Adaptive computation of fluid flow in porous media

Abstract

We face the mathematical modeling and numerical simulation of fluid flow through porous media. This kind of fluid motion is of interest in many scientific fields and industrial activities, like for example, in the modeling of aquifers and petroleum reservoirs. When the fluid velocity is moderate to high, the relationship between pressure and velocity is no longer linear and the Darcy-Forchheimer model, which takes into account inertial effects, is used. The main goal of this work is to devise adaptive algorithms based on a posteriori error indicators for the numerical solution of the Darcy-Forchheimer model. We first study a simple nonlinear equation in divergence form with a strongly monotone and Lipschitz-continuous operator. We propose a new stabilized mixed finite element method to solve that kind of problem and obtain a new a posteriori error indicator which is proven to be reliable and locally efficient. Next, we consider the primal-mixed formulation of the Darcy-Forchheimer model and derive a reliable a posteriori error indicator. In all cases, we provide numerical experiments that support the theoretical results. The methods developed in this work can be applied in particular to simulate the hightemperature drying of porous materials, like wood or wood composites. We propose an adaptive finite element method to predict fluid flow through porous materials with an abruptly changing permeability. We compare the numerical simulations to experimental results, obtaining a good correspondence. Besides, we provide a semi-analytical method for estimating an apparent permeability for composite materials. Finally, we propose a new augmented finite element method for the Darcy-Forcheimer model based on pressure-jumps stabilization. We develop the numerical analysis of the discrete problem and perform several numerical experiments, including the coupling of the Darcy- Forchheimer model with a heat equation.