A nonstationary geophysical inversion approach with an approximation error method for imaging fluid flow

We present a new methodology for imaging the evolution of electrically conductive fluids in porous media. The state estimation problem is formulated in terms of an evolution-observation model, and the estimates are obtained via Bayesian filtering. The approach is based on an extended Kalman filter algorithm and includes an approximation error method to model uncertainties in the evolution and observation models. 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 evolution model we employ is a simplified model for simulating flow through partially saturated porous media. The complete electrode model (with Archie's law relating saturations to electrical conductivity) is used as the observation model. We propose to account for approximation errors in the evolution and observation models 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 the approximation error method incorporated. Finally, the approximation error method allows for the use of a simplified and computationally efficient evolution model in the state estimation scheme. This work was supported, in part, by the Finnish Funding Agency for Technology and Innovation (TEKES), projects 40285/05 and 40347/05, and by the U.S. Dept. of Energy under Contract No. DE-AC02- 05CH11231.