Numerical Investigation of Shear-Thinning Fluid Flow in Geological Fractures with Variable Aperture Using Finite-Volume and Finite Element Methods

The flow behavior of engineered fluids within geological fractures, characterized by variable aperture, often displays shear-thinning tendencies that contribute to flow localization in channels. To simplify the 3-D problem, lubrication theory is commonly employed, transforming it into a more manageable 2-D formulation and offering a computationally viable approximation. The resulting generalized Reynolds equation for shear-thinning fluids becomes highly nonlinear, owing to the fluid rheology and the heterogeneity in the aperture field. Addressing the resulting p-Laplacian governing equation poses significant numerical challenges. Balancing computational efficiency and solution accuracy is crucial, particularly when investigating this phenomenon within a stochastic framework involving multiple realizations. In this study, we employ two purpose-built numerical methods for solving the p-Laplace equation: a Finite-Volume Newton-based (FVN) method and a finite element DMK-optimization-based approach. A comparative analysis is conducted, evaluating the methods in terms of convergence, accuracy, and robustness. Subsequently, these approaches are utilized to explore the impact of adopting different rheological models (e.g., power-law versus Ellis) on the Quantities of Interest (QoI) in the problem. The investigation specifically focuses on the model's performance in challenging scenarios, such as those involving very shear-thinning fluids with near-zero rheological indices.