Groundwater Flow Equation - Two-dimensional Groundwater Flow

Two-dimensional Groundwater Flow

The above groundwater flow equations are valid for three dimensional flow. In unconfined aquifers, the solution to the 3D form of the equation is complicated by the presence of a free surface water table boundary condition: in addition to solving for the spatial distribution of heads, the location of this surface is also an unknown. This is a non-linear problem, even though the governing equation is linear.

An alternative formulation of the groundwater flow equation may be obtained by invoking the Dupuit assumption (or Dupuit-Forcheimer assumption), where it is assumed that heads do not vary in the vertical direction (i.e., ). A horizontal water balance is applied to a long vertical column with area extending from the aquifer base to the unsaturated surface. This distance is referred to as the saturated thickness, b. In a confined aquifer, the saturated thickness is determined by the height of the aquifer, H, and the pressure head is non-zero everywhere. In an unconfined aquifer, the saturated thickness is defined as the vertical distance between the water table surface and the aquifer base. If, and the aquifer base is at the zero datum, then the unconfined saturated thickness is equal to the head, i.e., b=h.

Assuming both the hydraulic conductivity and the horizontal components of flow are uniform along the entire saturated thickness of the aquifer (i.e., and ), we can express Darcy's law in terms of integrated discharges, Q_x and Q_y:

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). By incorporating the correct definitions for saturated thickness, specific storage, and specific yield, we can transform this into two unique governing equations for confined and unconfined conditions:

(confined), where S=S_sb is the aquifer storativity and

(unconfined), where S_y is the specific yield of the aquifer.

Note that the partial differential equation in the unconfined case is non-linear, whereas it is linear in the confined case. 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. For heterogeneous aquifers with no recharge, Potential flow methods may be applied for mixed confined/unconfined cases.