Markov Chain Monte Carlo Methods for Fluid Flow Forecasting in the Subsurface.

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 work we focus on the transport of contaminants in a heterogeneous permeability field of a aquifer. In our problem the limited data comes in the form of contaminant fractional flow curves at monitoring wells of the aquifer. We then employ a Karhunen-Loeve expansion to truncate the stochastic dimension of the permeability field and thus the expansion helps reducing the computational burden. Aiming to reduce the computational burden further, we code our numerical simulator using parallel programming procedures on Graphics Processing Units (GPUs). In this paper we mainly present a comparative study of two well-known MCMC methods, namely, two-stage and DiffeRential Evolution Adaptive Metropolis (DREAM), for the characterization of the two-dimensional aquifer. A thorough statistical analysis of ensembles of the contaminant fractional flow curves from both MCMC methods is presented. The analysis indicates that although the average fractional flow curves are quite similar, both time-dependent ensemble variances and posterior analysis are considerably distinct for both methods.