20th October – Iuliu Sorin Pop: Non-equilibrium models for flow in porous media

We discuss various aspects related to non-equilibrium mathematical models for porous media flow. Typically, such models assume that quantities like saturation, phase pressure differences, or relative permeability are related by monotone, algebraic relationships. Under such assumptions, the solutions satisfy the maximum principle. On the other hand, experimental work published in the past decades report that phenomena like saturation overshoot, or the formation of finger profiles have been observed whenever the flow is sufficiently rapid. Such results are ruled out by standard, equilibrium models.

This is the main motivation to consider non-equilibrium models, where dynamic or hysteretic effects are included in the above-mentioned relationships. The resulting models are nonlinear evolution systems of (pseudo-)parabolic and possibly degenerate equations, and involving differential inclusions. For such problems, we present first some results concerning the derivation of such models from the pore scale to the Darcy scale, the existence and uniqueness of weak solutions, and discuss different numerical schemes. This includes aspects like the rigorous convergence of the discretization, and solving the emerging nonlinear time-discrete or fully discrete problems.

This is joint work with X. Cao (Toronto), S. Karpinski (Munich), S. Lunowa (Munich), K. Mitra (Hasselt), F.A. Radu (Bergen)