# 2.2 Hydraulic Head Gradient as a Manifestation of Other Variables and Conditions

Hydraulic gradient (Δ*h**/*Δ*L*) is often expressed mathematically in differential format as *dh/dL*. Rearrangement of Darcy’s law using this formulation shows that hydraulic gradient is a function of *Q*, *K*, and *A*:

(4) |

Therefore, a change in any one of these variables will manifest as a change in hydraulic gradient:

(5) |

(6) |

(7) |

The negative sign is due to the fact that water flows in a direction from higher head to lower head, as described previously. The term *–**dh/dL* represents the slope of the head decline in the direction of flow (the “steepness” of the hydraulic gradient).

Figure 5 summarizes this concept using three different, yet spatially-uniform scenarios. The hydraulic gradient, which is commonly measured by way of water levels in wells, is not the controlling parameter that dictates flow. Rather, hydraulic gradient is a manifestation of the combined effects of the system geometry, hydrogeologic properties and the flow rate imposed on the system.

In the example shown in Figure 6, *K*_{2}*<K*_{1} whereas *Q* and *A* are constant. *Q* is the same at every location along the tube because mass is conserved. Therefore, as indicated by Darcy’s law, the gradient in the region of* K*_{2} must be steeper than in the other regions. This simple scenario is an example of a heterogeneity; in this case, hydraulic conductivity is not uniform.

# Example Problem 1

Sketch the horizontal hydraulic head gradient along the length of the apparatus shown here in a manner similar to the way the gradient is shown in Figure 6 (there is no need to know the actual head values, so you can create your own relative values).

Click here for solution to Example Problem 1