Effect of Finite Grain Size on the Simulation of Fluid Flow in Porous Media

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Effect of Finite Grain Size on the Simulation of Fluid Flow in Porous Media

G. Kohring

To cite this version:

G. Kohring. Effect of Finite Grain Size on the Simulation of Fluid Flow in Porous Media. Journal de

Physique II, EDP Sciences, 1991, 1 (2), pp.87-90. �10.1051/jp2:1991148�. �jpa-00247511�

J 1fi~s. ^II ~1

(1991)

87-90

FtvRIER1991,

PAGE 87

Classification

1fi~sksAbstnwts

05.50~7.55M

Shom Communication

Effect of Finite Grain Size

_on

the Simulation of Fluid Flow in Porousmedia

GAKohring

Institut far Theoretische

Physik,

Universit3t _zu

Knin, 2illpicherstra6e 77,

D15~K©

KoIn,

ERG

(Received13 Lkcember199« accepted19 Lkcember1990)

Abstract. Effects caused

by

the

necessarily

finite grain size used in simulations of flew m porous media are

systematically

Studied _in Dvo dimensional

hydrodynamic

cellular automata vnth systems of up to 88 million sites, The

permeability

^of the media,

K,

18 found _to be a function of the grain size, R, and an

extrapolatiqn

to

physically

realistic grain sizes is given,

The last few years ^{have seen renewed interest} in the

problem

of fluid flow in

porous mjdia

[1

5J.

^{Thin is due in}

part

to the

importance

attached to various

physical applications,

_e,g., esti-

mating

the

properties

of oil and gas

flowing through

_porous rocks in

underground reservpirs;

and in

part

to

thp _emergence

of _{a new} numerical

technique, namely,

the

lattice-gas (or hydrodynamic)

cellular automata

[6].

^{As noted}

by

many

researchers,

the

lattice-gas

cellular automata

approach

is

particularly

well suited for the

problem

of flow in porous ^{media because of the} ease at which

complicated _pore geometries

can be introduced _~[3].

Previous work in this field has concentrated _on

proof-of-principle

^studies

(e,g., demonsjrating

the emergence ^of

Darcy's

law at low flow velocities

[1, 2])

^and on

qualitative

calculations of the

permeability [4, 5].

^{Given the}

relatively

low accuracy of the

lattice-gas technique

on

present

com-

puters,

^{there is}

justification

in not

pressing

such calculations any ^further

[7j. However,

a moment's consideration leads _one to the conclusion that

systematic

_errors in these calculations _are not

yet

under control.

For _a real fluid

flowing

in _a porous

medium,

the size of the individual

grains,

which at fixed

porosity

is related to the size of the rock pores, are several orders of

magnitude larger

than the

mean free

path

of _a

jypical

^{fluid molecule.}

By

contrast, ^the

lattice-gas

simulations

typically

use

grain

sizes

which

are of the same order of

magnitude

as the

particle

mean free

path.

In

principle,

this _can lead to

spurious, unphysical

effects since _true

hydrodynamic

flow

through

the pores, ^does not have _a chhnce _to

develop-

in such small confines

[11].

^This paper ^addresses this

problem,

which

was first

pointed

out

by

Rothman

[2], by systematically examining

the effect of different

grain

^sizes

at fixed

porosity.

The lattice gas ^{model used in the}

present

simulations is _a variant of the six bit FHP model _on a

triangular

lattice

[6].

^{An efficient}

implementa,tion

of this model for scalar and vector

computers

88 JOiJRNALDEPHYSIQUEII N°2

has been discussed in _a

previous publication _[5].

For

studying

the flow

properties media,

obstacles are

placed randomly

in the center half of _a

pipe

of

length

L and the

particles

flow from left to

right. (Such

a simulation mimics the way in which

experiments

are

performed.)

In order to insure that the flow has the

property expected

^of a very

long pipe,

the first column of the lattice is con-

stantly replaced by particles having

a

velocity

distrAution

corresponding

to that of

fully developed

Poiseuule flow

[4].

^The

particle density

in all the simulations

presented

here was, d _m 0.15

parti-

cles

per

^{link at the}

inlet,

or p ^m 0.9

particles _per

site. The maximum

particle velocity

at the inlet is about 0,15 sites per ^unit

time, I-e-,

small

enough

for nonlinear

velocity

effects to be

neglected.

These

parameters yield

a mean free

path

of several lattice sites in the free fluid _case

[2].

The

shape

of the obstacles are, for

convenience,

chosen to be rhombi with sides of

length, R,

but the exact

shape

does not appear ^to make any difference in the results _[8~

4, _5J.

In

particular

for _our lattice structure, circular obstacles would have

quite rough boundaries,

should therefore be avoided. lvhen

placing

the rhombi _on the lattice

they

_were allowed to

overlap, creating

_very

irregular

structures. After the rhombi _are all in

place,

the

porosity

^of the medium is calculated

as the ratio of

unoccupied

^sites to the total number of sites in the middle half of the

pipe.

^For a

random medium this affords _a

good

definition of

porosity, #.

In the first

part

^{of the} simulations the size of the rhombi varied while the

porosity

was held fixed.

Figure

I shows

typical

results for the variation of the reduced

permeability

as a function of

grain

size at constant

porosity,

in this _case

#

₌ 0 8 IS + 0 0 IS. The

figure

shows data for

system sizes,

2048 _x

640,

4096 _x

1344,

8192 _x 2688 an16384 _x 5376. The small

systems

were run on

the HLRZ'S

Cray-YMP/832 using

the

single _{pro~essor program}

dhcussed

previously.

The

largest system

size

required

I.4

Giga-bits,

or most of the machines shared memory, ^and was therefore

run in the

autotasking

mode

using _up

to 8

processors.

^{In this}

mode,

the

program

^achieves a

peak speed

of _over 1 000 million site

updates

per ^second.

The

permeability

of the

sample,

_~, was determined

by dividing

^the average

velocity

over the last

I/4

of the

pipe by

the

pressure

^difference across the

sample.

^{Note that there} are no clear indications of

systematic

effects due to the finite

system sizes,

but there is reasonable evidence for

believing

that the

permeability

scales

linearly

with

grain

size _{over a}

large

range ^{of sizes.}

Only

for the

largest grain

sizes do _{we see} indications of the

theoretically expected

_{Jc «}

^R~

behavior ₁₁

0, 12].

This effect may ^{be due to the}

development

of Poiseuille flow inside the

larger

_pores

ill].

Figure

I also

provides

evidence that it may ^be

posskle

to

extrapolate

to the

physically

inter-

esting region

^of

large _pore

size

by scaling

~

/R

as a linear function of

I/R.

The data indicate that this linear

extrapolation

should be carried out to the

regime

of I

/R

~

10~~

where _{a cross over} behavior appears ^{to set}

in,

and

~/R~

- coast. Note the results obtained for R ₌

I,

where the

system

^is

just

_a collectibn of

point

obstacles _{or a}

"dusty"

medium.

Although

such dust has been used in other work

[1, 9],

the

present

^{data indicate that} the dust

regime

is far from the

physical regime.

Figure

2

presents

a

plot

of the

extrapolated permeabilities

_{as a} function of the

porosity using

a linear I

/R extrapolation

of the data.

(Since only

for the

porosity

of

figure

I did _we have

enough computer

^time to simulate lattices

large enough

to see a

crossover.)

From this

figure

it _can be _seen hat _a

good

fit to the reduced

permeability

is

given by:

_~

/R

«

e6

5 *

/(1 #).

Such _a functional form

appears

^{to hold over a}

large

_range of

porosities, although

we avoid here the

complications

which arise when

studying porosities

_near the

percolation

threshold. The functional form of the

permeability

^{obtained here} in the limit of

large grain

sizes is not inconsistent with that of

previous

work

[3, 4],

^{but the}

quantitative

results we obtain _are very ^{differenL These} new data should be closer to real

experimental

numbers and it would be of interest to see how well the lattice gas

simulations

compare

to

experiments

for this

problem.

In summary, we have studied the effects which _are due _to the finite

grain

size used in lattice gas simulations of flow in porous

media,

and we find evidence for _{a crossover} behavior _as the

grain

size

N°2 SIMULA3TONOFFLIJIDFLOWINPOROUSMEDIA 89

10~ "

Ol 10°

Ct ~

~ 53 ,

O X o

[

10~~

o

~ CP

10'~

lo' ~ 10~~ 10~~ 10°

~-i

Fig,

I. Plot of the reduced

permeability, ~/R~ _wnusl/R

_{at porosity, #}

= 0 815 + o 015. The open

squares are for 2048 _x 640, the solid circles for 4096 x 1344, the _crosses for 8192 _x 2688 and the open

tnangle

for 16384 x 5376. Jlecause of the

crowding

of the data

only

a

single typical

_error bar is shown. The

units of

~/R~

_{are: unit time/ unit mass.}

lo'

cr~'

~ s-

1

§2

Fig.

^~

Semi-log plot

of the scaled and

extrapolated

^reduced

permeability (in

un lattice constant x unit time

/

unit

mass),

(1

#) _(~ /R),

_wnus #. The solid line _{is a} least squares ^{fit to the data.}

increases.

Figure

I has shown

however,

that it is still

possible

to extract

meaningful

information about the limit of

physical grain

^sizes

by

a

proper

^{finite size}

scaling analysh

of

systems having

small

grains.

For future

work,

it would be very

interesting

to carry ^{out these} calculations _near the

percolation

threshold and for _a fluid

moving

at nonlinear velocities.

9o

'JOURNAL

DE PHYSIQUE II ^' N°2

Acknowledgements.

It b _a

pleasure

to thank D. Stouffer and S. Zaleski for useful discussions. A

grant

^{bom tile BMFT}

(# 0326657D)

^for

partial _support

of tllh

project

is

gratefully acknowledged

as well _{as a}

grant

^of

about 100 hours of

computer

^time on tile HLRZ'S

Cray-YMP/832

at the EEA Julich.

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