Bayesian Estimation of Multiscale Structures in a Binary Medium from Sparse Observations

We present a multiscale Bayesian method to reconstruct the permeability field of a binary medium. The reconstruction is conditioned on measurements of permeability and tracer test breakthrough times, observed at a limited set of locations. The medium consists of a spatially variable distribution of inclusions, which are too small to be individually resolved at the grid scale. The unknown inclusion proportion is modeled using a multivariate Gaussian, represented using a truncated Karhunen-Loève transformation to reduce dimensionality. An upscaling model is used for the permeability, which is parameterized by the inclusion size. Along with a Darcy flow model, we formulate a Bayesian inverse problem for the Karhunen-Loève modes’ weights. The posterior distribution is calculated using an adaptive Markov chain Monte Carlo method and demonstrates that breakthrough times contain information on the small-scale structures. The inclusion sizes can be estimated accurately in certain cases. By selecting a few members of an ensemble of permeability fields consistent with the data, breakthough times at the sensor points are predicted. We combine them using Bayesian Model Averaging and find that the model-averaged ensemble can be predictive over the domain.