As shown in Equation 16, the ratio of ∆h and ∆L (the hydraulic head difference divided by the length of the sample or the distance separating two head locations) can be generalized into a differential called the hydraulic gradient, dh/dl as in Equation 21.

 (21)

where:

 dh = ∆h over an infinitesimal interval (L) dl = ∆L over an infinitesimal interval (L)

 (22)

Hydraulic gradient can also be represented in three dimensions when flow is not aligned with a coordinate axis, x, y, or z (Equation 23), so there is a component of flow in each axis direction as shown in Equation 23. In this case, in order to determine the direction of flow, each hydraulic gradient component is calculated with h2 being located at a larger coordinate position than h1.

 (23)

The partial differential (∂) representation (∂\∂x) is used because the gradient is partially dependent on the conditions in each of the coordinate directions. The resulting gradient vector (the overall magnitude and direction of the gradient) is dependent on the magnitudes of all three components of gradient in the x, y, and z directions.

The hydraulic gradient is commonly represented using the letter “i” such that Darcy’s law is often written as shown in Equation 24.

 Q = – KiA (24)

where:

 i = hydraulic gradient (Δh/ΔL or dh/dl) (dimensionless: L/L)

The difference in the hydraulic head over a distance along the flow path is defined as the hydraulic gradient, ΔhL. This gradient of mechanical energy is the driving force of groundwater flow. If water is not moving, the gradient is zero, and the value of head is the same everywhere. In this situation, hydrostatic conditions exist. Under hydrostatic conditions the elevation head and pressure head at any location in the porous media combine to form the same hydraulic head value (Figure 24).

Groundwater flow occurs in the direction of the decreasing head, independent of the position of the porous medium in space (Figure 20). Local gradients are impacted by the magnitude of the hydraulic conductivity, flow rate, and cross-sectional area as stated by Darcy’s Law (Equation 22) and illustrated in Figure 25.