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<p align="justify"><span>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. D</span><span>e</span><span>spite 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 </span><span>finite-element-type</span> <span>code</span><span> 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).</span></p>

Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec ($\mathcal {RTN}_{0}$ R T N 0 ) space, which meets the requirements of accurate particle tracking. The projection is based on minimizing the difference in the hydraulic gradients at the element centroids between the standard FEM solution and the hydraulic gradients consistent with the $\mathcal {RTN}_{0}$ R T N 0 velocity field imposing element-wise mass conservation. Using the conforming velocity field in $\mathcal {RTN}_{0}$ R T N 0 space on triangles and tetrahedra, we present semi-analytical particle tracking methods for divergent and non-divergent flow. We compare the results with those obtained by a cell-centered finite volume method defined for the same elements, and a test case considering hydraulic anisotropy to an analytical solution. The velocity fields and associated particle trajectories based on the projection of the standard FEM solution are comparable to those resulting from the finite volume method, but the projected fields are smoother within zones of piecewise uniform hydraulic conductivity. While the $\mathcal {RTN}_{0}$ R T N 0 -projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller.