Conservative multiscale strategies for multiphase flow on highly heterogeneous petroleum reservoirs using fully unstructured grids

Abstract

Simulation is fundamental to the management of subsurface oil reservoirs. The ability to predict the behaviour of multiphase flow in highly heterogeneous porous media allows to optimise recovery rates and maximize profits. Techniques such as history matching and optimisation make extensive use of simulations to better understand and predict the different scenarios and their respective impacts on production curves. However, recent advances in geological characterization have made it possible to integrate petrophysical data at a scale orders of magnitude higher than the feasible scale of standard petroleum reservoir simulators can handle. To deal with this discrepancy between scales, the family of approximate and conservative Multiscale Finite Volume (MsFV) methods was developed. These methods project the fine-scale system of equations onto a coarse space where it is solved and projected back. In this way, the high-resolution data is integrated into the simulator model, allowing fast and robust solutions at the price of a small loss of accuracy. Nonetheless, the MsFV family is not suitable for simulations on non-k-orthogonal grids. In this work, techniques for generalising methods of the finite volume and multiscale finite volume families have been investigated and developed in order to extend their applications to general unstructured grids. To this end, we have investigated the three main problems that prevent standard MSFV methods from being compatible with unstructured grids: 1) the lack of a consistent flux approximation for general grids, 2) the lack of a definition of multiscale units, and 3) the development of multiscale operators for unstructured grids. As a result, we developed the Algebraic Multiscale Solver for Unstructured Grids by proposing a new approach to create primal and dual coarse grids, developing a novel technique to avoid basis function leakage, and coupling the Algebraic Multiscale Solver (AMS) with a Multipoint Flux Approximation with a Diamond Stencil (MPFA-D). Another product of this work is the Flux Limited Splitting method, a novel repair technique that splits the flux of MPFA methods in terms of TPFA and Cross Diffusion Terms (CDT), where the letter is bounded by a relaxation parameter that is calculated nonlinearly to obtain a solution that satisfies the Discrete Maximum Principle (DMP).