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In this work we are interested in the (ill-posed) inverse problem forabsolute permeability characterization that arises in predictive modeling ofporous media flows. We consider a Bayesian statistical framework with apreconditioned Markov Chain Monte Carlo (MCMC) algorithm for the solution ofthe inverse problem. Reduction of uncertainty can be accomplished byincorporating measurements at sparse locations (static data) in the priordistribution. We present a new method to condition Gaussian fields (the log ofpermeability fields) to available sparse measurements. A truncatedKarhunen-Lo\`eve expansion (KLE) is used for dimension reduction. In theproposed method the imposition of static data is made through the projection ofa sample (expressed as a vector of independent, identically distributed normalrandom variables) onto the nullspace of a data matrix, that is defined in termsof the KLE. The numerical implementation of the proposed method isstraightforward. Through numerical experiments for a model of second-orderelliptic equation, we show that the proposed method in multi-chain studiesconverges much faster than the MCMC method without conditioning. These studiesindicate the importance of conditioning in accelerating the MCMC convergence.

Conditioning by Projection for the Sampling from Prior Gaussian Distributions