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:

(4-1)

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).

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The above equation formulation is called linear Darcy law. The applicability of this formulation is discussed in the chapter About Darcian and Non-Darcian Flow in the Porous Media Flow Module User’s Guide.

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Permeability can be defined in different ways, either directly or based on predefined permeability models. These are described in detail in the chapter Permeability Models.

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For a detailed overview of the permeability models available in each product visit https://www.comsol.com/products/specifications/.

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

(4-2)

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.

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. Darcy’s law then has the form:

(4-3)

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See the section Gravity Effects in the Subsurface Flow Module User’s Guide.

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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 https://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.