A two dimensional semi-continuum model to explain wetting front instability in porous media

Abstract

Modelling fluid flow in an unsaturated porous medium is a complex problem with many practical applications. There is enough experimental and theoretical evidence that the standard continuum mechanics based modelling approach is unable to capture many important features of porous media flow. In this paper, a two-dimensional semi-continuum model is presented that combines ideas from continuum mechanics with invasion percolation models. The medium is divided into blocks of finite size that retain the nature of a porous medium. Each block is characterized by its porosity, permeability, and a retention curve. The saturation and pressure of the fluids are assumed to be uniform throughout each block. It is demonstrated that the resulting semi-continuum model is able to reproduce (1) gravity induced preferential flow with a spatially rich system of rivulets (fingers) characterized by saturation overshoot, (2) diffusion-like flow with a monotonic saturation profile, (3) the transition between the two. The model helps to explain the formation of the preferential pathways and their persistence and structure (the core and fringe of the fingers), the effect of the initial saturation of the matrix, and the saturation overshoot phenomenon.

CFD study of the water production in mature heavy oil fields with horizontal wells

“…At low initial water saturations, the displacements are characterized by well-defined fingering patterns. However, the fingers grow slower and wider at fully saturated conditions, disappearing in a diffusion-like displacement front, as pointed out by Kmec et al [ 69 ].…”

“…This change in the flow patterns is attributed to a transition to a diffusion-like regime at increasing initial water saturations, causing a wider and broader water front. This feature has been characterized at the microscale on experimental and numerical studies of Viscous Fingering [ 68 , 69 ]. At low initial water saturations, the displacements are characterized by well-defined fingering patterns.…”

Richards' equation reconsidered

“…As soon as 1989, Glass and Yarrington 15 proposed a cellular automaton under the title of "mechanistic modelling", or "Macro Modified Invasion Percolation". In Kmec et al 16,42 , the authors introduced a semi-continuum model which will be described in this section. Let us consider a long narrow vertical test tube of cross-section A [m 2 ] and height L [m] filled with homogeneous and isotropic porous medium.…”

“…However, none of these modifications is able to reproduce all the observed features of finger-flow 2 . A different attempt to overcome these drawbacks is the combination (3A) proposed in 15,16,41,42 . It has been shown that some of these models describe diffusion-like flow, finger-like flow, and the transition between the two regimes 16,42 .…”

The difference between semi-continuum model and Richards’ equation for unsaturated porous media flow

“…Therefore, especially in the field of soil physics and soil hydrology, many attempts were made to modify the RE so that it may also capture finger-like flow 34 – 44 . Another attempt is the combination (3A) proposed in 4 , 18 , 45 , 46 . It has been shown that some of these models describe diffusion-like flow, finger-like flow, and the transition between the two regimes 4 , 46 .…”