Finite-Volume Flux Reconstruction and Semi-Analytical Particle Tracking for Finite-Element-Type Models of Variably Saturated Flow in Porous Media

Particle tracking is the most direct and a computationally efficient method to determine travel times and trajectories in subsurface flow modeling. Accurate and consistent particle tracking requires element-wise mass conservation and conforming velocity fields, which ensure continuity of the normal-flow component on element boundaries. These conditions are not met by standard finite-element-type methods. Despite this shortcoming, finite-element-type methods are often used in subsurface flow modeling because they continuously approximate the potential-head field and can easily handle unstructured grids and full material tensors. Acknowledging these advantages and the wide-spread use of finite-element-type models in subsurface flow simulations, we present a novel postprocessing technique that reconstructs a cell-centered finite-volume approximation from a finite-element-type primal solution of the variably-saturated subsurface flow equation to obtain conforming, mass-conservative fluxes. Using the resulting velocity fields, we derive a semi-analytical, parallelized particle tracking scheme applicable to triangular prisms, which leads to consistent and mass-conservative trajectories and associated travel times. Compared to other postprocessing schemes, our flux reconstruction is stable, robust, and fast as it only solves a linear elliptic problem on the order of the number of elements, whereas the original flow problem was transient and non-linear. The methods are implemented as postprocessing codes and linked to the finite-element-type code HydroGeoSphere, but could also be linked to any other software yielding a solution of variably saturated flow in porous media on triangular prisms. The postprocessing codes can handle catchment-scale models including heterogeneous materials, geometries, and boundary conditions, and facilitate to track a million particles through a catchment in just a few minutes on a Standard-PC in Matlab. The approach is described by Selzer et al. (2021).