Cooperative effects due to interpore surface tension in unstable displacement in porous media

Chaos, Solitons & Fractals Vol. 6, pp. 481~185, 1995

Pergamon 0960-0779(94)00294-0

Copyright © 1995 ElsevierScience Ltd Printed in Great Britain. All rights reserved 0960-0779/95 $9.50 + .00

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 between 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 Ca , 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 Ca. They also find roughness on length scales ranging from the mean finger width ~ to the bead size ao . They find ~ scales with C a a , 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 Ca 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 Lc IP

growth takes

place. The scaling of L c with Ca 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 + "~kmax + A p / 2 + po

(1)

where p is the pressure and satisfies V 2 p = 0, therefore V 2 p ' = 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 displacing 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]. ~kmax 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].

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483

Fig.1 Upper part of the figure shows a cluster for q=20, and r=32. For the lower p a r 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 •

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R. RANGEL

(3)

where Pb (R) represents in the algorithm the probability a releasing a walker from the site R at the interface and P1 is the probability of releasing a walker from a line just above the aggregate [5,6]. Here r = r'kmax _

_

q -- -A-ill2 -

,

ao(Vp)

(4)

zkmax

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

Pb(R) = 2 1 1 _ 2Pc(R) ~ PI Ap ) where

r =- -

,

(Vp) = Po / l

(5)

(6)

ao(Vp)

One can use Darcys' low v = - 0 ¢ / I g ) V p , for the interface velocity v and obtain r = (1¢Ap) / (ao)C a ), One has ~¢2 = a2, and A p = 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 D L A 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 zkmax 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 N W D 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 w a y , when Pc(R) is small, then z(R) should be small and the contrary. We say Pc(R) and

7:(R) are positive correlated ( W D ) . 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 L2 = 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 I P 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 a-co ' 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. 7.

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