Modeling nanoparticle-stabilized foam flow in porous media: mathematical analysis and uncertainty quantification

Abstract

This work proposes a local equilibrium model and a population balance model to study foam flow assisted by nanoparticles in porous media. In both cases, we include the nanoparticle effect reducing foam mobility in a version of the Stochastic Bubble Population model, based on literature experimental data for silica nanoparticles. The population balance nanoparticle-stabilized foam model, besides including foam texture as a variable, incorporates particle retention and the resulting permeability reduction. The local equilibrium model is a non-strictly hyperbolic system of conservation laws. We obtain the global analytical solution, which allows us to quantify the effect of nanoparticles on foam flow. When only gas is injected, the breakthrough time and water production increase with the nanoparticle concentration, but this effect is less pronounced for high concentrations. Counterintuitively, during water-gas co-injection for a certain parameter range, adding nanoparticles yields a negligible effect on water production. We also investigate a simplified version of this model (using quadratic relative permeability functions), allowing for algebraic expressions to construct the solution profiles. We conduct uncertainty quantification and sensitivity analysis for the quadratic model, focusing on breakthrough time, water production, and pressure drop. Nanoparticles significantly reduce uncertainty propagation due to foam stabilization. Our findings indicate that the effect of nanoparticles exceeds the model’s uncertainty, suggesting that measuring it experimentally is statistically feasible. We achieve convergence even using the Monte Carlo method, evidencing how analytical solutions drastically reduce computational costs. For the population balance model, we obtain a semi-analytical solution under steady-state conditions. We study foam flow considering nanoparticle concentrations of 0.1, 0.5, and 1.0 wt%, with and without NaCl, using retention parameters from experiments. Higher concentrations increase foam viscosity, reduce water saturation, and improve sweep efficiency, while 0.1 wt% shows minimal impact. Neglecting nanoparticle retention generally underestimates pressure drop, especially in scenarios with significant retention. However, while retained nanoparticles increase pressure by reducing permeability, the loss of suspended nanoparticles decreases pressure by reducing the foam’s apparent viscosity. Consequently, when considering both nanoparticle loss and reduced permeability, the pressure drop is higher than in models that ignore retention. In contrast, omitting retention effects on permeability, the pressure drop is lower.