---
title: "Variable head test, two soils"
subject: "Soil Mechanics"
chapter: 6
key: "semester_4/soil_mechanics:variable-head-two-soils-permeability"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/variable-head-two-soils-permeability
site: BCT Engineering
---

# Variable head test, two soils

Soil Mechanics · Chapter 6: Principle of Effective Stress, Capillarity and Permeability

- Asked in: 2075 Baisakh, 7 marks
- Video: [Falling Head - Variable Head Permeability Method](https://www.youtube.com/watch?v=SS7o6sKw2Xc) — Elementary Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/variable-head-two-soils-permeability
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-6

## Question

In a variable head permeability test on a soil of length $L_1$, the head of water in the standpipe takes 5 seconds to fall from 900 to 135 mm above the tail water level. When another soil of length $L_2$ = 60 mm is placed above the first soil, the time taken for the head to fall between the same limits is 150 seconds. The permeameter has a cross sectional area of 4560 mm² and a standpipe area of 130 mm². Calculate the permeability of the second soil.

## Short answer

Permeability of the second soil is found by subtracting time for the first soil alone to obtain the flow resistance in series. Result: k2 = 0.0224 mm/s (2.24×10−5 m/s).

## Answer

#### Principle

For a variable head test, $k = \dfrac{aL}{A\,t}\ln\dfrac{h_1}{h_2}$, so for given $a$, $A$, $h_1$, $h_2$, the time is $t = \dfrac{aL}{Ak}\ln\dfrac{h_1}{h_2}$, proportional to $L/k$. When the second soil is placed on the first, the two act in series, so the flow resistances ($L/k$) add and the total time is the sum of the times for each soil taken alone.

#### Given

$h_1 = 900$ mm, $h_2 = 135$ mm, $a = 130$ mm², $A = 4560$ mm², $L_2 = 60$ mm.

- Soil 1 alone: $t_1 = 5$ s
- Soil 1 + soil 2: $t = 150$ s
- Time due to soil 2 alone: $t_2 = 150 - 5 = 145$ s

$$
\ln\frac{h_1}{h_2} = \ln\frac{900}{135} = \ln 6.667 = 1.897
$$

#### Permeability of soil 2

$$
k_2 = \frac{a\,L_2}{A\,t_2}\ln\frac{h_1}{h_2} = \frac{130 \times 60}{4560 \times 145} \times 1.897 = 0.0224\ \text{mm/s}
$$

**Answer: $k_2 = 0.0224$ mm/s $= 2.24\times10^{-5}$ m/s.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the variable head test, for fixed a, A, h1, h2, time t is proportional to:
   A. k/L
   B. L/k
   C. L k
   D. k
   Answer: B. t = (aL/(A k)) ln(h1/h2), proportional to L/k. (Key point: Time proportional to L/k)

2. With soil 2 placed on soil 1, the total time is:
   A. The difference of the two times
   B. The sum of the times, since resistances L/k add in series
   C. The average of the times
   D. Equal to t1
   Answer: B. Series layers add flow resistance, so times add. (Key point: Series resistances)

3. The time attributable to soil 2 alone is:
   A. 150 s
   B. 155 s
   C. 145 s
   D. 30 s
   Answer: C. t2 = 150 - 5 = 145 s. (Key point: Time for soil 2)

4. k2 = (a L2)/(A t2) ln(h1/h2) is about:
   A. 0.0224 mm/s
   B. 2.24 mm/s
   C. 0.224 mm/s
   D. 0.00224 mm/s
   Answer: A. (130 x 60)/(4560 x 145) x 1.897 = 0.0224 mm/s. (Key point: Result)

## Related in Soil Mechanics

- [Average permeability of layered soil](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Alayered-soil-average-permeability) — Soil Mechanics, chapter 6
- [Consolidation test, 90% time, k, Cc](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconsolidation-test-50-90-percent-time) — Soil Mechanics, chapter 9
- [Constant head test, k at porosity 39%](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconstant-head-permeability-porosity) — Soil Mechanics, chapter 6
- [UCS test soil type and parameters](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aunconfined-compression-soil-type-mohr) — Soil Mechanics, chapter 10
- [USCS classification of soils A and B](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Auscs-classify-soils-a-b) — Soil Mechanics, chapter 3
- [Constant head permeameter, seepage velocity](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconstant-head-permeameter-seepage-velocity) — Soil Mechanics, chapter 6

## Related in other subjects

- [Settling basin 60 m³/s, 0.15 mm, head 300 m](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Asettling-basin-60cumec-300m-015mm-turbulence) — Hydropower Engineering, chapter 5, asked 2×
- [Time to lower level in two-orifice tank](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Atwo-orifice-tank-lowering-time) — Fluid Mechanics, chapter 6
- [Settling basin 60 m³/s, 0.15 mm, head 100 m](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Asettling-basin-60cumec-head-100m-015mm) — Hydropower Engineering, chapter 5
- [River crossing: minimum clearance above water](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Ariver-crossing-clearance) — Transmission and Distribution System Design, chapter 4