Abstract
A semi-continuum model for fluid flow in saturated-unsaturated porous medium in one spatial dimension is presented. The model is based on well-established physics, measurable parameters and material characteristics. The porous material is characterized by porosity, intrinsic permeability, main wetting and draining branches of the retention curve, and the saturation dependence of the relative permeability. The fluid is characterized by its density and dynamic viscosity. The only physics involved is the mass balance of fluid in porous media together with the Darcy-Buckingham Law for fluid flow in unsaturated porous media. The model is a cellular automaton based on the Macro Modified Invasion Percolation concept of dividing the porous medium into blocks which are not infinitesimal and are assumed to retain the characteristics of a porous medium. The cellular automaton repeats three successive rules: saturation update in each block, pressure update in each block, and flux update between neighboring blocks. The model tracks the evolution of the relative saturation, the fluid capillary pressure, and the fluid flux. The model is shown to reproduce qualitatively and quantitatively all features of one dimensional saturation overshoot behavior reported in the literature.
Highlights
Porous media flow is described in the framework of continuum mechanics[1]
In view of the two preceding paragraphs, it is crucial how we model the conductance at the finger tip, where ∇P is large, and S changes abruptly from small values in front of the fingertip to large values inside the fingertip
If we perform the limit dt → 0, dx → 0, the model converges to the Richards’ Equation (RE) and, the overshoot behavior disappears. This convergence is theoretically clear from the statement of the model and has been tested numerically. This model is interpreted as a cellular automaton with continuous levels of saturation
Summary
Porous media flow is described in the framework of continuum mechanics[1]. There are several persistent and important flow regimes in unsaturated homogeneous porous media (UHPM) which are not captured by continuum-mechanics based modeling[3]. The key feature of this regime is the non-monotonicity of the saturation: At certain points (through which the finger tip passes) saturation is a non-monotone function of time, first it increases abruptly as the over-saturated fingertip arrives, and it gradually decreases again as the under-saturated fingertail passes. This effect is called the saturation overshoot. The main experimental results can be summarized as follows: www.nature.com/scientificreports/
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