A hyperbolic-elliptic PDE model and conservative numerical method for gravity-dominated variably-saturated groundwater flow
arxiv(2022)
摘要
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.
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