Information-theoretic approach for upscaling
Keywords: Energy Resources Engineering.

Recent advances in hardware, algorithms, and scientific computing open new possibilities to revisit long-standing problems with additional tools in our arsenal. Examples of such problems, which are explored in this study, are inverse problems and upscaling of dynamic models. Both problems include nonlinear and functional minimization of the discrepancy between data and model predictions. This study develops an information-theoretic approach for upscaling of dynamic models affected by uncertainty. An optimal, time-dependent, probabilistic characterization of the macroscale model is obtained to yield minimum discrepancy with respect to assigned quantities of interest provided by the microscale solution. Hard data forming a training set are obtained at the fine scale from repeated synthetic simulations or from observations. Fine-scale information is transferred to the coarse scale via minimization of a loss function that consists of the cumulative average discrepancy, regularized by the cumulative exchange of information measured via mutual information. This general procedure is applied to transient flow processes in heterogeneous media. Upscaling of the mean uniform transient flow in heterogeneous formations yields a time-dependent effective conductivity. As a result, the corresponding optimization of the aforementioned loss function is not trivial and, therefore, a flow simulator is developed with machine-learning software that uses state-of-the-art optimization methods. This methodology provides results that coincide with results obtained with traditional methods and further expands them in two major ways. First, it yields a probabilistic distribution of the upscaled parameter rather than a single value (the ensemble mean); this allows one to quantify prediction uncertainty of the upscaling procedure. Second, our methodology does not impose any physical constraints and limitations, such as the assumption of mild heterogeneity of a porous medium that underpins the perturbation-based strategies for conductivity scaling. In addition, our information-theoretic methodology is further expanded to tackle multi-dimensional transient flows. The change of direction of the mean flow induces anisotropy in the upscaled conductivity tensor, whose full effects cannot be captured with a two-point flux-approximation simulator. For this reason, a multi-point flux-approximation simulator is developed using the machine learning software. This allows us to predict the temporal evolution of all the components of the upscaled conductivity tensor. Finally, this study examines the use of neural networks as a surrogate model for Markov chain Monte Carlo (MCMC). For this purpose, a two-dimensional encoder-decoder convolutional neural network (CNN), which has been previously developed for inverse problems, is modified. The specific application considered in this thesis is a thermal-hydrologic-chemical model. The main challenges that arise in this setting are the large number of inputs and outputs needed to describe the problem. We show that the CNN surrogate is able to capture all the complex physical processes and is accurate enough to be used for MCMC. In addition, since all the gradient information is known, it enables the use of more evolved MCMC algorithms such as Hamiltonian Monte Carlo