

times,

the pressure disturbance

has only

been

propagated a short distance. It

has

not been felt out at the outer boundary.

Any

solution

then to this problem during early times should

correspond

to the

constant-term inal-pres sure solution

for an

infinite

system. This

effect is

shown on

Fig. 4.

1

PRESSURE

INNER

BOUNOARY

'.

-------

l-

CONST NT

PRESSURE

T OUTER

BOUNO RY

UTER

BOUNOARY

FIG.

4 -

PRESSURE PROFILES IN AQUIFER.

~ O I E T Y

O}- PETRO LEI1M

E ~ G I ~ E E R S

JO l iRNAL

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After

the

pressure disturbance has reached

the

outer

boundary,

the

flow condition soon stabil izes,

with as

much water

entering

the aquifer

through

the outer boundary as

is

leaving through

the

inner

boundary.

When such a condition

is reached, the

dimensionless production function becomes linear

with dimensionless time. The

system

is truly at

a steady state; e.g.,

the

pressure profile remains

unchanged

with

time.

Figs. 5, 6 and 7 give

the

results for the

radial

constant-terminal-pressure

case with a

constant

pressure

at the outer boundary.

The curve on these

figures

which

has

lowest

value s of

dimensionless

efflux

for all shown

times is

that for a system with

a radius ratio of 100. This aquifer

system

never

comes

to

a

steady state

for

the times shown on the

figure.

The new solutions for

the linear system will be

presented in

the

discussion of the nonhomogeneous

aquifer

results.

DIMENSIONLESS NONHOMOGENEOUS

AQUIFER

A problem

involving

the performance of a homo-

geneous

aquifer

can

be

solved

if

quantities called

response functions are available. The two problems

that

can be solved are

the

following:

(1)

determining

the

amount

of water that would flow

out

of

an

aquifer in response

to

a

pressure decline

at the

inner boundary;

and (2)

determining the

amount

that

the pressure at the inner

boundary

would decrease

in

response

to

a

withdrawal

of

water.

The

response

function required for the solution

of the first

problem

is

a

relationship between

'

'

'

'

-

z

o

2

'

:

z

u

z 1

-

'

S

.

c

-

,

e

e

z

2

;;

IMENSIOILESS

TIME

FIG. 5-EF F LUX

RESPONSE

FUNCTION

FOR

A RADIAL

HOMOGENEOUS

AQUIFER

WITH A

FIXED PRESSURE

AT

THE

OUTER BOUNDARY -CONSTANT

TERMINAL

PRESSURE AT

INNER-BOUNDARY RANGE OF

DIMEN-

SIONLESS TIME 0

TO

1.8.

MARCH,

1962

dimensionless water efflux

and

dimensionless

time.

The response function required

for

solution of the

second

problem

is

a

relationship between

dimension-

23

22

2

20

9

0

17

;;

16

15

2

-

11

'

10

.

c

I

-

0

2

'

FIG.

6-EFFLUX RESPONSE FUNCTION FOR A RADIAL

HOMOGENEOUS AQUIFER WITH A

FIXED PRESSURE

AT THE OUTER BOUNDARY-CONSTANT

TERMINAL

PRESSURE

AT INNER-BOUNDARY RANGE OF DIMEN-

i 1

:;:

1

-

z

o

'

2

z

2

-

'

,.

c

-

,

'

-

z

o

2

SIONLESS TIME 0 TO 18.0.

IMENSIONLESS TIME

FIG. 7-EF F LUX

RESPONSE

FUNCTION

FOR

A RADIAL

HOMOGENEOUS

AQUIFER WITH A

FIXED PRESSURE

AT THE OUTER BOUNDARY -CONSTANT

TERMINAL

PRESSURE AT

INNER-BOUNDARY

RANGE OF DIMEN-

SIONLESS

TIME

0 TO 180.0.

37

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less

pressure

drop and dimensionless time.

The particular shape of

these response

functions

depends

on

the

geometry

of

the system,

the

boundary

condition

imposed

at

the inner and outer

boundary,

and

in the case of

the

radial system on the ratio

of

the

outer

boundary radius

to

the inner boundary

radius. All the dimensionless quantities computed

in deriving the dimensionless

functions

refer to the

properties

at

the inner boundary, as well as the

aquifer

itself.

This

is

true

because

the homogeneous

aquifer

is

defined

as

one whose

properties are

constant throughout. The character or shape

of

the

particular dimensionless function

depends only

on

the boundary conditions or

the

radius

ratio

of

a

particular

aquifer system.

The argument is

put

forth here that, so

long

as

the

dimensionless

quantities

are put in terms of the properties of the

inner boundary,

a

set of dimensionless response

curves

can

be derived

which

predict the

behavior

of an aquifer

whose

physical

properties

of

permea

bility,

thickness, viscosity

and compressibility

vary back in the aquifer itself. Each of the dimen-

sionless

response curves

for these

nonhomogeneous

aquifers was obtained

for

one particular system.

However,

the curves

can

be

used for

any aquifer

system whose

properties vary

inside the system

in

the

same

ratio as those

aquifers

from

which

the

response

functions were calculated.

L

'

'

:>

'

'

'

L

'

'

..J

Z

o

.,

z

I

;;

z

::>

,.

'

..J

=>

.,

'

>

..J

w

w

>

..J

::>

I

Hicks, Weber and Ledbetter 5

have reported

on

derivation of such dimensionless

response

curves

by means of the electric

analyzer.

They

have

called their

results

influence functions . Their

reported results were given

only

as example systems,

and

presented

no detail

on

the

aquifer properties.

They

are

of no

value

in solving a particular

aquifer

problem because they

give no

quantitative

data.

The reasoning for the use of

such

response

functions with nonhomogeneous aquifers is

based

on

the properties

of

the governing differential

equation.

The governing

differential

equation for

non-

homogeneous aquifers

is

linear in terms of the

dependent variable

P.

t

is a

second-degree, partial

differential

equation. As

a result of

the linearity

of the equation, two properties

which

have been

shown 5 to

be

associated with such equations have

been used to demonstrate the general usefulness

of the dimensionless pressure-drop

and

dimension

less production response curves. These two prop

erties of

linear

second-order

differential

equations

are

as follows.

1 Any solution of the equation may

be multiplied

by

a

constant

and

still

remain

a

solution.

2 Any sum

of

two

solutions of the diffusivity

equation is

a

solution.

As

a result

of

these

consequences of the linearity

of

the aquifer equation, it is proposed to obtain by

digital

computing

methods

the

dimensionless pres-

L IIIIIIIIIII

;

0:001

DIWENSIONLESS TIME

FIG. 8-EF F LU X RESPONSE FUNCTION

FOR

A LINEAR AQUIFER,

CONSTANT

PRESSURE

AT

OUTER BOUNDARY

CONSTANT TERMINAL PRESSURE AT INNER BOUNDARY. BETA IS RATIO OF

O UT ER - T O

INNER-BOUNDARY

TmCKNESS.

38

SO : IETY

O F

P E T R O L E U M

ENGINEERS JOURNAL

8/10/2019 SPE-1604-G

7/11

sure-drop

and dimensionless efflux

response

func

tions for a set of hypothetical aquifers

with

vary

ing permeability,

viscosity, thickness, porosity

and

compressibility properties.

RESPONSE FUNCTIONS

FOR

NON

HOMOGENEOUS

AQUIFERS

Referring to the

aquifer

Eqs. 1 and 2, two groups

of

variables can

be

a function of the space

variable:

khlll

variable

group

inside

of

space

derivative; and

ch variable group multiplying time derivative.

The first variable

to

be studied is the

change

of

h or the

aquifer

thickness. Of all aquifer

properties,

the thickness is the easiest to measure. Almost all

well-surveying

tools give

information on

formation

thickness,

whereas other properties of aquifers

and

their contained

water must be

measured

indirectly

and, sometimes, inferred from the aquifer behavior.

The variation

in thickness

to be studied here

will

be

a linear variation with

respect

to either

the radius

in the radial

aquifer or

to

the length in

the linear aquifer. This type of thickness variation

is

commonly

observed

in natural systems. It is

caused

by

several sedimentary processes.

To characterize

an aquifer system whose thick

ness

is

varying

linearly with distance, define a

parameter f as follows.

e

'

z

0

;:

..

::>

'

II:

-

''

-

z

0

;;;

z

w

::

z

::>

e

II:

...

'

;:

-

::>

'

II:

-

0

II:

'

I I :

::>

'

'

II:

-

'

'

-

z

0

;;;

z

w

'

A

0.001

0.01

where hI = thickness at

inner

boundary,

and

b

o

= thickness at

outer boundary.

Therefore, the

factor f

is given the meaning

of

a

dimensionless ratio

of

outer-boundary

thickness

to inner-boundary thickness.

When the basic

differential

equations of the

linear aquifer system are

put

in dimensionless

form only

one

equivalent dimensionless system

exists for

all length aquifers. This

is

not true

for

the radial system

where

each dimensionless

aquifer has a characteristic

size

parameter, the

ratio

of outer- to inner-boundary

radii.

As a con

sequence

in

the linear

aquifer

system,

a

solution

for one particular variation of the

thickness

can

be

applied to any length

of

natural system

where the

variation of thickness occurs in the same

way as

it

does

in the original system.

In

addition, only a

finite

number

of

combinations of

outer

and

inner

boundary conditions

exist that

can be applied to

the linear aquifer. It can be seen that four

distinct

combinations

can be

made

of

the

two

boundary

conditions

found

most frequently in linear aquifers

at the

inner

and outer

boundaries.

These response

functions are reported

on

Figs.

8

through

11.

Fig.

9 reports on the pressure-drop

response

function for a closed linear aquifer with the constant

terminal-rate

condition. The pressure-drop

response

function changes logarithmically with dimensionless

time

until the initial pressure disturbance 1S felt

DIMENSIONLESS TIME

FIG

9 -

PRESSURE-DROP RESPONSE FUNCTION

FOR A

LINEAR AQUIFER, CLOSED OUTER

BOUNDARY

_

CONSTANT

TERMINAL

RATE AT INNER BOUNDARY. BETA IS THE

RATIO

OF O UT ER - T O

INNER-BOUNDARY

THICKNESS.

MARCH 1962

39

8/10/2019 SPE-1604-G

8/11

out at the outer

boundary.

After this

time,

the

pressure profile in the aquifer

stabilizes

and all

pressures in the

system decline

at

the same rate.

This

condition

is

often referred to

as

a condition

of quasi-steady

state,

because the

shape

of

the

UJ

....

( I 0

A

1.0

A

n

UJOO

'

::::>...J

00 Z

00 0

UJ

-

''''

1. Z

W

00 ::1

00-

WA

...J

Z ....

_z

00::::>

Z

w

00

w

w

'

'

::::>

00

>

UJ

...J

cr

LL

(1.

UJ

00

'

W

>

...J

z

W

-J cr

... IL

...

W '

'

W

>

-J

_

:z

.... 0

-J

'

=>

:z

'

W

=>

:>

;:;

'

'

::

-J

Z

Z =>

0

en:z

:>

w

0

2

cr

;:;

...

I

, , - -c

I

.0

;

,

f

i i.

.,

I

i

I

iH

:

...

1--1

I

0 .1

i

/ .

..

.Y

i

~ t ; O i I '

I?

~

I

. ,

. ,

, ...

L

,

:

I

d

i

,

...

1:

j

0 .0

1

,

,

2 . '

3

4

3 ,

0 .001

,

IT I

1 1 1 - E I

i

tJ-; - ~ I

rid

. . .

G

h'..

,0

J..

.:

:;-.:'

'

,

:

.. .

II

/1"\

~ l n ,

:. :.

J Y ~

i? :TI'.

,1

11=

,

... 1 1 jt .

'

IT

f,++ i

e

: ",

, , t -

IT

i

I ~ :

' I . . :

','

IF

I

,'r"

,

~ , , - , \ i

r '

rT

-

:-:

1:-'

.

J

.

~ ~

. . .

I i

i:

" I

:

I

,

t

....

i

----:

.._.

i

I

1'

1

..

. . . .

reT

i

I

i

,

1

i

I

---

+. to ..

.. :.

t

-I

I

t i

;

j

a a ,

,.

,

20 3

. 3 6 7 8 1

,.

, 2 . ' .

. .

a

Q 1

,.

,

2.15 3

0 .01

0 .1

1 .0

10.0

DIMENSIONLESS TIME

FIG. 12 -EFFLUX

RESPONSE

FUNCTION

FOR

A LINEAR AQUIFER, CLOSED OUTER

BOUNDARY-

CONSTANT

TERMINAL PRESSURE AT INNER BOUNDARY. BETA

IS

THE RATIO OF OUTER- TO

INNER-BOUNDARY

PERME

MEABILITY OR

RECIPROCAL VISCOSITY.

10

...

lil

w

a:

iil

w

:'

::l

w

0

:

1 .

.

;;

=

:>

,.

0

a:

...

'

>-

-'

:>

:::

a:

"

>

-'

>

>-

-'

:>

,.

:>

"

:::

0

'

,.

;;

0 01

0.1

1.0

10 0

100 0

IMENSIONLESS

T M

FIG. 1 3 - E FFLUX

RESPONSE

FUNCTION

FOR

A LINEAR AQUIFER, CLOSED

OUTER

BOUNDARY - CONSTANT

TERMINAL PRESSURE AT INNER BOUNDARY. BETA IS RATIO OF O UT ER - T O

INNER-BOUNDARY COMPRESSI

BILITY

OR POROSITY.

M A R C H , 1 9 6 2

41

8/10/2019 SPE-1604-G

10/11

and II .

The most

common

aquifer

problem to

be solved

by a practicing engineer is the prediction

of

the

quantity of water

that

will

flow out

of

a closed

aquifer

if

the

pressure history is known

at

the

inner boundary.

The

efflux response function is

required

to solve

this problem. In the

work

reported

here, this aquifer system was solved for

the linear

system

with varying linear properties.

The

parameter

f

retains

the same definition

as

when

it

was

used

to characterize the

thickness

variation.

It

is the

ratio of the

val

ue of a property

of

the system

at

the

outer boundary to the value of the

property

at

the

inner boundary.

It

can be seen that, in the two

groups of

variables

khl/1 and ch only the factors

kip.

and c can be varied independently.

In

the

case

of

kl/1 f refers

to the

ratio

of

l

at the

outer boundary to the value at the inner boundary;

with c f refers to the ratio of c at the outer

boundary to

the value

at

the

inner

boundary.

Two

closed, linear

aquifer

systems were

studied

with the

digital method.

The response functions

are

given on

Figs.

12

and

13

for

aquifers

whose

properties

of

kl/1

and

c vary. On Fig. 12, it

can

be

seen

that

the

response

function

for all

permea

bility combinations

terminated

at some dimension

less

time with the same

value of

efflux. This

phenomenon results from

the

fact that a

change in

permeability has not changed the total vol ume of

water available for effl ux but,

rather,

merel y shifted

the time axis when

this

water

will flow from the

aquifer.

It can be seen on Fig. 12 that, in the

aquifer

system with a f of 1/10

(a

system which thins

away

from

the inner boundary), the response function

is at a maximum which is 25 per cent

different

from

the

homogeneous-aquifer

response function. Even

with

this

severe thinning, this relatively small

change in the

response

function

would

probably be

undetected

in engineering

calculations.

The aquifer

with

a f of

10.0

(a

system

which thickens

away

from the inner

boundary)

has a shift

in

the response

FIG.

14 -

EFFLUX

RESPONSE

FUNCTION FOR

A

RA

DIAL AQUIFER, CLOSED OUTER

BOUNDARY

- C O N

STANT

TERMINAL

PRESSURE

AT

INNER BOUNDARY.

RATIO

OF

O UT ER - T O

INNER-BOUNDARY

RADIUS

IS

10.0. BETA IS THE RATIO OF OUTER- TO INNER-

BOUNDARY THICKNESS.

42

curve

which

is

more severe. It can be seen that the

curve at certain times

is

one-third the value

of the

response function of the homogeneous aquifer.

In

the radial

aquifer, the ratio

of the outer-

to

inner-boundary radius is

a

number

which character

izes the size

of the

aquifer. Thus, to solve for all

possible radial configurations would

require

an

infinite

number of solutions. It

was

decided to solve

here

the response

functions

for

the

closed radial

aquifer with

a

condition of constant

terminal

pressure

at the

inner

boundary and a radius ratio of

10.0.

This aquifer

size

is typical

of

many aquifers that

surround important

oil

reservoirs in the

United

States. The f

parameter

again is defined in the

the

same manner

as in the linear aquifer. Fig.

14

gives

the

efflux response function

for an aquifer

whose thickness changes; Fig. 15 gives the efflux

response function for a

radial aquifer

whose

per

meability or reciprocal of viscosity

varies;

Fig.

16

gives

the efflux response function for a radial

aquifer

whose compressibility

or

porosity varies;

and

Fig. 17

is the pressure-drop response function

for a radial

aquifer

whose thickness varies.

SUMMARY

Digitally

derived

response

functions have

been

presented

that would enable one to predict the

quantity of water efflux

and,

in

certain cases, the

pressure drop for both

radial and

linear

aquifers

whose properties vary linearly inside the aquifer.

The se response functions should prove useful for

evaluating the

effects

of nonhomogeneity

on

aquifer

performance with the

van Everdingen,

Timmerman

and

McMahan method.

REFERENCES

1. Hurst, William:

Water

Influx Into

a

Reservoir and

Its

Application to the

Equation

of Volumetric

Bal

ance , Trans.

AIME

(1943)

Vol. 151,

57.

2.

van

Everdingen,

A.

F.

and Hurst,

W.:

The Applica

tion of

the Laplace Transformation to Flow Problems

.0

0.

.0

DIW M810MLESS TIME

FIG. 15 - EFFLUX RESPONSE FUNCTION FOR A RA

DIAL

AQUIFER,

CLOSED

OUTER BOUNDARY - CON

STANT TERMINAL PRESSURE AT INNER BOUNDARY.

RATIO OF O U TE R- T O

INNER-BOUNDARY

RADIUS

IS

10.0.

BETA

IS

THE

RATIO

OF OUTER- TO INNER

BOUNDARY P ERMEABILITY OR

RECIPROCAL

VIS-

COSITY.

SOCIE T Y O F P E T R O L E U M

E ~ G I i E E R S JOlJR iAL

8/10/2019 SPE-1604-G

11/11

w

'

>

IJ

IJ

W

0::

Q

IJ

IJ

W

..J

z:

0

IJ

z:

W

2

z:

=>

2

0

0::

'

:

..J

=>

IJ

W

0::

>

=>

..J

W

W

>

.J

=>

2

=>

IJ

'

..J

z:

0

IJ

Z

W

2

Q

1 0

.

10.0

100.0

10,000.0

DIMENSIONLESS TIME

FIG. 16 -

EFFLUX

RESPONSE FUNCTION FOR A RADIAL AQUIFER, CLOSED

OUTER

BOUNDARY -CONSTANT

TERMINAL PRESSURE AT INNER

BOUNDARY.

RATIO OF

OUTER-

TO

INNER-BOUNDARY

RADIUS

IS

10.0. BETA

IS

RATIO OF O UT ER - T O

INNER-BOUNDARY

COMPRESSIBILITY

OR

POROSITY.

FIG. 17 - PRESSURE-DROP

RESPONSE

FUNCTION

FOR A RADIAL AQUIFER, CLOSED

OUTER

BOUND

ARY -

CONSTANT TERMINAL

RATE

AT INNER

BOUNDARY.

RATIO OF OUTER- TO

INNER-BOUND

ARY

RADIUS IS 10.0.

BETA IS

THE

RATIO

OF

OUTER-

TO

INNER-BOUNDARY

TlllCKNESS.

M,, RCH, 962

in Reservoirs ,

Trans

. AIME (1949) Vol. 186, 305.

3. van Everdingen,A. F.,

Timmerman,

E.

H.

and McMahan,

J. J.: Application

of

the

Material Balance

Equation

to a Partial Water-Drive Reservoir ,

Trans.

AIME

(1953)

Vol. 198, 51.

4. Muskat, Morris:

Physical

Principles o/Oil Production

McGraw-Hill Book

Co.,

Inc., N.Y. (1949) 58.

5. Hicks,

A. L.,

Weber,

A. G.

and Ledbetter, R. L.:

Computing

Techniques

for Watel -Drive Reservoirs,

Trans. AIME

(1959) 216, 400.

6.

Mortada,

M.: A Practical

Method

for Treating

Inter

ference in

Water-Drive

Reservoirs ,

Trans. AIME

(1955) Vol. 204, 204.

7.

Hurst, William:

Unsteady

Flow

of

Fluids

in

Reservoirs , Physics Uan., 1934) 5.

Oil

43