Assessment of unsteady Brinkman’s model for flow in karst aquifers

The Brinkman’s equation simplifies the numerical modeling of karst aquifers by allowing the use of a single transport equation to model the flow of fluids in both the free-flow and porous regions, in effect reducing the error arising from improper modeling of the interface between the two regions. Most equations available to model flow within karst aquifers deal with steady flow conditions. This may not be accurate in aquifers where unsteady conditions exist. We considered the effects of unsteady flow conditions in karst aquifers by assessing the addition of an unsteady flow term to the Brinkman’s equation. We solved the coupled mass conservation-transport equations that models unsteady fluid transport in karst aquifers and studied the effects of unsteady flow conditions on tracer transport in two different sample aquifers and compared to the results obtained from the steady flow Brinkman’s equation. The solution method adopted is sequential and it involves solving the unsteady Brinkman’s model first, followed by the advection-diffusion-adsorption equation using the cell-centered finite volume approach. The first example presented here is a simple aquifer model consisting of a single conduit surrounded by porous regions. The second example is a complicated structure consisting of complex geometrical caves embedded in a highly heterogeneous porous media. The results show that, inside the caves, the unsteady Brinkman’s model yielded lower tracer concentrations at early times when compared to the steady flow model. At longer times, both models produced almost similar results. In particular, the results obtained from the simplified example case (Example 1) indicate that the velocity profiles for unsteady flow within open conduits do not instantly yield a parabolic shape expected from the Brinkman’s equation, but gradually develops into one starting from a linear profile. Results obtained also show that the addition of unsteady flow term to the Brinkman’s model does not affect the flow of tracer within porous media in any significantly observable manner.