A three-scale index for flow in karst conduits in carbonate rocks

Abstract

A three-scale model for flow in karst conduit networks in fractured carbonates is rigorously constructed based on a reiterated homogenization procedure. The first upscaling, performed from the high-fidelity flow model, is based on sequential partially and fully topological model reduction procedures considering two discrete networks of solution-enlarged fractures and conduits. The subsequent macroscopization procedure projects the reduced model into the cells of a coarse computational grid, where homogenized equivalent properties are numerically constructed. Such a two-level upscaling gives rise to a macroscopic flow model characterized by mass-transfer functions between the geological structures. A notable consequence of the approach proposed herein is the appearance of a new karst index concept, whose underlying physics relies on the generalization of the traditional Peaceman’s theory of well index. Such a concept rules the mass exchange between conduits and matrix and can be extended to the general scenario of coupled flow in multi-branch karst conduit systems, displaying general non-circular cross sections and surrounding damage zones. The downscaling representation for the karst index can be further explored to improve accuracy of the exchange coefficient between the geological objects. Numerical experiments are carried-out showing the magnitude of the index for certain conduit and fracture arrangements, along with illustrating the impact upon flow patterns.