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Physica A 191 (1992) 253-257 North-Holland
Tracing interfaces
in porous media
Rafael Rangel’ and J. River0 Centro
de Fisica, IVIC,
Apdo.
21827, Caracas
1020A,
Venezuela
The geometry of the interface of clusters growing under both pressure gradients and capillary forces in porous media is mapped into a single value function by tracing the surface of the aggregate and recording the Y coordinate of the position of a walker moving along the perimeter of the clusters as a function of the arc length 1. We find a crossover behavior in the Hurst exponent of the self-afline function Y(I). For small scales, the Hurst exponent corresponds to invasion percolation with trapping (IPT) (0.73); for larger scales to diffusionlimited aggregation (DLA) (0.60). This is consistent with a previously found crossover length L, from IP to DLA (Phys. Rev. Lett. 67 (1991) 2958).
1. Introduction
Recently there has been an enormous interest in the dynamics and geometry of surfaces defined by interfaces in fluid-fluid displacement in porous media [l]. Recent experiments of processes which are far from equilibrium are reproduced by two types of computer simulations of these processes, namely diffusion-limited aggregation (DLA) and invasion percolation with trapping (IPT) models. Consider a nonviscous fluid injected into a porous medium which is occupied by a viscous fluid. Both fluids are incompressible. It is found that, other parameters being equal, varying the injection rate produces two very different types of flow regimes in the fluid. One regime obtains at high injection rates, for which the DLA model is appropriate. In this model, the interface is assumed to advance with a velocity u given by Darcy’s law, u mVp, where u and p are respectively the velocity and the pressure. The incompressibility condition V- u = 0 then implies that the pressure satisfies Laplace’s equation. The DLA model is therefore governed by viscous forces, and capillary and surfacetension effects are neglected. Laplace’s equation is modelled by random walks in the defending (viscous) fluid, the pressure being constant in the invading fluid. The regime is called the open branching regime, due to the absence of encircled pockets of the defending fluid. 1 Tel: (58-2)5011312, Fax: (58-2)5011148, E-mail:
[email protected]. 037%4371/92/$05.00 0
1992 - Elsevier Science Publishers B.V. All rights reserved
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At low injection rates, in contrast, the flow is dominated by capillary forces; viscous forces are neglected. This regime is modelled by IPT. Here, the interface is advanced wherever the capillary pressure (modelled by a grid of random numbers) is largest. Trapping (pockets of encirclement of the defending fluid) may, and in general does, occur. A previous work [3] established the existence of a crossover length L, between the IP and open branching regimes, i.e. from small-scale capillaryforce controlled growth to large-scale viscous fingering in porous media. The model considered describes the case of nonwetting invasion [4,5] where the meniscuses in the throats most probably do not touch each other, suggesting a small value of surface tension. In another work [6] we will consider the role of interpore surface tension. In this work we concentrate on the study of the geometry of the cluster. This is done by tracing the perimeter of an aggregate by plotting the Y coordinate of the actual position of the walker moving along the perimeter as a function of the distance made by the walker. This length dependent signal constitutes a random fractal [7]. It was analyzed by Barabasi and Vicsek [8] for pure DLA clusters. We use the same procedure they have used, i.e., we consider the perimeter of the clusters as the set of empty sites that are nearest to the aggregate. 2. Model The model used to generate the aggregates has been described in ref. [3]; for completeness we recapitulate it here. The algorithm is basically the DLA algorithm wherein random walkers traverse a square lattice. A random number p,, uniformly distributed between Apl2 and -Ap/2, is assigned to each point in the lattice when the simulation is initialized, and represents the capillary pressure in the porous medium. Walkers may be released from a site at the top of the lattice, or from a randomly chosen site on the interface. The corresponding probabilities may be inferred from the following argument: let pO be the injection pressure at the bottom of the lattice. Since p, the pressure in the defending fluid, satisfies Laplace’s equation, then p’ = -p + p,, + Ap”l2 does also. At the top of the lattice, where we take p = 0, p’ = p. + AF/2. At a site R on the interface, p’ = Ap”/2 -p(R). The probabilities of releasing walkers from top or interface sites are taken to be proportional to the corresponding values of p’. Thus, let Pb be the probability of releasing a walker from a given bond (link) at some site R on the interface, and let P, be the probability of releasing a walker from the top boundary. These probabilities are related by P
2
_
pt -
A17/2 -p”(R) p,, -t Ap”/2
’
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It is desirable to avoid walking above the highest point in the aggregate. Therefore, walkers are released from a point just above the top of the aggregate, which is 1 units from the top. Let a represent a “typical” length scale, such as the length of a pore. In our model, a is the lattice constant. Let P[ be the probability of releasing a walker from the line just above the aggregate. Then P[ = aP,lI; the mean pressure gradient is (Vp) = poll. Substituting this into (1) above and letting both 1 and pO be large while poll remains constant, gives
(2) where
The only parameter in the problem is Y. It can be related to the capillary number Ca. Using Darcy’s law for (VP), taking the permeability K - a’, and Ap”- r/u, where r is the surface tension parameter, we find r - Ca-‘. The model can handle lattices as large as 1024 x 1024. Fracture of the aggregate is not permitted. Once an aggregate has been grown, the perimeter is found using a recursive algorithm to “walk” along the interface.
3. Results Typical results of DLA runs for different values of r are shown in fig. 1 of ref. [3]. We analyse a typical cluster and investigate the scaling behavior of the average of the standard deviation on an interval of length L, CT(L) =
((i g1 vi - e2)li2j7
(4)
where Y is the mean value of Y on a segment of length L and ( ) denotes averaging over many such samples of length L. The width of the function u(L) is expected to scale as c+(L) - LH on the basis of the traces (see fig. lb). a(L) can therefore be considered as a self-affine function [9]. We first calculate (+ for pure DLA and IPT clusters [lo] and find H= 0.60 and H = 0.73, respectively. We also calculate the box dimension of the interface and find D, = 1.66 and 1.30 respectively. The last value is in accord with the value 1.37 found by Furuberg et al. [ll] for IP with trapping and by Grossman and Aharony [12]. For the pure DLA case in ref.
R. Range1 I Tracing interfaces in porous media
256
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Fig. 1. (a) shows the perimeter for a typical aggregate. (b) shows the function Y(I) which results from walking along the interface as a function of the arc length 1. (c) shows the plot of In a(L) versus In L. For small length scales there is a good fit with V(L) - LH with H = 0.73. For larger L the best linear fit gives H = 0.60. Observe that the fluctuations increase as L increases. This is because the number of samples decreases as L increases, so that the statistics of the average worsens.
(81 a scaling argument was given which relates H and D, as H = 1 ID,. However, this relation is known to hold for general fractional Brownian motion, in particular, the relation holds for the values found above for H and D, for DLA and IPT. Next, we calculated H for a typical cluster. The In a(L) versus In L plot shown in fig. lc shows for small length scales a behavior like the pure IPT clusters. For different values of r we always find Hz0.73. For larger length scales the best fit gives always H = 0.60. This is in contrast with the pure DLA or IPT cluster where for all scales only one Hurst exponent is found. The box dimension of the interface (the perimeter) is given by D, = 1.3 2 0.1 for length scales 1 such that 8 5 1~ L,/2. This is consistent with the relation H = l/D, with H = 0.73 for small length scales. We are not able to resolve clearly the D, dimension for larger length scales (LJ2 4 E< L).
B
R. Range1 I Tracing interfaces in porous media
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On the other hand, because the In CT(L) versus In L plots contain a decreasing number of points as L increases, it is difficult to localize the crossover length in this kind of plots. This and other topics will be discussed in detail in other work [13].
4. Conclusions A crossover behavior in the geometry of the interface of clusters growing under both pressure gradients and capillary forces in porous media is found. The geometry is characterized by the Hurst exponent of the self-affine curve defined by the perimeter (next nearest neighbors sites to the aggregate). For small length scales the exponent H-0.73 corresponds to invasion percolation with trapping. For larger length scales the exponent corresponds to DLA, H = 0.60. The behavior is in accord with the behavior of the crossover length previously found [3].
Acknowledgements
I am indebted to Dr. Ricardo Paredes and Professor Julio F. Fernandez for helpful discussions. I would like to thank Dr. Rafael Angulo and Dr. Ernest0 Medina (INTEVEP) for helpful remarks.
References [l] F. Family and T. Vicsek, eds., Dynamics of Fractal Surfaces (World Scientific, Singapore, 1991). [Z] H.E. Stanley, A. Bunde, S. Havlin, J. Lee, E. Roman and S. Schwarzer, Physica A 168 (1990) 23. [3] J.F. Femandez, R. Range1 and J. Rivero, Phys. Rev. Lett. 67 (1991) 2958. [4] J.P. Stokes, D.A. Weitz, J.P. Gollub, A. Dougherty, M.O. Robbins, P.M. Chaikin and H.M. Lindsay, Phys. Rev. Lett. 57 (1986) 1718. [5] R. Lenormand, J. Phys.: Condens. Matter 2 (1990) SA79. [6] R. Rangel, to be submitted. [7] R. Voss, Random fractals, in: Dynamics of Fractal Surfaces, F. Family and T. Vicsek, eds. (World Scientific, Singapore, 1991) p. 39. [8] A.-L. Barabasi and T. Vicsek, Phys. Rev. A 41 (1990) 6881. [9] B.B. Mandelbrot, in: Dynamics of Fractal Surfaces, F. Family and T. Vicsek, eds. (World Scientific, Singapore, 1991). [lo] R. Paredes, unpublished IPT data. [ll] L. Furuberg, J. Feder, A. Aharony and T. Jossang, Phys. Rev. Lett. 61 (1988) 2117. [12] T. Grossman and A. Aharony, J. Phys. A 19 (1986) L745; 20 (1987) L1193. [13] R. Rangel, in preparation.
