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

Abstract

Semi-continuum modelling of unsaturated porous media flow is based on representing the porous medium as a grid of non-infinitesimal blocks that retain the character of a porous medium. This approach is similar to the hybrid/multiscale modelling. Semi-continuum model is able to physically correctly describe diffusion-like flow, finger-like flow, and the transition between them. This article presents the limit of the semi-continuum model as the block size goes to zero. In the limiting process, the retention curve of each block scales with the block size and in the limit becomes a hysteresis operator of the Prandtl-type used in elasto-plasticity models. Mathematical analysis showed that the limit of the semi-continuum model is a hyperbolic-parabolic partial differential equation with a hysteresis operator of Prandl’s type. This limit differs from the standard Richards’ equation, which is a parabolic equation and is not able to describe finger-like flow.

Semi-continuum modelling of unsaturated porous media flow to explain the Bauters' paradox

“…Let us recall the 2D semi-continuum model that was introduced by Kmec et al (2021). Here, we use the same model with an appropriate scaling of the retention curve with the block size (Vodák et al, 2022). The porous medium is represented as a rectangular grid of N × M square blocks of uniform size ∆x × ∆x.…”

“…A crucial idea behind the semi-continuum model is the appropriate scaling of the main branches of the retention curve which was first introduced by Vodák et al (2022). The scaling of the retention curve is based on the fact that the shape of the retention curve depends on the size (especially the height) of the sample on which the measurement is made (Larson and Morrow, 1981;Hunt et al, 2013;Silva et al, 2018).…”

“…The key feature of the semi-continuum approach is to account for the block size. This is done by an appropriate scaling of the retention curve with the block size (Vodák et al, 2022).…”

“…its derivative decreases) while keeping the hysteresis effect constant. See Vodák et al (2022) for more details and a physical justification.…”

“…A two dimensional version of the model was shown to correctly capture the transition between finger flow and diffusion-like flow with increasing initial saturation (Kmec et al, 2021) for uniform infiltration imposed on the entire top boundary. Vodák et al (2022) examined the limit of the semi-continuum model as the block sizes go to zero. They report a limit version of the model in the form of a partial differential equation with a Prandtl-type hysteresis operator (Visintin, 1993) under the derivative.…”