![]() ![]() Recently, X-ray imaging has allowed us to image the physical heterogeneity of carbonate rocks at different scales: at the core (cm) scale using medical-CT imaging 26, 27, at the pore (mm) scale using micro-CT imaging 28, 29, 30, 31, and at the nano (μm) scale using nano-CT imaging 32, 33, 34, 35. Moreover, flow measurements at the core scale are often not representative of flow at the reservoir scale due to large-scale heterogeneities present in the reservoir that are not captured in a small cm-scale core sample (e.g. Furthermore, it provides little insight into the local multi-scale structural influences on permeability, which is a well-recognized and significant challenge for upscaling carbonates 21, 22, 23, 24, 25 because bulk measurements do not provide enough information to accurately extrapolate flow properties to other samples based on structural information. This method can be expensive and time consuming. Traditionally, permeability has been measured in core flood experiments using Darcy’s law 20, which is based on bulk porosity measurements and the pressure drop across the core during flow. These porous structures are often heterogenous and range several orders of magnitude in scale, particularly in carbonate rocks which are an abundant geological material for both oil and gas reservoirs 15 and carbon storage sites 16, 17, 18, 19, making prediction of permeability in these cases especially difficult. The internal structure of a porous media defines its ability to transmit fluid and therefore it’s permeability. Predicting flow through porous media is pivotal for a broad range of scientific and engineering endeavors including fuel cells 1, 2, 3, oil and gas recovery 4, 5, 6, geologic carbon storage 7, 8, geothermal energy 9, 10, 11, material composites 12, and nuclear waste disposal 13, 14. The Kozeny–Carman model was a poor predictor of upscaled permeability in all cases. 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. All numerical simulations were performed using GeoChemFoam, our in-house open source pore-scale simulator based on OpenFOAM. Ten test cases of 360 3 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. 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. (Earth Arxiv,, 2019) was used to assign permeability values to the cells containing microporosity. The microporosity–porosity–permeability relationship from Menke et al. ![]() A micro-CT image of Estaillades limestone was divided into small 60 3 and 120 3 sub-volumes and permeability was computed using the Darcy–Brinkman–Stokes (DBS) model. We present a novel method for upscaling multimodal porosity–permeability relationships using machine learning based multivariate structural regression. We have used this integrated approach to tackle the challenge of upscaling multimodal and multiscale porous media. 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. It is the connectivity both within and between these fundamentally different structures that ultimately controls the porosity–permeability relationship at the larger length scales. micro-porosity, cavities, fractures) are interacting. The upscaling is particularly challenging for rocks with multimodal porosity structures such as carbonates, where several different type of structures (e.g. Database-driven numerical solvers for large model domains can only accurately predict large-scale flow behavior when they incorporate upscaled descriptions of that structure. pore-size distribution, porosity, coordination number). The permeability of a pore structure is typically described by stochastic representations of its geometrical attributes (e.g. ![]()
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