Computational methods for statistical solutions of inverse problems for flow in porous media
James G. Glimm, Hongjoong Kim · 2000
We present a numerical study of the upscaling problem for the fractional flow and total mobility functions in the Buckley-Leverett and Darcy equations for flow in porous media. We then formulate and analyze a probability model for the numerical solution error from upscaling. The upscaling problem is to define an averaged equation by local spatial averages, mapping from a micro-physical description to a meso-physical description and from a fine grid to a coarser one. Upscaling leads to the closure problem, which defines the nonlinear terms in the averaged equation, as these terms are not respected by the averaging process. Properties of the upscaled equations are given in terms of the geostatistical parameters which define the ensemble of permeabilities. Numerical studies presented here show that the layering and the large heterogeneity (CV) are main factors in both hyperbolic and elliptic renormalizations. The viscosity ratio has a strong effect on the elliptic upscaling. The Hurst exponent increases the upscaling effects. Given the probability error model, we explore the extent to which the coarse grid oil production rate is sufficient to predict future oil production rates. We find that very early oil production data is sufficient to reduce the prediction error in oil production by about 30%, relative to the prior probability prediction. We obtain 5% and 95% uncertainty bounds of about ±15% for this prediction neglecting finite sample size effects in the numerical error modeling and ±25% including them.