Darcy’s Law — Equation Formulation

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the structure of the porous medium:

(7-7)

In this equation, κ (SI unit: m2) denotes the permeability of the porous medium, μ (SI unit: kg/(m·s)) the dynamic viscosity of the fluid, p (SI unit: Pa) the pressure, and u (SI unit: m/s) the Darcy velocity. The Darcy’s Law interface combines Darcy’s law with the continuity equation:

(7-8)

In the above equation, ρ (SI unit: kg/m3) is the density of the fluid, ε (dimensionless) is the porosity, and Qm (SI unit: kg/(m3·s)) is a mass source term. Porosity is defined as the fraction of the control volume that is occupied by pores. Thus, the porosity can vary from zero for pure solid regions to unity for domains of free flow.

If the Darcy’s Law interface is coupled to an energy balance, then the fluid density can be a function of the temperature, pressure, and composition (for mixture flows). For gas flows in porous media, the relation is given by the ideal gas law:

(7-9)

where R= 8.314 J/(mol·K) is the universal gas constant, M (SI unit: kg/mol) is the molecular weight of the gas, and T (SI unit: K) is the temperature.