Question 1: Can we upscale single phase permeability in microporous rocks?
Labelling of the connected components of the primary porosity in a sub-volume of the Estaillades sample. (A) The raw micro-CT image, (B) the 15-phase segmentation of a sub volume extracted from the microCT image (red box), (C) the segmented primary porosity, and (D) the uniquely labelled connected primary porosity. Images in this figure were rendered using SciKitImage. Figure Credit: Dr. Hannah P. Menke.
The upscaling workflow: A 360 cubed voxel block (A) is cut into 60 cubed sub-volumes (B). The permeability for each sub-volume is then solved using DBS for numerical upscaling, the features extracted for machine learning upscaling, and the porosity calculated for Kozeny–Carman upscaling. The upscaled permeability for each model is then assigned to each upscaling Darcy block (C), and then the Darcy permeability is solved on the upscaled volume (D). Images in this figure were rendered using Paraview. Figure Credit: Dr. Hannah Menke
Significance: The permeability of a pore structure is typically described by stochastic representations of its geometrical attributes (e.g. pore-size distribution, porosity, coordination number). Database-driven numerical solvers for large model domains can only accurately predict large-scale flow behavior 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 type of structures (e.g. micro-porosity, cavities, fractures) are interacting. It is the connectivity both within and between these fundamentally different structures that ultimately controls the porosity–permeability relationship at the larger length scales. Recent advances in machine learning techniques combined with both numerical modelling and informed structural analysis have allowed us to probe the relationship between structure and permeability much 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 micro-CT image of Estaillades limestone was divided into small 603 and 1203 sub-volumes and permeability was computed using the Darcy–Brinkman–Stokes (DBS) model. The microporosity–porosity–permeability relationship from Menke et al. 2022 was used to assign permeability values to the cells containing microporosity. Structural attributes (porosity, phase connectivity, volume fraction, etc.) of each sub-volume were extracted using image analysis tools and then regressed against the solved DBS permeability using an Extra-Trees regression model to derive an upscaled porosity–permeability relationship. Ten test cases of 3603 voxels were then modeled using Darcy-scale flow with this machine learning predicted upscaled porosity–permeability relationship and benchmarked against full DBS simulations, a numerically upscaled Darcy flow model, and a Kozeny–Carman model. All numerical simulations were performed using GeoChemFoam, our in-house open source pore-scale simulator based on OpenFOAM. We found good agreement between the full DBS simulations and both the numerical and machine learning upscaled models, with the machine learning model being 80 times less computationally expensive. The Kozeny–Carman model was a poor predictor of upscaled permeability in all cases.
