Finite-volume flux reconstruction and semi-analytical particle tracking on triangular prisms for finite-element-type models of variably-saturated flow

摘要

Consistent particle tracking relies on conforming velocity fields that ensure local mass conservation on elements. Cell-centered finite-volume and mixed finite-element methods result in conforming velocity fields but this is not the case for continuous Galerkin methods, such as the standard finite element method (FEM). Nonetheless stan-dard FEM is often used for subsurface flow modeling because it yields a continuous approximation of hydraulic heads, and it naturally handles unstructured grids and full material tensors. Acknowledging these advantages and the wide-spread use of finite-element-type simulations, we present a postprocessing method that reconstructs a cell-centered finite-volume solution from a finite-element-type solution of the variably-saturated subsurface flow equation to obtain conforming, mass-conservative fluxes. Using the linear average velocity field derived from these fluid fluxes, we employ element-wise analytical solutions for triangular prisms to compute particle trajecto-ries and associated travel times. As a result, we can compute consistent particle trajectories for variably-saturated flow solutions generated by node-centered methods, such as finite element or finite difference methods, that do not yield conforming velocity fields. Our flux reconstruction solves a linear elliptic problem whose size is on the order of the number of elements, which is computationally much faster than solving the initial, non-linear transient variably-saturated flow equation. Compared to other postprocessing schemes, our flux reconstruction is numerically stable, fast to compute, and does not induce severe numerical artifacts when applied to heteroge-neous domains with strongly varying velocities. However, these advantages come with a comparably high coding effort and the necessity of solving a global system of equations. We show that the results of our flux reconstruction are close to the node-centered primal solution for variably saturated three-dimensional flow with heterogeneous material properties.