Predicting nonlinear-flow regions in highly heterogeneous porous media using adaptive constitutive laws and neural networks

• Heterogeneous Permeabilities and Fluid Velocities: In porous media with varying permeabilities, a range of fluid velocities can result in significant inertial and frictional effects, which may not be adequately addressed by Darcy’s law alone. • Darcy–Forchheimer Law: For regions with high-speed fluid flows, the nonlinear Darcy– Forchheimer law, which includes an additional term to account for these effects, is more suitable but computationally expensive to apply globally. • Adaptive Model for Law Selection: A recent model adapts dynamically to select either Darcy’s law or Darcy–Forchheimer law based on local fluid velocity, optimizing computational resources by restricting the more complex law to necessary subdomains. • Neural Network Training for Domain Partitioning: To improve efficiency, the study uses data from the adaptive model to train neural networks to classify each mesh cell as requiring Darcy or Darcy–Forchheimer law, based on input parameters like boundary conditions and inertial coefficients. • Validation and Results: The approach is validated with two test cases, and various performance metrics such as cost functions, parity plots, precision-recall plots, and receiver operating characteristic curves are analyzed to demonstrate the effectiveness of the neural network-based classification. In a porous medium featuring heterogeneous permeabilities, a wide range of fluid velocities may be recorded, so that significant inertial and frictional effects may arise in high-speed regions. In such parts, the link between pressure gradient and velocity is typically made via Darcy’s law, which may fail to account for these effects; instead, the Darcy–Forchheimer law, which introduces a nonlinear term, may be more adequate. Applying the Darcy–Forchheimer law globally in the domain is very costly numerically and, rather, should only be done where strictly necessary. The question of finding a prori the subdomain where to restrict the use of the Darcy–Forchheimer law was recently answered in [1] by using an adaptive model: given a threshold on the flow’s velocity, the model locally selects the more appropriate law as it is being solved. At the end of the resolution, each mesh cell is flagged as being in the Darcy or Darcy–Forchheimer subdomain. Still, this model is nonlinear itself and thus relatively expensive to run. In this paper, to accelerate the subdivision of the domain into low- and high-speed regions, we instead exploit the adaptive model from [1] to generate partitioning data given an array of different input parameters, such as boundary conditions and inertial coefficients, and then train neural networks on these data classifying each mesh cell as Darcy or not. Two test cases are studied to illustrate the results, where cost functions, parity plots, precision-recall plots and receiver operating characteristic curves are analyzed.