Groundwater flow equation

The groundwater flow equation is often derived for a small representative elemental volume (REV), where the properties of the medium are assumed to be effectively constant.

It is simply a statement of accounting, that for a given control volume, aside from sources or sinks, mass cannot be created or destroyed.

This mathematical statement indicates that the change in hydraulic head with time (left hand side) equals the negative divergence of the flux (q) and the source terms (G).

This equation has both head and flux as unknowns, but Darcy's law relates flux to hydraulic heads, so substituting it in for the flux (q) leads to Now if hydraulic conductivity (K) is spatially uniform and isotropic (rather than a tensor), it can be taken out of the spatial derivative, simplifying them to the Laplacian, this makes the equation Dividing through by the specific storage (Ss), puts hydraulic diffusivity (α = K/Ss or equivalently, α = T/S) on the right hand side.

Especially when using rectangular grid finite-difference models (e.g. MODFLOW, made by the USGS), we deal with Cartesian coordinates.

However, it has an option to run in a "quasi-3D" mode if the user wishes to do so; in this case the model deals with the vertically averaged T and S, rather than k and Ss.

The Laplace equation can be solved using techniques, using similar assumptions stated above, but with the additional requirements of a steady-state flow field.

A common method for solution of this equations in civil engineering and soil mechanics is to use the graphical technique of drawing flownets; where contour lines of hydraulic head and the stream function make a curvilinear grid, allowing complex geometries to be solved approximately.

An alternative formulation of the groundwater flow equation may be obtained by invoking the Dupuit–Forchheimer assumption, where it is assumed that heads do not vary in the vertical direction (i.e.,

Assuming both the hydraulic conductivity and the horizontal components of flow are uniform along the entire saturated thickness of the aquifer (i.e.,

), we can express Darcy's law in terms of integrated groundwater discharges, Qx and Qy: Inserting these into our mass balance expression, we obtain the general 2D governing equation for incompressible saturated groundwater flow: Where n is the aquifer porosity.

The source term, N (length per time), represents the addition of water in the vertical direction (e.g., recharge).

For unconfined steady-state flow, this non-linearity may be removed by expressing the PDE in terms of the head squared: Or, for homogeneous aquifers, This formulation allows us to apply standard methods for solving linear PDEs in the case of unconfined flow.