PHYSICAL REVIEW A VOLUME 43, NUMBER 8 Interface fluctuations in random media 15 APRIL 1991 David A. Kessler Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120 Herbert Levine and Yuhai Tu Department of Physics and Institute for nonlinear Science, University of California, San Diego, La Jolla, California 92093 (Received 19 November 1990) We study a stochastic model of an interface moving through a random medium. This model diAers from the standard Kardar-Parisi-Zhang equation by the fact that the fluctuations are quenched random variables. We find an intermediate scaling regime with roughness exponent ap- proximately equal to 0.75; this compares favorably with recent experiments [M. A. Rubio, C. A. Edwards, A. Dougherty, and J. P. Gollub, Phys. Rev. Lett. 63, 1685 (1989)] on multiphase Aow through a bead pack. w(L, t ~) -L'. (4) The exponent a depends on the dimension d, but is in- dependent of the coupling constants (at least over some finite range). This scaling behavior has been shown to agree with that observed for a number of other interface models, most notably the Eden and ballistic aggregation models. In the case of d =1 the exponent a is exactly 2 . There has been much recent interest in the scaling be- havior of nonequilibrium interfaces broadened by noise. Progress has occurred on both the theoretical and simula- tional fronts towards understanding the various scaling re- gimes as functions of the substrate dimensionality and the strength and correlation properties of the noise. In this context, a recent experiment on the displacement of one Auid by another in a porous medium' is worth noting. Specifically, the observed roughness exponent of the two- Auid interface does not correspond to any known theoreti- cal model. In this paper, we propose a simple model that reproduces the qualitative features of the experiment and shows the observed unusual scaling, ~-L' with a — 0.73, over some range of length scales. We shall argue, though, that the true asymptotic scaling behavior is however not anomalous, and is that of the Kardar-Parisi-Zhang (KPZ) equation (or equivalently, ballistic aggregation or the Eden model ) in I+ I dimensions. The KPZ equation is the simplest continuum descrip- tion of a stable nonequilibrium interface roughened by noise. The equation governs the time development of the height y(x, t) of a surface above a d-dimensional sub- strate, y(x) =DV y+XIVyl +rtKpz where the noise gKpz is &function correlated in substrate position and time: &riKpz(x, t)riypz(x', t')& =SB (x x')8'(t — I') . The interface width, averaged over a region of linear di- mension L, w (L, t ) — = (&y & — &y& ) ' saturates at long times to a value which scales as a power law in L: The physics of the experiment diff'ers from that embo- died in the KPZ equation in two regards. First, the noise is quenched, as the irregularities of the bead pack consti- tuting the porous medium are fixed during any particular experiment. This is in contrast to the noise in the KPZ equation, which is uncorrelated in time. This effect should serve to broaden the interface; if the interface encounters a point where it becomes temporarily pinned, it will tend to experience that particular (large) value of the random force for a relatively long time, in fact until it becomes depinned. This is clearly different than being subject to a noise field which is completely uncorrelated in time. The second major difference is the nonlocal nature of the liow field. The result of this is a relative stabilization of the in- terface. This can be seen in the linear stability of the pla- nar interface where the decay rate of a perturbation is proportional to the magnitude of the wave vector of the perturbation, as opposed to the square magnitude of the wave vector in the KPZ case. This effect would tend to inake the interface less broad, decreasing the exponent. This can be seen explicitly in a simulation of the interface width using a time-reversed diffusion-limited aggregation (DLA) algorithm. Since we are interested in understand- ing how the exponent can be larger than the KPZ value of 2, we shall concentrate in this paper on the first mecha- nism, namely the effect of the quenched noise. The model we study in this paper is dt =DV y+F+ ri(x, y), where ri(x, y) represents the quenched noise with &ri& =0 and the correlation function &ri(x, y) ri(x', y') & =g b(x — x')b(y — y'). F is the pushing force, and D is the sur- face tension parameter. This model is exactly the KPZ equation, except for the nature of the noise and the ab- sence of the nonlinear X term. The latter term is, in fact, induced automatically in a renormalization-group sense since the noise is a nonlinear function of the interface po- sition, y. The above model has, in fact, been written down previously in two different contexts, domain walls in mag- netic systems and segregating Auids in porous media. In both of those studies, the emphasis was on the pinning- depinning transition. Let us first analyze the model in two different limits: 4551