Permeability is a parameter that measures the resistance that fluid faces when flowing through a porous medium. Usually, this parameter is determined in routine laboratory tests by applying Darcy’s law. Those tests can be complex and time-demanding, and they do not offer a deep understanding of the material internal microstructure. Currently, with the development of new computational technologies, it is possible to simulate fluid flow experiments in computational labs. Determining permeability with this strategy implies solving a homogenization problem, where the determination of the macro parameter relies on the simulation of a fluid flowing through channels created by connected pores present in the material’s internal microstructure. This is a powerful example of the application of fluid mechanics to solve important industrial problems (e.g., material characterization), in which the students can learn basic concepts of fluid flow while practicing the implementation of computer simulations. In addition, it gives the students a concrete opportunity to work with a problem that associates two different scales. In this work, we present an educational code to compute absolute permeability of heterogeneous materials. The program simulates a Stokes flow in the porous media modeled with periodic boundary conditions using finite elements. Lastly, the permeability of a real sample of sandstone, modeled by microcomputed tomography (micro-CT), is obtained.

Predicting permeability from Nuclear Magnetic Resonance (NMR) data is a fundamental yet challenging task in reservoir characterization, primarily due to the uncertainty associated with surface relaxivity ( $$\rho $$ ρ ) parameters. In this work, we investigate the feasibility of using Machine Learning (ML) to estimate permeability from $$T_2$$ T 2 distributions and quantify how $$\rho $$ ρ uncertainty affects predictive accuracy. To address this, we generated a dataset of 15,000 synthetic 3D porous media representing granular sedimentary rock samples. We employed efficient in-house implementations of a Random Walk algorithm (governed by Bloch-Torrey physics) to simulate magnetization decay and obtain $$T_2$$ T 2 distributions, alongside a Finite Element Method (FEM) solver for the Stokes equations to compute absolute permeability, assuming 100% water saturation. The study comprises three computational experiments designed to isolate the impact of $$\rho $$ ρ . In the first experiment, we simulated $$T_2$$ T 2 distributions by assigning a constant $$\rho $$ ρ value for all synthetic porous media. In the second experiment, we applied a different $$\rho $$ ρ value to each synthetic porous medium to emulate real-world uncertainty, representing the scenario where $$\rho $$ ρ is unknown. The third experiment extends the second by converting the $$T_2$$ T 2 distributions into surface-to-volume ratio distributions using the specific $$\rho $$ ρ value assigned in the second experiment to each medium. We systematically compared the Multilayer Perceptron (MLP) performance against the industry-standard Schlumberger-Doll-Research (SDR) model. Overall, the MLP yielded strong predictive performance. The second experiment presented a performance drop for both models, confirming the impact of $$\rho $$ ρ uncertainty. The main contribution of this work is the systematic quantification of the sensitivity of predictive permeability models to $$\rho $$ ρ , establishing a controlled benchmark that addresses and reduces existing uncertainties. Additionally, these findings demonstrate that the MLP provides a robust and competitive alternative for permeability estimation in scenarios where $$\rho $$ ρ is unknown.