Optimal path for the flow of yield stress fluid

Numerous industrial and environmental applications rely on an understanding of fluid behavior within porous materials, from remediating groundwater contamination to carbon dioxide storage. Yield stress fluids, which resist flow until a critical stress threshold is exceeded, present both unique challenges and opportunities. Capillary forces, responsible for separating immiscible fluids, significantly influence multi-phase flow in porous structures. This thesis embarks on a theoretical exploration of non-linear flow within porous media, where both forces contribute to the overall rheology, interacting with the geometrical disorder at the pore level. The present study investigates the steady-state flow of yield stress blobs in a capillary fiber bundle filled with Newtonian liquid, as well as the flow of a single Bingham fluid in a tree-like pore network. We derive expressions for the pressure threshold and the average flow rate as a function of the applied pressure drop. In both problems, the non-linearity observed stems from the sequential opening of flowing paths with increasing pressure drop, and the distribution of pressure thresholds related to these paths defines the specific flow rate law. Furthermore, we explore the transition from viscous fingering to foam during immiscible drainage in a two-dimensional porous medium. We characterize this transition through numerical simulations using a dynamic pore network model, measuring the transition location, overall flow rate, and local pressure gradient as a function of imposed global pressure and viscosity ratio. We discuss potential mechanisms leading to this transition, considering local flow rate fluctuations, which we also measure and characterize. In all three cases studied, the porous medium’s heterogeneity emerges as a pivotal factor.