SPE-1305-PA Chatas a.T. Unsteady Spherical Flow in Petroleum Reservoirs(1)

7/24/2019 SPE-1305-PA Chatas a.T. Unsteady Spherical Flow in Petroleum Reservoirs(1)

1/13

_ .9+.. ~05 . --

Unsteady

Spherical Flow in

Petroleum Reservoirs

A. T. CHATAS

MEMBER A ME

ABSTRACT

A description of the geometrical characteristics

of spherical reservoir systems, a discussion of

unsteady-state flow of such systems and examples

of erzgirzeering

applications are presented as

background materiaL The

{undameital

differential

equation,

a description of average spherical

permeability and the introduction of the Laplace

transformation serve as tbeoret ical foundations,

Engineering concepts are irwestigated to indicate

particular solutions of interest, which are analyti-

tally obtained with tbe aid of the Laplace transform.

These are numerically evaluated by comput e~ and

presented in tabular form.

INTRODUCTION

A

tractable mathematical analysis of unsteady

fluid flow through porous media generally requires

incorporation

of

a

geometrical symmetry. The

simplest

forms include the linear, cylindrical

(radiaI) and spherical. Most analytical endeavors

have concentrated on cylindrical symmetry because

it occurs more often in petroleum reservoirs.

Nevertheless, some

r eservoi r

systems do exist that

are better approximated by spherical geometry.

Review of technical literature revealed but a

single reference to unsteady spherical f~ow in

petroIeum reservoirs. ~ The motive and purpose of

the present work was to remove this gap in technical

information? and to provide the practicing engineer

with some useftd analytical tools. The mathematical

details associated with the partictdar solutions of

interest involved use of thti Laplace transfor-

mation. Hurst and van Everdingen previously

demonstrated the efficacy of this operational

technique; mrd in many respects the present

treatment was patterned after their earlier work. 2

PRELIMINARY CONSIDERATIONS

GEOMETRICAL

CHARACTERISTICS

. .

*

I

I

1

I

I

i

,

~

i

t

Ge~metricaIIy, a spherical

reservoir

system ;is

Jig&+

.

~TF..()___

dZfinZii _iiiZ@-i%tGiiF% f-&ii~- ti~~~cgn~e~ki~--- ---

- .- -.--S-- .-.-.e +,. ,-... ._,_- (-lJ---=---

.

Original manuscript received in Saciet y

of

Petroleum Engineers

But to do. this, a reservoir system must contain

offlCe SetIt.

27, 1965. Revised manuscript of SPE 130Sreceived

APrii S, 1966.,

either art ideal fluid, which impIies a vanishing

preferences given at end of PaPer,

viscosity,

or an @compressible fluid, which

102

SOGIETYOF.PETROLEUM

ENGINEERS JOURNAL

:; 1-.. -fi.:i:-=:- .-.-.-:---: .-.; -.=. --- : :---: ---T- :. .- -~= :-: :~k-: :-:= ~ - .-,-.=;- ~:---:i :-: i:::::::: ;-:~-- -~. ,,:;:-::~.-~ --:; :::

-

-. ..

.::b-: -:=: .. -- --- -.. : .7 -

. . . .

-- .

,:.

.-. -.

...--.. .+. ..=; -, . . .. ..-~.

. --- .,. . . ,.

..-, . ... .,-- . ... . . .- ..-. .. -.,- . ..-.

.- ..... . . . . .. . . . ..--. .> 7..-.... .-; 1..,..

. -;-.-:, .-, .

-.

..-. .

... ..-3%-:

=4. .: _. . -----

IRANIAN OIL EXPLORATION & PRODUCltiG CO,

TEHRAN, IRAN

hemispheres whose physical properties of interest

vary

only with the radial distance. Every physical

property is thus restricted to be a space function

of only one variable: the distance along a radius

vector emanating from the center.

Such a system is composed of an outer region

and an inner region, separated by a defined internal

boundary. The inner region simply extends inward

from this boundary, whereas the

outer region

extends

outward from it to an external boundary. The

position of the internal boundary is presumed fixed,

so that the,

size

of the inner region remains constant.

On the other hand, the position of the external

boundary at any given instant of time is determined

by the distance into the system that a sensible

pressure reaction has occurred. Thus, the external

boundary may change position with time.

It jnitially emerges from the inner region and

advancea outward to its ultimate position. When

this ultimate position coincides with a geometric

limit, the reservoir system is said to be Iimited.

When it coincides with points subject to pressure

gradients furthest removed from the internal

boundary, yet short of a geometric limit, the system

is aaid to be unIimited. In this investigation two .

different boundary conditions are imposed at the

ultimate. boundarie a of limited systems. The first

requires that no fluid fIow occur across this

boundary; the second that the pressure remain

fixed at this boundary. s-s

UNSTEADY-STATE FLOW

10 a strict sense virtua~ly alI flow phenomena

associated with a reservoir system are unsteady-

state. The transient behavior of these phenomena ,

requires accounting, however, only when time

maist be introduced as an explicit variabIe. Other-

wise, steady - state mechanics

tiay be used.

AnaIyticaIIy,

steady-state conditions prevail

in a reservoir system only over that portion of

its history when this relation is satisfied: .


2/13

. . . .

-----

implies a vanishing compressibility; or it must have

pressures fixed with time such that the time-

derivative vanishes. Evidently, strict steady-state

conditions are virtually impossible to attain, since

these provisions are abstractions of the mind and

not properties of physical systems. From a practical

standpoint, however, this fact does

not

exclude

application of steady-state mechanics, because in

many situations Eq. 1 is closely approximated. 3-5

The significant physical properties that determine

the extent of transient behavior in spherical

reservoir systems are exhibited by the so-called

readjustment time which is approximated by:

~ _ Crez

.

r

2k/p

(2)

These factors are the size of the system, its

compre ssibi Iity and its mobility. When they combine

to yield a large readjustment

.ne, unsteady +tate

mechanics should be used wless pressures are

in~ariant-3,5

ENGINEERING APPLICATIONS

~en a water drive field is characterized by

bottom-water encroachment,

the hydrocarbon

accumulation usually fills only a portion of the

total thickness of the reservoir

formation

and is

entirely underlain by water. Flow of water into the

pay zone results from a gradual and uniform rise

of the underlying water.

Of particular interest to the reservoir engineer

are methods, fotmally independent of materiaI

balance principles, for determining the water

influx into bottom-water drive fields. First, rhese

methods afford determination of a number of

reservoir properties

through an analysis of the

past reservoir history iYy an adjunctive use with

other reIatIons. Secondly, by independently yielding

the water influx they provide means of predicting

future reservoir performance. Many bottom-water

drive fields Iend themselves to the imposition of

spherical geometry; hence, solutions of the funda-

mental flow equations appropriate ro this symmetry

can be used to analytically determine the water

infIux for this class of reservoir.4~ b

Although many wells are completed after the drill

has passed entirely through the pay formation, some

are purposely completed after only partial penetration

has been effected. Sometimes such wells are

completed after th~ rop surface of the reservtiir is

merely tapped by the drill, in which case they are

termed non-penetrating weIls.

Non-penetrating wells that occur in a relatively

thick formation can be treated as spherical systems.

They can be analytically investigated by using

damaged sand conditions. Also, although the

analytical soIutions strictly appfy ordy to the

single-phase flow of compressible liquids, the results

can sometimes be used (with proper interpretation)

the fIow of gases when pressure drops are small,

and to the simultaneous flow of oil and gas upon

imposition of drastic assumptions.s q~ ~

THEORETICAL CONSIDERATIONS

FUNDAMENTAL DIFFERENT IAL EQUAT ION

The fundamental differential equation

governing

the dynamics of the flow of compressible liquids

through spherical reservoir systems can be written

as:

where the porosity, compressibility and mobility

are interpreted as fixed averages, and where the

effects

of gravity are rreg~ecred.

Define a

dimen-

sionless length ratio, dimensionless time ratio and

sionless length ratio, time ratio and pressure-drop

ratio, respectively, as foIlows:

(4)

e d = . . . . . . . . . . . . . . . . .

kt

td=

pcr

(5)

P~ =

P~ (~~t @ =

Pi p(rEI D)

(6)

Pi+l,tj) - .

Introduction of these relations into Eq. 3 permits ir

to

be rewritten as:

which represents

the fundamental differential

equation in dimensionless form appropriate to

reservoir systems

characterized by spherical

symmetry .z-%s

AVERAGE SPHER ICAL PERMEAB ILITY

Available evidence indicates that the uermeabiIitv

.

of porous media constituting rerervoir sys,tems is

not isotropic in character, As a rule the vertical

permeability is less than the horizontal, and in

some inswnces the difference is profound. Since

spherical symmetry embraces a three-dimensional

geometric space, it was felt necessary to include

the effects of this anisttopy here. The radial perme-

ability in a spherical porous medium characterized

by uniform vertica and horizontal permeability

components can be analytically described by:

appro~iate solutions of the fundamental flow

-.-._. ___ .. ____ . . . . _____

.-.

---~ _. .._ _______ ..._

=

[

equations corresponding to spherical symmetry. -

These investigations

include flow

calculations,

1

=

k; k:

sin2a+ cosza. . . . , . . , (8)

snaly sis of drawdown and bui id-up tests, determin-

40 . ~

i

ation of static bottom-hole pressure, productivity

indices, effective permeabilities and evaluation of

The average spherical permeability can then be

obgained with th,e volume integral:

JUNE,,1966

.-

10,?

. . . . . . . . . . . . . . . . . ._____ . .. . . . . . . . . . _. ..-


3/13

. -.

F+=

(2/3)

m(r~3 -?:)

,..

. (9)

fffi ~ sin a drda~fl

Oorw

which, upon evaluation, gives:

3kbku

k= , . ,

(lo)

kh+2ku

the average spherical permeability.

AP PLICATION OF THE

LAPLACE TRANSFORMATION

Th e fundamental differential equation for a

spherical reservoir system has been expressed in

dimensionless form by Eq. 7. Define the product:

h=rDpD . . . . . . . . . . . . . .. (11)

Then Eq. 7 can be written in the alternative form:

d2b db

= .. . . . . . . . . . . .. (12)

&D2 atD

The Laplace transform of

b is

given by the

definite integral:

~= J_-bexp(-stD)dtD . . . . , . . .

(13)

o

Multiplication by the nucIeus of the transform and

integration over all time converts Eq. 12 from a

partial to the ordinary differential equation:

dz~

~F

=

{14]

drD2

The general solution of this subsidiary equation

can be written at once:

z=

Clexp(-rD@) -t C2 exp(rD@) , . . . (15)

where Cn k an arbitrary constant.

2,9-11

Particular soIutions to the subsidiary equation

corresponding to specifically imposed boundary

conditions are obtained upon appropriate evaluation

of the constants that appear in its general solution.

These particular solutions would represent the

Laplace transforms of the required particular

sohirioris to Eq. K?. The Iattei are determined by

effecting the inverse transformation of their Laplace

transforms. This procedure will be used to develop

the particular solutions of interest.

S ELECT ION OF PARTICULAR SOLUT IONS

system. But due to the generality introduced, it

becomes necessary to relate certain physical

quantities associated with absolute units of

measurement to functions of the &lmensionless

variables in Eq. 7.2*5

The macroscopic radial velocity at the internal

boundary of a spherical rdservoir system is given

by Darcys law: 2-4

k

()

u=-

/L Tr; @)

Introduction of the reIstions defined by Eqs.

4

through

6

yields:

k A p

(r,fl, t)

()

8pD

.

,. . . . . .

L%D ~

.

(17)

P

~w

which relstes the actual velocity with the dimen-

sionjess function (@D /&D)l. The rate of fluid

influx at the internal boundary is given by:s, g

()

= -

J* Jmr2u

sin

adadO=2trr~~ $

00

fw

. . . . . . . . . . . . . . . . . . (18)

Then, introduction of Eqs. 4 through 6 yields:

which relates the actuaI fluid influx rate with the

dimensionless function

- (13p~/dr~)l.

The cumulative fluid influx at the internal bound-

ary Up to any time t is given by: 2

t [dp

F . $t edt =2trr~~ f

o

()

o ~ ,Wzt

Similarly, introduction of Eqs. 4 through

.()

D apD ~t

F = -2 Prqicr$ ilp(rw, 2){ ~- ~

D1

. . .

(20)

6

yields:

, . .

(21)

which relatea the actual cumulative fluid influx

with the time integral of che dimensionless function

- (dpD/8@ )l. Upon proper interpretation, Eqs. 17,

19 and 21 can be used to determine the fluid flow

and p r e s su r e behavior in a spherical reservoir

system, and also to indicate the appropriate choice

of particular soltitions co Eq. 7.

Ttio

distinct cakes

arise: the so-called pressure and rate cases. 2,5

The Pressure Case

The pressure case presumes know~edge of the

ptessure conditions at the internal boundary of a

-_re*rv~irfiyUadp~dtie4~~i.ga4QL~f__*&_ _.~

-.

R&dw~cion--of--E~ =%*e*e -dimensiorrles-s-fcwia-

depicted by Eq. 7 was effected, because the corn-

fIuid flow behavior. Consider a spherical reservoir

plete dimensioolessness of Eq. 7 renders the numer-

system characterized by. dimert sioriless properties.

icaI

va[ues associated with

its

particular soIucions

Ler this system be charged to a unit dimensionless

entirely independent of the actual magnitudes of

pressure, and at zero time let the pressure at the

the physical properties of any given reserv-oir

internal boundary vanish and remain zero. This

104

SOCIETY OS PETROLIiuM ENGINEERS. Jou.WAL .

... .. .. . . .. . .. . . .. ------ . .. -. -. .

. .,.. . . U.

L-: -.-::: .-

. - ~ . , ., , ,

- .,,. -,- --

~- ~~ ~ ,-J .-.

-

. . - -=. ? ---


4/13

*

condition represents the distinctive feature of the

pressure case. The problem then remaitr~ to

determine the dimensionless rate and cumulative

fluid influx at the internal boundary as functions

of dimensionless time. This dimensionless descrip-

tion of the fluid flow behavior and its uanslation

into absolute units of ~easurement constitutes the

pressure case,2, s

Under the precepts of the pressure case, the

dimensionless fluid influx rate is defined by:

()

p~ :

eD=eD(ljtD) =-~ , . . . . . .

.(22)

D1

and the dimensionless cumulative fluid influx by:

()

D &lD

FD=FD(I, tD)=- J_

dad . . . . .

O arD ~

SymbolicaUy, the actuaI veIocity, rate and cumula-

tive fluid influx majj now be expressed in terms of

eD and

FD

as follows:

u = U(rw, t) =

1 Ap(rw,O)eD(l,tD).

prw

e = e (rw, t) = 2rrr

3 .3p(rw,0)eD(l,tD).

UP

F = F(rw, t) = 2rr4Jcrm~Ap(rw,O)FD (l,tD).

. (24)

. (25)

.

(26)

Eqs. 24 through 26 express the facets of fluid flow

behavior in terms of field data and the dimensionless

functions eD and FD. By application of the super-

position principle (Duhamels theorem) these

functions can also be used to treat time-varying

pressure histories.

The

Rate Case

The rate case presumes knowledge of the fIuid

flow conditions at the internal boundary and permits

determination of the pressure behavior. Consider a

dimensionless spherical reservoir system charged

to a unit dimensionless pressure, and from zero-time

onward Iet a unit dimensiotdess fluid influx rate be

imposed. This condition, which expressed analyti-

cally is:

()

?p~ ~

_.

.

(27)

~rD 1: ._

for all time tD, represents the distinctive feature of

the rate case. The problem here is to determine the

dimensionless pressure drop distribution in the

system,

and the pressure drop at the internal

boundarv under the conditions txescribed bv Ea.

. .

p(r,t) = pi - --&& pD(rD, CD). . . . .

. (28)

w

Similarly, the actual pressure

at the internal

boundary is given by:

P = p(~w, t) = pi _~PD(l>tD) . . . (29)

w

These symbolic relations express the pressure

behavior in terms of field data and the dimensionIesp

functions PD (tD. tD) and PD (1, tD). Likewise, by

appli~ation of the superposition principle, these

functions can be used t o treat time-varying rate

histories. ,

DESCRIPTION OF PARTICULAR SOLUTIONS -

UNLIMITE D SYS TEM

By definition the external boundary of an unIimited

system continuously recedes from the internal

boundary without reaching a geometric limit. Under

these conditions the product rD

pD

vanishes and Eq.

15 becomes:

Z= Clexp(-rD@). . . . . . . . . . .(30)

The precepts of the pressure case require that

a dimensionless pressure drop of unity be maint-

ained at the internal boundary, and since the

Laplace transform of unit is 1/s, it foUows that:

z.

+exp[-fi(rD-l)l , . . . . .

. (31)

which is the subsidiary equation a ppto priate to

the pressure case for an unlimited system. The

dimensionless fluid influx rate e D can be rewritten

in terms of /7:

Then the Laplace transform of eD, utilizing Eqs.

31 and

32,

ia:

1

~D -

-++= . . . .. (33)

whose inverse transformation can be written at

once as:

eDml-l- (~@-1i2,-. . . . . . . .. (34)

which is the dimensionless fluid influx rate of ,

an unlimited system. The Laplace transform of

F ~

(dimensionless cumulative fluid : .Hux) is

simply:

2ZT%k-iiirn-Sin%lonIess descs~pdofi-of->~~wsu-;~ -_ -- - --- ----- -

.. .. _

behavior and its translation into- absolute units of

.~D=:

1 J-

(35)

measurement constitutes the

rate case. 2; 5

3 7-2 +- So*** .

Under the precepts of the rate case, the actual

pressure distribution in the system is given by:

whose inverse transformation can likewise be

J UNE , 1 9 6 6 .

.-

1 0 s


5/13

.-

.-

. .

written at once as:

()

D 1 / 2

FD=t~+2~ , . . . . . . . .. (36)

which is the dimensionless cumulative fluid

influx of an unlimited system. g) 11)13,

14

The

precepts of the rate case require that a

dimensionless rate of unity be maintained at the

internal boundary, which can be wk= in terns of

b as:

-(- )1=-( -6)1= o --()

Using Eq. 30 it. foHows that:

~ =

eXp [- @

(;D- 1)]

. (38)

S(l+fs)

which is the subsidiary equation appropriate to

the rate case for an unlimited system. The inverse

transformation is available from integral transform

tables. This result divided by rD yields:

[()

-1

pD(rD*tJ =

$ erfc

K -

exp (tD + rD

-efi 1

TD- I

+fi . .(39)

26

which is the dimensionless pressure-drop distribu-

tion of an urdimited system. Upon placing rD at

unity, Eq. 39 reduces to:

pD=l

exp(tD) erfc (t&), . . . . . . .

(40)

which. is the dimensionless pressure drop at the

internaI boundary of an unlimited system.2 g, 11,1% 14

At this juncture some significant observations

can be made. First, the least upper bound of the

dimensionless pressure drop is unity. Consequently,

under the conditions of constant rate the pressure

drop at the internal boundary of an unlimited

.

apherlcd system can never exceed s fixe-d finite

value. Secondly, the greateat lower bound of the

dimensionless rate is also unit~. Hence, the rate

engendered by a single pressure drop impes ed at

?eio time ,at the internal boundary of an unlimited

spheric-i- system ciin never b& less thafi a fixed

non-vanishing value. h either situation, it appears

that an unlimited spherical tesetwoir syatern

approaches steady-state conditions as dimensionless

time assumes excessively Iarge values. This

limit. At this limit, a system with a closed external

boundary can sustain no fluid flow across it. Hence,

the normal pressure derivative there must vanish.

Introduction of this condition into Eq. 15 gives:

,[

D

(-)ex +(b-;j

=C exp(-r @ +

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

Under the precepts of the pressure case and by

subsequent conversion to hyperbolic functions, Eq.

41 becomes:

;.

sinh[@r~~r D)] {S rDCOSh[{S(rD- rD)]

s{sinh[{s (7D-1)]-@ rD cosh[{s (rD -l)]]

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

(42)

which is the subsidiary equation appropriate to the

pressure case for a closed limited system. The

LapIace transform of eD, using Eqs. 32 and

42,

ia:

fs(rD-l)cosh[@(rD~l)] +(srD~l)sinh[fs (rD~I)]

s ffs rDcosh

[fs (rD-l)] - sinh[(s (rD-l)] I

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

(43)

The inverse transformation of the relation may be

obtained with the aid of Mellins inversion theorem,

and -is given by the foIlowing in tegttd in the

compIex pfane:

which for the function at hand may be evaluated by

converting it to a closed contour integral and then

applying the calculus of residues. ThuB, by virtue

of Cauchy s integral formula:

where

R. is

the residue corresponding to the

singularity at the origin and Rn the residues

corresponding to the orher singular points. Evalua -

tion of Eq. 45 yields the dimensionless fluid influx

rate for a closed limited spherical system, aa

follows:

w wn2rD2+ (rD~l)2

[ 1

n%o

D=+

tizl rD2 -

(~~~1) Xp

(,;2.1)

. . . . . . . . . . . . . . . . . . . (46)

roperty, strangely enough, is not enj eyed by

unlimited. linear or cylindrical (radial) sy~ms. 2,5.

. -where -wn-are-th-roots.of. the equation ~.---- .___

LIMITED SYSTEM WITH

R

CLOSED

EXTlitRhJAL BOUNDARY

tan w

. . ...+ ., ...

(r; 1)

. .

(47)

In a limited reservoir system the externaI

w

boundary evenruaIIy coincides with a geometric .

The Laplace transform of

FD is:


6/13

. . .

.

..-

.. .

J

F&$

@(rD~l)cosh@(rD~l) +(srDX)sinh@~Ll)

= ~2[@rDcosh W (rD-l) - sinh @ (~D-I)l

. . . . . . . . . . . . . ...*.

. (48)

By virtue of previous arguments, the inverse

transformation of Eq. 4S yields the dimensionless

cumulative fluid influx for a C1OSed limited system:

where Wfl a re also the roots of Eq. 47. z, 10 llsls-lfJ

Under the precepts o: the rate case, Eq. 41

becomes, upon conversion to hyperbolic functions:

b=

~rDcosh\tJG (r~-rD) -sinh fi(rD-rD )

.S[@YO-l )cosh@D-l) + ( srD-lh+inh@rD-l )]

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

. . (50)


rate case for a closed limited system. As before,

the

inverse. transformation

of Eq. 50 is given by the

sum of the residues, and since b is r~PD, there J

foHows :

. . . .

. . . . . . . . . . . . . . ..?

(49)

TABLE 1 - UNLIMITED SYSTEM

Dimensionless Dimensionless Dimensionless Dimensionless

Tim.

Rate Influx Pressure- Drop

(t~) (GD)

(fD)

(PD)

0.001

0.002

0,003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.02

0.03

0.04

0.05

0,06

0.07

0.08

0.09

0.10

0.20

oo3r

0.40

0.50

0.60

0.70

0.80

O*9O

1.0.

2,0

300,

4,0

500

6,0

7.0

8,0

18.84124

13.61566

11.30065

9.92062

8.97885

8.28366

7.74336

7.30783

6,94708

6,64J90

4.98942

4.25735

3.82095

3,52313

3,30329

3.13244

2.99471

2,88063

2.78412

2.26157

2.03006

1.89206 ..

1.79788

1.72837

1.67434

1.63078

1059471

.1.56419

1.39894

1.32574 .,

1.28209

1.25231

1.23033

1.21324

1.19947

0.03668

0.05246

0,06430

0.07536

0,08479

0.09340

0,10141

0.10893

0,11605

0.12204

0.17958

0.22S44

0.26568

0.30231

0.33640

0.36854

0.39915

0,42851

0.45682

0,70463

0.91804

1,11365

1.29788

1.47404

1.64407

1.80925

1.97047

2.12830

%59577

4W95441

6.25676

7.$2313

8?76395

9.98541

11.19154

0.03471

0,04853

0.05892

0.06755

0.07504

0.08174

0,08782

0.09343

0.09865

0.10354

0,14152

0.16894

0.19098

0.20962

0.22588

0,24036

0.2534S

0,26540

0,27642

0,35621

0.40798

0,44639

0.47684

0,50198

0,52330

0.54175

0.55798

0.57242

t16638Q

0.71266

0.74460

0.7676$

0,78534

0.79946

0.81109

Dimenshrless

Time

(tD)

60.0

70.0

80.0

90,0

100.0

200.0

300.0

400.0

500.0

600,0

700.0

BOO*O

900.G

1,000.0

2,000.0

3,000.0

4,000.0

S,ooooo

6,000.0

7,000.0

8,000,0

9,000.0

10,000.0

20,000.0

30,000,0

40,000.0

50,000.0

;60,000.0

70,000.0

80,000.0

90,000.0

100,000.0

200,000.0

300.000.0

Dimensionless Dimensionless Dimensionless

Rote

Influx

(6D)

(FD )

Pressure.DrOp

J,07284

1.06743

1,06308

1.0s947

1.05642

1.03989

1.03257

1,02821

. 1.02523

1,02303

1,02132

i .01995

1,01881

1.01784

1.01262

1.01030

}.00892

1,00798

1.00728

1.00674

1,00631

1.00595

1,00564

1000399

1.00326

1.00282

1.00252

1}00230

1,00213

1,00199

1.00188

1,00178

1.00126

LOO1O3

1,00089

68 ,7

79 .4

90.1

100.7

111.0

216.0

320.0,

423.0

525,0

628oO

730.0

832,0

934,0

1,036,0

2,050.0

3,062.0

4,071.0

5,080.0

6,087,0

7,094;0

8,101,0

9,107.0

10,113.0

20,160.0

30,195.0

40,226.0

50,252.0

60,276,0-

70,299.0

80,319.0

90,339.0

100,357.0

200,505.0

300,618,0

400,714.0

0.92595

0.93103

0.93512

0.93851

0.94139

0.95703

0.96408

0.96835

0.97131

0.97352

0.97526

0.97668

0.97787

0.97888

0.98453 ~

0.98714

0,98874

0.98984

0.99067

0,99132

0.991 8S

0.99229

0.99267

0.99473

0,99566

0.99623

,.

0.99662

-0.99690

0.99713

0.99731 .

0.99746

0.99759

0,99829

0.99860

0.99878

..

loLIoom

1000 1.17841

13.56825 0,82927

500~8i0 0.99891-?

600;000.0

1.00073

600,874.0

20,0

0.99900

1.12616

25.04626

0.87624 700,000,0

1:00067 7oo,944bo

0,99908

30.0

1. 0301 36,18039 0,89770

800,000,0

1.00063

801,009,0

0,99914

40.0

L08921 47,13650

0.91060

900,000,0

50.0

1.00059

901,070.0

1.07979

%999 19

57.97885 0.91943

1,000,000,0

1,00056

1,001, 128*G

0.99923

-107

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

. . . ... . . . . . . . . . . . _. -.-

.

. ..- .:-r . . .. . . .-.-.--.+ . . . . . . .. . . . ...

. .

,=. ,.. . . . . . . . . . .. .. . . . . . .. . . -

..

JUNE. 1966


7/13

.

1

rD-l) ~ (rD-l)4+ 27D(r~~l)%3 rD2 r~

[2cos(wain(.% le--

.

Wnx[wnDcos Wu+ (rD% 1) sin wn ) -

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

(51)

where Wn are here the roots oh

Ctn w

1

TD

- =

w

W2

(r~-l)* - . . . . . . . (52)

The expression embodied by Eq. 51 represents

the dimensionless pressure-drop distribution for a

CIOSed limited sphtiical system. Upon placing rjy

at unity and simplifying, there follows at once the

dimensionless pressure-dropat me internal boundary:

PD=

[(

1

(r; l)2+3rD ~ rD-@ (2rD+1) +t~

-

+(rD

[

1)2 *(rD

1

-l)2+7D

[

1

?D-* TD-4+2r~-1 2~+3r~2

[w?

~%(rD-l) 2]

-2 (rD-

1)3 2 2 ,2

*=I

w ~[w ~ TD +( FD2+~D+l)(~D~ 1)2]

~ =(r~=- 1)2

?

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

(53)

where Wu are still the roots of Eq. 52.

----- -----

LXMITED SYSTEM

WITH OPEN EXTERNAL BOUNDARY

It w iIl be recalled that a limited reservoir

system is characterized by the arrestment of

growth of the extema~ boundary when the latter

.-

Under the precepts of the pressure case and

conversion to hyperbolic functions, Eq. 54 becomes:

sinh @ (?D - fD)

j=

. . . (55)

s[sinhfi(~D- l)] .


pressure case for an open limited system. The

Laplace transform of e D using Eq. 55, is:

1 cosh @ (r~ - 1)

7D =;.+

@ [sinh ~ (tD- I)] . . . . 56)

The inverse transformation is available from integral

tables in the form:

and upon expanding the Theta function this becomes:

Dp

x Xp[ - - o

W

D=

+

D

-l rD-l fz=l

which is &e dimensionless rate for an open limited

system. As before, the Laplace transforms of FD is:

D

1

cosh @(rD- I)

D=~=~+

, - (59)

ssfl [sinh @ (rD- I )]

,whose inverse transformation was obtained with. the

aid of the Faltung convolution theorem as:

the dimensionless cumulative fluid infIux for an

open limited system. ~-n, 1320

Under the precepts of the rate case, Eq. 54

becomes:

.=

coincides with the geometric limit of tb? system.

of Eq. 61 was again obtained by Mellin Js inversion

For the case- open boundary it is presumed

th@orem~a9-@WhXxs lyex~ldtreiiti~us~~-

that at this hit (r ~f) the ~ system suffera no

pressure-drop- distribution is given by:

pressure drip. - ~ . .

--~~ction of this condition into

Eq.. 15 gives:

laa.. .

SOCIETY OF PETROL1

F=

ainh @ (rD- rD)

,.. (61)-

s[~cosh @(rDLl )+ sinh @(rD-l)]


rate case for a Iimited system with a fixed pressure

at the external boundrw. The inverse transformation

.. .. .. . .. . .. .-. , . .

-.. . . . . . . . .. .

J.- ..... . . . ... . . .

RUM ENGINEERS

J OURNAL

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

...-.-


8/13

.

.-

TABLE 2

LIMITED SYSTEMS

Clased External Boundary

Dlmenslonless Functions

lmmnsianlass Functions

Time Rate Influx

Pressure Drop

Time

(tD)

1.0

2*O

3.0

4.0

5,0

6.0

700

8,0

9,0

10,0

20,0

30.0

40.0

SO*O

60.0

70.0

80.0

90.0

100.0

200,0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

20,0

30,0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

Rate

Influx

Pressure Drc.p

(e~) _

(FD)

6%)

Dimsnsfmdess External Radius r; = 5

1.5642 2.128 0,5724

1.3986

3,596

0.6638

1.3216 4.953

0.7133

L2673

6.246

0.7479

1,2203

7.490

0,7764

1,1766

8*688

0.8024

1.1348

9.843 0,8273

1.0946

10.958

0,8518

1.0558 12.033 0.8761

1.0184

13.070 0,9004

0,7103 21.621 1,1424

0.4954

27.585

1.3843

00345s 31.744

Lgjz;

0.2410 34.646

0.1680 36.669

2.1101

0,1172 38.080

2.3520

0.0s18 39.064

2.5940

0.0570 39.751

2.8359

(eD) _

(FD)

PD)

Dimensionless External Radius r~ = 2

0.07

0.08

0.09

0.10

0.20

0030

0.40

0.50

0,60

3.7(J

0,80

0.90

1.0

3.1324

2.9947

2.8806

2,7839

2.2411

1.9342

0.3685

i7.3992

0.4285

0,4568

0.7040

0.9120

1.0927

1.2503

1,3879

1.5080

1.6128

1.7044

0.2404

0.2534

0.2654

0.2764

0,3567

0.4120

0.4591

0.5033

0.5467

0,5897

0.6326

0.6755

1 . 685S

1 . 4713

1 , 2844

1 . 1212

0 . 9788

0 . 8544

0 . 7459 1.7843

2.1921

2,2970

2,3240

2.3309

2,3327

2.3332

2.3333

2.3333

2.3333

0.7184

1.1469

1.5755

2.0041

2.4327

2.8612

3.2898

3.7184

4.1469

4.5755

2.0

3.0

4,0

5.0

6.0

7.0

8.0

9.0

10.0

0.19~6

0,0491

0,0127

0.0033

0.0000

0,0002

0.0001

0.0000

0.0000

0:0398

40.230

0.0000 41.333

Dimension less Externu l Radius

.1.3989 3,596

1,3255

4.954

1,2807

6.256

1.2477

7.520

1,2201

8.753

T.1951

9.961

1.1714

11.144

1.1487

12.304

1.1265 13.441

0.9283

23,683

0.7650

32,123

0.6304

39.078

0.5195

44.810

0.4281

49,534

0.3528

53.426

0.2907

56,634

0.2396

59.277

0.1974

61.455

0.0285 70.191

0,0041

71.453

0.0006

71.636

3.6778

6.5068

t;=6

Dimenskmless External Red l us r;=3

0,6638

0.7127

0.7449

0.7687

0.7881

0.8051

0.8207

0,8356

0,8501

0.9903

1.1298

1.2693

1.4089

0.2

0.3

0.4

O*5

0.6

0.7

0.8

0.9

2.2616 0.7046

2.0301 0,9180

1,8920

1.1136

1.7972 1.2978

1.7261

1,4739

1.6688 1.6435

L6 199 1.R079

0.3562

0.4080

0.4464

0.4769

0.5021

0.5236

0,5425

0.5595

0.5750

0,7012

0,8171

0.9325

100479

1.1633

L2787

1:5764

1.9677

1,5363

2,1233

1.2114

3.4891

0.9586

4.5692

0.7586

5,4239

0.6004 6.1004

0.4751

:::;;

0.3760

0,2975 7*3944

0,2354

7.6598

0.1863 7.8698

0,0180 8,5899

0,0017

8,6593

0,0002 8.6659

0.0000

8,6666

0.0000 8.6667

Lo

2*O

3.0

4.0

5.0

, 6.0

7.0

8.0

9.0

1 , 5484

L6B79

1 , 8275

1 . 9670

2 , 1065

3 . 5019

4 8 9 7 2

6 . 2926

7 , 6879

9 , 0833

1;3941

1.5095

1.6249

2,7787

3,9325

5.0864

6.2402

7*394 1

10.0

20.0

30.0

40.0

50,0

60.0

0.0001

71,662

0.0000

71.666

0,0000

71.666

0.0000

7 1.66?

10 , 4786

11 , 8740

,

Dime.tsienless Ewernal Radius rj = 4

Dlmensionlew

External Radius

1,6743

;:$;

0.5233

1,6308

0.5418

1.5946

1.970

0.5580

1.5640

2.128

0.5724.

1.il369

3,592

0.6655

r55

4,921

0.7234

6.147

0.7734

...-.

1

7.279

1.0049 8.325

1.32,57

1.2820

1.2519

1,2289

1,2099

1*1933

1,1780

1.1636

1.0354

4 , 95

6 . 26

7 , 52

8 . 76

9 , 98

11 , 18

12 , 37

13 , 54

24 . 52

0.7127

0,7446

0,7678

0.7857

0,8004

0.8131

0.8244

0.7

,.

0.8

O*9

1.0

2.0

3.0

3.0

4.0

5*O

6.0

7.0

::

10.0

20.0

0,8216

0,8693

0,8348

0.9255

---

6.0

30.0

40.0

50,0

60.0

77;

90,0

100.0

200.0

0.92i3

0,8216-

0.7318

0.6518

.0.5806

0.5172

0.4607

0,4104

0.1290

34*3O

43.01

50.76

57,68

63,83

6.,31

74.20

78.55

102,86

1.0133

;:;;;;

L2765

1.3642

1,4519

1,S396

1,6273

2,5045

9.0

0.7922 110008 1.0122

tn.n n.72m

11.770

110599..

;;?3::

1.5361

290122

i

20.144

Z@:

7 20.613

, . .- _ -

80.6

ti0028

2W64-

-

--=-----

4

90.0

0,0013

:

2- -

~@o-

oioYl7-

l-12~91

- -=--------4.2598

. . . . . .

. . . .. . ...- 114*OO

1}000.0

0.000:

114.00

%,6449

9.522?


9/13

.

-.

.-

TABLE 2- LIMITED SYSTEMS (ccmthued)

Dimensionless Functions

Dimensionless Functions

Time

Rote

Influx

Pressure Drop

Time

R.to

Infl Ux

Pressure Drop

(t)J)

(t) (e~)

(FD)

(pD)

mD) _

FD)

(I%)

Din

30.0

40.0

50.0

wnsionloss

1.1030

1,0892

1.0796

1,0724

1.0664

1.0611

1.0562

1.0516

1.0088

0,9681

0.9291

0.0916

0.8S57

0,8212

0.7881

0.7563

0.7259

0.4810

0.3187

0.2111

0.1400

0.0929

0.0616

0.0408

0.0270

0.0179

0.0000

External Rodlus

36.2

47.1

58.0

68,7

79,4

90, I

100,7

111.2

214,2

313s0

407.9

498,9

586,2

670,1

750s

827.7

901,8

1,496,9

1,891.2

2,152.4

2,325,7

2,440.5

2,516,7

2,567.1

2,600,6

2,622.7

2,666.3

20

Dimensionless

1.2821

1.2S23

1.2302

1.2128

1.1983

1.1859

1.1747

1.0860

1.0078

0.9354

0.868

0.8056

0.7477

0.6939

0.6440

0.S976

0.2s32

0.1342

External Radius

6.26

7.52

8.76

9.98

11.19

12.38

13.56

,

D .0.8977

0.9106

0.9199

0.9270

0.9328

0.9378

0.9423

0.9465

0.9851

1.0229

1.0604

1.0979

0.7446

0.7678

0.7854

0.7996

0.8115

0.8216

0.8306

0,8971

0.9561

1,0148

1.073s

4.0

5*O

6.0

7.0

8.0

9*O

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100,0

2Q0.O

300.0

400.0

p:

80.0

90.0

100.0

200.0

300.0

400.0

S30.0

600.0

700.0

800.0

900.0

1,000.0

.2,000.0

3,000.0

4,000s0

5,000.0

6,000.0

7,000.0

8,000.0

9,000.0

10,000.0

20,000.0

24.85

35.31

45.02

54.04

62.40

70.17

77.37

84.06

90.26

132.37

152.34

1.13s4

1.1729

1.2104

1.2479

1.2854

1.6604

1.1322

1.1910

1,2497

1.3084

1.3671

1.9542

2,0355

2,4105

2.78s6

3,1606

3,5357

3.9107

4.2858

4,6608

8.4113

2.5412

0.0637

161.80

0.0302 166.29

0.0143 168*41

0.0068

169.42

0.0032 169.90

0.0015 170.13

0.0007 170.24

0.0000

170.33

3,1283

3.7154

4.3025

4,8896

5.4767

500.0

600.0

700.0

800.0

900.0

1,00000

2,000.0

6.0638

6.6508

12.5218

Dimensionless

80.0

1.0631

90.0 1.0595

100.0 1.0564

200.0

1,0381

300,0

1,0254

400.0 1.0133

500.0

1.0014

600.0

0,9895

700.0 0.9780

800.0

0.9665

900.0 0,95S2

1,000.0 0.9439

2,000.0 0.8388

3,000.0

0,7453

4,000.0

0,6622

5,000.0

0,5884

6,000.0 0.5228

mal

Rod[us r~ = 30

90.1

100.7

111.3

215.9

319.1

421.0

521.7

621.3

719.7

816.9

913.0

1,007.9

1,898.3

2,689.4

3,392.4

4,017.0

$,572.0

5,065.2

5,503.4

5,892,7

6,238.6

8,153,6

8,739.1

8,919.7

8,975.1

8,992.1

8,997.3

8,998.9

8,999.4

8,999.6

0.9351

0.9385

0,9414

0.9000

0.9724

0,9840

0.9954

1.0068

1,0179

1.0290

1.0401

1.0512

1,1623

1,2735

1.3846

1.4957

1,cW68

1.7179

1.8290

1.9401

2.0s13

3.1624

4.2736

5.3847

6.4959

7.6070

8.7182

9.8293

10,940s

12.0516

DlmensionIess External Radius ?; = 9

5.0

6.0

7.0

8.0

1.2523

1,2303

1,2132

1.1993

7.52

8.76

9.99

0.7676

0,7853

0.7995

0,8112

0.8211

0,8296

0.8848

0.9271

11.19

12.38

13.57

24.98

35.70

46.04

55.83

65.11

9.0

10.0

20.0

30.0

40.0

50.0

60.0

70,0

80.0

90.0

100.0

200.0

300,0

400.0

500.0

600.0

700.0

800.0

900.0

1,000.0

2,000.0

1.1877

1.1776

1,1094

1.0539

1.0015

0.9518

0.9045

0;9684

1.0096

1.0508

0,8596 73,92

0,8169 8230

; 0.7763

90.27

0.7378 97.84

0.4433 155,64

0,2663 190.37

1.0920

1,1332

1.1745

1.2157

Zooo,o

8,000,0

9,000.0

10,000.0

20,000.0

30,000.0

40,000.0

50,000.0

60,000.0

70,000.0

80,000.0

90,000.0

100,000.0

0,4646

0,4128

0,3668

0,3259

,6278

2.0398

0.1000

0,0307

0,0094

0,0029

0,0009

0.0003

0.0001

0.0000

0.0000

0.1600

0,0962

0,0578

0,0347

0.0209

211.24

223,79

231.32

- 235.85

238.57

2.4519

2.8640

3.2761

3.6882

4.1003

0,0125

0.0075

0,0000

246,21

241.19

242.67

External Radius r;= 10

8.76

9*99

11.19

12.38

13.57

25.02

36.01

46.60

4.5124

4,9245

9.0455

Dime.

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

1,000.0

2,000.0

3,000,0

4,000.0

5,000.0

6,000.0

7,000.0

8,000.0

9,000.0

10,OW,O

20,000.0

._30,000,0

40,000.0

50,000,0

60,000.0

70,000.0

80,000,0

90,000.0

100,000.0

200,000,0

Sionless

1.0564

1.0398

1.0320

1.0262

1.0210

LO 160

1.0110

1.0060

1.0011

0.9962

0.9485

0.9031

0.8S98

0.8186

0.7794

0.7421

0.7066

Extwnal RadIUS

11100

216,0

320.0

422.0

525.0

627,0

728.0

829.0

929,0

1,029.0

2,001.0

2,927,0

3,808.0

r;= 40

Dimenslonlmss

1.2303

1.2132

1.1995

1.1880

1.1783

1.1196

c 1.0783

1.0398

1.0027

0.9669

0,9325

0.8992

0.8672

0,8362

0.5816

0.9414

0.9570

0.9653

0.9715

0.9769

0.9820

0.9871

0.9925

0.9972

1,0019

.1.0488..

1.0957

L 1425

h 1894

1.2363

I . 2832

1.3301

L3769

1.4238

1.8926

_2.3613

2,8301

3.2988

* 3.7676

4.2363

4.?051

5.1739

5s6426

10,3302

6.0

7.0

8.0

0.7853

0.7995

0.s112

0,8210

0.8295

0.8797

0.9124

9.0

10.0

20:0

30.0

40.0

50.0

0.9427

0.9728

1.0028

,1,0329

1.0629

1,0929

1.1229

1.4232

56.81

60.0

70.0

80.0

90.0

100.0

200.0

300.0

400.0

500.0

;g~f

800.0

66;66

76.15

85.31

94.14

102.66

172.78

4;647.0

5,446.0

6,207.0

L931.O

0:404s

0.2813

0,1954

C$:;W;

0:0659

00458

0.0319

0,0008

0.0000

0.0000

%:2

279.07

=29549 ----

306.90

314,85

1.7235

2.0238

2.3241 -

Zf6244

2.9247

3,2250

0;6727

0.6405

0,3920

.:;:: 3;.

0:0901

0.0552

0.0338

0,0207

0.0127

0,0078

0,0000

~620.O

8,277.0

13,339.0

s~3&o+

18,333.0

19.496.0

ii

.-

900.0

1,0000

2,000,0

3,000.0

4,000,0

320.37 3.5253

3.82S6

6.8287

9.8317

12.8347


10/13

-.

-.

TABLE 2-

DhnmslonlmssFunctlmm

The Ret,

Influx

PressureDrop

(t~)

bv]

eD) Q .

Dlmmdwhss E+tarnot Radius r : = SO

LIMITED SYSTEMS (continued)

Dimensionless Functirms

Timo

Rate Influx

Pres.wro Drop

(t,,)

(p) _

(Fn)

@D)

Dimensionless

900.0

1,000.0

2,000,0

3,000,0

4,000,0

5,000.0

yw::

8:000.0

9,000.0

10,000,0

20,000.0

30,000.0

40,000,0

50,000.0

60,000.0

70,000.0

80,000.0

90,000.0

100,000.0

2W,000,0

300,000.0

400,000.0

500,000.0

. 600,000.0

700,000.0

800,000.0

900,000.0

1,000,000.0

2,000,000.0

External Rndlus r ;= 60

200,0

300.0

400,0

500,0

600,0

700,0

800.0

900.0

1,000.0

2,000.0

3AO0.O

4,CQ0.O

5,W0.O

6,00cbo

7,000.0

8,000.0

9,000.0

10,000.0

20,000.0

30,000.0

40,W0.O

50,000.0

60,000.0

70,000.0

80,000.0

90,000.0

100,000.0

200,000.0

3Q0,000,0

400,000.0

500,000,0

1.0399

1.0325

1.0280

1.0246

1.0217

1,0190

1.0164

. .0139

1.0113

0,9865

0.9622

0.9385

0.91$5

0.8930

0,8710

0.0496

0,8287

0,8083

0.6241

0.4912

0.2828

0.2984

0,2326

0.1818

0.1418

0.1106

0.0862

0.0072

0.0006

0.0000

0,0000

216,0

320.0

423.0

52%0

628.0

730.0

831.0

933.0

1,02-4.0

2,033.0

3,007.0

3,958.0

4,884,0

5,789.0

6,671.0

7,531,0

8,370,0

9,188.0

16,344.0

21,921.0

26.269.0

29,658.0

32,299.0

34,357.0

35,967.0

37,222.0

38,201.0

,41,37s.0

41,642,0

41,664.0

41,666,0

0.9570

0.9641

0.9693

0.9732

0.9765

0.9795

0.9823

0.9856

0.9880

1.0120

1.0360

1.0600

1.0840

1.1080

1,132+3

1.1560

1.1800

1.2040

1.4440

1,6840

1,9240

2,1640

2.4040

2.4440

2,8840

3.1240

3.2640

5,7641

8.1641

10.5641

12,9641

1.0188

1,0178

1.0108

LO047

0.9987

0,9927

0.9868

0.9809

0,9750

0.9692

0.9634

0.9073

0.8545

0.8048

0.7579

0.7138

0.6723

0.6331

0,5963

0,5616

0.30s1

0.1697

0.0933

0.0512

0.0281

0,0155

0,0085

0.0047

0.0026

0.0000

9:. .0 -

1,036,0

2,05%0

3,057.0

4,059.0

5,055,0

6,045.0

7,028,0

8,006.0

8,978,0

9,945,0

19,29S,0

28,102.0

36,396.0

44,207,0

51,S64.0

58,493.0

65,018,0

71,163.0

76,950,0

119,176,0

142,333?0

155,096.0

162,112,0

165,967.0

168,084.0

169,248.0

169,887,0

170,238.0

170,666,0

0.9779

0,9791

0.9875

0.9944

1.0009

1.0068

1,0127

1.0185

1.0244

1.0302

1.0361

1,0947

1.1533

l,21i9

1,2705

1.3291

1,3877

1.4463

1,5049

1.5634

2,1494

2.7353

3.3213

3,9072

4.493 I

S,0791

5.6650

6.2510

6.8369

12,6963

lmmtsi.anlessExtamal Radlusr~=60

300.0

400.0

500.0

600.0

700.0

800.0

900.0

l,oco.o

2,000.0

3,000,0

4,000.0

S,ooo.o

6,000.0

7,000.0

8,000.0

9,000,0

10,000,0

20,000.0

30,000.0

40,000,0

50,000,0

60,000,0

70,000.0

80,000.0

90,000,0

100,000.0

200,000,0

300,000.0

400,000.0

500,000.0

600,000.0

700,000.0

800.000.0

1.0326

L0282

1.0252

1.0228

1.0209

1.0192

1.0176

1.0160

1.0015

0.9872

0.9732

0.9594

0.9457

0,9323

0.9190

0.9060

0.8931

0,7739

0.:3.::

0,5036

0.4363

0.3780

0.3275

0,2838

0,2458

0>0690

0.0141

0.0034

0.0008

0.0002

0.0000

0.0000

320,0 -

0.9641

423.0 0.9684

525.0

0.9717

628.0

0.9743

730,0

0.9765

832,0 0.9784

934,0

0,9801

1,035.0

0.9818

2,0440

0,9969

3,038,0

1.0117

4,019.0

1,0256

4,985,0

1,0395

5,937,0

1,0533

6,8?6.0

1.0672

7,802.0

1.0811

8,714.0 1.0950

9,614.0

1.1089

17,935.0

1.2478

25,145,0

\:w6;

31,393.0

36,808.0

1:6645

41,499.0

1.8034

45.564.0 1.9422

49.086,0

2.0811

52,137.0 2.2200

34,78 Lo 2.3589

67,87%0

3.7478

71,014.0

5.1367

71,764,0 6,5256

71,942,0 7,914s

71,986.0 9,3034

71,996,0 10.6923

72,000,0

12,0812

Dimensi.anlncsEat.wnal Radiusr~, 90

1,000.0

2,000,0

3,000.0

4,000.0

5,000.0

6,000.0

7,000.0

8,000.0

9,000.0

Io,ooo.o

20,000.0

30,000.0

40,000.0

50,000.0

60,000.0

70,000.0

80,000.0

90,000.0

100,000.0

200,000.0

300,000.0

400,000.0

500,000.0

600,000.0

700,000.0

800,000.0

900,000.0

1,000,000.0

2,000,000.0

1,0178

1,0119

I.0075

1.0033

0.9991

0,9949

0,9907

0.9866

0,9824

0.9783

0,9381

0,8995

0,8625

0.8270

0,.7930

0.7603

0.7291

0,6W1

0,6703

0,4402

0.2890

0.1905

0,1253

0.0824

0.054 I

0,0356

0.0234

0,0154

0,0000

1,036,0

2,050,0

3,060.0

4,065.0

5,066,0

6,063,0

7,056.0

8,045.0

9,029.0

10,610.0

19,590,0

28,776.0

37.585.0

0.9789

0,9864

0,9914

0,9965

1.0006

1.0047

1.0088

1.0129

1.0170

1,0212

1,0623

1,1035

1.1446

46;031,0

54.129.0

61,895,0

69,341,0

76,480.0

83,326,0

138,050.0

173,987,0

197,575,0

213,137,0

223,370,0

230,097.0

234,519,0

237,425,0

239,335,0

243,000.0

1,1858

1.2269

1,2681

1,3092

1.3504

1.3915

1.8031

2.2146

2,6261

3.0376

3,4491

3,8607

4.2722

.

4.6637

5,0952

9,2104

Dimensionless

700.0 1.0213

800.0

1.0198

900.0 1.018S

1,000.0

1.0174

2,000,0 1.0079

3,000.0

0.;2.;;

4,000.0

5,000.0

0.981 I

6,000.0

0.9723

7,0W.O

0.9635

8,000.0 0.9550

9,000,0. 0.9405.

10,000.0 %9380

20,000,0

0.8575

30,000.0

0.7838

40,000.0

0,7165

50,000.0 0,6549

60,000.0

0,5987

70,000,0

0.5472

80,000.0

00WQ2

90,000.0

0.4572

.0,4179

Q%,&%:_

f7,1zo.7_

300,000.0 0.0692

400,000.0

0.0280

500,000.0 0.0113

600,000.0

0.0045

700,000.0 0.0018

800,000.0 0.0007

900,000.0

o,aoo3

1,000,000.0

0.0001

2,000,000.0 0.0000

ExtarrmlRadius r;= 70

730,0

832.0

924,0

1,036,0

2,048,0

3,052.0

4,046,0

5,032.0

6,008.0

6,976.0

7,926.0

8,88o.O

9,829.0

18,800.0

27,W2.O

, 34.498,0

41,351.0

47,615.0

53,341.0

58,37S.0

63,359.0

67,732,0

_9$ 396.o

106,570.0

111,168.0

113,042.0

113,807.0

l14,11&o

114,245.0

114,297.0

l14,31&o

114,333.0

0.9753

0,9769

0.9785

0.9799

0.9906

1,0005

1.0093

1,0180

1.0268

1.035s

1.0443

y::;g.

1:1492

1,2367

1.3241

Dimemsianles$External Radius r~= 100

1,000;0

2,000.0

3,000.0

4,000.0

5,000.0

6,000.0

7,000.0

8,000.0

9,000.0

10,000.0

-20,000.0.

30,000.0

40,000.0

,60,000.0

,60,000.0

70,000.0

80,000.0

90,000.0

.100,000.0

200,000.0

300,000,0

400;oooio-

500,000.0

600,000,0

700,000.0

800,W0.O

900,000,0

1,000,000,0

2,000,0000

1.0178

1,0123

1.0090

1.0058

1,0028

0.9997

0.9967

1,036.0

2,050,0

3,061.0

4,068.0

5,073.0

6,074,0

7,072.0

8,067.0

9,059.0

10,048.0

..19,775.0

39,208,0

38,358,0

47,232.0

55,839,0

64,187.0

72,283.0

F30,136.0

87,753,0

1S2,377.0

1%9,972.0

::;;: ;f:

279:S49:0

293,918.0

304,287.0

311,938.0

317,558.0

333,333.0

0,9789

0.98s3

0.9894

0,9930

0.9964

1,0000

1.0030

1.0060

1.0090

1.0120

1.0420

1.0720

I.102O

1.1320

1.1620

1.1920

1.2220

1.2520

1.2820

1.5820

L8820

y4:;;

2:7820

3.0820

3,3820

3.6820

3.9820

6.9820

0,9936

0.9906

0.9876

0,9578

0,9290

0.9010

0,8739

0.8476

0.8221

0.7974

-

i.4116

1.4991

1.586s

1,6740

1.7615

1,8489

-_2z7236

3,59K2-

4,4728

s.~7.

6.Z:

7A968

7.9714

8.8460

9.7207

l&4677

0.7733

0.7501

0.s5?5

0,4068

-0299s

0.2210

0.1633

0s1203

0.0887

0.06S3

0.0482

0,0000

-

.l-~,


11/13

.

ABLE 3- LIMITED SYSTEMS

Open External Boundary

Dfmensfenless Functlc.ns

T mm

Rata Influx

Pressure Drcm

Dlmensienloss Functions

Rate

Influx

Prassure Drop

(@/J)

(FD)

fla~)

Time

(tL))

3*O

4,0

5,0

6.0

7.0

8.0

9.0

10,0

20.0

30.0

40,0

50,0

4,0

5*O

6,0

7,0

8*O

9*O

10.0

20,0

30.0

40.0

50,0

60.0

70,0

5,0

6.0

7.0

800

9,0

10*O

20.0

30.0

40.0

5000

60.0

70.0

80.0

6,0

7,0

8,0

9.0

10,0

20.0

30.0

40,0

50.0

60.0

70.0

80.0

90.0

30,0

40.0

50.0

60.0

70;0

80.0

90.0

100.0

200.0

300,0

t D)

0,07

0,08

0,09

0,10

0020

0.30

0,40

O*5O

0.60

0.70

0,80

0..;

2.00

3*OO

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.7

0.8

0,9

1

2,0

3*O

4.0

5.0

6.0

7.0

8,0

9.0

10.0

20,0

1.0

2.0

3.0

4.0

5*O

6.0

7.0

8,0

9.0

10.0

20,0-

30.0

(e~)

Dfmenslonless External Radius r:= 2

3.1324

0.3685

0.2404

2.9947

0.3992

0,2534

2.8007 0.4285 0.2654

2.7043 0,4568

0,2764

2.2786 0.7052

0,355s

2.1036

0.9228

0.4048

External Radius rfi= 7

4*9544

6,2568

7,5234

8,7649

9,9881

11.1977

12.3969

13.5883

25,3283

37.0000

48,6666

60.3333

1,3257

1,2822

L2527

1.2315

1,2157

1,2039

1*1950

L11?82

1.1681

1.1668

1.1667

L 1667

0,7127

0.7446

0,7676

0,7851

0.7988

0.8098

0,8187

0,8258

0,8531

0,8566

0.8571

2,0386

2,0144

2.0054

2,0020

2.0007

2.0003

2.0001

1.1294

103319

1.5328

1.7331

1,9333

2.1333

2,3333

4.3333

6.3333

0.4370

0.4582

0,4723

O*4B17

0.4878

0.4919

0.4947

0.4999

0,8571

2.0000

2,0000

Dlmenslanless

External

Radius r~=

8

0:5000

L2821

1.2523

1.2305

1.2136

1.2003

1.1897

1.1811

1.1479

6,2568 -

7,5231

8.7640

9,9857

11.1925

12.3873

13.5725

0,7446

0.7676

0,7853

0.7944

0,8109

0.8205

Oimensianless External

2.2616

2,0301

1,8921

1.7984

1.7302

1,6788

1.6393

1.6087

1.5849

1,5072

1.5006

Radius r;= 3

0.7046

0,9180

1.1137

1,2979

1.4742

1.6445

1.s103

1.9727

2.1323

3.6638

5.1664

6.6666

8.1667

9.6667

11,1667

0,3562

0.4080

0.4464

0.4768

0,5019

0,5230

0,5412

0.5568

0s704

0.6407

0.6597

0.6648

0.6661

0.6665

0.6666

0,8285

0,8653

0.B730

0,B746

0.8749

0.8750

0.8750

25.1652

36.6157

;::;;:

70.9048

82,3333

1.1435

1.1429

1.1429

L 1429

1.1429

External

Radius r;= 9

1.5001

1.5000

1.5000

1.5000

1.2523

1.2303

1.2133

1,1996

1.1884

1.1790

1.1364

1.1274

1.1255

1,1251

1.1250

1.1250

1.1250

7.5231

8.7640

9.9854

11,1917

12,3855

13.5690

25.0925

36.4008

47.6633

5s.9159

0.7676

0.7853

0,7995

008111

0.8209

0,8292

0.8717

0.8838

0.8874

D[mens[anIess

1.6743

1.6308

1.5948

1.5643

1.4078

1.3582

1.3416

.1.3361

1,3343

1.3336

1,3334

1.3334

1.3333

External Radius r~= 4

1:6441

1.8093

1.9705

2.1284

3.5988

4.9773

6.3258

7,6641

8.9992

10.3331

11,6666

13,0000

14.3333

0,5233

0.5418

0.5580

0.5724

0.662S

0,7054

0.7272

0.7383

0,7440

0.7469

0.7484

0.7492

0.7496

0.7500

0.8884

0.8888

0,8S89

0.8089

70,1665

8104166

92.6667

External Radius r~= 10

0.764

9.985

;;:;;

13,568

25,063

36,286

47.431

58,551

69.665

80.777

91,889

103.000

1,2303

1.2132

1.1995

1.1881

1.1785

1.1306

0.7853

0.7994

0,8111

0.8209

0,8293

0,8747

0,8906

0.8965

0.8987

0.8995

0.8998

0,8999

0,9000

1.3333

Dimensionless

1.5642

1.3992

1.3289

1,2924

1.2729

1.2623

1,2567

1.2536

1.2519

1.2510

1.2500

1.2500

27,6667

Radius r~ = 5

2.1284

3.S958

4.9558

6,2646

7.5462

8.8133

10.0725

11,3275

12,5802

13.8316

26.3333

External

.

1.1169

101128

1.1116

1.1113

1,1112

1.1111

1,1111

0.5724

0.6638

0.7121

0.7422

0.7618

0.7748

0.7833

0,7890

0.7927

0,7952

0,7999

0.8000

Dimensionless Externai Radius r~= 20

101030

36.180 0.8977

1.0892

47. I37

0,9106

1.0799 570980

0.9193 .

1.0732 68,743 0,9261

1.0682

79,449

0.9312

1.0645

90.112

0.9351

1,3989

3.S95B

0.663B

1.0616

100,741

0.9382

1.3259 4.9545

0,7126. 1,0595

111.346

0,9406

1.2832

6,2573

067444 1.0531

216.843

0,9486

1,2557

7,5258 0.7669 1.0527

322.122

0,9495

1.2375

8,7718

0.7834

400.0

.x._._ Zs::L

1,0526

427.386 0.9497

_._.--.102252_-

10,0028

__Q:Z&-

_@&g ___LQ&

532,K49

09499

1,2170

11.2236

637.912

0.9500

9*O

1.2115

J2.4377

0.8119

10.0

1.2077

.13.6471 0,8172

,..

20.0

1)2001

25.6663

: 0.8324

30.0

1.2000

37.6667

0,8333

40,0

1.2000

49,6667

0.8333

3B.8333

Dimenslenloss External Radius rD = 6

2,0

3.0

4.0

5.0

6.0


12/13

rD-r~

2 (q.j-

1) =

PD(rD~tD~ = ~ +

[ 1

.

TD ?@

-

-

@;q2

. [ - 1; 4 1 = 2 TD: 2~rDL:

h:ch is the concIudmg resuk.9 11,17

Wn[rD

(rD- 1) + tun2] cos W* NUME~CAL COMPUTATION OF

PARTICULAR SOLUTIONS

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

...0. .

(62)

Nine

particular solutions to Eq, 7 obtained with

where Wn are the roots of the equation:

the aid of the LapIace transformation were numeri-

cally computed. Specifically, these included the

tan w

1

(63)

functions Mined by

Eqs. 34, 36, 40, 46, 49, 53,

= -~ ,

w

. . . . . . . . . .

D-l

58, 60

and 64.

The numericaI computations were carried out

Upon pIacing ?D at unity in Eq.

62

a n d s im p l ify in g ,

wit h t he a id of IBM 1401 and 1620 computer systems.

the dimensionless pressure drop is obtained:

Programming was in FORTRAN. The functions for

Dlmenslonless Functions

TABLE 3 - LIMITED SYSTEMS (Cwstinued)

Time

(tD)

80,0

90,0

100.0

200.0

300,0

400,0

500.0

600.0

700.0

800.0

900.0

1,000.0

100.0

200.0

300.0

400.0,

500.0

600.0

700,0

BOW

900.0

1,000,0

2,000,0

200,0

300.0

400.0

500.0

600,0

700.0

800.0

900.0

1,000,0

2,000,0

3,000,0

300.0

g:~:

600.0

JUNE 1966

Rate

Influx

Pressure Drop

(eD) (FD)

kD)

.

Dimensionless External Radius FL= 30

1.0631

1,0595

;:::::

1.0365

1,0351

1.0347

1.0345

1.0345

1.0345

?-80345

1,0345

Dimensionless

1.0564

1,0399

1,0330

1.0295

1.0276

1.0267

L0262

1,0259

1,0258

1,0257

1,0256

Dlmansimslsss

1,0399

1.0326

1,0283

1.0256

1.0239

1.0227

1.0219

1.0214

1,0211

1,0204

1.0204

90,093

100*705

111.284

216,001

319,838

423.406

526,891

630.351

733.803

837,252

940.701

1,044.149

0,9351

0;9385

0.9414

0.9576

0,9629

0.9649

0.9656

0.9660

0.9662

0,9664

0.9665

0.9667

External Rodius r:=

111*28

215,96

319,56

422.67

525.51

628.22

730,86

833,47

936,05

1,038.63

2,064.28

40

0,9414

0,9570

0,9646

0.9686

0.9708

0.972 I

0.9729

0.9734

0.9737

0.9739

0.9750

Externol Rad[us

215,96

319s4

422,5B

525,27

627,74

730.06

r;= 50

832o29

934,46

1,036,58

2,057, i5

3,077.56

0,9570

0,9641

0.9688

0.9718

0.9739

0.9754

0,9764

0,9771

0,9?76

0.9794

0.9800

Dimension es$ Exterqol Redius rj = 60

1.0326 319*54 o*9~41

1.0282 422,57

0.%684

1.0253

528.23 r,9716

ii0232

627.65

0.9740

Dimenslcinless Functions

Time

t~)

700.0

800.0

900.0

1,000.0

~ooo.o

3,000.0

4,000.0

5,000,0

6,000,0

900,0

1,000.0

2,000.0

3,000.0

4,000.0

5,000.0

6,000,0

7,000.0

1,000.0

2 000 0

3 000 0

4 000 0

5 000 0

6,000.0

7,000.0

8,000,0

1,000.0

2,000,0

3,000.0

4,000,0

5,000,0

6,000iO-

7,000.0

8,000.0

9,000.0

Rate

Influx

Pressure Drop

(eD) (FD) (PD)

Dlmen4ior rless Externa l Radius r~=, 70

1.0213

730.0 0,9753

1.0200

832.0

0.9767

;::;:9

934.0

0.978 I

1,036.0

0.9795

1.0150

2,052.0

0.9838

1.0146

3,066.0

0.9848

1.0145 4,081.0

0.9853

1,0145

5,095.0

0.9857

1.0145 6,110.0

0.9857

Dimensionless Exte/

1,0188

1,0179

100137

1.0129

1 . 0127

L0127

1 , 012 :

1 , 0127

rrral Radius

934,0

1,036.0

2,051.0

3,064.0

4,077.0

S,090.O

6,102.0

7,1 15*O

0.9779

0.9794

0,9847

0.9862

0.9868

0.9872

0.9875

0.9875

Dlmenslonles

1.0178

100131

1,0118

1,0114

1.0113

1,0112

1.0112

s Extel

1.0112

Dimensionless

1*OT78

1,0128

100111

LO1O5

1*O1O2

1.0101

1,0101

1*O1O1

1,0101

,no l Rad ius

1,036.0

2,051.0

3,C63.O

4,074.0

5,086.0

6,097.0

7,108,0

8,120.0

0.9789

0.98S0

0.9870

0,9B78

0.9883

0.9886

0.9889

0.9889

External Rod I us I

1,03600

2,051.0

3,062.0

4,073.0

5,083.0

6,094.0

t 7.104.0

8;1 14.0

9,124,0

,,

D=

100

0.9789

0.9846

0.9874

0.9885

0,989.1

0,9894 -

0.9897

0.9899

009900

t

.

-.-.. ...

..z

1 1 3


13/13

.

.

.-

the unlimited system were computed first over the

dimensionless time range 0.001 to 1,000,000. Then

tables of the trigonometric relations described by

Eqs. 47, s,2 and 63 were developed from which the

roots w (with n =

6) were

obtained. Finally,

numerical vaIues of the functions for limited

systems were computed over the range of external

radii (rD) 2 to 100.

The range of dimensionless

time (tD) for these functions was chosen to begin

with the points of divergence from the unlimited

system envelope and to end with steady-state valwes.

These numerical results are included in tabular

form to foster practical application of this work.

NOMENCLATURE

Cl, C2 = arbitrary constants

F = cumulative fluid influx

FD = dimensionless cumulative fluid influx

~D = Laplace transform of FD

,RO = residue of singularity at origin

Rv = residues of singularities at z=

b = dimensionless product of pressure drop

and radial distance

~ = LapIace transform of b

c = compressibilir~

e = rate of fhtid influx or fluid rate

D

=

dimensionless rate of fluid influx

TD = Laplace transform of eD

k = permeability

kb =

horizontal permeability

k, =

radial

permeability in spherical system

kv = verticaI permeability

n = element of domain of positive integers

p = pressure

pi = initial pressure

~D = dimensionless pressure drop

r

.

radial distance, length of radius vector of

sphere .

re =

radius of external boundary

rw =

radius of

intemal

boundary

rD = dimensionless radial distance

rD = dimensionless Sadius of external boundary

s =

Lap lac e t ra n sform param et er , a com plex

var iable

t = t im e

t r = rea d ju s t m en t t im e

t o = d im en s ion le s s t im e -.

t = m axim um t im e

U

.

m a crosc op ic ve lo cit y in p oro ,u s m e dia

w= arbit ra ry r ea l va r iable

z

.

compl ex variable

a

=

c ola t it ud e a ngle , s ph er ic al c oo rd in at es

p = viscos it y

+ = poros it y

J = cum ula t ive p re s su re d r op

ACKNOWLEDGMENTS

Grateful acknowledgment is made to A. S. Odeh

of Mobil Oil Co.s Field Research Laboratory es who

reviewed this work, critically checked the mathe-

matics and offered some valuable criticisms, The

author wishes to express his appreciation to Deno

Ladas of IBM Corp. for his help in programming the

analytic functions and to William Chichester for

his help in their computation. Thankfui acknowledg-

ment is aIso made to H. L. Smith of the U. S. Corps

of Engineers for his practical suggestions ahd

encouragement t to publish this paper.

1 .

2.

3,

4.

5.

6.

7.

8.

9.

10.

11.

12 .

1 3 .

14.

1 5 .

1 6 .

1 7 .

1 8 .

REFERENCES

Hurst, W,: Water Influx into a Reservoir and Ita

Application to the Equstion of Volumetric Balancer ~,

Twwzs., AIME (1943) Vol. 151, 57.

Hurst, W. and van Everdingen, A. F.: ~l e Application

of the Laplace Transformation to .F1ow Problems in

Reservoirs?, Trans., AfME (1949) Vol. 186, 305.

Musk~t, M.: The Flow o}

Homogeneous Fluids Through

Porows Media, J, W. Edwards, Ann Arbor (1946).

Muakat, M.: Physical Principles of Oil Production,

McGraw-Hill Book Co.-, New York, N. Y. (1949).

Chatas, A. T.: ~~APraciical Treatment Of Nonsteady -

State Flow Problems in Resewoir Systema3~, Pet.

.En& (May, June and Aug., 1953) 25,

Muakst, M.: The Performance of Bottom-Water Drive

Reaervoirs~), Trans., AIME (1947) VO1. 170, 81.

Hurst, W.: The Skin

Effect

and, Ita Impediment to

Fluid Flow irrto a Wellbore ~, Pet. Erzg. (Oct., 1953)

Vol .

25, B-6.

Eisenhart, L. P.: An Introduction to Di//erential

Geometry with tbe Use o/ the Tensor Calculus,

Princeton

U, Press, Princeton, N. J. (1947).

Churchill, R. V.:

Modem Operational Matbernatics in

Engineering, McGraw-Hill Book Co,, New York, N, Y.

(1944j, ,

Widder, D. V.: Tbe LapJace Transform, Princeton U.

Press, Princaton, N. J. (1946).

Carslaw, H, S. and -Jaegqw, J, C.: Operational Methods

in pplie Mathematics,

Dover, New York (1963). c

Bush, V:: Operational Circuit Analysis, John Wiley

& Sons, fnc., New York, N, Y. (1929).

Abrarnowits, M. end Stegun, I. A.:

Handbook oj

Matbernatical

Functions, U. S. Government Printing

Office, Washington, D. C, (1964).

Erdelyi, A. et

aL: Tables

of

Integral Tran s fo rms ,

McGrew-HiU Book Co,, New York, N,Y. (1954) Vol. 1.

Hildebranrf,

F. B,: Advanced Cafcuhs /or Engineers,

Prentice-Hall, .I I~, Inglewood .Cliffs, N. J. (1948).

Mumaghan, F. D.:

introduction to Applied

Mafbe.

tnatics, over NewYork (lg63).

Churchill,

R.

V.: lntrodriction to Complex Variabfes

and Applications, McGraw-Hill Book Co., New York,

N. Y. (1948).

Kern, G.

A.

and Kern,

T. M.: Mathematical Handbook

_,. for S~i@tists and Engineers,.

McGqiw-Wll

BocIk CO.,

.

y-=ab~eis~ti-o~ ~otivergence---

Me%Yolii;KY7[i961),

l

~ = arbitiary parameter

1 9 .

Erdslyi, A.

et dt

Higher Trcinscendetzta~ Functions,

@ =

lo n git a din a l a n gle ,

spherical coordinates

McGraw-Hill Book Co., New York, N.Y. 1 9 5 3 ) Vol. 11.

(34 =

Jacobian theta function, also denoted by

20. Cs rs la w, H . S. and Jaeger, J . C.: Corzductfon o/ Heat

,

@oord

in Solids,

Oxford U.

Prees, Oxford, E ngla n d 1 95 9).

+++

SPE-1305-PA Chatas a.T. Unsteady Spherical Flow in Petroleum Reservoirs(1)

Documents

Transcript of SPE-1305-PA Chatas a.T. Unsteady Spherical Flow in Petroleum Reservoirs(1)