Coupled Geomechanics and Flow Simulation on Corner-Point and Polyhedral Grids

Abstract

Abstract Reservoir geology has traditionally been described using corner-point grids. For such grid and general polyhedral grids, two-point flux-approximation (TPFA) schemes, or modifications thereof, are simple and robust methods which, despite grid-orientation errors and a general lack of consistency, have been considered sufficient for simulation in light of the inherent geological uncertainty. However, there is no equivalent methods to simulate elasticity, and this has hampered flexible coupling between grids made for flow simulations and grids made for geomechanical calculations. Using the newly developed virtual element method (VEM) for mechanics on polyhedral grids, we show how geomechanics simulations on corner-point grids can be coupled with traditional discretizations (and solvers) used in black-oil and compositional simulators. The new VEM method can be seen as an extension of finite element methods to general grids. In a first order VEM method, the energy of the non-linear basis functions is not computed exactly, as in FEM, but approximated by introducing a regularisation term. This may result in large error, especially for grids with high aspect ratio. We discuss the implications of the use of VEM for applications to reservoir modeling. We will primarily use Biots type coupling restricted to linear elasticity. However, we also discuss how nonlinear coupling can be practically introduced using Automatic Differentiation (AD), which is the numerical technique of choice in several modern research simulators. The AD framework typically enables automatic evaluations of the derivative of vector functions. The difficulty in the case of nonlinear couplings with mechanics is that many of the important physical quantities involved, such as the mechanical properties and the stress, naturally live on cells, so that the discretization operators themselves becomes nonlinear quantities that have to be differentiated. Accurate and robust treatment of these challenges are discussed. Computational cost is a major challenge when coupling geomechanical effects with flow simulation. First of all, the mechanical system needs to include the overburden and the rocks surrounding the reservoir. Secondly, mechanics in its simplest form is described by a vector-Laplace equation, which results in an additional elliptic system that is approximately three times larger than the pressure part of the flow equations. Several decoupling techniques have been introduced to overcome this challenge. We show how such methods can easily be implemented directly on the corner-point or polyhedral grids with the proposed VEM discretization. To showcase the coupling of geomechanics and flow simulation, we use a flexible open-source prototyping framework, which has been benchmarked against state-of-the-art commercial and research simulators.