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Microscopic (pore-scale) properties of porous media affect and oftendetermine their macroscopic (continuum- or Darcy-scale) counterparts.Understanding the relationship between processes on these two scales isessential to both the derivation of macroscopic models of, e.g., transportphenomena in natural porous media, and the design of novel materials, e.g., forenergy storage. Most microscopic properties exhibit complex statisticalcorrelations and geometric constraints, which presents challenges for theestimation of macroscopic quantities of interest (QoIs), e.g., in the contextof global sensitivity analysis (GSA) of macroscopic QoIs with respect tomicroscopic material properties. We present a systematic way of buildingcorrelations into stochastic multiscale models through Bayesian networks. Thisallows us to construct the joint probability density function (PDF) of modelparameters through causal relationships that emulate engineering processes,e.g., the design of hierarchical nanoporous materials. Such PDFs also serve asinput for the forward propagation of parametric uncertainty; our findingsindicate that the inclusion of causal relationships impacts predictions ofmacroscopic QoIs. To assess the impact of correlations and causal relationshipsbetween microscopic parameters on macroscopic material properties, we use amoment-independent GSA based on the differential mutual information. Our GSAaccounts for the correlated inputs and complex non-Gaussian QoIs. The globalsensitivity indices are used to rank the effect of uncertainty in microscopicparameters on macroscopic QoIs, to quantify the impact of causality on themultiscale model's predictions, and to provide physical interpretations ofthese results for hierarchical nanoporous materials.

Causality and Bayesian network PDEs for multiscale representations of porous media