Multiscale Sampling for Subsurface Characterization

In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian framework combined with a preconditioned Markov Chain Monte Carlo (MCMC) for the solution of the inverse problems. Reduction of uncertainty can be accomplished by incorporating measurements at sparse locations (static data) in the prior distribution. The first contribution of this work is a new method to condition Gaussian fields (the log of permeability fields) to available measurements. A truncated Karhunen-Lo`eve expansion (KLE) is used for dimension reduction. In the proposed method the imposition of static data is made through the projection of a sample (expressed as a vector of independent, identically distributed normal random variables) onto the nullspace of a data matrix, that is defined in terms of the KLE. Through numerical experiments for a model second-order elliptic equation we show the importance of conditioning in accelerating MCMC convergence. The second contribution of this dissertation is the introduction of a new multiscale sampling strategy. This is a new algorithm to decompose the stochastic space in orthogonal complement subspaces, through a one-to-one mapping onto a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by Gibbs sampling: we apply a KL expansion locally, at the subdomain level. The effectiveness of the proposed framework is tested also in the solution of inverse problems related to elliptic partial differential equations. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. We propose a new method to speed up MCMC studies of subsurface flow problems as the third contribution of this dissertation. We formulate a multiscale perturbation method for uncertainty quantification problems. The new procedure is presented for (linear) contaminant transport problems in the subsurface. The method, however, may be applicable to non-linear problems, such as two-phase immiscible displacements in petroleum reservoirs.