# A Multiscale CNN-Based Intrinsic Permeability Prediction in Deformable Porous Media
## 1 Introduction  
The study of single- and multiphase flow in deformable and possibly fractured porous materials is a topic of interest in various fields, such as petroleum engineering, hydrology, soil science, and biomechanics. Related applications and references can be found in, e.g., [PERSON] [1], [PERSON] et al. [2], [PERSON] et al. [3], [PERSON] et al. [4], [PERSON] [5], [PERSON] and [PERSON] [6], [PERSON] [7], [PERSON] et al. [8], [PERSON] and [PERSON] [9], [PERSON] et al. [10], [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [12], [PERSON] et al. [13]. The understanding and accurate mathematical description of fluid flow in these media is crucial for optimizing extraction processes, predicting contaminant transport, improving soil irrigation techniques, and predicting the stability of geotechnical structures, among others. However, the inherent heterogeneity and anisotropy of porous media pose a major challenge to the accurate prediction of their macroscopic properties, such as permeability. This underlines theneed to incorporate the microscopic information through an accurate but computationally efficient multiscale approach. In recent years, machine learning (ML), particularly artificial neural networks (ANN) has emerged as a promising tool to tackle several problems in multiscale material modeling. Using, e.g., deep neural networks (DNN), constitutive models based on lower-scale simulations can be found in different research works, such as within crystal plasticity [14; 15], elasto-plasticity for multiphase, composite materials [16], fracture mechanics [17; 18], hyperelasticity with enforced constitutive restrictions, as the symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency [19], and other applications in [20]. A recent review on ML applications within solid mechanics was presented by [PERSON] et al. [21]. Within multiscale modeling of porous materials, ANN, including convolutional neural networks (CNNs) and regression models, have demonstrated their ability to learn complex patterns and relationships. This made them suitable for predicting macroscopic properties and material models, such as anisotropic permeability, retention curves of unsaturated porous materials, and inelastic stress-strain relationships (see, e.g., [22; 23; 24; 25; 26; 27] for an overview). Specifically, the CNN approach holds significant capabilities in linking information across scales and in model parameter prediction within computational mechanics, as highlighted, for example, by [PERSON] et al. [28], [PERSON] and [PERSON] [29]. The application of ML in this context has the potential to significantly reduce the computational costs associated with traditional multiscale simulation methods while ensuring a higher degree of accuracy in comparison with phenomenological models. However, the good performance of these surrogate models requires a sufficient amount of data, e.g., small-scale CT images or numerical data. If the available image database is small, data enhancement techniques can help to overcome this challenge. For example, [PERSON] et al. [30] have used generative adversarial networks (GAN) together with actor-critic (AC) reinforcement learning to synthesize realistic and controlled 3D microstructures, which are also used in this work. For the macroscopic simulation of multiphase, heterogeneous, and deformable porous materials, the theory of porous media (TPM) can be employed, which represents a reliable and robust framework, as has been demonstrated in many works, e.g., [6; 31; 32; 33; 34; 35]. In this context, important parameters, such as deformation-dependent intrinsic permeability, relative permeabilities, and the degree of saturation in relation to capillary pressure, determine flow behavior in continuum multiphase porous media models. Unlike in phenomenological models and their assumptions, these parameters and the associated constitutive formulations can be estimated based on lower-scale flow simulations.