Modeling Stochastic Heterogeneities

The appearance of geological patterns bearing somewhat random features, or complicated, but periodic, well-defined physical structures, very often raises questions among flow modelers regarding possibilities for quantitative flow simulation. The reservoir description process—the ability to describe geological structures accurately—is certainly not useful unless the ability to simulate flows is equally well developed. This chapter presents analytical methods drawing on the literature from other areas of continuum mechanics. The best known attempts at simple continuum models are the dual porosity approaches for naturally fractured reservoirs. Mathematical geostatisticians often develop their reservoir models by minimizing suitably defined error functions that are consistent with measured statistics, subject to auxiliary boundary constraints. These functions are typically positive definite, so that a solution to the minimization process exists. The differential equation methods used in modeling and the optimization approaches used in geostatistics are closely related. The similarities are explored in variational calculus, a well-known mathematical specialty that relates differential equations to global minimization problems.