Algorithms and discretization schemes for efficient simulation of multiphase flow and transport in porous media
Keywords: Energy Resources Engineering.

Multiphase flow and transport in porous media are described by highly nonlinear partial differential equations. Implicit finite volume discretizations -- used to numerically simulate these processes -- introduce additional strong nonlinearities that make the coupled algebraic system of equations challenging to solve efficiently. When both a gas and liquid phase are present, the discrete problem is particularly difficult due to greatly contrasting fluid properties, large buoyancy forces and sizeable capillary forces. Hence, standard simulation techniques frequently fail to meet the efficiency and accuracy requirements for processes such as geologic carbon sequestration and natural gas production. As these industries grow, the need for multiphase flow and transport simulators that meet their demanding requirements intensifies. In response, this dissertation presents several strategies to reduce the computational cost of simulation. The first chapters focus on a discretization scheme that allows the nonlinear solver (Newton-Raphson) to solve the governing system of flow and transport equations more efficiently and robustly. Discontinuities in the derivative of the problem's solution space can lead to slow convergence of Newton method's linearized updates. To avoid such complications, Chapter 2 proposes a smooth total velocity hybrid upwinding scheme with weighted averaging. First, we extend the favorable transport (sub-)problem properties of hybrid upwinding to problems with three or more phases. Then, by applying phase-by-phase weighted averaging to the mobilities determining the total velocity, the scheme yields a flow subproblem that is smooth with respect to changes in the sign of phase fluxes (common in counter-current flow regimes), and is well-behaved when phase velocities are large or when co-current viscous forces dominate. On a series of challenging test cases with realistic fluids, the proposed scheme consistently outperforms existing schemes, yielding benefits from 5% to over 50% reduction in nonlinear iterations. Additionally, Chapter 2 presents a second version of the scheme using a total mass formulation that shows promising results. Overall, based on the current results, we recommend the adoption of the proposed total velocity hybrid upwinding scheme with weighting averaging as it is highly efficient and robust. Chapter 3 investigates strategies to further improve hybrid upwinding by removing the strong nonlinearity induced by the method. Specifically, a discontinuity in the flux derivative occurs in the treatment of the viscous term when the total velocity changes sign. To design a novel method, criteria for a consistent, smooth and monotone scheme are stipulated. We conclude that the requirements have conflicts and thus such a scheme is not possible. We additionally support this conclusion visually by mapping the regions of the flux function that satisfy the monotonicity requirements. A different approach to improve efficiency and accuracy of multiphase flow simulations is investigated in Chapter 4; we aim to achieve higher accuracy on coarse grids for domains with spatially heterogeneous constitutive relationships. An efficient and robust strategy for multiphase discrete interface conditions is proposed. By using phase capillary pressures as the primary variable of the local problem that computes the flux, the novel method overcomes previous limitations and extends naturally to three or more phases. This also allows us to investigate a three-phase problem's solution space with capillary boundaries (entry- and maximum-pressures) for the first time. We show how different flow regimes are delineated by the heterogeneous capillary constitutive relationships. Finally, to ensure that the local problem can be solved for complicated fluid properties, we propose a robust update damping strategy for the local nonlinear solver. The last chapter proposes an efficient preconditioner for non-M matrices that frequently arise in geomechanical problems or multipoint flow stencils. Specifically, we enhance the widely implemented multiscale restriction-smoothed basis function method with a filtering strategy. Through applications to porous media flow and linear elastic geomechanics, the method is proven to be effective for scalar and vector problems with multipoint finite volume and finite element discretization schemes, respectively