Enhanced multiscale mixed methods for two-phase flows in high-contrast porous media - Rocha, Franciane Fracalossi (2020)
This thesis proposes new methods for the numerical solution of two-phase flows in high-contrast porous media typical of petroleum reservoirs. An operator splitting strategy is used, where the saturation of one of the phases and the velocity field are updated sequentially. We focus on approximating the velocity field by multiscale methods, which allow for the global solution to be computed on coarse meshes (large scale), while detailed basis functions are defined locally (usually in parallel) in a much finer grid (small scale). The methods developed here are based on the Multiscale Robin Coupled Method (MRCM), a domain decomposition method that generalizes other well-established multiscale mixed methods and adds great flexibility to the choice of interface spaces as well as in the boundary conditions for coupling of local solutions. We find that the coupling of nearest neighbor subdomains through the imposition of a continuous pressure (respectively, normal fluxes) is the best strategy in terms of accuracy to approximate two-phase flows in the presence of high (resp., low) permeability channels (resp., regions). Thus, we introduce a new adaptivity strategy for setting the Robin algorithmic parameter of the MRCM, that controls the relative importance of Dirichlet and Neumann boundary conditions in the coupling of subdomains. The new strategy presents accurate approximations in challenging, high-contrast permeability fields. Then, it is used to improve the accuracy of the MRCM by considering alternative choices for the interface spaces other than the classical polynomials since they are not optimal for high-contrast features such as high permeability channels and barriers (low permeability). We introduce new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of high permeability channels and barriers, accommodated respectively, by the pressure and flux spaces. We show that the proposed interface spaces produce solutions significantly more accurate than polynomial spaces for problems with high-contrast permeability coefficients. We investigate different techniques to enhance the approximation of two-phase flows in terms of computational efficiency. We formulate a new procedure, the Multiscale Perturbation Method for Two-Phase Flows (MPM-2P), to speed-up the solution of two-phase flows. A modified operator splitting method is presented, where we replace full updates of local solutions by reusing basis functions computed by the MRCM at an earlier time of the simulation. We show that the MPM-2P reduces drastically the computational cost of two-phase flow simulations, without loss of accuracy. The MRCM is also investigated in a sequential implicit scheme for two-phase flows, that allows for the use of arbitrarily large time steps when compared to explicit time integration methods, improving the efficiency of the simulation. We show that the MRCM produces accurate and robust approximations when combined with different hyperbolic solvers, including implicit techniques. Our numerical simulations of two-phase flows with the MRCM present an unprecedented accuracy for realistic problems when compared to some standard multiscale methods. Moreover, the MRCM can take advantage of state-of-the-art supercomputers to efficiently simulate two-phase flows in high-contrast porous media.