Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow

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.

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Acknowledgment

We like to thank Ole Klein from the Interdisciplinary Center for Scientific Computing at Heidelberg University for a very insightful discussion on finite element methods and the numerics behind a possible \(\mathcal {RTN}_{0}\) projection. Furthermore, we are grateful to Max Allmendinger for proofreading the mathematical notation and several joyful discussions on mathematical topics.

Funding

This work was funded by the German Research Foundation (DFG) within the Research Training Group RTG 1829 “Integrated Hydrosystem Modelling” and the Collaborative Research Center CRC 1253 “CAMPOS-Catchments as Reactors.”

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Appendix: Using barycentric coordinates to find the starting element of a particle trajectory

Appendix: Using barycentric coordinates to find the starting element of a particle trajectory

Every point \(\hat {\mathbf {x}}_{s}\) within a simplex can be described by a linear combination of the coordinates of the nodes, \(\hat {\textbf {x}}_{i}\), and a barycentric, nodal weight, βi ≥ 0, respectively, such that

$$ \hat{\mathbf{x}}_{s}=\sum\limits_{i\in\mathcal{N}_{E}}{\upbeta}_{i}\hat{\mathbf{x}}_{i}, $$
(57)

with

$$ \sum\limits_{i\in\mathcal{N}_{E}}{\upbeta}_{i}=1,\quad\text{and}\quad{\upbeta}_{j}=1-\sum\limits_{i\neq j\in\mathcal{N}_{E}}{\upbeta}_{i}, $$
(58)

in which j is the index of a node and βj is its associated weight, such that every point in the simplex can be described by the coordinates of its nodes and d + 1 weights. Combining (57) and (58) leads to

$$ \begin{array}{@{}rcl@{}} \mathbf{T}\boldsymbol{\upbeta} & = & \hat{\textbf{x}}_{s}-\hat{\textbf{x}}_{j}, \end{array} $$
(59)
$$ \begin{array}{@{}rcl@{}} \Rightarrow\boldsymbol{\upbeta} & = & \mathbf{T}^{-1}\left( \hat{\textbf{x}}_{s}-\hat{\textbf{x}}_{j}\right), \end{array} $$
(60)

in which T is a transformation matrix only depending on the coordinates of the nodes; T− 1 can easily be evaluated analytically; β is the vector of the d independent, barycentric, nodal weights; and \(\hat {\textbf {x}}_{j}\) is the vector of coordinates of node j of the element.

We exploit the concept of barycentric coordinates in the search for the element in which the starting point of our particle tracking scheme resides. To do so, we set \(\mathbf {x}_{p}=\hat {\mathbf {x}}_{s}\) and solve equation (60) for every element. If a particle lies within an element, all weights β = (βi)i= 1,...,d+ 1 are within the interval 0 ≤βi ≤ 1 and sum up to unity. We stop the search at the first instance at which both criteria are met.

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Selzer, P., Cirpka, O.A. Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow. Comput Geosci 24, 1605–1624 (2020). https://doi.org/10.1007/s10596-020-09969-y

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Keywords

  • \(\mathcal {P}_{1}\) Galerkin finite element method
  • Lowest-order Raviart-Thomas-Nédélec space
  • Local mass conservation
  • Simplices
  • Groundwater flow

Mathematics subject classification (2010)

  • 76M10
  • 76M12
  • 76R05
  • 76S05