Implicit hybrid upwinding and advanced nonlinear solver for multiphase flow and transport in porous media [electronic resource]

The partial differential equations governing multiphase flow and transport in heterogeneous porous media are highly nonlinear. Therefore, in the fully implicit finite-volume method, solving the algebraic systems at each time step is challenging and accounts for the majority of the simulation cost. We present numerical schemes and solution strategies applicable to general-purpose simulation that reduce the computational cost by drastically improving the nonlinear convergence. In the discretized transport problem, the interfacial approximation of the functions of saturation -- i.e., phase relative permeability and capillary pressure -- has a strong impact on the strength of the nonlinearities. We generalize and analyze an approximation method tailored to the underlying three-phase flow physics and based on Implicit Hybrid Upwinding (IHU) that results in fast nonlinear convergence. This is achieved with a differentiable and monotone numerical flux for two- and three-phase transport obtained from separate evaluations of the viscous, buoyancy, and capillary fluxes. Then, using IHU, we construct an efficient physically based discretization scheme for the mixed elliptic-parabolic problem in which the flow is coupled to the transport of species. The resulting IHU scheme is more general and more efficient than the saturation damping methods previously used to improve the nonlinear convergence. In addition, to correctly represent the trapping mechanisms associated with capillary heterogeneity, the proposed finite-volume scheme accounts for spatially discontinuous relative permeabilities and capillary pressure at the boundary between different rock types. Specifically, we design a robust numerical method that combines IHU with discrete transmission conditions between rock regions to improve the accuracy of the flux approximation compared to the standard Phase-Potential Upwinding (PPU) scheme. Importantly, we present a mathematical analysis that places this new fully implicit finite-volume scheme on a strong theoretical foundation. The mathematical study is supported by challenging heterogeneous two- and three-phase numerical tests. These computational examples demonstrate that the IHU scheme results in significant reductions in the number of nonlinear iterations compared to the commonly used PPU scheme for viscous-, buoyancy-, and capillary-dominated flow. Finally, we extend a nonlinear solution strategy in which the flow variables (pressure) and the transport variables (saturation and composition) are updated sequentially. The saturation and composition update relies on a reordering of the control volumes exploiting the directionality of the transport problem and is done locally in decreasing order of potential. We show that this procedure leads to a drastic improvement in the nonlinear convergence properties compared to Newton's method with damping for immiscible, Black-Oil, and compositional problems with strong buoyancy.

Keywords: Energy Resources Engineering.
Advisor: Tchelepi Hamdi primary advisor. Thesis advisor
Durlofsky Louis advisor. Thesis advisor
Mallison Bradley advisor. Thesis advisor