Modelling fluid flow in karst reservoirs using Darcy Model with estimated permeability distribution

The existence of free-flow regions (fractures, vugs and caves) inside porous media at multiple-scales affects the flow paths in aquifers and underground reservoirs. To simulate advection-dominant phenomena with a reasonable level of accuracy, the Stokes-Brinkman and the Darcy-Stokes models have been separately proposed to model the simultaneous flow of fluids in the free-flow regions and the porous regions of karst reservoirs. However, the computational cost associated with the use of Stokes-Brinkman is much higher when compared with the use of the Darcy's model while Darcy-Stokes requires the implementation of the interface conditions. This paper introduces an approximation technique that assigns different permeability values to the grids in the free-flow region. The technique provides a way for computing apparent permeability values for different grid locations in the free-flow regions. This method makes it possible for the Darcy's model to closely approximate the Stokes-Brinkman's model. This proposed approach makes it possible to replace the Stokes-Brinkman model with the Darcy's model without significant loss of accuracy in modelling flood fronts. The values of the apparent permeabilities of the grids inside the free-flow region and the surrounding porous media are calculated from the analytical solution of Stokes-Brinkman's equation. Four examples are presented to illustrate the effectiveness of this technique. In the first three examples, the principal axes of the cave (free-flow region) are aligned with those of the porous media. The fourth example consists of a more complex scenario in which the principal axes of the cave are not in alignment with those of the porous media.