The Multiscale Perturbation Method for Two-Phase Flows in Porous Media

This dissertation is focused on creating new multiscale mixed methods that can help reduce the computational cost and improve the accuracy of the numerical solution to the two-phase, immiscible, incompressible flow problem obtained by the operator splitting technique. The two-phase flow governing systems of equations consists of two partial differential equations:(i) a second order elliptic equation or a Poisson equation, (ii) a hyperbolic conservation law. Typically in an operator splitting technique the elliptic and the hyperbolic equations are solved sequentially. The procedure to solve the elliptic equation numerically is computationally very expensive once the size of the linear system to be solved is large. Considering that, this dissertation focuses on the development of fast and efficient solvers which are naturally parallelizable to compute the solution of the second order elliptic equation that is approximated numerically. At first, the Multiscale Perturbation Method for second order elliptic equations (MPM) is presented . This method is based on the Multiscale Mixed Method (MuMM). The MuMM, like most multiscale methods solves the elliptic equation numerically by first computing a set of local multiscale mixed basis functions (MMBFs) with special boundary condition (Robin in this case) and then solves the global problem. The MPM proposes a new algorithm that can reuse the MMBFs computed at an initial time as well as take advantage of a good initial guess by using classical perturbation techniques. Secondly, the MPM is then incorporated into the operator splitting algorithm to create a new modified algorithm (MPM-2P:Multiscale Perturbation Method for two-phase flows) that solves the two-phase flow equations numerically. A relative cost reduction which is the gain in terms of the computational time is computed between the solution obtained via the new MPM-2P and by using simply the MuMM. The results show an exceptional speed up - a reduction in computational cost from 60.8% to 96.7% - for the elliptic equation of realistic reservoir flow problems indicating that a large number of MuMM solutions in a traditional operator splitting method can be easily replaced by the inexpensive MPM-2P solutions. This makes it the most important contribution of this dissertation. Finally, a new multiscale mixed method with overlapping domain decomposition (O-MuMM) is formulated and a parallel algorithm that implements the O-MuMM computationally is proposed. Numerical results for all three methods are discussed and analyzed to prove the validity of the new methods proposed.