Hydraulic Head - Hydraulic Gradient

The hydraulic gradient is a vector gradient between two or more hydraulic head measurements over the length of the flow path. It is also called the 'Darcy slope', since it determines the quantity of a Darcy flux, or discharge. A dimensionless hydraulic gradient can be calculated between two piezometers as:

where

is the hydraulic gradient (dimensionless),

is the difference between two hydraulic heads (Length, usually in m or ft), and

is the flow path length between the two piezometers (Length, usually in m or ft)

The hydraulic gradient can be expressed in vector notation, using the del operator. This requires a hydraulic head field, which can only be practically obtained from a numerical model, such as MODFLOW. In Cartesian coordinates, this can be expressed as:

$\nabla h = \left( {\frac{\partial h}{\partial x}}, {\frac{\partial h}{\partial y}}, {\frac{\partial h}{\partial z}} \right) = {\frac{\partial h}{\partial x}}\mathbf{i} + {\frac{\partial h}{\partial y}}\mathbf{j} + {\frac{\partial h}{\partial z}}\mathbf{k}$

This vector describes the direction of the groundwater flow, where negative values indicate flow along the dimension, and zero indicates 'no flow'. As with any other example in physics, energy must flow from high to low, which is why the flow is in the negative gradient. This vector can be used in conjunction with Darcy's law and a tensor of hydraulic conductivity to determine the flux of water in three dimensions.

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