Singularity distributions for the analysis of multiple-fluid flow through porous media

In Part 1 the simultaneous flow of fluids of different properties is treated by substituting these fluids by one hypothetical fluid and applying singularities at those points where the properties of the actual fluids change. Their magnitude is chosen so that the specific discharges in the hypothetical fluid are everywhere identical to the specific discharges in the actual fluids. The flow in the hypothetical fluid can be determined by potential theory from the transformed boundary conditions and the influence of the singularities. For the determination of the discharge a stream function is used which contains singularities in the form of vortices. For the determination of the fluid pressures a multiple-fluid potential is defined which contains singularities in the form of source and sink distributions. The stream and the potential functions each combine with auxiliary, many-valued functions to form complex potentials. These permit solutions in the form of one integral in complex variables, valid for any point in the entire field, irrespective of the fluid present. The solution for the transition zone between fluids as well as the abrupt interface is elaborated. In Part 2 the two-dimensional example of an infinite, confined aquifer with an initial vertical interface between two fluids of different specific weight is elaborated, giving as a result the movement of the fluids in the entire field at the first moment and a first approximation for the rotation of the interface around the center as a function of time. These results are verified by a parallel plate model and an electric resistance model. In the latter model the vortices are replaced by sources for the tracing of streamlines and by source-sink combinations forming doublets for the potential lines.