A Probability Model For Errors in the Numerical Solutions of a Partial Differential Equation

Abstract

We consider numerical solutions of the Darcy and Buckley-Leverett equations for flow in porous media. These solutions depend on the geology of the porous media, which is given here as a realization of a random field to describe the reservoir permeability. We measure the solution error as the difference between the oil production rates (oil cut) for fine grid and upscaled coarse grid solutions of the equations. The main content of this paper is to formulate and analyze a probability model for this error. On the basis of this error model, we explore the extent to which the coarse grid oil production rate is sufficient to distinguish among geologies or their correlation lengths and to choose correctly the geology (random permeability field) or its correlation length defined on a fine grid, and to predict the future oil production rates. We find that our prediction methodology is effective in distinguishing ensembles defined by different correlation lengths, but that has limited power to d..