SME
Soil Mechanics · Problem type
Falling-head test
Use this for “In a falling-head test on a silt 10 cm long and 40 cm² in area, the head in a 0.5 cm² standpipe falls from 100 cm to 50 cm in 600 s. Find k.”
Chart
h(600) = 50
This is an example. Type your own values to start a new problem.
k (cm/s)
=
a (cm²)
×
L (cm)
A (cm²)
×
t (s)
×
ln(
h₁ (cm)
÷
h₂ (cm)
)
Formulas
k
=
(
a
×
L
A
×
t
)
×
ln(
h₁
h₂
)
1.4441 ×
10
-4
=
(0.5
×
10
40 × 600
)
×
ln(
100
50
)
Assumptions
•
For fine soils (silts, clays), where a constant-head flow is too small to measure.
•
Heads h₁ and h₂ are measured above the outflow level.
•
The sample is saturated and the flow through it is laminar (Darcy’s law).
Step-by-step
WE KNOW
a = 0.5 cm², A = 40 cm², L = 10 cm, h₁ = 100 cm, h₂ = 50 cm, t = 600 s
FIND
coefficient of permeability (k)
Step 1 · Find coefficient of permeability (k)
Use k = (a × L ÷ (A × t)) × ln(h₁ ÷ h₂)
The
water
falling
in
the
standpipe
is
the
water
through
the
sample:
−
a
dh
dt
=
k
(
h
L
)
A
;
integrating
from
h₁
to
h₂
gives
the
ln.
k
=
(0.5
×
10
40 × 600
)
×
ln(
100
50
)
k
=
0.5
×
10
24,000
×
ln(2)
k
=
0.5
×
10
24,000
×
0.69315
k
=
5
24,000
×
0.69315
k
=
0.00020833
×
0.69315
k
=
1.4441 ×
10
-4
cm/s
Check
Put the numbers back in. Both sides should match.
1.4441 ×
10
-4
=
(0.5
×
10
40 × 600
)
×
ln(
100
50
)
✓
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