High resolution adaptive implicit method for reactive transport in heterogeneous porous media

Abstract

The Adaptive Implicit Method (AIM) reduces the computational cost related to simulations of field scale displacements in porous media. Standard AIM uses a mixed implicit/explicit time discretization that overcomes the time step size limitations of purely explicit approaches and considers single-point upwinding to approximate both the implicit and explicit numerical fluxes. Use of high-order fluxes improves accuracy but leads to increased stencil size and more complex Jacobians in the implicit regions. To reduce numerical diffusion and improve the accuracy of AIM while avoiding the larger stencils and nonlinearities introduced by an implicit treatment of high-order schemes, we introduce a scheme that in addition to blending implicit and explicit time discretizations, deliberately blends single-point upwind and a high-order flux-limited total variation diminishing approximation of the numerical fluxes. The proposed scheme does not interfere with the discretization of the implicit terms and the structure of the matrices that need to be solved is the same of standard AIM, making the scheme easy to apply in existing simulators. The details of the scheme are presented and it is applied to single-phase reactive transport problems in 1-, 2-, and 3-dimensions with simple homogeneous decay reactions, mixing-limited reactions, and the calcium carbonate system. For mixing-limited reactions, that are highly sensitive to numerical diffusion, accurate results depend on properly representing the sharp reactive fronts and a strategy to avoid an implicit treatment inside the reactive fronts is presented. The numerical results indicate significant gains in accuracy at the additional expense of slightly more involved flux computations in the explicit regions, that represent a small fraction of the total CPU cost.