Homogenization-informed convolutional neural networks for estimation of effective transport properties in porous and fractured media
Keywords: Sustainability.
Energy Resources Engineering.

Understanding flow and mass transport properties of porous and fractured media is beneficial for many industrial, environmental, and biological applications. Such systems are difficult to model accurately and efficiently due to their multiscale nature, i.e. their macroscopic responses are dramatically impacted by the fine-scale characteristics of the microstructure. Advancement in imaging techniques has enabled both the detailed characterization of porous and fractured media at unprecedented resolution and the calculation of effective parameters (e.g. conductivity, permeability, dispersion) from direct numerical simulations at the fine-scale by correlating fluxes and driving forces. Yet, such simulations are computationally expensive since they must be performed on macroscopic domains while fully resolving the entire microstructure. In order to mitigate this issue, machine learning (ML) tools have been used to make computationally efficient predictions of effective microstructural properties once training is completed. However, the computational burden to develop datasets to train ML algorithms is significant since it requires the fine-scale solution of macroscopic flow and transport problems. This leads to cost-accuracy trade-offs: computational burden can be contained, for example, by reducing the size of the training set, at the expense of predictive capability. Alternatively, the topological complexity of the pore-scale structure can be restricted to more idealistic geometries to speed up simulations; however, this approach limits the scale-range of material parameters that can be accurately predicted. In this dissertation, we marry rigorous homogenization theory and ML techniques to significantly cut training costs by 2-3 orders of magnitude. We achieve this by determining the effective parameters through the solution of a local boundary value problem, i.e. closure problem, on a representative unit cell of the material, and training convolutional neural networks (CNNs) on such local solutions. This approach allows us to efficiently train CNNs on sets of two-dimensional computer-generated images of unprecedented size ($\sim10^5$ images) and microstructures with highly varying geometrical properties and realistic topological complexities to be applicable in a wide range of applications. The CNN is tested against both two-dimensional and three-dimensional structures and is capable of predicting effective properties that span 3-4 orders of magnitude. We specifically investigate effective conductivity of realistic structures of battery electrodes, as well as permeability and dispersion of porous and fractured media. First, we discuss an application to lithium-ion battery modeling, where a CNN is trained to estimate the effective diffusion and electric conductivity coefficients for macroscale electrochemical battery models. This CNN is trained using a dataset of 100,000 computer-generated images on which a partial differential equation (PDE) closure problem was solved. The images varied significantly in morphological properties to represent various electrode chemistries. The CNN is shown to achieve sufficiently high accuracy while being robust to noise and applicable to real electrodes images of various chemistries. Next, we describe a novel algorithm, named $\tau^2$-SIMPLE, for accurately solving the PDE closure problem for permeability and dispersion derived by homogenization theory. This algorithm, which introduces artificial time scales to enforce global constraints, converges to the unique solution for arbitrarily complex geometry and flow conditions and is used to generate training data for the remaining projects in this work. Next, we develop a CNN to predict both permeability and effective dispersion given an representative unit cell image of a porous material and a P\'eclet number to describe the flow regime. We explore how augmenting the network with physics-based inputs such as image transformations can improve learning. Finally, we apply this methodology to a new dataset of images depicting fractured media systems. Once again, the CNN accurately predicts effective permeability and dispersion tensors for a variety of fracture configurations across multiple flow regimes. In each work, we demonstrate that by framing the estimation of effective macroscopic parameters in the context of homogenization theory, we can expand the CNN training data by orders of magnitude, compared to current practices, and, consequently, encompass a wide variety of porosity, specific surface area, and other morphological characteristics, while not compromising on the total computational costs for training. This leads to the development of trained CNNs whose predictive capabilities are much more robust across a significantly wider ranges of parameters and dynamic conditions, spanning up to four orders of magnitude.