Small- and Waiting-Time Behavior of a Darcy Flow Model with a Dynamic Pressure Saturation Relation

We address the small-time evolution of interfaces (fronts) for the pseudoparabolic generalization ∂u ∂t = ∂ ∂x ( u ∂u ∂x + u ∂2u ∂x∂t ) of the porous-medium equation, identifying regimes in which the local behavior remains fixed for some finite time and others in which it changes instantaneously. A number of phenomena beyond those exhibited by the porous-medium equation are elucidated, including retreating fronts and novel types of local behavior. Related results for the important limit case ∂u ∂t = ∂ ∂x ( u ∂2u ∂x∂t )