A hyperbolic–elliptic PDE model and conservative numerical method for gravity-dominated variably-saturated groundwater flow

IF 4 2区 环境科学与生态学 Q1 WATER RESOURCES
Mohammad Afzal Shadab , Marc Andre Hesse
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Abstract

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.

重力主导型变饱和地下水流的双曲-椭圆 PDE 模型和保守数值方法
理查兹方程常用于表示不饱和多孔介质中的两相流体流动,当其中一相比另一相更重、更粘稠时。然而,由于毛细管压力项的退行性,如果不进行专门处理,该方程无法描述某些毛细管函数的完全饱和流动。在数学上,重力主导的可变饱和流非常有趣,因为其控制偏微分方程从非饱和区的双曲方程转换为饱和区的椭圆方程。此外,湿润前沿的存在引入了强烈的空间梯度,往往会导致数值不稳定。在这项研究中,我们建立了一个稳健的多维数学模型,并针对这种可忽略毛细力极限下的可变饱和流,实施了一种著名的高效、保守的数值方法。通过在非饱和单元中使用固定水头边界条件求解仅对应于饱和单元的简化系统,饱和区域的椭圆问题被有效地集成到我们的框架中。总之,这个耦合双曲-椭圆 PDE 框架为双曲理查兹方程提供了一个基于物理的高效扩展,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,涉及一维和二维饱和流动。这些简单的问题及其相应的分析解,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。对于提出的问题,我们的数值解与分析结果进行了很好的比较。最后一个测试问题是分层异质土壤中的二维渗透,显示了多个互不相连的饱和区域的形成和演变。
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来源期刊
Advances in Water Resources
Advances in Water Resources 环境科学-水资源
CiteScore
9.40
自引率
6.40%
发文量
171
审稿时长
36 days
期刊介绍: Advances in Water Resources provides a forum for the presentation of fundamental scientific advances in the understanding of water resources systems. The scope of Advances in Water Resources includes any combination of theoretical, computational, and experimental approaches used to advance fundamental understanding of surface or subsurface water resources systems or the interaction of these systems with the atmosphere, geosphere, biosphere, and human societies. Manuscripts involving case studies that do not attempt to reach broader conclusions, research on engineering design, applied hydraulics, or water quality and treatment, as well as applications of existing knowledge that do not advance fundamental understanding of hydrological processes, are not appropriate for Advances in Water Resources. Examples of appropriate topical areas that will be considered include the following: • Surface and subsurface hydrology • Hydrometeorology • Environmental fluid dynamics • Ecohydrology and ecohydrodynamics • Multiphase transport phenomena in porous media • Fluid flow and species transport and reaction processes
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