Application of the finite difference heterogeneous multiscale method to the Richards' equation

This paper extends the finite difference heterogeneous multiscale method (FDHMM) to simulate transient unsaturated water flow problems in random porous media. The numerical method is based on the use of two different schemes for the original equation, at different grid levels which allows numerical results at a lower cost than solving the original equations. The main feature of FDHMM is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain. The generated code is verified by applying the linearization model of the Richards' equation. Considering two different constitutive relationships, this method is applied to several test examples with different soil textures and boundary conditions. Both the Dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by FDHMM. The numerical experiments demonstrate that FDHMM can effectively simulate the transient unsaturated water flow in the specific soils. The numerical experiments also demonstrate that FDHMM can achieve accurate global mass balance and is a globally convergent algorithm, and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method.