Hydraulic Gradient Calculator (i = Δh/L, flow direction and critical gradient)

Understanding the hydraulic gradient calculator — concept and principle

This calculator finds the hydraulic gradient between two points in groundwater, the head drop per unit length of flow path, along with the flow direction and, for soils, the critical gradient and the factor of safety against piping. Hydrogeologists and geotechnical engineers use it for seepage, dewatering and well studies.

A 1.2 m head drop over 60 m is a gradient of 0.02, far below the 1.031 that would make the soil boil.

How it is calculated

i = (h₁ − h₂) ÷ L

Gradient (%) = i × 100; slope angle = atan(i)

Critical gradient i_c = (G_s − 1) ÷ (1 + e)

Factor of safety against piping = i_c ÷ i

h₁ and h₂ are total heads in metres (water levels in piezometers or wells), L is the flow-path length in metres.
G_s is the specific gravity of the soil solids and e the void ratio.

The principle

Groundwater moves from high to low total head, which is elevation head plus pressure head. The hydraulic gradient is how steeply that head falls along the path, and Darcy's law says the flow velocity is proportional to it: v = k i. When water flows upward through soil, it drags on the grains. At the critical gradient this seepage force equals the soil's submerged weight, the effective stress drops to zero and sand can boil up: the quick condition, which can lead to piping.

Worked example

Default inputs: h₁ = 102.5 m, h₂ = 101.3 m, L = 60 m, G_s = 2.65, e = 0.6.

Δh = 102.5 − 101.3 = 1.2 m.
i = 1.2 ÷ 60 = 0.02, or 2%; slope angle = atan(0.02) = 1.146°.
Flow goes from point 1 to point 2.
i_c = (2.65 − 1) ÷ (1 + 0.6) = 1.031.
Factor of safety = 1.031 ÷ 0.02 = 51.56.

Each point is the calculator re-run with only “Total head at point 1 (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Total head at point 1 (m)	Hydraulic gradient i
25.6	1.2612
51.2	0.8342
76.9	0.4071
102	0.02
128	0.4471
154	0.8742
205	1.7283

How to read the result

Natural groundwater gradients are often small, of the order of a few thousandths to a few hundredths. The critical gradient for most sands is about 1. A factor of safety well above 1 means upward seepage is far from causing a quick condition; values close to 1 are dangerous near excavations, dams and levees.

Common mistakes

Using the difference in ground elevation instead of water levels.
Measuring L horizontally when the flow path is much longer or inclined.
Applying the piping check to downward or horizontal flow.
Mixing up void ratio and porosity.

Keep in mind: Seepage and piping assessments for dams, levees and excavations need site investigation, applicable codes and a qualified geotechnical engineer.

Frequently asked questions

What is total head?

The sum of elevation head and pressure head, read as the water level in a piezometer.

How do I get the flow velocity?

Multiply i by the hydraulic conductivity k (Darcy's law); divide by porosity for the average velocity through the pores.

What is a quick condition?

Upward seepage strong enough to cancel the effective stress, so sand behaves like a liquid.

Sources: Standard soil-mechanics formulas: hydraulic gradient in Darcy's law and critical gradient (G_s − 1)/(1 + e)

Checked 9 Nov 2026. Hydraulic gradient i = Δh/L in Darcy’s law and the critical (quick-condition) gradient i_c = (G_s − 1)/(1 + e) from effective-stress analysis of upward seepage — standard soil-mechanics formulas.

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