Note About the Groundwater Flow Equation
In the area of hydrogeology, the groundwater flow equation is commonly used to describe the groundwater flow through an aquifer. In COMSOL Multiphysics, equation-based models of physical processes are solved by ensuring a balance of modeled quantities, such as mass or energy.
The derivation of the groundwater flow equations starts with the mass balance in a representative elementary volume (REV). This entails ensuring that the fluxes into and out of the REV are in equilibrium with the mass stored within the volume. This relationship can be expressed as follows:
with S denoting the storage within the volume.
Figure 2-2: Fluxes through an elementary volume.
The flux in the x direction is:
or in differential form
The flux balance in the y and z directions are defined accordingly and the total flux balance in the volume is
(2-1)
The specific storage Ss is defined as
where Δh denotes the change in hydraulic head, a specific measure of fluid pressure. The rate of change in volume of water Vw then is
together with Equation 2-1, the definition of the flux q = ρu, and accounting for source/sink terms Qm this leads to the formulation of the groundwater flow equation
(2-2)
In the COMSOL Multiphysics® software Equation 2-2 is formulated in terms of pressure within the Darcy’s Law interface (See Pressure, Pressure Head, and Hydraulic Head Formulation). Use the Storage Model to include the Storage term on the left hand side of the equation and add a mass source to include the sink term on the right hand side.
Hydraulic Conductivity
The velocity u in Equation 2-2 is commonly described by the formulation of Darcy law in terms of the gradient of the hydraulic head as follows:
(2-3)
where K(m/s) is the hydraulic conductivity which is primarily used in hydrology applications where the fluid is water. It is related to the permeability according to
Note that the hydraulic conductivity is a property of both the porous matrix and the fluid, whereas the permeability describes solely the properties of the fluid.