Efficient Solvers for Field-Scale Simulation of Flow and Transport in Porous Media

Abstract

The first part of the thesis serves as an introduction to modelling of flow and transport in porous media from the perspective of reservoir simulation. We discuss generation of computational grids and discretization strategies, and look at some of the main components that make up a reservoir simulator: discrete operators, linearization strategies, linear solvers, and nonlinear solvers. The second part consists of nine papers that present novel work on efficient solution strategies applicable to field-scale reservoir simulation. These strategies are primarily based on sequential splitting of the governing equations into a flow and transport subproblem. In the first two papers, we consider novel techniques for generation of high-quality Voronoi grids that conform to 2D surface constraints representing geological features like faults and fractures, and 1D line constraints representing complex well trajectories. The flow problem has a strong elliptic character, and the first and third paper also study consistent discretization methods for elliptic (Poisson-type) flow equations with emphasis on numerical errors and computational efficiency. Flow equations can be solved efficiently using so-called multiscale methods. The fourth paper describes a dynamically adaptive, iterative multiscale method with improved convergence that uses additional coarse partitions to target features in the geological model and/or adapt to dynamic changes in the flow field. The transport problem typically has a strong hyperbolic character. In the fifth and sixth papers, we use this to devise robust adaptive damping strategies for Newton’s method that delineate different contraction regions in the residual function. In the seventh paper, we also exploit unidirectional flow properties to develop a local nonlinear solver that topologically sorts the grid cells according to the flow direction. By traversing the grid cells in this order, the nonlinear transport subproblems can be solved locally in a highly efficient manner. We apply this to accelerate the simulation of the widely used black-oil equations, discretized by first- and second-order discontinuous Galerkin methods. The eighth paper extends the method to compositional problems, and combines discontinuous Galerkin methods with a simple adaptive dynamic coarsening strategy to further accelerate the simulation of fine-scale transport equations. The last paper present a robust and efficient framework for adaptive dynamic coarsening, and combines this with our local nonlinear solvers.