| We present two new formulations for the equations of immiscible two-phase flow through a porous medium. Both of these formulations implement a Kirchoff transformation of the capillary pressure, which overcomes some of the numerical difficulties associated with capillary pressure effects. We present a fully-implicit model for compressible fluids, and an implicit pressure, explicit saturation (IMPES) method for incompressible fluids.; We also present a multilevel preconditioner for use with the IMPES method above, analyze the method using standard tools from linear algebra, and prove that this preconditioner produces a bound for condition number, dependent only on the physical properties of the model in the multilevel case, and additionally on the ratio of grid sizes in the two-level case. The novelty of this preconditioner is that it is proven for the semi-definite elliptic linear system with homogeneous Neumann boundary conditions; and with non-conforming approximating spaces.; Numerical results confirm the theoretical effectiveness, even in the presence of highly heterogeneous and anisotropic coefficients.; We extend the Kirchoff transformation to other systems of equations of similar type, without implementation or analysis. Equations are presented for the three-phase “Black Oil” model, and for the highly compressible two-phase “Air/Water” model. |
