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Chaos, Solitons & Fractals Vol. 6, pp. 481~185, 1995 Copyright © 1995 Elsevier Science Ltd

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Cooperative Effects due to Interpore Surface Tension in Unstable Displacement in Porous Media

RAFAELRANGEL

Departamento de Ffsica. Universidad Simon Bolivar.Apdo. 89000, Caracas 1080 A. Venezuela, and Centro de Fisica. Instituto Venezolano de Investigaciones Cientificas. Apdo 21827, Caracas 1020A. Venezuela.

Abstract. We model fluid-fluid displacement in d=2 by a diffusion limited aggregation

(DLA) algorithm which takes interpore surface tension and capillary forces into account.

The invading fluid is non-viscous. Cooperative effects are important when the ratio be-

tween capillary forces and tension forces q is ~ 1. In that case we introduce a simple

rule that considers the wetting displacement (invading fluid wets more ) and the non

wetting case ( displaced fluid wets more ). We find qualitative agreement with the

experiments of Stokes et. al. ( Phys.Rev. Lett. 57, 1718 (1986)). Furthermore, we

study the tree trunk thickness of the patterns generated as a function of the control

parameter r oc C~1, (Ca= the capillary number) and study the geometry o f the interface.

We compare the results with the experiments.

INTRODUCTION

The study of fluid displacement in porous media where one fluid displaces another fluid, involves a

number of parameters like the viscosity ratio of the fluids, the permeability of the medium, the surface tension

between the fluids and the relative ease with which the two fluids wet the medium. These parameters define the

shape of the patterns formed in experiments. Stokes, Weitz, Gollub, Dougherty, Robbin, Chaikin and Linsay

[1] investigated the wetting behavior in experiments of inmiscible displacement in the case of unstable

displacement, i.e., a fluid of very low viscosity /21 displaces a fluid of much higher viscosity P2- In their

experiments the viscosity ratio was M =122/#1 ~ 200. This means that the pressure was dropped almost

exclusively in the displaced fluid. Stokes et. al find that if the invading fluid preferentially wets the medium,

i.e., wetting displacement (WD) or imbibition, the width of the typical finger is always found to be much

larger than the pore size and to follow a scaling law that depends on the flow rate, the surface tension and the

permeability of the medium. On the other hand, if the displaced fluid preferentially wets the medium, i.e., non

wetting displacement (NWD) or drainage, these authors find that the finger width is of the order of the pore size. For the case of WD, and low capillary number Ca, (Ca=I22U/7 , 7 is the interface surface tension, U

is the velocity of the displaced fluid far away from the interface boundary(IB)). They find that a single finger develops, and with increasing C a , tip-splitting instabilities increases together with a decreasing of the mean

481

482 R. RANGEL

finger width (to). o9 scales with the permeability of the medium ~ and the with C a. They also find roughness

on length scales ranging from the mean finger width ~ to the bead size a o . They find ~ scales with Caa ,

with ct=0.51 4-0.1. The NWD corresponds closely with the experiments of Lenormand et. al.[2]. They find at

low Ca patterns which are similar to Invasion Percolation (IP) [3] aggregates, while at high C a the aggregates

can be described by the Diffusion Limited aggregation model (DLA) [4]. In fact, in a previous paper,

Fernandez et al. [5], demonstrated that for NWD a crossover length L c exists and that for length scales much

larger than L c DLA patterns are obtained, while for length scales much smaller than L c IP growth takes

place. The scaling of L c with C a was found by scaling arguments and by simulations. In this paper interpore

surface tension was not considered. We also pointed out in the previous paper [6] that in NWD the

meniscuses in the throats most probably do not touch each other, suggesting a small value of the interpore

surface tension and therefore to neglect it was justified. It is the purpose of this paper to introduce the interpore

surface tension in a model of fluid-fluid displacement in d=2 by a diffusion -limited aggregation algorithm

which considers random capillary forces at the pore level. As explained above, the interface boundary condition

between the two fluids is crucial to the nature of WD. These facts were studied in a beautiful work by Weitz,

Stokes, Ball and Kushnick [7]. They find a very particular behavior of the capillary pressure Apc(v) with a

pinning effect at low local velocity v. Apc(v) changes sign and at higher velocities and shows a non-linear

behavior. We believe the model approaches correctly the continuos equations at the pore level. The model also considers a realistic boundary condition that we expect models the behavior of Apc(v).

MODEL AND RESULTS

There are two basics ideas which define the model: Laplacian growth combined with a boundary condition

at the interface [5,6]. The Laplacian growth is realized through a DLA type algorithm which incorporates through

the interface boundary condition the pressure drop across the interface, which physically is defined by: a) the random capillary pressure drop in the throats and is represented by random numbers pc(R), where R defines a

site at the interface, and by b) ~(R)k(R), where z is the random interpore surface tension and k he curvature at

R. We use the definition of curvature on a lattice given by Femgmdez and Albarr~a [8].

We define"

p' = - p + "~kma x + A p / 2 + po (1)

where p is the pressure and satisfies V2p = 0, therefore V2p ' = 0. The definition of p' ensures that it will be

positive everywhere. Po is the pressure at the bottom of the cell and is the pressure everywhere on the displa-

cing fluid of negligible viscosity. A p / 2 is the half width of the dispersion of the p--cnumbers uniformly

distributed between - A p / 2 and + A p / 2 [5]. ~kma x is the maximum value of the product of the surface tension

and the maximum value of the local curvature on the lattice. The definition of the curvature is [8], k = -(zimzlL / a o). Here Am = +_/, when a site is added or removed from the interface and zlL = 4-1 is the

increment in the length of the interface boundary when a site is added or removed respectively. We stress that

this definition was proved to be the one which correctly agrees with scaling arguments [8].

Unstable displacement in porous media 483

Fig.1 Upper part of the figure shows a cluster for q=20, and r=32. For the lower par t , r=128, q=1.5. To

the left, there is a cluster grown with positive correlation. To the right, a cluster with negative correlation.

At the interface we have on the viscous side

p' = ( zkma x - 7:( R)k(R)) + (A p / 2 - p--~( R )) (2)

because p ( R ) = T(R)k(R)+ Pc(R)+ Po. On the non-viscous side p = Po. "r is a random function like Pc

defined on each lattice site. In this paper z is taken from a uniform distribution in the unit interval [0,1]. We

finally arrive at the following key relation of our model •

484 R. RANGEL

(3)

where Pb (R) interface and

Here

represents in the algorithm the probability a releasing a walker from the site R at the

P1 is the probability of releasing a walker from a line just above the aggregate [5,6].

_ _ A-ill2 r = r'kmax , q -- - - (4) ao(Vp) zkmax

In the limit where q >> 1, we arrive at Eq. 2 in [5,6], where Eq. 3 reduces to

Pb(R) =211_ 2Pc(R) ~ PI Ap )

(5)

where r = - - , (Vp) = Po / l (6) ao(Vp)

One can use Darcys' low v = -0¢ / Ig )Vp , for the interface velocity v and obtain r = (1¢A p) / (ao)C a ),

One has ~¢2 = a 2, and Ap = y / a o [1,7]. The parameter r in Eq. 3, control cluster growth and is similar to

the one defined in [8], where the limit r ~ 0 defines the DLA limit, corresponding to a vanishing surface

tension or infinite high driving velocity U . On the other hand, the expressions inside the brackets define

rearrangements of the interface. The parameter q measures the relative strength of the capillary forces to the

interpore surface forces. If the meniscuses at the interface are such that the invading fluid has an obtuse angle

like mercury in glass, then zkma x is expected to be very small and therefore one has the extreme non-wetting

limit with q >> 1. This is the limit studied in [5,6]. On the contrary, for q << 1, i.e., when the interpore

smfaces forces dominates over the capillary forces, one has a medium like the homogenous medium studied

in [8]. The mean finger width ~ scales with Ca-0"5. There is essentially no internal structure and basically

compact growth occurs(see results below). However, when both forces compete, i.e., q ~ 1 we have to

differentiate WD from NWD at the pore level. This distinction is of no relevance for the limiting cases

discussed above. We want also to describe pinning at low local velocities, i.e., APc(V) < 0 , because the

porous medium imbibing the invading fluid, and Apc(v) > 0 for higher velocities, i.e., an additional pressure

in needed to push the interface through the porous medium. In this case Apc(v ) acts stabilizing against the

viscous fingering instability. APc(V) varies in random way along the interface since v varies in a random

way. In this way , when Pc(R) is small, then z(R) should be small and the contrary. We say Pc(R) and

7:(R) are positive correlated (WD) . In NWD we expect that when Pc(R) is small then z(R) should be

big, and the contrary (negatives correlation). A typical result for r - - 3 2 and q =20 , and size

L 2 = 1024X1024 is shown in the upper part of Fig. 1. Here, there are no differences in growing clusters

Unstable displacement in porous media 485

wlm negative ana positive correlation as we chscussexl above, tn me lower part ot the tlgure, to the lett, we

show a cluster grown for r = 128 and q = 1.5, L 2 = 5 1 2 X 5 1 2 and positive correlation. This figure should

be compared with Fig. 1 (b) of ref. 1. There is a small amount of trapping, however, most of the cluster shows

compact growth. Tracing the interface of this cluster [6], reveals roughness at large scales with a Hurst

exponent H=0.61. At small length scales H=0.73, i.e., the two regimes D L A and IP are present in the

interface[6]. This in qualitative agreement with the experiments[l]. The mean finger width is o) =53.5.

Clusters for which q << 1 show similarities with this case. The cluster on the right ( lower part of fig. 1), has

the same values for q and r, but was grown with negative correlation. It shows a large amount of trapping

and seems similar with the one in the upper part, where q >> 1. We have measure the box dimension and

obtain for the smallest boxes D i = 1.54, while for the bigger boxes we obtain 1.82, 1.87 and 1.87. Tracing

the interfaces gives shows again the two regimes for H ( H--0.66 and H=0.80). ~ =36 for this cluster. This

figure should be compared with Fig.l(c) of [1]. The whole dependence of og(r,q) with r and q remains an

open problem. A numerical verification would involves an enormous amount of computer work. However,

we can make the following statement: for q << 1 there will be essentially a dependence of to with C -co a '

where c~ ~ 1 / 2. For q >> 1 a crossover of the thickness to the tree trunk thickness defined in [5] should take

place. We expect also a dependence with CatX, and where tx = 2 / 2 + D s , where D s = 1.3 is the fractal

dimension of the interface at small length scales (tx --0.606). Another point open is how to measure the

parameter q in experiments. We will discuss open points in another work.

Acknowledgment: We are greatly indebted to J. F. Fern~dez helpful remarks in the initial stages of this effort and his sustained

encouragement.

REFERENCES

1. J.P. Stokes, D.A. Weitz, J.P. Gollup, A. Dougherty, M.O. Robbins, P.M. Chaikin, and H. M.

Lindsay. Interracial Stability of Inmiscible Displacement in a Porous Medium. Phys. Rev. Lett. 57,

1718 (1986)

2. R. Lenormand and C. Zarcone. Invasion Percolation in an Etched Network: Measurement of the

Fractal Dimension. Phys. Rev. Lett. 54, 2226 (1985). R. Lenormand. Liquids in Porous Media.

J. Phys. Condens. Matter 2(1990) SA79.

3. D. Wilkinson. Percolation Effects in Inmiscible Displacement. Phys. Rev. A34, 1380 (1986).

4. For a Review see: L. M. Sander. Growth and Aggregation far from Equilibrium. I n " Solids

Far From Equilibrium ". Edited by C. Godreche. Cambridge University Press 1992.

5. Julio. F. Fernfmdez, Rafael Rangel and Juan Rivero. Crossover Length from Invasion Percolation

to Diffusion-Limited Aggregation in Porous Media. Phy. Rev. Lett.67, 2958 (1991).

6. Rafael Rangel and J. Rivero. Tracing Interfaces in Porous Media. Physica A191,(1992) 2535.

7. D.A. Weirs, J.P.Stokes, R.C. Ball, and H.P. Kushnick. Dynamic Capillary Pressure in Porous

Media: Origin of he Viscous-Fingering Length Scale. Phys.Rev. Lett. 59, 2967 (1987).

8. Julio F. Fem~indez and Jestas M. Albarr~n. Diffusion-Limited Aggregation with Surface Tension:

Scaling of Viscous Fingering. Phys. Rev. Lett. 64, 2133 (1990).