A nonlocal interface equation is derived for two-phase fluid flow, with arbitrary wettability and viscosity contrast c = (µ1 − µ2)/(µ1 + µ2), in a model porous medium defined as a Hele-Shaw cell with random gap b0 + δb. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, non-conserved and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which scale with capillary number as ℓ1 ∼ b0(cCa) −1/2 and ℓ2 ∼ b0Ca −1 . Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments.PACS numbers: 47.55. Mh, 68.35.Ct The displacement of a fluid by another in a porous medium is a problem of fundamental interest in nonequilibrium physics as a paradigm of interface dynamics in disordered media [1,2]. Experiments on bead packs in Hele-Shaw cells [3] in particular, have stimulated considerable theoretical efforts, but the problem has consistently revealed itself rather elusive [1,2]. More recently a new surge of interest has arisen with the recognition of the inherently non-local character of the problem as a key ingredient [4,5], and the realization of a new series of experiments in Hele-Shaw cells with random gap [6,7,8]. Roughening exponents of the proposed nonlocal equations have been explored by means of Flory-type scaling arguments [4] and phase field simulations [5,6]. While the specific properties of noise are known to be crucial to determine the universal aspects of interface roughening, fluctuations are usually modeled at a phenomenological level, and including only local capillary effects. Noise related to the non-Laplacian viscous pressure due to quenched disorder in the permeability has been so far neglected. While this may be justified for imbibition experiments [5], other situations, such as forced fluid invasion, do require a quantitative assessment of this point. In addition, it would be desirable to have a unified formulation for general conditions of viscosity contrast c = (µ 1 −µ 2 )/(µ 1 +µ 2 ) and wettability given the rich variety of phenomena that the experimental evidence has unveiled as a function of those parameters [2].Here we address the general problem of fluid displacement in a Hele-Shaw cell with random gap, as a simple model of a porous medium. This model system has the great advantage that no coarse-graining procedure must be invoked in the theoretical formulation, thus allowing us to derive ab initio a general and complete interface equation, quantitatively accurate, with explicit dependence on 'bare' parameters, and including all noise sources. On the experimental side, the system is also appealing since a direct control of the disorder is locally possible on the microscopic scale [6,7].A complete description of interface fluctuations must contain three basic physical effects of a porous matrix on its motion, namely local variations of (i) capillary pressure, (ii) permeability, and (iii) available vol...
