The Xxxxxxxx Equations Sample Clauses

The Xxxxxxxx Equations. ‌ The Xxxxxxxx equations describe the flow of a viscous fluid in cavity and porous media. It was initially proposed in [2, 3] as a homogenization technique for the Navier-Xxxxxx equations. Since in real applications the number and the locations of the Xxxxxx-Xxxxx interfaces might not be known a priori, the unified equations in the Xxxxxxxx model represent an advantage over the domain decomposition methods coupling the Xxxxx and the Xxxxxx equations. Mathematically speaking the Xxxxxxxx model is a parameter-dependent combina- tion of the Xxxxx and Xxxxxx models. Letting Ω be a bounded domain in Rd with a regular enough boundary ∂Ω, the steady Xxxxxxxx equations read div u = 0 ∀x ∈ Ω  −ν ∆u + k(x)u + ∇p = f (x) ∀x ∈ Ω  Bu = g on ∂Ω (1.2) where ν ≥ 0 is the fluid viscosity and k(x) is the inverse permeability of the medium. If ν = 0 the Xxxxxxxx equations reduce to the Xxxxx ones. The challenge of this prob- lem is when the coefficient k = k(x) admits extreme large variations between different parts of Ω. In Fig. 1.4 we show, as an example, an actual permeability field used in a oil reservoir simulation benchmark (xxx.xxx00.xxx). Note that the permeability field admits jumps of several orders of magnitude between distinct layers of soil. The upper part is a Xxxxxx formation consisting of massive homogeneous highly permeable sandstones, while the lower part is a Ness formation, a heterogeneous mixture of sand- stones, mudrocks, and coal.