A Bayesian Non-Stationary Inversion Approach for Imagine Fluid Flow

Abstract

We present a new methodology for imaging the evolution of electrically conductive fluids in porous media. The inversion problem is formulated as a state estimation problem. The approach is based on an evolution-observation model and is solved using an extended Kalman filter algorithm. The example we consider involves the imaging of time-varying distributions of water saturation in porous media using time-lapse electrical resistance tomography (ERT). The complete electrode model (with Archieís law relating saturations to electrical conductivity) is used as the observation model. The evolution model we employ is a simplified (approximate) model for simulating flow through partially saturated porous media. We propose to account for approximation errors in the evolution model by constructing a statistical model of the differences between the accurate and approximate representations of fluid flow, and by including this information in the calculation of the posterior probability density of the estimated system state. The proposed method provides improved estimates of water saturation distribution relative to traditional reconstruction schemes that rely on conventional stabilization methods (e.g., using a smoothness prior) and relative to the extended Kalman filter without incorporating the approximation error method. Finally, the approximation error method allows for the use of a simplified and computationally efficient evolution model in the state estimation scheme. The methodology presented here for unsaturated flow through porous media may be extended for applications of nonisothermal multiphase flow in fractured geothermal reservoirs using a variety of geophysical methods.