Incorporating Pore-Scale Data in Field-Scale Uncertainty Quantification: A Multi-Scale Bayesian Approach

Pore-scale modeling is recently become an important tool for a deeper understanding of complex transport phenomena in porous media. However its direct usage for field-scale processes is still hindered by limited predictive capabilities. This is due to the large uncertainties in the micro-scale parameters, in the pore geometries, in the limited number of available samples, and in the numerical errors. These issues are often overlooked because it is usually thought that the computational cost of pore-scale simulation prohibits an extensive uncertainty quantification study with large number of samples.

In this work we propose an computational tool to estimate statistics of pore-scale quantities. The algorithm is based on (i) an efficient automatic CFD solver for pore-scale simulations, (ii) a multi-scale Bayesian theoretical framework, and (iii) a generalized multilevel Monte Carlo to speed up the statistical computations. Exploiting the variance reduction of the multi-level and multi-scale representation, we demonstrate the feasibility of the forward and inverse uncertainty quantification problems. The former consists in quantifying the effect of micro-scale heterogeneities and parametric uncertainties on macro-scale upscaled quantities. Given some prior information on the pore-scale structures, the latter can be applied to (i) assess the validity and estimate uncertainties of macro-scale models for a wide range of micro-scale properties, (ii) match macro-scale results with the underlying pore-scale properties.