Modeling and simulation of fingering instabilities in groundwater flow
In this project, we aim to develop a mathematical model and perform numerical simulations for gravity-driven fingering instabilities that occur during the flow of groundwater in soils and aquifers. Fingers form in response to gravitational forces that derive from the difference in density between invading water and displaced air. These forces can destabilize an otherwise stable, planar wetting front, leading to a periodic array of rapidly propagating fingers. These instabilities can play a very important role in the study of both contamination and remediation of soils, because fingers provide a route for rapid entry of rain or other surface water sources into the subsurface. Numerous models have been proposed for capturing fingered flow phenomena, which are mostly based on the well-known Richards equation for flow in porous media in combination with appropriate constitutive equations for soil properties. In this project, we are interested in studying three specific models that employ different approaches for incorporating hysteresis and non-equilibrium effects in the dynamic equation that governs capillary pressure.
Our aim in this project is to take the basic computational framework we have already developed in Matlab for Model A [1] and to extend it by implementing models B and C. The governing equations in each case are a coupled nonlinear system of partial differential equations of parabolic type. Based on the simple, rectangular geometry of the typical experimental soil sample, we will restrict ourselves to a two-dimensional rectangular domain and use a standard finite difference discretization in space. The time discretization will be handled using a method-of-lines approach with the ODE solvers built-in to Matlab. We will then be prepared to perform a comprehensive comparison of the ability of all three models to simulate gravity-driven fingering phenomena, and to evaluate the advantages and disadvantages of each. No such comparison has yet been done because models B and C [2,5] are relatively recent; neither has model B or C yet been systematically compared to the wealth of experimental data available in the literature (see the references in [1] for examples).
The student will spend roughly the first month reviewing the two models mentioned above and implementing an algorithm for the numerical solution in Matlab. The code development will be aided by the fact that we already have a working model for capillary hysteresis [1], which is primary complication in the code. We will then perform a series of numerical simulations to validate the results and compare to other related models. The aim is to determine which model most accurately and reliably captures observed fingering behavior.
