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:

(8-14)

In this equation, u is the Darcy’s velocity or specific discharge vector (SI unit: m/s); κ is the permeability of the porous medium (SI unit: m2); μ is the fluid’s dynamic viscosity (SI unit: Pa·s); p is the pore pressure (SI unit: Pa) and ρ is the density of the fluid (SI unit: kg/m3).

For large-scale applications it might be necessary to take gravity effects into account. Darcy’s law then applies when the gradient in hydraulic potential drives fluid movement in the porous medium. Visualize the hydraulic potential field by considering the difference in both pressure and elevation potential from the start to the end points of the flow line. According to Darcy’s law, the net flux across a face of porous surface is

(8-15)

In this equation g is the vector of gravitational acceleration. Furthermore, in the Subsurface Flow Module, g can be defined as g = − g∇D where g is the magnitude of gravitational acceleration (SI unit: m/s2), D is the elevation (SI unit: m), and ∇D is a unit vector in the direction over which the gravity acts. Here the permeability κ represents the resistance to flow over a representative volume consisting of many solid grains and pores.

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Note that gravity effects are only included in some modules. For a detailed overview of the functionality available in each product visit http://www.comsol.com/products/specifications/.

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Gravity effects are not active by default. Select the check box to activate the acceleration of gravity. Setting the elevation D to zero also turns off gravity effects.

The Darcy’s Law interface combines Darcy’s law with the continuity equation

(8-16)

In the above equation, ρ is the fluid density (SI unit: kg/m3), εp is the porosity, and Qm is a mass source term (SI unit: kg/(m3·s)). 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.