Application of the Cell Method to the Simulation of Unsaturated Flow

摘要

The present work shows an alternative to the classical methods to solve the Richards' Equation (RE), used to model flow in unsaturated porous media. This alternative is named Cell Method (CM). The CM is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometrical content of the original problem, and is made possible by the existence and properties of a common mathematical structure of field theories. The goal is to maintain the focus, both in the modelling and discretization steps, on the physics of the problem. The present work derives the discrete formulation of the RE. Because of the non-linearities involved, RE is often solved using low-order numerical approximation methods, such as Finite Difference (FDM) or Finite Element Methods (FEM). These types of solution methods are used in many of the existing unsaturated flow codes. We show how the CM can be applied in this problem. We have solved a number of test cases, available in literature, to verify the ability of our model to reproduce these results. We have used the Newton-iterative methods which use iterative linear solvers, such as the Bi-CGSTAB. Numerical results, as it is possible to see in the verification exercise section, show the CM to be effective compared with the classical approaches (FDM and FEM) to solve the flow in unsaturated porous media. The procedure presented here is not peculiar to groundwater hydraulics but also applicable in fluid dynamics, solid mechanics, heat conduction and electromagnetism.