A mixed finite element solver for natural convection in porous media using automated solution techniques

Abstract

IntroductionNatural convection in porous media is an important subject with applications in many geophysical and engineering fields, such as CO2 sequestration in brine aquifers. Numerical modeling plays an important role in understanding the dynamics of these flows and making suitable decisions. However, the traditional process of developing numerical solvers is time-consuming and error-prone. ObjectivesThis paper aims to present the design, implementation and verification of a fully coupled finite element solver for the simulation of natural convection in porous media, by making use of automated solution techniques. MethodsThe mathematical model is composed of the mass conservation equation for fluid flow, Darcy's law to relate pressure and velocity, and the advection-diffusion equation for the temperature/concentration field. In order to discretize the system of equations, a mixed finite element pair, namely a Brezzi-Douglas-Marini element and a discontinuous Galerkin element, is used to interpolate the velocity and the pressure field, respectively, while a discontinuous Galerkin element is chosen for the temperature/concentration. With the help of automated solution techniques, a readable and extensible code is developed for this class of multiphysics problems. The code is developed with FEniCS, an open-source framework for the automated solution of partial differential equations through the finite element method. ResultsAfter a convergence test using the method of manufactured solutions, the developed solver is validated by comparing the numerical results against commonly cited benchmarks. In all cases, results in this work are in good agreement with the benchmarks. ConclusionThe test results demonstrate the correctness of the implementation and the effectiveness of automated solution techniques for model development.