Liquid Conservation and Nonlocal Interface Dynamics in Imbibition

Abstract

The propagation and roughening of a liquid-gas interface moving through a disordered medium under the influence of capillary forces is considered. The system is described by a phase-field model with conserved dynamics and spatial disorder is introduced through a quenched random field. Liquid conservation leads to slowing down of the average interface position H and imposes an intrinsic correlation length j x ϳ H 1͞2 on the spatial fluctuations of the interface. The interface is statistically self affine in space, with global roughness exponent x Ӎ 1.25, and exhibits anomalous scaling.

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

“…A conserved noise term of this form was phenomenologically argued in Ref. [5]. On the other hand, the nonconserved noise term proportional toξ h results from the trivial lowest order contribution of permeability noise of the form Vξ h (k), plus a nontrivial local term of opposite sign coming from the expression of the bulk noise defined as δv ξ ≈ − b 2 0 12µ ∂δp ∂n which takes the form…”

“…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].…”

“…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].…”

“…(6) is that the problem has two characteristic length scales. ℓ 1 controls the well-known crossover between (deterministic) capillary and viscous forces [5]. The second length ℓ 2 is a newly identified one which defines a crossover between conserved and non-conserved noise.…”

“…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.…”