Correlation of Saturation Profiles in Slow Drainage in Porous Media

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Correlation of Saturation Profiles in Slow Drainage in Porous Media

C. Du, B. Xu, Y. Yortsos, M. Chaouche, N. Rakotomalala, D. Salin

To cite this version:

C. Du, B. Xu, Y. Yortsos, M. Chaouche, N. Rakotomalala, et al.. Correlation of Saturation Profiles in Slow Drainage in Porous Media. Journal de Physique I, EDP Sciences, 1996, 6 (5), pp.753-767.

�10.1051/jp1:1996240�. �jpa-00247212�

Correlation of Saturation Profiles in Slow Drainage in Porous Media

C. Du (~),

B. Xu

(I), Y-C- Yortsos (~'*),

M.

Chaouche (~),

N.

Rakotomalala (~)

and D. Satin

(~)

(~)

Department

of Chemical

Engineering

and Petroleum

Engineering Program, University

of Southern

California,

Los

Angeles,

CA

90089-1211,

USA

(~) Laboratoire

Fluides, Automatique

et

SystAmes Thermiques,

Bitiment 502,

Campus Universitaire,

91405

Orsay Cedex,

France

(Received

30

January

1995, revised 15 December 1995, accepted ²²

January1996)

PACS.05.40.+j Fluctuation

phenomena,

random _processes, and Brownian motion PACS.05.90.+m Other topics in statistical

physics

and

thermodynamics

PACS.06.30.Bp Spatial

dimensions

(e.g., position, lengths, volume, angles, displacements, including

nanometer-scale

displacements)

Abstract. We

study

the

spatial

correlation of one-dimensional

I-D)

saturation

profiles

ob-

tained

during

slow

drainage

in porous media, where

capillary

effects

predominate. Using

_prop-

erties of Invasion Percolation in _an uncorrelated

medium,

_we compute the correlation structure of the

profile.

At small values of the

lag (but

for

sufficiently large systems)

the

profile

_ap-

proaches

the structure of the record of _a

fractional

Browman motion

(mm)

with Hurst exponent H

=

(D -1) /2,

where D is the fractal dimension of the

percolation

cluster.

Sufficiently

far from

percolation,

the

profile

is

a white _noise. Noise measurements in

displacement

experiments

using

an acoustic

technique

_are

reported

for the 3-D _case. The

theory

is confirmed with 2-D simula- tions. An apparent mismatch in the 3-D _case is attributed to finite-size effects. The

approach

is

generalized

to correlated

percolation.

R4sum4. Nous _avons 6tud16,

th60riquement

et h l'aide de simulations

num6riques,

les _cor-

r6lations

spatiales

des

profils

de saturation obtenus _{au cours} de l'invasion d'un milieu poreux par une

phase

_non mouillante

(drainage).

La saturation

(concentration volumique)

de la

phase

envahissante est

moyennde

dans la direction

perpendiculaire

h l'dcoulement; cette saturation est ensuite dtudide dans des conditions ok les elfets capillaires sont dominants. Ce _processus est bien ddcrit _par la Percolation d'Invasion. Pour _un milieu _sans corr41ation, _on montre que la

structure des corr61ations approche _un mouvement Brownien fractionnaire

(fractional

Brownian

motion

(mm))

d' exposant de Hurst H

=

(D -1)/2,

ok D _est la dimension fractale de I' _amas de

percolation.

Suffisamment loin du seuil de

percolation,

la structure des corr61ations du

profil

de saturation est d6crite _{par un} bruit blanc. Des _mesures de bruits _{au cours}

d'exp6riences

de d6-

placement

1l'aide d'une

technique

acoustique ont dt6 elfectu6es. A 2-D, la th60rie est confirmde par des simulations. A 3-D, la diff6rence entre th60rie d'une part et simulations et

expdrience

d'autre part, est attribude h des effets de taille finie. Les r6sultats sont ensuite

g6n6ralis6s

_{au cas}

de la

percolation

corr416e.

(*)Author

for correspondence

(e-rnail: yortos@euclid.usc.edu)

@

Les

(ditions

_de

_Physique

₁₉₉₆

1. Introduction

The slow

displacement

of _a

wetting phase

from _{a porous} medium

by

the

injection

of _{a non-}

wetting phase,

immiscible to the former, has been

analyzed

in much

detail,

due to the relevance of the process ^to

engineering applications,

such _as oil recovery and

groundwater remediation,

but also because of its interest from a statistical

physics viewpoint.

The well-known _processes of Invasion Percolation

(IP) _[1,

_2] and of Invasion Percolation in _a Gradient

(IPG) [3,4]

have

been used to describe these processes in uncorrelated random _porous

media,

in the absence

or presence of _an external

gradient, respectively. Recently,

the

properties

^of

percolation

in correlated

media,

the correlation function of which

decays algebraically,

_e-g-

C(p)

r~~

p~~/,

where

-)

_<

il

_{< o and}

v is the correlation

length

exponent, ^{have also been} uncovered

[5].

Typically, laboratory displacement experiments

in porous media _are

performed

in geome- tries of _a uniform cross-section in the direction of

displacement

_x

(e.g.

rectilinear _or

cylindrical cores).

The main

quantity

^{of interest is the saturation of the}

non-wetting (invading) phase, Six, t),

which denotes the volumetric fraction of the pore-space

occupied by

the

non-wetting phase, transversely averaged

_over the cross-section

(yz).

This

quantity

_can be measured

using

various

diagnostic methods, including acoustics, X-rays,

etc., ⁱⁿ order to infer various char- acteristics of the

displacement.

For

example,

in the _case of

IPG,

the

slope

of the saturation

profile

_near the front

gives

_{a measure} of the

gradient applied [6-8].

For the _case of

IP,

the fluctuation of the saturation

profile

furnishes additional information _on the medium and the process. This information would be of

particular interest,

for

example,

if it indicates _{a cor-} related pore-space

[9,10].

In _an earlier

investigation using

a continuum

model,

Yortsos and

Chang _[10] pointed

out that the saturation

profile

_can reveal the characteristics of the

underly- ing

_pore structure,

provided

that the

displacement

is controlled

by capillarity

and the _process is away from

percolation.

To _our

knowledge, however,

_a

study

of the correlation _structure of

the

profile

_near

percolation

has not been

attempted.

The

objective

of this

investigation

is to

study

the structure of the fluctuation of the saturation

profile during drainage,

both _near and far from

percolation. Because,

in _our

context,

slow

drainage

is

equivalent

to

IP,

the

problem

is also

equivalent

to the

analysis

of the fluctuations of the

occupation profile

of IP. In this _{paper, we} first

develop

_a

theory

based _on the correlation

properties

^of

percolation

_processes, which is

subsequently

tested ~vith the numerical simulation

of the _process in 3-D and 2-D lattices.

Experiments

in uncorrelated _porous media _are also

conducted and

provide

a test of the

theory

in 3-D.

Theory

and simulations _are then extended to correlated

percolation problems,

where the correlation function

decays algebraically.

2.

Theory

2.1. 3-D GEOMETRIES. We consider slow

drainage (invasion percolation)

in _a rectilinear porous

medium, represented

as a 3-D cubic lattice of extent L _x L _x N

(N

_>

L) (Fig. I).

No-flow

boundary

conditions _are

applied

at the lateral

sides,

the

displacement being

forced in the direction _x.

During

the invasion process, a front

develops

at the

leading edge

of the

displacement,

the extent of which increases until it becomes

comparable

to the lateral extent L. Within this

region,

the _mean saturation varies with both

time,

t, and

position,

_x,

leading

to _a

non-stationary profile.

Of interest to this paper is the

spatial

fluctuation of the saturation in the

region following

^{the front, where the} mean saturation is

approaching

_a

steady-state,

and

the

profile

is closer to

being stationary.

Let

9(r)

denote _an indicator

function, equaling

I _or o when the site is _or is not

occupied,

w MM w

w

20

15

io

~

~ 0 5 lo t5 20 ²⁵

30

Fig.

I.

Snapshot

of Invasion Percolation in _a 3-D lattice.

respectively. Then,

the transverse saturation is

~j _9(r)

~~~~

_~ ^rEAu) _~~~

L2

From

percolation theory,

_we know that _< 9 _>r~~

L~~~,

where D is the fractal dimension of the

percolation

cluster and E is the

dimensionality

of the

system. Hence,

_we obtain the

following scaling

of the saturation

< S _>r~~

L~~~

_<

S~

_>r~~

L~l~~~l ₍₂₎

To estimate the correlation structure of this

profile,

_we consider the autocorrelation function

R(h

e<

S(x)S(x+ h)

_>. where all distances have been normalized with the _{mean pore}

length,

I. To

develop

_an

expression

for

R,

_{we use} the correlation function

g(r)

=<

91r')9(r'+ r)

>

(3)

which denotes the

probability

that _a site at _a distance _r

=

jrj

from _an

occupied

site is also

occupied (it belongs

to the _same

cluster). Then,

~ _g(r)

<

S(x)S(x

₊

h)

>" ^~

~~

(4)

the _sum

being

_over all _r that connect every

point (yo, zo)

on the

plane

_x with _every

point (y, z)

_on the

plane

_{x +} h

(Fig. 2a).

We know that _near

percolation, g(r)

behaves _as

g r~J

Crl~-~l (5)

where the

prefactor

scales _as C _r~~< 9 _>r~~

Ll~~~)

_the

_exponent

_iii

(5) gives

the

scaling

for _an IP

problem

and differs

by

_a factor of 2 from the OP exponent used in _our

previous

~ h _{_} L

y,z

L ~Y0'~0

~---~- ~---

"

' ' '

' ' '

/ ' /

a)

^{~ ~ ~~~}

1,o i,i

c b

yo>zo

d _a

b)

^{0,0 1,0}

Fig.

2. Illustrations of the notation used:

a)

3-D lattice,

b)

Partition of the _yz plane for the

evaluation of the

integral.

publication _[11]. Hence,

to evaluate the _sum in this

regime

is

equivalent

to

computing £~ r~°,

where _we have denoted _a

#

(D 3) /2 (note

that -o.235 _<

a <

o).

To

proceed,

_we

replace

the _sum

by

_an

integral

K _m

o,zo,y,z r~°dydzdyodzo ₍₆₎

and note that

r~

= p~ +

h~ (7)

where

P~ ₌ (Y Yo)~ +

(z _zo)~ [8)

Next,

_we rescale all distances

by L,

the rescaled variables

varying

in

[o,I] (Fig. 2b),

and

proceed by keeping

the

previous

notation for the sake of convenience.

Then,

the

integral

ⁱⁿ

(6)

reads

K =

L~~+~ / / / /

(p~

+

e2)"dydzdyodzo

₊

L~°+~I(e) (9)

yo,zo,y,z

where _we introduced the

variable,

_e

= h

IL.

To evaluate the

quadruple integral

I we

partition

the

plane

_{yz as} shown in

Figure

2. After _some

algebra,

I reduces to

I(E)

= 4

/ / _{(1- v)(i z)(v~}

+ z~ ₊

e~)"dydz (lo)

~ ~

Before _we

proceed

with the numerical evaluation of

(lo),

_we will evaluate its

asymptotic

be- havior in the limit of small _e. For this _we note that I consists of the _sum of the

following

three

integrals

~

Ii (e)

₌ ⁴

/ /

(y~

+

z~

₊

e~)°dydz (ii)

o o

12(e)

₌ -4

/ /

(y~

+ z~ ₊

e~)°dy~dz (12)

~ ~

13(e)

=

/ / ~ ~ (y~

+

z~

₊

e~)"dy~dz~ (13)

the

leading

behavior of which in the limit of small _{e can} be

readily

obtained. After _some

calculations,

_we find

~~~~ ~~~~

(° )

i) ^~~~~~

^~

^~~~~~

^~~~~

I~(e)

₌

i~(o)

₊

o(e2a+3) (is)

ere

we

1

_~2a+2

where

~ ~

I(o)

= 4

(1 y)(I z)(y~

+

z~)°dydz (18)

which _can be also

expressed

in terms of

incomplete

Beta

functions,

if necessary. We note

(by converting

in radial coordinates and

making

_use of the result 2a +1 >

-1)

that the

integral

in

(18)

exists. Back substitution in

equations (4)

and

(9) gives

^the

asymptotic

result

<

S(x)S(x

+

h)

>r~~

CL~°I(o) (I

^b

~)) ^~~~)

^{+ O}

() ~l(19)

where _we defined the

positive

constant

~r

j

=

(a

₊

i) _I(o~ ⁽²⁰⁾

Equation (19)

is

consistent,

in the limit h _-

o,

with the result _<

S~

_>r~~

L~l~~~l

of

(2), given

the

scaling

with L of the

prefactor

C.

Typically,

_one is interested in the

semi-variogram, SV,

which in _terms of the

original

defini-

~~°" ~~~~~

sv +<

(s(x

+

h) s(x))2

_>m~ i

j ₍₂₁₎

Its

asymptotic

behavior in the small _e limit _can be obtained from

(19).

We find

<

(S(x

+

h) S(x))~

>r~~ 2b _<

S~

_> ^~

(22)

L

~~

0-'

~ _- -4

a) ~°~ _~~~~~

-1

d3

-2.5 -2 -1 .5 -1 -0.5 0

b)

^{~°~ ~~~~~}

Fig.

3. Numerical evaluation of the

semi-variogram plotted

_vs the dimensionless

lag

h

IL

for inva- sion

percolation: a)

3-D

geometries, b)

2-D geometries. ^The dashed lines _are the asymptotic

theory predictions

with

slope

2a + 2 ₌ 1.53 for 3D and 2a +1= 0.896 for

2D, respectively.

This behavior is similar to the

(self-affine)

trace of _a fractional Brownian motion with Hurst

exponent [12]

H =

~

~

(23)

which in the

present percolation problem (with

_or without

trapping)

reads H m 0.765. This exponent is different from the _one

reported

ⁱⁿ

[11],

^where a different

scaling

of the correlation function _was used.

The numerical evaluation of

(21)

is

plotted

in

Figure

3a. The

asymptotic

behavior at small

e is consistent with

(22), provided

that the dimensionless

lag

is smaller than

approximately

0.01. The rather small range of the

validity

of

(22)

must be taken into consideration in the

interpretation

of the numerical simulations

reported

below.

The above is

applicable

_near

percolation.

For _a system away from

percolation,

the

pair-

connectedness function takes the

dependence,

_g

r~~

exp(-r If).

where the correlation

length (

is assumed much smaller than L. We _can calculate the correlation function

by proceeding

_as

before and write

1(e, b)

= 4

j j _{ii y)ji z)}

exp

(-j(y2

^{+ z2 +}

^{2)1/2j dydz 124)}

~ ~

To derive the

asymptotic

behavior of I in the limit b _e

)

~ 0 we

apply

tvice the

Laplace

method for the

asymptotic

evaluation of

integrals [13]

to obtain

1(E b)

₌

4(27rb)~/~ (~(i

^{~/)(~/~ +}

^E~)~/~

exP

-1(~/~

+

~)~/2)

dv

=

87rbexP 1- II (25)

The final result reads

<

S(x)S(x

+

h)

>r~~<

S~

_>

exp(-

^~

(26) f

which _can also be rewritten _as

<

(S(x

₊

h) S(x))~

_>r~J 2 <

S~

_>

I

exp(- ()j ⁽²⁷⁾

This expresses the

semi-variogram

of _a

signal

correlated _over distances of order

(.

For suffi-

ciently large h, therefore,

the

profile

has the correlation structure of _a white

noise,

_as

expected.

2.2. 2-D GEOMETRIES. A similar

approach

_can be

applied

for 2-D

geometries. Consider, first,

the behavior _near

percolation. Now,

the calculation of the

integrals

is

easier;

for

example,

the

integral corresponding

to

(10)

reads

1(E)

" 2

/

(1

Y)(Y~ +

f~)°dy ₍₂₈₎

where _a is defined _as

before,

but with E

= 2. Before the full numerical

evaluation,

_we consider the

leading

behavior of

(28) by writing

I(f)

-

~~a

+

/~a

+

i~

^{+ 2}

~ (i

_{Y) i(Y~ +}

f~)° _Y~°i dY (29)

and

taking

the small _e limit. The result is

~~~~

(20

+

~(a

+

1)

~~~~~~~

_~

~~~~

~~~~

In the

above,

_we took into consideration that 2a +1 < 1, and introduced the

positive

constant

a =

/ _(v~°

(v~

+

i)a) dy (31)

Since _a <

0,

the

integral

converges.

Thus,

_one arrives at the

following expression

^for the autocorrelation function

<

S(x)S(x

+

h)

>r~~<

S~

_> I _c

~

l+ ^{O(e) (32)}

L

~~~~

where _we introduced the

positive

constant

c =

2a(20

₊

1)(a

₊

1) (33)

Equivalently,

we may write

<

(s(x

+

h) s(x))2

_>r~~ 2c _<

s2

_>

((

^~~

₍₃₄₎

where the Hurst

exponent

is _now, H

=

(D

E

+1) /2

₌

(D -1) /2. Hence,

we

expect

^H m 0.4I for IP with

trapping

and H _m 0.448 for

Ordinary

Percolation

(OP)

_or for IP without

trapping.

As in the 3-D _case, this result differs from Du et al.

ill].

Figure

^3b shows _a

plot

of the SV for the 2-D _case. The

asymptotic dependence (34)

is well satisfied for values of the dimensionless

lag

smaller than

approximately 0.I,

which is _a factor of10

larger

than in the 3-D _case. This behavior should also be considered when

interpreting

the simulation results below.

Away

from

percolation,

_a similar

approach applies.

We consider _an

exponential pair-

connectedness function _as before.

Then,

the behavior of the

integral

1(e, b)

₌ ²

~ (1 y)

_exp

(-b~~(y~

⁺

e~)~/~j dy (35)

is

sought

in the limit b

=

( IL

_« I. Its evaluation is

straightforward using Laplace's

method

[13]

leading

to the result

I(e, b)

₌

2(2~rbe)~/~

_exp

(- _{)) (36)}

The end result is identical to the

previous equation (27).

2.3. CORRELATED PERCOLATION. The

previous approach

_can be

generalized

to correlated

percolation

_[5]. We will restrict _our attention to

percolation

in lattices with _a correlation function of

algebraic decay, C(p)

r~~

p~~,

where

-)

_<

^k

_<

0,

and _v is the correlation

length exponent

^of uncorrelated

percolation.

In such _cases, results _are available for the

ordinary

_per-

colation

problem,

but not for the

corresponding

invasion

percolation problem.

The correlation function still scales _as in

(5),

where the fractm dimension is obtained from D

= E

(,

_where

)

_{and P}

are

percolation

and correlation

length exponents, respectively,

of the correlated _per- colation

problem.

Isichenko _[5] has shown that

exponent j

_{is identical to that of} uncorrelated

percolation

in the

corresponding dimension, j

=

fl,

but that exponent P is not, if

)

_<

^fl

_{< 0,}

where it is

given by

P

=

). Nonetheless,

the

previous approach

is still

applicable. Proceeding

as

above,

_we find that the exponent ^H

describing

the

early

_part of the SV for this

problem

is

H =

~~

+ l

(37)

in 3-D and

H

=

~~

+

l/2 (38)

in

2-D, respectively. Therefore,

in the correlated

percolation

_case, H varies in the interval

[0.765, lj

in 3-D

geometries,

and in the interval

[0.41, 0.5j

^{in 2-D}

geometries.

This

suggests

^the

possibility

that at least in the

appropriate

_range, the trends of the 3-D

profile (where

H >

0.5)

are

persistent,

while those of the 2-D

profile (where

H _<

0.5)

_are

anti-persistent [2j.

^This conclusion _was also reached in

[I ii, although

with different values of the

exponent. Away

from

percolation,

_we

expect

the correlation structure of the

underlying

lattice to emerge.

3. Simulations

To test the

theory,

numerical simulations of

ordinary

and invasion

percolation,

in

2-D,

_un-

correlated _or

correlated,

lattices and in 3-D uncorrelated lattices _were conducted. For the case of correlated

percolation,

the correlated lattice _was created

using

the method of

Midpoint

Displacement

and S~ccessive Random Additions

[14].

^{The various results} are summarized in

Table I. Hurst

exponent (H ) for

various _{processes near}

percolation

Process

Dimension/Type ^Fieldk

Latticesize Simulation

Theory

random 100 _x 100 _x 500

l10 _x l10 _x 500 0.55

site random _{x x}

300 _x

site _x

2D _x

IP 300 _x 600 0.30

2D random _x

500 _x 0.410

2D

/

site

(no trap) k

= 0.0 400 _x 800 0.51 fBm 0.500 fBm

2D

/

site

(no trap) k

= -0.2 400 _x 800 0.47 fBm 0.486 fBm

2D

/

site

(no trap)

H

= -0.5 400 _x 800 0.40 fBm 0.465 fBm

300 x 300 x 300 0.57 fBm 0.765 fBm

x x

2D

/

site

k

= 0.0 512 _x 512 0.54 fBm 0.500 fBm

OP 2D

/

site

k

= -0.2 512 _x 512 0.48 fBm 0.486 fBm

2D

/

site

k

= -0.2 1024 x 1024 0.49 fBm 0.486 fBm

2D

/

site H

= -0.5 512 _x 512 0.45 fBm 0.465 fBm

2D

/

site

k

= -0.8 512 _x 512 0.40 fBm 0.448 fBm

Table II. Hurst

exponent (H ) for

_{vartous processes}

far from percolation

Process Dimension Field

k

_{Lattice Size < S > Simulation}

_Theory

3D

lo _x 10 _x 500

~~~d°m x x

4° _x 40 _x 500

°.50 fGn

x

IP x

2D random 300 x 600 0.5 fGn

50 _{x x} 500

x x 0.49

~

o_50 fGn

random 9°° ~~

°'~°

Tables I and II.

Typical

saturation

profiles

_near and far from

percolation

for the 3-D IP _case

are shown in

Figures 4top

and

4bottom, respectively.

Because the above

theory

is based _on the

assumption

of _a

well-developed percolation

state, in all IP

problems

_we

analyzed

the

part

0 50 100 150 200 250 300 350 400 450 500 Position X

50 loo 150 200 250 300 350 400 450 500

Position X

Fig.

4. Saturation profiles ^from Invasion Percolation in _a 3-D lattice:

(top)

at

breakthrough

of the

invading phase, (bottom)

at conditions _away from

breakthrough.

of the

profile

_away from the

invading

front and from the inlet to minimize finite-size effects.

In _an

analogous study _[8],

_a

length-to-width

ratio of two _was

used,

and the middle

part

of the

profile

_was

analysed.

The fluctuations _were

analyzed

in various ways,

by constructing

the

spectral density,

the

semi-variogram

or the

R/S

_curve

[2,9]. However, consistently

reliable results _were obtained

only

from the

semi-variogram,

the other _two indicators often

suffering

from

large

^local correlations.

Figure

5 shows the

plot

of the

numerically

determined exponent

Hn~m

vs its theoretical value for _a

profile

_near

percolation.

The _raw data used _are described in detail in Table I. The data

reported

are ensemble _{averages over} 10 realizations

(in

the standard

case).

It is apparent that

theory

and simulation agree

reasonably

well for the 2-D _case, the

agreement being

better for the OP _case

(where

the

profile

of the

largest

cluster at

percolation

_was

analysed).

In

general.

however,

the numerical results

underpredict

the theoretical value. In

3-D, only

random lattices

were used. For this case, there is _a noticeable difference between theoretical and numerical

predictions.

We attribute this mismatch to finite-size

effects,

which appear ^to diminish

slowly

with increased size

(compare

the first three entries in Tab.

I).

Finite-size effects _are also

present

in

2-D,

the trend

being

in the

right direction,

_as the size increases

(Tab. I).

The _reason for the difference between theoretical and numerical results becomes clear when

we compare the

corresponding semi-variograms. Figure

6 shows

typical plots

^of the SV of the numerical simulation results in 3-D and 2-D

geometries.

Because of

computational

lim-

itations, the 3-D simulations _are restricted to

generally

small

sizes,

hence the

corresponding

dimensionless

lag,

h

IL,

is

typically larger

than 0.01. In that range,

however,

^the behavior of the theoretical SI~ does not

strictly

follow

(22),

_as shown in

Figure

3a. As _a

result, attempting

~ ^x

~ +

fl

xx

x

0 O-1 0.2 03 04 0.5 06 07 0.8 09

H

Fig.

5.

Comparison

of numerical

(Hnum)

and theoretical

(H)

values of the Hurst exponent ^for an

mm

signal

for _a system _near

percolation ((x)

denotes 2-D

results, (+)

denotes 3-D

results).

to

interpret

these simulations with _a

power-law scaling

leads to a value smaller than

expected,

as indeed shown in

Figure

5. On the other

hand,

in 2-D

geometries

the lattice sizes _are

larger (Fig. 6b),

and the dimensionless

lag

is within the

region

where the

asymptotic

behavior

(34)

holds

(compare Fig.

3b and

Fig. 6b).

A closer

agreement

^{is thus}

obtained,

_as shown in

Figure

5.

Table II shows results

corresponding

to a

system

_away ^from

percolation.

A

generally good agreement

_was found between

theory

and

simulation, although

results _are better for OP rather than IP. The transition from fBm to fGn

type

noise _{as we} move away from

percolation,

is

clearly

observed in all _cases studied

(compare

Tab. I and

II). However,

the rate of

approach

to an fGn

signal

_as the

percolation probability

increases was found to be much

slower, namely

to

occur at

higher

_mean saturation

values,

in IP

compared

to OP processes.

4.

Experiments

Slow

drainage experiments

were next conducted. To allow for the simultaneous determination of the correlation structure near and far from the

percolation regimes,

the

experiments

were

conducted in the _presence of _a small amount of

gravity, corresponding

to Invasion Percolation in _a Gradient. In

previous

studies

[4-8],

it _was shown that IPG shares _many of the features of Gradient

Percolation, namely

it contains _a

leading

front of width _aft

r~~

Bj"/l~~"I,

where the

properties

of IP _are

exhibited,

followed

by

_a

compact

pattern. The correlation structure is

expected

to

correspond

to IP at

percolation conditions,

near the

front,

and to IP far from

percolation,

upstream from the

front,

thus _we

anticipate

the two

previous scalings

to emerge

near and far from the

front, respectively.

In the

experiments,

a

lighter density non-wetting

fluid

(nonane)

_was

injected

at very low

rates

(with

the

capillary

number, Ca

=

qp/~i, equal

to

10~~,

where _q denotes the flow

velocity,

p the

viscosity

^and _~i ^the interfacial tension between the

fluids)

to

displace

_a heavier

wetting

-1 .5

~.5

_{-2 -1.5 -1 -0.5 0}

a)

^~°~~~~~~

> 0

$

G-1

~~3

_{-2.5 -2 -1 .5 -1 -0.5 0}

log (h/L) b)

Fig.

6.

Typical semi-variograms

of saturation

profiles

from numerical simulations:

a)

3-D Invasion Percolation in _a lattice 200 _X 200 _X 400,

b)

2-D Invasion Percolation in

a lattice 1000 X 2000. Solid lines denote fitted

straight

lines with

slope

1.19 for 3D and 0.896 for 2D

respectively,

where slope ₌ 2H.

fluid

(water)

in the direction from

top-to-bottom.

The porous medium consisted of

glass

§eads

of diameter _a

= 100 _pm

(permeability

5

Darcy) packed

in _a rectilinear

geometry

^of dimensions 4 x 2 _x 30

cm~ corresponding

to a 400 _x 200 _x 3000 network.

Hence,

the effective size for the

analysis

of the

experimental

correlations is about 200. This should be taken into consideration in the

interpretation

of the

experiments.

Care _was taken

during packing

to _ensure

good mixing

^of the beads in order to avoid

long-range

correlations. The

corresponding gravity

Bond

number, Bg

=

~~~~~,

_where

_Ap

_{is the}

_density

_{difference and}

g the acceleration of 'f

gravity,

_was estimated _at

Bg

₌

^10~~

Saturation data _were obtained

by

_an acoustic

technique

described elsewhere

[15],

^which allows for _a

spatial

resolution of1 _mm with _an accuracy of

10~3

_in saturation. The far from

percolation regime

is ensured

by

_a

capillary barrier,

which

allows _a

large

increase in the saturation up ^to S

= 0.60.

Typical

saturations

profiles corresponding

to the two

regimes

are shown in

Figure

7 with the

corresponding

_power spectra ^shown in

Figure

8.

Here,

_we relied _on

spectral analysis

to

determine the

scaling

of the correlation of the fluctuations. We recall that the _power

spectral density

^{function scales} as [16]

S(f)

r~~

f~~~~~

for _an fBm

signal

and _as

S(f)

r~~

f~~~~

for _an fGn

signal.

The data away from the threshold

(open

circles in

Fig. 8)

show _a

slope

close to

nearly

_zero

corresponding

to uncorrelated

noise,

in

agreement

with the

theory.

These results

0.8

0.6 S 0.4

0.2

0 0.2 0.4 0.6 0.8

x~

Fig.

7T- Saturation

profile

from

displacement experiment:

bottom _curve at

breakthrough

of the

invading

fluid; top curve away from

breakthrough.

",,

~,~~

~

^',

i~ ",

~ ',

4J .,.

't3 o

- o

t4 '<,

b

° ° Q

jIn°~

Q

4~ '.

+ ~

o.oi o.i

wave vector

Fig.

8.

Log-Log plot

of the

spectral density

of the saturation _versus the

spatial

wave-vector. Full squares

correspond

to data in the

vicinity

of the

percolation threshold,

_open circles

correspond

to the

regime

_away from

percolation.

The dashed line of slope -2

corresponds

to H _m 0.5 mm. The

horizontal line corresponds to H _m 0.5 fGn.

tend to confirm the uncorrelated _structure of the

underlying glass

bead

pack.

On the other

hand,

_near

percolation,

the

logs log f

data

(squares

in

Fig. 8)

can be fitted with _a

straight

line of

slope

_m -2

(dashed line) corresponding

to an fBm

signal

with H _m .5. This is in

contrast to the 3-D theoretical

prediction

of H _m .765

(although

in better

agreement

^with

the

simulations).

As in the latter _{case, we} attribute this difference to finite-size effects. As _a result of the small size of the

experimental

_cross

section,

the dimensionless

lag

in the

analysis

of the

experiments

varies in _a range

exceeding 1/200,

which is not

sufficiently

^small to allow for _a critical test of the

asymptotic prediction (compare

with

Fig. 3a). Thus, interpreting

the available data with _a power law leads to an

underprediction

of the

exponent,

^{much like} in the numerical simulation _case.

Experiments

in correlated media _were not conducted due to lack of

a model _porous medium of _a controlled correlation structure.

Progress

in this direction would be

desirable,

_{as many} natural _porous media _are correlated at various scales.

5. Conclusions

In this _{paper, a}

theory

_was

developed

that

predicts

the correlation structure of

transversely averaged

saturation

profiles during

slow

drainage,

where

percolation

_{processes are} involved.

The

theory

indicates that _near

percolation,

the

asymptotic

behavior of the

semi-variogram

at small values of the dimensionless

lag

has the

general

characteristics of the record of _an fBm

signal,

^with trends that _are

persistent

for the _case of 3-D

geometry

^and

anti-persistent

for the

2-D _case,

regardless

of the correlation of the

underlying

lattice. The

theory

_{_was} confirmed

with numerical simulation in 2-D lattices. Simulations and

experiments

in 3-D

lattices,

how- ever,

underpredicted

the

asymptotic exponent.

This difference _was attributed to the small network sizes

used,

which result in dimensionless

lags

that _are not

sufficiently

small for the true

asymptotic

behavior to

develop.

Assuming

that

experiments

in

larger

_{systems can} be

conducted,

the results should be useful to the identification of the correlation structure of the pore-space in porous

media,

based _on saturation measurements under conditions of slow

drainage,

where

capillary

effects

predom-

inate. The results _near

percolation

in this paper

complement

those of reference [10] ^{in the} continuum

regime,

far from

percolation.

Acknowledgments

The work of

C-D-,

B-X- and Y-C-Y- _was

partly supported by

DOE Contract No. DE-FG22- 93BC14899. The collaboration between the two groups was facilitated

by

NATO Grant No.

CRG 901053 and

by

_a limited

appointment

^{of Y-C-Y-} as Associate Researcher of the CNRS.

All these _sources of

support

are

gratefully acknowledged.

We would also like to thank J.-F.

Gouyet

for

pointing

out _an

important

ommission in the

original

draft of the

manuscript.

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