SCIENCE · INVESTIGATE AND EXPLAIN

Use hydraulic head to determine flow direction

Interpret a shared head datum and distinguish a driving drop from the slope of a head profile.

Learning outcome: Calculate the signed head gradient and explain why head is neither terrain height nor depth to water.

Starting recap

Hydraulic head expresses water energy per unit weight as an equivalent height. In this constant-density model it combines elevation and pressure head.

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Try

Left head 14 m and right head 10 m over 100 m give a driving drop per length of +0.04; the mathematical head slope to the right is -0.04.

Open the experiment

Predict

If left head falls to 6 m while right head remains 10 m, which direction will the model flow?

Your prediction and reason

Experiment

Compare left heads 14, 10 and 6 m at the same conductivity. Inspect the linear head profile and explain its slope, the driving drop and the flow direction.

Compare head, conductivity and groundwater flow

Worked teaching model · synthetic values · Synthetic steady, one-dimensional saturated column: length 100 m, bulk cross-section 20 m², effective porosity 0.25. Right hydraulic head is maintained at 10 m; the left head is adjustable relative to the same arbitrary datum. The column remains saturated by assumption. Uniform conductivity, constant-density water and laminar Darcy flow are assumed. Positive rates mean left to right. Hydraulic head is not terrain elevation or depth below ground. No recharge, pumping, transient storage, dispersion, diffusion or sorption is modeled.

Left hydraulic head: 14 m Hydraulic conductivity: 2 m/day

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Head, discharge and ideal advective travel
Quantity	Calculation or condition	Result
Head drop left minus right	14 − 10	4 m
Signed driving head drop / length	(14 − 10) / 100	0.040 m/m
Head slope dh/dx	(10 − 14) / 100	-0.040 m/m
Darcy flux q = −K dh/dx	2 × 0.040	0.080 m/day
Discharge Q = q A	0.080 × 20	1.600 m³/day
Average pore-water velocity v	0.080 / 0.25	0.320 m/day
Ideal advective transit	100 / \|v\|; from higher-head endpoint	312.50 days

Bulk volume is 2,000 m³; the assumed connected pore-water volume is 500 m³. Constant boundary heads maintain steady flow. Transit uses average pore velocity and excludes spreading, diffusion, sorption and changing conditions. Equal heads give zero net advective flow here, not a claim that all molecular motion stops.

Left to right. Discharge Q = 1.600 m³/day; Darcy flux q = 0.080 m/day.

Average pore-water velocity v = q / 0.25 = 0.320 m/day. Ideal 100 m advective transit from the higher-head endpoint: 312.50 days. This is a conditional water-transport calculation, not a contaminant arrival prediction or safe withdrawal estimate.

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Understand

A horizontal column can have a driving difference

Imagine a saturated porous column extending 100 m horizontally from left to right. Its physical elevation does not slope, but water pressures at its ends can differ. The investigation assigns a left total hydraulic head that you control and a fixed right total head of 10 m. The boundaries maintain those heads throughout the selected steady case.

Hydraulic head expresses energy per unit weight as an equivalent length. For the constant-density groundwater treatment here, it is elevation head plus pressure head; the small velocity-head contribution is neglected. Pressure head represents pressure divided by water’s weight per unit volume. The same total head can therefore combine different elevation and pressure contributions. USGS groundwater glossary.

The head graph is not a ground-surface profile, a water-table cross-section or a map of depth below land surface. The model keeps the column saturated for every setting. It does not simulate drainage, an unsaturated zone or dewatering when a boundary head changes.

Declare the coordinate and datum

Position x increases to the right, from 0 to 100 m. Flow to the right is positive. A datum is the reference elevation used to express both heads; comparisons require a common datum. Changing that reference shifts both head numbers together without changing their difference.

At the default, left head is 14 m and right head is 10 m. The drop in the positive flow direction is 14 − 10 = 4 m. Dividing by 100 m gives a driving drop per length of +0.04 m/m. Water flows from the higher left head toward the lower right head under the stated conditions.

Now calculate the mathematical slope of the head graph as x increases: (10 − 14)/100 = −0.04 m/m. The slope and the driving drop have opposite signs because their subtraction order differs. This distinction explains the minus sign in the usual expression q = −K dh/dx; it is not a disagreement between two measurements.

Draw and read the profile

On paper mark the endpoints (0 m, 14 m) and (100 m, 10 m), labeling horizontal position and hydraulic head separately. Connect them with a straight line. At the midpoint, head is 12 m. At 25 m it is 13 m. Compare these values with the model’s evenly spaced profile samples.

The linear profile follows the homogeneous, one-dimensional steady setup with maintained endpoint heads. It is not a measured sequence through a real aquifer and it is not a time series. Moving from one point on the line to another changes spatial position, not elapsed time.

Change the left head to 6 m while retaining the right head of 10 m. The mathematical slope becomes +0.04, the driving drop per length becomes −0.04, and flow reverses. Sketch that case using the same axes and scales. Explain why the direction follows the head difference rather than the physical slope of the horizontal column.

Interpret equality and reference shifts

Set left head to 10 m. The profile is flat and the modeled advective flow is zero. Conductivity can remain positive: a material able to transmit water does not create a head difference by itself. This statement concerns the selected Darcy flow, not every microscopic water movement.

For a paper-only reference change, subtract 20 m from both default heads. They become −6 and −10 m relative to the new datum. Their difference is still 4 m and the inferred flow direction is unchanged. Negative head relative to a datum is not automatically negative physical water volume or an unsaturated column. USGS explanation of hydraulic head and its reference elevation.

Transfer to an unfamiliar pair

A separate horizontal column has heads 18 and 15 m over 150 m, left to right. Find the driving drop per length and the mathematical head slope. State the direction without relying on terrain.

Check the hydraulic-head transfer

The driving drop per length is (18 − 15)/150 = +0.02 m/m. The mathematical slope is (15 − 18)/150 = −0.02 m/m. Under the same saturated constant-density assumptions, flow is left to right, from higher to lower head.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

State the datum and sign convention, distinguish head from the physical column elevation, and explain why equal heads imply zero modeled advective flow.

Your explanation and limitations

Continue

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Previous: Choose the boundary before counting water flows Next: Calculate Darcy flux and volumetric flow

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