An analytic element model for transient axi-symmetric interface flow

Abstract

The problem of axi-symmetric flow of two immiscible fluids is solved by use of the analytic element method. A sharp interface divides the flow into two domains with different but homogeneous fluids. The sharp interface is realistic for truly immiscible fluids, and is an approximation for miscible fluids such as fresh and salt water in coastal aquifers. A specific discharge potential is introduced which is discontinuous across the interface. The jump in the specific discharge potential depends both on the difference in fluid viscosity and fluid density. The discontinuous specific discharge potential is realized by use of a singularity distribution along the interface. For the case of axi-symmetric flow, the singularity distribution may be approximated by a set of either sink rings or vortex rings. The approach is implemented in a computer program, which is validated by simulating the movement of a spherical inclusion of one fluid in an infinite domain of another fluid, and comparing the results with an exact solution. Finally, the practical use of the program is demonstrated by simulating a saltwater upconing problem underneath a partially penetrating well. The critical pumping rate for which the interface remains stable is determined and compared with both a laboratory experiment and numerical calculations reported in the literature.