Variable-density flow and transport in porous media

Flow and solute transport through porous media is of interest in both industrial engineering and environmental contexts. For example, the flow of groundwater through an aquifer can be treated in this way. The solute being transported may then represent the spread of a pollutant through groundwater supplies. Often these pollutants change the density of the fluid they are dissolved in, leading to complicated fluid dynamics.

The University of Texas CFDLAB is exploring these processes using its in-house numerical simulation code, MGFLO. MGFLO's modular and flexible approach makes it simple to describe governing equations (and their discretization) which differ from the Navier-Stokes equations used in other applications.

To simulate variable-density flow and transport in a porous medium, MGFLO solves a pair of coupled partial differential equations which may be highly nonlinear. Different constitutive equations may be considered for e.g. fluid viscosity, the relation between solute concentration and fluid density, and hydrodynamic dispersion. MGFLO also supports fully anisotropic porous media.

The Elder problem is one of the main benchmark tests used to compare codes which simulate variable-density flow and transport in a porous medium. Unfortunately, despite laboratory experiments and many numerical studies, it is still unclear what its true solution is. Nevertheless, the Elder problem is still of interest as it demonstrates how sensitive some nonlinear transient simulations can be to small changes in mesh design and numerical solution technique.

The domain of the Elder problem is a vertical cross-section. No fluid flows in or out of any of the boundaries, excepting the two upper corners, which are held at a constant pressure of zero. Centred along the top boundary is a constant concentration boundary of value 1 unit, representing a source of solute such as a landfill site or salt lake. The bottom boundary is held at a constant concentration of zero. Initially, the concentration throughout the region is zero and the pressure field is hydrostatic.

Solute plumes form under the high-concentration boundary along the top. These grow and merge with time. The exact shape and development of these plumes depends on the simulation code, mesh and timestepping scheme used.

The following figures show some sample results from the left half of the domain after four years of simulation time, as calculated by MGFLO. The first plot is calculated using variable timesteps, according to an ABTR scheme. The second is calculated using fixed timesteps of one month.

An animation depicting one set of transient results can be found here .

Given the limitations of the Elder problem and other similarly flawed benchmarks, the CFDLAB is developing better verification problems derived from analytic solutions and newer laboratory work.

Also of interest, especially to those simulating hydrogeological fieldsites, is the question of heterogeneity. Aquifer properties may vary by many orders of magnitude over short distances and a code needs to be robust enough to handle this. The CFDLAB is at present investigating how this robustness may be encouraged and evaluated.

Future work may include the development of MGFLO into a multiscale model. This is because many hydrogeological features--such as rock fractures, wells and sources of pollutant--are very small compared to the extent of the region of interest, which may be an aquifer several hundreds of kilometeres in extent. Multiscale modelling, possibly employing different physics at different scales, offer the possibility of accurate simulation without resorting to massive (and largely redundant) fine-scale simulation of large areas.