Uncertainty Quantification of Parameters in SBVPs Using Stochastic Basis and Multi-scale Domain Decomposition

Abstract

Quantifying uncertainty effects of coefficients that exhibit heterogeneity at multiple scales is among many outstanding challenges in subsurface flow models. Typically, the coefficients are modeled as functions of random variables governed by certain statistics. To quantify their uncertainty in the form of statistics (e.g., average fluid pressure or concentration) MC methods have been used. In a separate direction, multi-scale numerical methods have been developed to efficiently capture spatial heterogeneity that otherwise would be intractable with standard numerical techniques. Since heterogeneity of individual realizations can drastically differ, a direct use of multi-scale methods in MC simulations is problematic. Furthermore, MC methods are known to be very expensive as a lot of samples are required to adequately characterize the random component of the solution. In this study, we utilize a stochastic representation method that exploits the solution structure of the random process in order to construct a problem dependent stochastic basis. Using this stochastic basis representation a set of coupled yet deterministic equations is constructed. To reduce the computational cost of solving the coupled system, we develop a multi-scale domain decomposition method utilizing Robin transmission conditions. In the proposed method, enrichment of the solution space can be performed at multiple levels that offer a balance between computational cost, and accuracy of the approximate solution.