Knowlpedia

Definition

Incompressible porous media equation

Density transport coupled to a Darcy velocity and zero divergence.

In a normalized two-dimensional convention, the incompressible porous media equation (IPM) is

∂tρ+u⋅∇ρ=F,u=−∇p−ρe2,div⁡u=0,e2=(0,1).\partial_t\rho+u\cdot\nabla\rho=F,\qquad u=-\nabla p-\rho e_2,\qquad \operatorname{div}u=0, \quad e_2=(0,1).

Here ρ\rho is density, pp is pressure, and the second relation is the normalized Darcy law. The and act in space. When Euclidean L2L^2 projection is applicable, u=−P(ρe2)u=-\mathbb P(\rho e_2) with the , making this an .

Model conventions

Positive physical constants can be included for permeability, viscosity, and gravity before normalization. Gravity direction fixes the density sign. Domain and boundary conditions matter when recovering pressure. A sourced equation allows F≠0F\ne0; the unforced model sets F=0F=0.

References