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.

中文翻译：

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对用于地下水流的精确颗粒跟踪的标准有限元速度场进行后处理

粒子跟踪是一种计算上有利且快速的方案，可用于确定地下水文学中的行进时间和轨迹。精确的粒子跟踪需要逐元素质量守恒的一致速度场。标准线性Galerkin有限元方法（FEM）无法满足此条件。我们提供了一个投影，该投影将在三角形和四面体上计算出的不符合元素的给定速度场映射到最低阶Raviart-Thomas-Nédélec（\（\ mathcal {RTN} _ {0 } \））空间，可以满足精确粒子跟踪的要求。该投影基于最小化标准FEM解决方案中与单元应力一致的单元质心处的水力梯度差异以及与\（\ mathcal {RTN} _ {0} \）速度场，进行逐元素质量守恒。利用三角形和四面体上\（\ mathcal {RTN} _ {0} \）空间中的顺应速度场，我们给出了发散和非发散流的半解析粒子跟踪方法。我们将结果与通过为相同元素定义的以单元为中心的有限体积方法获得的结果进行比较，并与考虑水力各向异性的分析案例进行测试。基于标准FEM解决方案的投影的速度场和关联的粒子轨迹与有限体积法得出的速度场和轨迹相当，但是在分段均匀水力传导率区域内，投影场更平滑。而\（\ mathcal {RTN} _ {0} \）投影的标准FEM解决方案因此更加精确，以单元为中心的有限体积方法的计算成本大大降低。