Capsule Network for prediction of Permeability Tensor from X-ray images of porous media

Accurately determining the absolute permeability of heterogeneous and anisotropic porous materials, such as sedimentary deposits, is a critical step in pore-scale studies. In recent years, various machine learning models, particularly Convolutional Neural Networks (CNNs), have been employed to predict absolute permeability from X-ray images of porous media. However, these efforts have largely focused on predicting scalar permeability in a single direction, without addressing the full permeability tensor. The CNNs often struggle to capture image orientation, posing challenges for the prediction of tensorial properties such as permeability. To address this limitation, we have developed 3D Capsule Network (CapsNet) regression models to predict permeability tensors from 3D grayscale and binary X-ray images of porous media. We compiled a dataset comprising 3D images from six sandstone types. Corresponding permeability tensors were computed using the Lattice Boltzmann Method (LBM). Subsequently, we customized the CapsNet for a 3D regression problem and trained the model using the generated dataset. Our comparative analysis revealed that CapsNet outperformed CNN, achieving an overall R² score of 0.91 compared to CNN’s 0.86. Importantly, CapsNet demonstrated greater consistency across various rock types and flow directions, whereas CNNs exhibited more variability and generally underperformed. To the best of our knowledge, this study represents the first application of Capsule Networks in the context of porous media analysis. Our findings highlight the superior predictive capability of CapsNets over CNNs, suggesting their potential as a robust alternative for characterizing porous materials in a wide range of applications, including carbon capture and storage, enhanced oil recovery, membrane design, and biomedical studies.