| Three-phase flows in porous media occur in applications of high socioeconomic impact, such as enhanced oil recovery, and environmental remediation of the vadose zone. This investigation addresses some of the unresolved issues in the mathematical and numerical modeling of such flows.; The traditional macroscopic description of three-phase flow relies on a multiphase extension of Darcy's equation. When capillarity effects are neglected, the mathematical model leads to a 2 x 2 system of conservation laws---the saturation equations---whose character depends exclusively on the relative permeabilities. It is well known that widely-used relative permeability models lead to regions in the saturation space where the system is elliptic, rather than hyperbolic---the so-called elliptic regions. It was concluded in some investigations that elliptic regions are unavoidable when sufficiently general relative permeability functions are employed. In this dissertation, we show that this conclusion is not quite correct.; We argue that elliptic regions are the artifacts of an incomplete mathematical model, and that they are not physically plausible. The key element of our analysis is to understand relative permeabilities as functionals of the various fluid/rock descriptors, and not as fixed functions of saturations alone (or even saturation history). We derive conditions that the relative permeabilities must satisfy, so that the system is everywhere strictly hyperbolic. These conditions depend, in an essential way, on the fluid viscosity ratios and the gravity number. They are supported by the physics of multiphase displacements, and are also in good agreement with experimental data.; After observing that an appropriate choice of the relative permeabilities leads to a strictly hyperbolic system, we derive the general analytical solution to the Riemann problem of three-phase flow. We present, for the first time, the complete catalogue of solutions that may arise, and conclude that the wave structure is restricted to only 9 solution types.; In the second part of this dissertation, we develop stabilized finite element methods for the simulation of miscible, two-phase, and three-phase flows. The key idea of the formulation is a multiscale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a method with enhanced stability, and not overly diffusive. The multiscale formalism, which is now dominant in fluid mechanics, is adopted and extended here for the simulation of three-phase flows. We illustrate the performance of the method with representative examples, which demonstrate its great potential for the numerical solution of complex multiphase compositional flows. |
