Averaging of flows with capillary hysteresis in stochastic porous media
Ben Schweizer · European Journal of Applied Mathematics · 2007
Fluids in unsaturated porous media are described by the relationship between pressure ( p ) and saturation ( u ). Darcy's law and conservation of mass provides an evolution equation for u , and the capillary pressure provides a relation between p and u of the form p ∈ p c ( u ,∂ t u ). The multi-valued function p c leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.