Hydraulic Gradient Calculator

Background.

Two monitoring wells stand 30 metres apart. Measured from the same datum, the water stands 2.4 metres higher in one than in the other, so groundwater is moving from the first towards the second, losing 2.4 m of head over 30 m of travel. That ratio, 0.08, is the hydraulic gradient: the quantity that sets groundwater moving, and the one Darcy's law multiplies by the ground's permeability to say how fast.

Head is the key idea. The hydraulic head at a point is the level water would rise to in a well or piezometer open there, which combines the point's elevation with the water pressure at it. Groundwater flows from high head to low head, which is not always downhill. Water at depth under pressure can rise towards the surface, as it does at springs and beneath many valleys, and two points at very different pressures can sit at the same head and exchange no water at all.

The gradient is the head lost per unit length of path, i = Δh ÷ L. Because both are lengths, the result has no unit: 0.08 means 8 cm of head lost for every metre travelled, or 80 m per kilometre. Gradients across regional aquifers are usually far gentler, often well under 0.01. Steep values point to something local, such as the cone of depression around a pumping well, seepage under a dam, a drained slope or a thin layer of clay that the water has to force its way through.

The result is a magnitude along the line you measured. It does not give a direction in three dimensions, and turning it into a flow rate needs a hydraulic conductivity measured or estimated for the ground in question.

What is hydraulic gradient calculator?

The hydraulic gradient is the drop in hydraulic head per unit distance along a flow path, i = Δh ÷ L. Hydraulic head is energy per unit weight of water, written as a height: elevation above a datum plus pressure head, the pressure divided by the unit weight of water. Velocity head is left out because groundwater moves too slowly for it to matter. The gradient is what drives flow through soil and rock; in Darcy's law the specific discharge is q = K × i, where K is the hydraulic conductivity. In a pipe or channel the same ratio is the friction slope, the head lost per metre of conduit.

How to use this calculator.

1. 01Take water levels in two wells or piezometers screened in the same aquifer, at about the same time, and convert both to elevations above one datum. Their difference is Δh.
2. 02Measure the distance between the two points along the flow path. If the wells do not lie on a flow line, the gradient along the line joining them understates the true one.
3. 03Enter both values in metres. Any single unit works, since the ratio is dimensionless, but do not mix feet and metres.
4. 04To estimate flow, multiply the gradient by the hydraulic conductivity for the Darcy flux, then divide by the effective porosity for the speed at which dissolved substances travel.

The formula.

i = Δh ÷ L, where Δh is the difference in hydraulic head between two points and L is the length of the flow path between them. Both must be positive, and the result is a positive magnitude. Darcy's law, from Henry Darcy's 1856 experiments on sand filters for the public fountains of Dijon, gives the flow: q = −K × dh/dl, where the minus sign records that water moves towards lower head and q is the discharge per unit cross-sectional area. Over a finite distance the derivative becomes Δh ÷ L. The relationship holds for the slow, laminar flow typical of groundwater and breaks down where flow becomes fast or turbulent, as in coarse gravel right next to a pumping well or in open fractures and karst conduits. Because hydraulic conductivity ranges over more than ten orders of magnitude between clean gravel and unfractured clay, the same gradient can mean metres a day in one material and almost no movement in another.

A worked example.

Frequently asked questions.

References& sources.

1. [1]USGS Water Science School, Groundwater Flow and the Water Cycle.
2. [2]Freeze and Cherry, Groundwater (1979), chapter 2 (OPEN TEXT), hydraulic head and gradient.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.