Kinetic Roughening in Two-Phase Fluid Flow through a Random Hele-Shaw Cell

Abstract

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...

Experimental study of stable imbibition displacements in a model open fracture. II. Scale-dependent avalanche dynamics

“…Since c diverges when Ca → 0, the capillary number provides a quantitative measure of the distance to criticality in imbibition displacements. Indeed, as shown in detail in Part I (accompanying paper [20]), the front dynamics is driven by spatially localized avalanches with various scaling relations expected when the critical point for depinning is approached as Ca → 0 [2,5,14,15].…”

“…Flows in porous and fractured media exhibit complex spatiotemporal dynamics [1][2][3][4][5][6][7]. Avalanches and non-Gaussian intermittent velocity fluctuations [8][9][10][11][12][13][14][15][16][17] can arise from the medium heterogeneous structure, which may involve a very wide range of spatial scales, from nanometer pore size to kilometer field scales.…”

Imbibition in disordered media

“…Although it was examined in the early 1990's [7,45] in connection with percolation theory and deviations from KPZ behaviour, recent work [82,99] pointing to the importance of fluid conservation, has led to a flurry of new experimental results [101,122,311,312,313] supported by further theoretical work [178,261]. Even though many details remain obscure, we feel that the general theoretical picture of roughening in imbibition is now well established.…”

“…This analysis was pushed further by Pauné and Casademunt [261]. They considered the specific problem of fluid flow in a Hele-Shaw cell, where the only disorder is through variation in the thickness of the cell b.…”

Fluctuations in Fluid Invasion into Disordered Media

“…Capillary disorder p c r P c p c r acts only at the interface, and permeability disorder r r controls the flux of liquid from the reservoir to the interface. If r 0 is the typical pore size and r 0 its deviation, then p c =r 0 and r 2 0 , with p c = p c = [22,25].…”