Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow
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ORIGINAL PAPER
Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow Philipp Selzer1 · Olaf A. Cirpka1 Received: 7 February 2019 / Accepted: 16 April 2020 / Published online: 24 June 2020 © The Author(s) 2020
Abstract 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-N´ed´elec (RT 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 RT N 0 velocity field imposing element-wise mass conservation. Using the conforming velocity field in RT 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 RT N 0 -projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller. Keywords P1 Galerkin finite element method · Lowest-order Raviart-Thomas-N´ed´elec space · Local mass conservation · Simplices · Groundwater flow Mathematics subject classification (2010) 76M10 · 76M12 · 76R05 · 76S05
1 Introduction Groundwater flow is commonly simulated by substituting Darcy’s law into the continuity equation, resulting in an elliptic (steady-state problem) or parabolic (transient problem), second-order differential equation of the hydraulic head [4, 29, 32]. Conservative solute transport is traditionally described by the advection-dispersion equation (ADE) employing the velocity field originating from the solution Olaf A. Cirpka
[email protected] Philipp Selzer [email protected] 1
Center for Applied Geoscience, University of T¨ubingen, H¨olderlinstr. 12, 72074 T¨ubingen, Germany
of the flow problem [4, 25]. The ADE may numerically be solved by Eulerian methods, such as finite volume methods (FVM), continuous Galerkin methods, such as the standard Galerkin finite element method (P1 Galerkin FEM) [4, 25], or discontinuous Galerkin (DG) methods [32, 33], which are all computat
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