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

Foam injection with nanoparticles can be useful in various subsurface applications, particularly in challenging field conditions such as oil production in the Brazilian Pre-Salt. By enhancing foam stabilization, nanoparticles improve their effectiveness as mobility-control agents in gas flooding.  We propose new models of nanoparticle-stabilized foam flow in porous media, accounting for nanoparticle transport and its effect on reducing foam mobility. The first model considers foam at local equilibrium and is governed by a non-strictly hyperbolic system of conservation laws. The existence and uniqueness of a global solution as a sequence of waves are proved using entropy conditions. The analytical solution is utilized to evaluate the impact of nanoparticles on key industrial parameters such as breakthrough time, water production, and pressure drop over time. We also perform sensitivity analysis and uncertainty quantification studies. The second and more complex model accounts for particle retention and permeability reduction. In this case, a steady-state semi-analytical solution is presented. It is then used to investigate the impact of nanoparticle retention on water saturation, foam's apparent viscosity, and pressure drop profiles. We also discuss two opposing effects of retention on pressure drop and how models that neglect one or both of these effects underestimate pressure.