Numerical Upcaling of Coupled Flow and Geomechanics in Highly Heterogeneous Porous Media

This paper shows that numerical upscaling of permeability and elastic stiffness tensors can be applied to a very heterogeneous and deformable reservoir system. Fluid flow in deformable porous medium is a multiphysics problem that considers flow physics and rock physics simultaneously. This problem is computationally demanding since we need to solve different types of governing equations such as the mass balance and the equilibrium equations. Numerical upscaling of the transport properties and the mechanical properties using flow and mechanics solvers will provide a coarse reservoir model that represents fine scale contribution of fluid flow and geomechanics. This would help us perform more efficient modeling and simulation of coupled flow and geomechanics in a petroleum reservoir. Introduction In the reservoir simulation community researchers want to incorporate more realistic physics while modeling and simulating the reservoir performance. At the same time, they have high demands for very efficient computation. Imagine that we obtained a detailed fine scale geologic description of a petroleum reservoir from geologists. Running flow simulation with this reservoir model is not practical due to an expensive computation cost. We can parallelize the reservoir simulator to achieve faster computation but the number of linearly independent equations and the number of memory to save during the simulation do not change. Furthermore, the number of nodes that can be used for parallel computation in a company or university is limited. When we are dealing with a reservoir system that needs to have a more accurate estimation of geomechanical impact on the reservoir performance such as an unconsolidated reservoir, we need to couple a geomechanics simulator to the reservoir simulator. This procedure makes the computation more demanding. To resolve these problems efficiently, we need to define different scales of the reservoir model (fine grid scale and coarse grid scale) and develop a method that effectively captures the fine scale effect on the coarse scale without directly computing all the small features. This process is called upscaling, which assigns equivalent properties to the coarse scale cells, which are determined by solving fine scale boundary value problems. Therefore, the upscaled model can represent the complex physics of the fine scale model using the coarse grid that contains the contribution of the fine scale physics. Upscaling technique can reduce not only the size of the global matrix but also the number of solutions and parameters to save, allowing an efficient computation to be achieved. The purpose of the numerical simulation is to obtain approximate solutions of the partial differential equations that describe physical phenomena on discretized points, namely, mesh. The upscaling procedure can coarsen the mesh so the number of discrete points is less than the original problem. Therefore, the most important work is to assign the most accurate equivalent properties to each discrete point after coarsening. Flow based numerical upscaling has been widely used since this can capture the complex flow physics using a pressure solver (Warren and Price 1961; Begg and Carter 1989; Durlofsky et al. 1991; King et al. 1995; King and