Hyperbolic-elliptic models for well-reservoir flow

We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This 2 × 2 system of conservation laws is coupled to the integral equation through natural coupling conditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incompressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reservoir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.