An Assessment of Markov Chain Monte Carlo Methods for Fluid Flow Forecasting in the Subsurface




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Accurate predictions in subsurface flows require the forecasting of quantities of interest by applying models of subsurface fluid flow with very little available data. In general a Bayesian statistical approach along with a Markov Chain Monte Carlo (MCMC) algorithm can be used for quantifying the uncertainties associated with subsurface parameters. However, the complex nature of flow simulators presents considerable challenges to accessing inherent uncertainty in all flow simulator parameters of interest. In this thesis we focus on the transport of contaminants in an aquifer with a heterogeneous permeability field. The original contributions of this thesis are two-fold. We, first, present a careful comparative study of known MCMC methods, namely, single-stage, two-stage, Differential Evolution (DE) and DiffeRential Evolution Adaptive Metropolis (DREAM), for the characterization of two-dimensional aquifers. A thorough statistical convergence analysis of ensembles of contaminant fractional flow curves using the Potential Scale Reduction Factor (PSRF) and the Multivariate Potential Scale Reduction (MPSRF) for MCMC methods is presented. We find that the convergence of single and two-stage MCMC procedures converge so slow that an analysis of the posterior distributions of quantities of interest can not be used as a reliable convergence criterion. The second and the most important contribution of this thesis is a new multi-step localized sampling strategy for high-dimensional problems. We propose a new framework for uncertainty quantification that aims at taking advantage of a spatial decomposition of the domain of interest. This algorithm is introduced based on the concept of maximum local acceptance: in the first (screening) step a search determines the subdomain where acceptance is the highest. In the second phase of the algorithm, a local search is performed in the selected subdomain. We move to smaller subdomains using this procedure recursively. We compare this new strategy with the two-stage and DREAM MCMC methods for a contaminant transport problem. Our results show that this new technique outperforms the known procedures both in acceptance and convergence rates.



Markov processes, Evolution equations, Differential equations, Convergence