Richards equation is often used to represent two-phase fluid flow in an unsaturated porous medium when one phase is much heavier and more viscous than the other. However, it cannot describe the fully saturated flow for some capillary functions without specialized treatment due to degeneracy in the capillary pressure term. Mathematically, gravity-dominated variably saturated flows are interesting because their governing partial differential equation switches from hyperbolic in the unsaturated region to elliptic in the saturated region. Moreover, the presence of wetting fronts introduces strong spatial gradients often leading to numerical instability. In this work, we develop a robust, multidimensional mathematical model and implement a well-known efficient and conservative numerical method for such variably saturated flow in the limit of negligible capillary forces. The elliptic problem in saturated regions is integrated efficiently into our framework by solving a reduced system corresponding only to the saturated cells using fixed head boundary conditions in the unsaturated cells. In summary, this coupled hyperbolic–elliptic PDE framework provides an efficient, physics-based extension of the hyperbolic Richards equation to simulate fully saturated regions. Finally, we provide a suite of easy-to-implement yet challenging benchmark test problems involving saturated flows in one and two dimensions. These simple problems, accompanied by their corresponding analytical solutions, can prove to be pivotal for the code verification, model validation (V&V) and performance comparison of simulators for variably saturated flow. Our numerical solutions show an excellent comparison with the analytical results for the proposed problems. The last test problem on two-dimensional infiltration in a stratified, heterogeneous soil shows the formation and evolution of multiple disconnected saturated regions.

中文翻译：

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重力主导的变饱和地下水流的双曲-椭圆偏微分方程模型和保守数值方法


当一相比另一相重且粘稠时，理查兹方程通常用于表示不饱和多孔介质中的两相流体流动。然而，由于毛细管压力项的简并性，如果没有专门处理，它就无法描述某些毛细管功能的完全饱和流量。在数学上，重力主导的可变饱和流很有趣，因为它们的控制偏微分方程从不饱和区域的双曲线切换到饱和区域的椭圆曲线。此外，润湿锋的存在引入了强烈的空间梯度，通常导致数值不稳定。在这项工作中，我们开发了一个稳健的多维数学模型，并针对可忽略的毛细管力极限下的这种可变饱和流实施了众所周知的高效且保守的数值方法。通过使用不饱和单元中的固定头边界条件求解仅对应于饱和单元的简化系统，饱和区域中的椭圆问题被有效地集成到我们的框架中。总之，这种耦合的双曲-椭圆偏微分方程框架提供了双曲理查兹方程的高效、基于物理的扩展，以模拟完全饱和区域。最后，我们提供了一套易于实现但具有挑战性的基准测试问题，涉及一维和二维的饱和流。这些简单的问题及其相应的分析解决方案可以证明对于可变饱和流模拟器的代码验证、模型验证 (V&V) 和性能比较至关重要。我们的数值解与所提出问题的分析结果进行了很好的比较。 最后一个关于分层、异质土壤中二维渗透的测试问题显示了多个不连续饱和区域的形成和演化。