Axisymmetric Flow in an Oil Reservoir of Finite Depth Caused by a Point sink above an Oil-Water Interface

The flow of a stratified fluid (e.g., oil/water) withdrawn from a vertically confined porous medium through a point sink is considered. The withdrawal tends to cause the oil-water interface to move upwards. So long as the interface is below the well, the less dense fluid (oil) is pumped into the well without the denser fluid (water) until a critical flow rate is reached. The flow is considered to be axisymmetric, and involves a nonlinear boundary condition along the free surface. A boundary-integral equation method (BIEM) is used to find the interface position for different pumping rates. For small flow rates, a small-parameter expansion is derived and the results are compared with numerical solutions to the problem. There exists a critical withdrawal rate beneath which the water does not break through into the sink, this rate depending on the sink location and bottom geometry.