What is the constant head permeability test formula?

k = Q·L / (A·h·t), where Q is the volume of water collected in time t, L is the length of specimen between the manometer taps, A is the specimen cross-sectional area and h is the head difference between the taps. It is the direct application of Darcy's law and is the method of ASTM D2434 for coarse-grained soils.

What is the falling head permeability test formula?

k = (a·L / (A·t))·ln(h1/h2), where a is the standpipe cross-sectional area, L the specimen length, A the specimen area, and h1 and h2 the heads across the specimen at the start and end of the elapsed time t. The same formula written with base-10 logarithms is k = 2.303·(a·L/(A·t))·log10(h1/h2).

When do I use a constant head test versus a falling head test?

Use constant head (ASTM D2434) for clean sands and gravels, roughly k above 1e-3 cm/s, where enough water passes to measure a volume accurately. Use falling head for fine sands and silts, roughly 1e-3 to 1e-6 cm/s. Below about 1e-6 cm/s use a flexible-wall (triaxial) permeameter per ASTM D5084, because rigid-wall sidewall leakage dominates in clays.

Why correct permeability to 20 °C?

Hydraulic conductivity is inversely proportional to the water's viscosity, and water viscosity drops about 2 to 3 percent per degree Celsius near room temperature. Reporting k at a standard 20 °C makes results from different days and labs comparable: k20 = kT·(μT/μ20). A test run at 10 °C reads about 30 percent low without the correction.

What hydraulic gradient should a constant head test use?

Keep the gradient low enough that flow stays laminar, since Darcy's law assumes it. Run the test at several heads and check that the flow rate is proportional to the gradient; if k falls as the head rises, the flow is becoming turbulent and the lower-gradient values should be used.

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Permeability Test Calculator — Constant & Falling Head

Reduce laboratory permeameter readings to hydraulic conductivity: the constant-head test of ASTM D2434 for sands and gravels, or the falling-head test for fine sands and silts. Results are corrected to 20 °C and reported in cm/s, m/day and ft/day.

Test type

Specimen diameter, Dcm

Length between manometer taps, Lcm

Head difference, hcm

Volume collected, Qcm³ (mL)

Elapsed time, ts

Water temperature, T°C

Specimen area, A—cm²

Hydraulic gradient, i = h/L——

Discharge velocity, v = Q/(At)—cm/s

k at test temperature, k_T—cm/s

Viscosity ratio, R_T = μ_T/μ₂₀——

k₂₀ = k_T·R_T—cm/s

k₂₀—m/day

k₂₀—ft/day

Typical soil at this k—

Defaults: constant-head test on a 4-in (10.16 cm) specimen, 5 cm of head over 15 cm between taps, 150 mL collected in 15 minutes, water at 24 °C. Switch to falling head for the second worked example below.

$$ \text{Constant head: } k_T = \frac{Q\,L}{A\,h\,t} \qquad \text{Falling head: } k_T = \frac{a\,L}{A\,t}\,\ln\frac{h_1}{h_2} \qquad k_{20} = k_T\,\frac{\mu_T}{\mu_{20}} $$

$$ \log_{10}\frac{\mu_T}{\mu_{20}} = \frac{20-T}{T+96}\left[1.2364 - 1.37\times10^{-3}(20-T) + 5.7\times10^{-6}(20-T)^2\right] $$

k hydraulic conductivity · Q volume collected in time t · L flow length (between manometer taps, or specimen length for falling head) · A specimen area · h constant head difference · a standpipe area · h₁, h₂ head across the specimen at start and end of t · μ water viscosity · T water temperature, °C. The viscosity ratio is the Kestin, Sokolov & Wakeham (1978) correlation, within 0.05% of tabulated values from 0 to 40 °C.

Which test for which soil

Laboratory permeability test selection
Test	Typical k range (cm/s)	Soils
Constant head, rigid wall — ASTM D2434	> 10⁻³	Clean sands and gravels, fines under about 10%
Falling head, rigid wall	10⁻³ – 10⁻⁶	Fine sands, silty sands, silts
Flexible wall (triaxial) — ASTM D5084	≤ 10⁻⁶	Clays, compacted liners and cores

In a constant-head test on a fine soil, so little water passes that evaporation and meniscus errors swamp the collected volume. In a falling-head test on clean sand, the standpipe drops too fast to time. That is why the method changes at around 10⁻³ cm/s. Below about 10⁻⁶ cm/s, water leaking down the gap between the specimen and a rigid mold can carry more flow than the soil, so fine-grained soils go into a flexible membrane under confining pressure (ASTM D5084).

Getting a trustworthy number

Saturate first. Trapped air blocks pores and can make k read several times low. De-aired water and flushing from the bottom up are standard practice.
Check laminar flow. Run the constant-head test at three or more heads. Q should rise in proportion to h. If k drops at higher heads, the flow is turbulent; report the low-gradient values.
Repeat readings. Report the average of several consistent trials, not a single reading.
Correct for temperature. Water viscosity changes about 2–3% per °C near room temperature. A test at 10 °C reads about 30% low without the correction.
Remember what a lab test represents. A remolded 4-inch specimen does not capture layering, root holes or fissures. Field k for stratified deposits is often an order of magnitude higher horizontally than a vertical lab test shows.

Worked examples

Example 1 — Constant head on a medium sand

Given: 10.16 cm (4 in) diameter specimen. Manometer taps 15 cm apart, 5 cm head difference between them. 150 mL collected in 15 minutes (900 s). Water at 24 °C.

A = π(10.16)²/4 = 81.07 cm²; i = 5/15 = 0.333

k_T = QL/(Aht) = (150 × 15) / (81.07 × 5 × 900) = 6.167×10⁻³ cm/s

R_T at 24 °C = 0.9091

k₂₀ = 6.167×10⁻³ × 0.9091 = 5.61×10⁻³ cm/s (4.84 m/day, 15.9 ft/day), a clean sand

Example 2 — Falling head on a silty sand

Given: Specimen compacted in a standard Proctor mold, D = 10.16 cm, L = 11.64 cm. Standpipe 1.0 cm inside diameter. Head falls from 100 cm to 50 cm in 10 minutes (600 s). Water at 15 °C.

a = π(1.0)²/4 = 0.7854 cm²; A = 81.07 cm²

k_T = (0.7854 × 11.64)/(81.07 × 600) × ln(100/50) = 1.303×10⁻⁴ cm/s

R_T at 15 °C = 1.136

k₂₀ = 1.303×10⁻⁴ × 1.136 = 1.48×10⁻⁴ cm/s (0.128 m/day, 0.419 ft/day)

For typical k ranges by soil type, the Hazen D₁₀ estimate, and hydrologic soil group limits, see the soil permeability reference card. To turn a lab k into a flow rate or seepage velocity, use the Darcy's law calculator.

References: ASTM D2434, Standard Test Methods for Measurement of Hydraulic Conductivity of Coarse-Grained Soils. ASTM D5084, Hydraulic Conductivity of Saturated Porous Materials Using a Flexible Wall Permeameter. U.S. Army Corps of Engineers, EM 1110-2-1906, Laboratory Soils Testing, Appendix VII, Permeability Tests. Kestin, J., Sokolov, M., Wakeham, W.A. (1978), Viscosity of liquid water in the range −8 °C to 150 °C, J. Phys. Chem. Ref. Data 7(3).

Related tools

Soil permeability (k) by soil type
Darcy's law — flow rate and seepage velocity from k
Grain-size distribution — D₁₀ for a Hazen estimate
Embankment seepage — Casagrande phreatic line and seepage rate
Filter criteria — Sherard / NRCS filter design