Averaging of flows with capillary hysteresis in stochastic porous media

Abstract

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.

Instability of gravity wetting fronts for Richards equations with hysteresis

“…Existence properties of the hysteresis system were studied in [23] for s-independent permeability and affine capillary pressure, neglecting gravity. The emphasis in that existence result was to generalize relation (1.2) to a Prandtl-Ishlinskii hysteresis relation in order to treat the system which is obtained after homogenization.…”

“…The emphasis in that existence result was to generalize relation (1.2) to a Prandtl-Ishlinskii hysteresis relation in order to treat the system which is obtained after homogenization. In order to specify the results of [23] to our context, we set Γ(x, .) ≡ δ γ (.)…”

“…≡ δ γ (.) and p c (σ) ≡ aσ + b, and read the results for s(x, t) := u(x, t), a := a * , b := b * , and k := K * , where the letters used in [23] appear on the right hand side of the four settings. Equations (1.8)-(1.10) of [23] with w(x, t) ≡ w(x, γ, t) then read…”

“…The general Prandtl-Ishlinskii hysteresis can be formulated equivalently, replacing the finite sums by integrals. A homogenization result is derived in [23] for linear laws p j : a porous medium which consists of different materials that exhibit the play-type hysteresis (1.2) (with different parameters) can be described in its averaged behavior by a Prandtl-Ishlinskii relation.…”

“…In Subsection 2.1 we recall existence results of [23]. In Subsection 2.2 we define our concept of stability.…”