Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media

We give an approach for using flow information from a system of wells to characterize hydrologic properties of an aquifer. In particular, we consider experiments where an impulse of tracer fluid is injected along with the water at the input wells and its concentration is recorded over time at the uptake wells. We focus on characterizing the spatially varying permeability field, which is a key attribute of the aquifer for determining flow paths and rates for a given flow experiment. As is standard for estimation from such flow data, we use complicated subsurface flow code that simulates the fluid flow through the aquifer for a particular well configuration and aquifer specification, in particular the permeability field over a grid. The solution to this ill-posed problem requires that some regularity conditions be imposed on the permeability field. Typically, this regularity is accomplished by specifying a stationary Gaussian process model for the permeability field. Here we use an intrinsically stationary Markov random field, which compares favorably to Gaussian process models and offers some additional flexibility and computational advantages. Our interest in quantifying uncertainty leads us to take a Bayesian approach, using Markov chain Monte Carlo for exploring the high-dimensional posterior distribution. We demonstrate our approach with several examples. We also note that the methodology is general and is not specific to hydrology applications.