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-Nédélec (mathcal {RTN}_{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} velocity field imposing element-wise mass conservation. Using the conforming velocity field in mathcal {RTN}_{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}-projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller.
Highlights
Groundwater flow is commonly simulated by substituting Darcy’s law into the continuity equation, resulting in an elliptic or parabolic, second-order differential equation of the hydraulic head [4, 29, 32]
We have proposed an RT N 0 projection of velocities obtained from solving the groundwater flow equation by P1 Galerkin finite element method (FEM)
Our projection yields physically reasonable flow fields preserving the smoothness of the P1 Galerkin FEM solution for isotropic test cases and mild anisotropy
Summary
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]. Cordes and Kinzelbach [11] introduced a scheme to reconstruct a conforming velocity field on linear triangular and bilinear quadrilateral Galerkin finite elements It is based on the mass conservation property of the internal A-cells [11, 37]. Sun and Wheeler [43] introduced a similar, element-wise approach of flux correction to obtain local mass conservation and continuous normal components of fluxes on element boundaries for non-conforming velocity approximations, originating from continuous Galerkin finite elements [35, 41]. We define a cell-centered finite volume method on simplicial centroids against which the results of the RT N 0 projector are evaluated On this basis, we present semi-analytical particle tracking methods for divergent and non-divergent flow, similar to those of Pollock [34]
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