Upscaling the porosity-permeability relationship of a microporous carbonate to the Darcy scale with machine learning

The permeability of a pore structure is typically described by stochastic representations of its geometrical attributes. Database-driven numerical solvers for large model domains can only accurately predict large-scale flow behaviour when they incorporate upscaled descriptions of that structure. The upscaling is particularly challenging for rocks with multimodal porosity structures such as carbonates, where several different types of structures are interacting. It is the connectivity both within and between these different structures that controls the porosity-permeability relationship at the larger length scales. Recent advances in machine learning combined with numerical modelling and structural analysis have allowed us to probe the relationship between structure and permeability more deeply. We have used this integrated approach to tackle the challenge of upscaling multimodal and multiscale porous media. We present a novel method for upscaling multimodal porosity-permeability relationships using machine learning based multivariate structural regression. A m-CT image of limestone was divided into sub-volumes and permeability was computed using the DBS model. The porosity-permeability relationship from Menke et al. was used to assign permeability values to the microporosity. Structural attributes of each sub-volume were extracted and then regressed against the solved permeability using an Extra-Trees regression model to derive an upscaled porosity-permeability relationship. Ten upscaled test cases were then modelled at the Darcy scale using the regression and benchmarked against full DBS simulations, a numerically upscaled Darcy model, and a K-C fit. We found good agreement between the full DBS simulations and both the numerical and machine learning upscaled models while the K-C model was a poor predictor in all cases.