Designing porous materials with specific properties requires advanced inverse design methods to handle the relationships between microstructure and material behavior. This thesis presents a conditional variational autoencoder (CVAE) model that learns these relationships using Minkowski functionals, geometric descriptors including porosity, surface area, and Euler characteristic, as conditioning inputs. By using the CVAE with convolutional neural networks (CNNs), the model captures morphological features of 2D porous microstructures and links them to the target Minkowski descriptors. A Statistical Representative Elementary Volume (SREV) approach is adopted in data preparation to ensure the descriptors are representative of bulk material behavior, allowing the model to account for natural variability in microstructures. The trained CNN-CVAE can generate realistic porous microstructures that meet specified descriptor targets, effectively performing inverse design. Key findings demonstrate that the model reproduces target Minkowski metrics with high fidelity and maintains plausible connectivity and pore structure. Notably, the CVAE exhibits strong generalization, accurately generating microstructures for descriptor combinations not seen during training. These results highlight the potential of Minkowski functionals as effective design descriptors in machine-learning-driven material design. In addition, they confirm that the CVAE is capable of generating and tailoring porous structures to meet targeted descriptor requirements.

The first part of the thesis explores the process of homogenization, particularly for the permeability properties of rock samples. The Hill-Mandel postulate of energy consistency throughout the transitioning of scales is revisited since traditional homogenization methods rely on applying specific boundary conditions to enforce energy consistency. However, it is shown that the applied boundary conditions influence the effective physical parameter and provide upper or lower bound estimations. Recently, it is shown that these boundary conditions influence a layer near the boundary of the sample and that homogenization applied on the subsample away from this boundary layer is not affected by the boundary conditions.

The research focuses on energy consistency by studying the evolution of the energy within the intrinsic subsample, away from the boundaries. With the help of Finite Element simulations of Stokes-flow through idealized structures, the energy of the fluid is traced without the influences of grain properties on the energy dissipation. By plotting the ratio of the energy dissipation of the macro- and micro-scale, it is shown that the energy consistency is not found within small subsamples. Yet, with a growing subsample, energy consistency is achieved naturally, without the enforcement of boundary conditions. As a result, it is concluded that the energy consistency is found at the Representative Elementary Volume (REV), which is a similar requirement as for traditional homogenization methods. The study of the natural energy consistency in idealized microstructures is extended to real microstructures, which include more natural heterogeneities, such as grain properties. It is shown that energy consistency is also found with the natural heterogeneities included, albeit with a slower convergence.

For the homogenization of the permeability of a sample, the energy ratio is now known to be unitary, which can be used as an accurate indicator to determine the size of the REV.

The second part of the thesis explores the process of determining the size of the REV, which is a common, yet essential practice in Digital Rock Physics. Currently, this is an extensive exercise, involving many and large-size simulations to trace the convergence of the physical property and requires a lot of computational resources and time. Numerical-statistical studies have shown that the convergence of the REV visualizes in a cone-like shape. By plotting the convergence for both the permeability and the energy dissipation ratio for idealized microstructures, this study analyses the shape and evolution of the cone of convergence. From this, the generic evolution law of the convergence is determined.

It is shown that the asymmetrical convergence cone is described with a log-normal distribution, with a stable mean throughout the evolution of the cone and a variance for each sample size. The evolution of the variance is described with the law of large numbers, taking into account a reference value. This makes it possible to determine the size of the REV. Since the statistical method applies, information about the error of the fit, the error of the determined homogenized property, and the error of the size of the REV is provided.

The study is extended to real microstructures to validate whether the determined evolution law applies when natural heterogeneities are included. It is shown that the evolution law still accurately describes the REV's convergence. Therefore, the REV's size of real rock samples can also be determined. Even when the REV is not included within the sample, the evolution law can provide an estimate of the size of the REV or the homogenized property.

By using the cone of convergence, it is not necessary to run simulations on the full sample to find the REV, which is computationally expensive, instead running a number of simulations on small subsamples is sufficient, which saves both time and computational resources. It also unlocks the possibility to find the REV for high-resolution samples, as splitting the sample into subsamples allows for smaller simulations.