Uncertainty Quantification for Subsurface Flow Problems Using Coarse-Scale Models

The multiscale nature of geological formations can have a strong impact on subsurface flow processes. In an attempt to characterize these formations at all relevant length scales, highly resolved property models are typically constructed. This high degree of detail greatly complicates flow simulations and uncertainty quantification. To address this issue, a variety of computational upscaling (numerical homogenization) procedures have been developed. In this chapter, a number of the existing approaches are described. These include single-phase parameter upscaling (the computation of coarse-scale permeability or transmissibility) and two-phase parameter upscaling (the computation of coarse-scale relative permeability curves) procedures. Methods that range from purely local to fully global are considered. Emphasis is placed on the performance of these techniques for uncertainty quantification, where many realizations of the geological model are considered. Along these lines, an ensemble-level upscaling approach is described, in which the goal is to provide coarse models that capture ensemble flow statistics (such as the cumulative distribution function for oil production) consistent with those of the underlying fine-scale models rather than agreement on a realization-by-realization basis. Numerical results highlighting the relative advantages and limitations of the various methods are presented. In particular, the ensemble-level upscaling approach is shown to provide accurate statistical predictions at an acceptable computational cost.