Dual Mesh Method in Upscaling

Abstract

Abstract We develop and test a dual mesh approach to upscaling. A fine grid model is divided into coarse blocks, each of which contains many smaller cells. Pressure is computed first on the coarse grid using upscaled average transmissivities. These pressures are then used to compute the pressure at the small scale within each coarse block. In this way an approximation for the pressure is obtained without ever having to solve the full fine-grid problem, saving both CPU time and memory. The velocity is everywhere continuous and used to transport fluids on the fine grid - this step is no different from a conventional finite-difference approach once the pressure field is known. We develop the method for multi-well two-phase incompressible flow with gravity in three dimensions - the method is an extension of the work of Verdière et al.1,2, and uses ideas developed by Gautier et al.3 in the context of streamline-based simulation. We discuss how this approach can be the platform for a fully decoupled dual mesh approach, where the fine grid transport problem is also decoupled into a series of coarse cell problems, resulting in a rapid but accurate method that can be applied in principle to grids with any number of cells. We test the approach on two and three-dimensional problems with gravity. One of the test cases is a model from the SPE 10th Comparative Solution Project on upscaling4. The dual mesh method gives results that are very close to the fine grid results, but uses less CPU time and memory. The results are good even when a very coarse mesh is used and the method is more accurate than using standard single-phase upscaling techniques.