# 7.3 Governing Equations for Unconfined Groundwater Flow

Unconfined flow equations are non-linear in that the transmissivity of the aquifer depends on the saturated thickness and the saturated thickness varies in the direction of flow because the water table slopes. By definition, a confined aquifer is fully saturated, so transmissivity values are constant for a confined aquifer of constant thickness. Under water table conditions the water table slopes, flow is parallel to the water table and the saturated thickness decreases in the direction of flow (Figure 55). If the slope of the water table is small, Darcy’s Law can be applied to develop governing equations by using the Dupuit simplification or Dupuit assumptions. Dupuit’s simplification uses the approximate gradient (*dh over the distance x*, –*dh*/*dx*) rather than the true gradient (*dh over the flow path length*, –*dh*/*dL*) by assuming the flow is horizontal (no vertical components of flow) as shown by the solid arrows in Figure 55 and by Equation 72.

rather than | (72) |

Expressing the flow area as the product of the height of the water table and the unit width of the system in the *y* direction, the flow is shown in Equation 73.

(73) |

If the bottom of the unconfined aquifer is used as the datum, then the head defines the saturated thickness. To include this dependency in the flow equations, Equation 70 is adjusted so that aquifer thickness, *b*, is replaced with *h*, and the varying value of *h* has to be inside the derivative. In addition, to represent unconfined flow, specific yield, *S*_{y}, is used as the aquifer storativity. Thus, *unconfined, two-dimensional (plan view), transient, anisotropic, heterogeneous* conditions of groundwater flow are represented by Equation 74.

(74) |

For *unconfined, two-dimensional (plan view), transient, anisotropic, homogeneous* conditions of groundwater flow, hydraulic conductivities do not need to be within the derivative, resulting in Equation 75.

(75) |

*Unconfined, two-dimensional, plan view, transient, isotropic, homogeneous* flow is represented using only one value of *K* as shown in Equation 76.

(76) |