A multiscale finite element method for coupled heat and water transfer in heterogeneous soils

Large-scale modeling of coupled heat and water transfer (CHWT) is challenging due to the spatial variabilities of soil hydraulic and thermal properties. A multi-scale finite element method (MsFEM) has been designed for simulating liquid water fluxes in unsaturated soils. In this study, the MsFEM approach is expanded as a new scheme that can handle CHWT in soils. Two groups of MsFEM basis functions are constructed to incorporate the heterogeneities in soil hydraulic conductivity and thermal conductivity, and a Petrov-Galerkin formulation is adopted to implement the proposed MsFEM scheme. The MsFEM scheme is also exploited as a sequential solver when heat transfer and water transfer are expressed in a partially coupled formulation (Wang et al., 2022a). Numerical examples illustrate that, without essentially increasing the computing load, the MsFEM scheme can improve the accuracy by up-to 30% compared to the standard finite element method (FEM), especially for thermally driven water transfer. Some fine scale spatial variations in soil temperature can only be revealed with the MsFEM scheme. If the MsFEM sequential solver is applied, the coupled heat and water transfer model can be rewritten into a sequence of modules, where liquid water transfer, heat transfer and vapor transfer are solved step-by-step, but the thermally driven liquid water is omitted due to its relatively small value (Wang et al., 2022a). With the MsFEM sequential solver, a flexible modeling architecture can be achieved at the cost of a relatively small increase of error (<5%). Therefore, the MsFEM scheme presented in this study is an effective numerical approach to simulating CHWT in heterogeneous soils.