Variably saturated flow described with the anisotropic Lattice Boltzmann methods

Abstract

This paper addresses the numerical solution of highly nonlinear parabolic equations with Lattice Boltzmann techniques. They are first developed for generic advection and anisotropic dispersion equations (AADE). Collision configurations handle the anisotropic diffusion forms by using either anisotropic eigenvalue sets or anisotropic equilibrium functions. The coordinate transformation from the orthorhombic (rectangular) discretization grid to the cuboid computational grid is equivalent for the AADE to the anisotropic rescaling of the convection/diffusion terms. The collision components (eigenvalues and/or equilibrium functions) become discontinuous on the boundaries of the computational sub-domains which have different space scaling factors. We focus on the analysis of the boundary continuity conditions by using anisotropic LB techniques. The developed schemes are applied to Richards’ equation for variably saturated flow. The anisotropy of the Richard’s equation originates from distinct soil conductivity values in both the vertical and horizontal directions. The method should on the interface between the heterogeneous layers maintain the continuity of the normal component of the Darcy’s velocity (total flux). Also the method should accommodate steep jumps of the moisture content variable (conserved quantity) resulting from the continuity of the pressure variable, a given non-linear function of the moisture content. The coupling between heterogeneity and the anisotropy is examined by using the distinct space steps in neighboring layers and tested against uniform grid solutions. Different formulations of the Richard’s equation illustrate the construction of distinct diffusion forms and their integral transforms via specification of the equilibrium components. Integral transforms are used to overcome the difficulties coming from the rapid change of the main variables on sharp fronts. The numerical assessment of the stability criteria and the interface boundary conditions extend the analysis of the Lattice Boltzmann schemes to nonlinear problems with discontinuous coefficients.