Density Dependent Flow in Porous Media

In this chapter we examine variable-density flow and corresponding solute transport in groundwater systems. Fluid dynamics of salty solutions with significant density variations are of increasing interest in many problems of subsurface hydrology. The mathematical model comprises a set of non-linear coupled partial differential equations to be solved for pressure/hydraulic head and mass fraction/concentration of the solute component. The governing equations and underlying assumptions are developed and discussed. The equation of solute mass conservation is formulated in terms of mass fraction and mass concentration. Different levels of the approximation of density variations in the mass balance equations are used for convection problems (e. g. the Boussinesq approximation and its extension, full density approximation). The impact of these simplifications is studied by use of numerical modeling. Numerical models for non-linear problems, such as density-driven convection, must be carefully verified in a particular series of tests. Standard benchmarks for proving variable-density flow models are the Henry, the Elder, and the salt dome problems. We studied these benchmarks using two finite element simulators — ROCKFLOW, which was developed at the Institute of Fluid Mechanics and Computer Applications in Civil Engineering, and FEFLOW, which was developed at the Institute for Water Resources Planning and Systems Research Ltd. Although both simulators are based on the Galerkin finite element method, they differ in many approximation details such as temporal discretization (Crank-Nicolson versus predictor-corrector schemes), spatial discretization (triangular and quadrilateral elements), finite element basis functions (linear, bilinear, biquadratic), iteration schemes (Newton, Picard), and solvers (direct, iterative). The numerical analysis illustrates discretization effects and defects arising from the different levels of the density approximation. We present results for the salt dome problem, for which inconsistent findings exist in literature (Kolditz et al. 1998).