Consistent particle tracking relies on conforming velocity fields that ensure local mass conservation on elements. Cell-centered finite-volume and mixed finite-element methods result in conforming velocity fields but this is not the case for continuous Galerkin methods, such as the standard finite element method (FEM). Nonetheless standard FEM is often used for subsurface flow modeling because it yields a continuous approximation of hydraulic heads, and it naturally handles unstructured grids and full material tensors. Acknowledging these advantages and the wide-spread use of finite-element-type simulations, we present a postprocessing method that reconstructs a cell-centered finite-volume solution from a finite-element-type solution of the variably-saturated subsurface flow equation to obtain conforming, mass-conservative fluxes. Using the linear average velocity field derived from these fluid fluxes, we employ element-wise analytical solutions for triangular prisms to compute particle trajectories and associated travel times. As a result, we can compute consistent particle trajectories for variably-saturated flow solutions generated by node-centered methods, such as finite element or finite difference methods, that do not yield conforming velocity fields. Our flux reconstruction solves a linear elliptic problem whose size is on the order of the number of elements, which is computationally much faster than solving the initial, non-linear transient variably-saturated flow equation. Compared to other postprocessing schemes, our flux reconstruction is numerically stable, fast to compute, and does not induce severe numerical artifacts when applied to heterogeneous domains with strongly varying velocities. However, these advantages come with a comparably high coding effort and the necessity of solving a global system of equations. We show that the results of our flux reconstruction are close to the node-centered primal solution for variably saturated three-dimensional flow with heterogeneous material properties.

一致的粒子跟踪依赖于确保元素局部质量守恒的一致速度场。以单元为中心的有限体积和混合有限元方法会产生一致的速度场，但对于连续 Galerkin 方法（例如标准有限元方法 (FEM)）而言，情况并非如此。尽管如此，标准 FEM 经常用于地下流动建模，因为它产生了水头的连续近似值，并且它自然地处理了非结构化网格和全材料张量。认识到这些优点和有限元类型模拟的广泛使用，我们提出了一种后处理方法，该方法从可变饱和地下流动方程的有限元类型解重建以单元为中心的有限体积解，以获得符合质量守恒的通量。使用从这些流体通量导出的线性平均速度场，我们采用三棱柱的元素解析解来计算粒子轨迹和相关的旅行时间。因此，我们可以为不产生一致速度场的以节点为中心的方法（例如有限元或有限差分方法）生成的可变饱和流解计算一致的粒子轨迹。我们的通量重建解决了一个线性椭圆问题，其大小在元素数量的数量级上，这比解决初始非线性瞬态可变饱和流动方程在计算上要快得多。与其他后处理方案相比，我们的通量重建在数值上稳定，计算速度快，并且当应用于速度变化很大的异质域时不会引起严重的数值伪影。然而，这些优势伴随着相对较高的编码工作量和求解全局方程组的必要性。我们表明，我们的通量重建结果接近于具有异质材料特性的可变饱和三维流动的以节点为中心的原始解。