A Probability Model For Errors in the Numerical Solutions

We consider numerical solutions of the Darcy and Buckley-Leverett equations for ow in porous media. These solutions depend on the geology of the porous media, which is given here as a realization of a random eld to describe the reservoir permeability. We measure the solution error as the diierence between the oil production rates (oil cut) for ne 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 suucient to distinguish among geologies or their correlation lengths and to choose correctly the geology (random permeability eld) or its correlation length deened on a ne grid, and to predict the future oil production rates. We nd that our prediction methodology is eeective in distinguishing ensembles deened by diierent correlation lengths, but that has limited power to distinguish among geologies drawn from a xed ensemble. Our results appear to be weakly sensitive to details in the choice of the error probability model. Our best results are obtained with Bayesian posterior prediction using a nonparametric error model derived from simulation data. 2 Glimm et al.