# A Multiscale CNN-Based Intrinsic Permeability Prediction in Deformable Porous Media
## 1 Introduction
For instance, the lattice Boltzmann method (LBM) can be applied to simulate the flow on the microscale through representative volume elements (RVEs) of the analyzed material. For model details and references, see, e.g., [11; 33; 36; 37]. The implementation of LBM is a common practice in computational fluid dynamics, especially for handling complex boundaries of heterogeneous porous materials. In this context, [PERSON] et al. [38] explored the integration of LBM and the discrete element method (DEM) to simulate fluid-particle interactions, highlighting the importance of iterative testing to ensure that synthetic data accurately reflect real-world permeability behavior. A study by [PERSON] et al. [39] reviewed current advancements in data-driven approaches for analyzing flow and transport in porous media. A special focus was placed on methodologies like image-based techniques, data-driven flow modeling, and physics-informed machine learning, offering insights into their applications and future research directions. [PERSON] et al. [40] explored four hybrid physics-machine learning (ML) methods for predicting fracture properties in porous media: _residual modeling_, which refines predictions by learning model errors; _integrated coupling_, where a physical model's output feeds into an ML model; _simple averaging_, which blends predictions from both models; and _bootstrap aggregating_, an ensemble approach leveraging multiple physical models. Their comparison found residual modeling to be the most effective, offering superior predictive performance. The paper by [PERSON] et al. [41] introduced a physics-informed CNN model that leverages transfer learning to simulate two-phase flow in porous media under time-varying controls. The approach enhanced simulation efficiency and accuracy, particularly in scenarios with dynamic boundary conditions. Within ML-based multiscale modeling of flow through porous materials, regression models using simple feed-forward neural networks (FFNN) are successfully utilized by [PERSON] et al. [23] and [PERSON] et al. [25] for capturing the time-independent permeability tensor. On the other hand, path-dependent responses, such as the retention model, could be captured using the recurrent neural networks (RNN) or the one-dimensional (1D) convolutional neural networks (1D-CNN), as thoroughly discussed in [23,25]. While the aforementioned FNN regression models use only the LBM datasets, in this work, we discuss the use of two-dimensional (2D) convolutional neural networks (CNN) to predict the macroscopic anisotropic permeability tensor at different deformation states. From an implementation perspective, the FFNN model can bypass the explicit determination of the permeability tensor and instead directly replace [PERSON]'s law within the macroscopic model of porous media. In contrast, the proposed CNN is designed to predict the permeability tensor as an effective property for macroscopic modeling. The supervised learning process in the proposed CNN model uses binarized CT images of real microgeometry as inputs, while the output is the second-order symmetric intrinsic permeability tensor. Figure 1 illustrates the major steps in this model, which include building the dataset based on CT images from Bentheim sandstone and training the 2D CNN model with the processed CT images as inputs and permeability tensors as outputs. This CNN-based treatment has shown its effectiveness in various studies, such as within multiscale magnetostatics [42] and multiscale modeling of heterogeneous materials [43]. After this introduction, the rest of the paper is organized as follows: Section 2 presents the fundamentals of macroscopic modeling of saturated porous media based on the TPM. This section covers the concepts of homogenization, kinematics, local balance relations, and essential constitutive relations for biphasic porous materials. Additionally, it reviews  
Figure 1: Overview of the major steps of the 2D CNN model for predicting the effective properties of porous media (mainly permeability) based on CT images.