Modeling subsurface water resource systems involving heterogeneous porous media using the variational multiscale formulation

This work compares two popular mixed finite element formulations used to model subsurface flow and transport in heterogeneous porous media, namely, the lowest order Raviart–Thomas and the variational multiscale stabilized formulations. Comparison is made based on performance for several problems of engineering relevance that involve highly heterogeneous material properties (permeability ratios of up to 1 × 105), open flow boundary conditions (pressure driven flows), and large scale domains in two dimensions. Numerical experiments are performed to show the degree to which mass conservation is violated when a flow field computed using either element is used as the advection velocity in a transport model. The results reveal that the equal-order interpolation under the variational multiscale formulation shows considerable mass production or loss for problems that involve flow tangential to layers of differing permeability. But the violation of local mass balance is marginal for problems in which flow is orthogonal to the layers of differing permeability. For problems involving pressure dependent viscosity, we show that models with a high degree of pressure dependence exhibit improved performance for the variational multiscale method. The results are useful in establishing rudimentary estimates of the error produced by using the variational mutliscale formulation for several different types of problems related to subsurface water resource systems.