SME
Hydrology
Groundwater flow (Darcy's law)
Use this for “Sand with K = 10 m/day carries water 400 m between two wells whose heads differ by 2 m, through a 1,000 m² face, porosity 0.25. Find the flow and the travel time.”
Diagram
i
=
Δh
L
=
2
400
=
0.005.
Q
=
KAi
=
50
m³/day.
v
=
q
n
=
0.2
m/day,
faster
than
q:
water
moves
only
through
the
pores.
t
=
2,000
days
to
cross
L.
Water flows from high head to low (well 1 to well 2).
Heights drawn × 50; lengths along the ground to scale.
This is an example. Type your own values to start a new problem.
Q (m³/day)
=
K (m/day)
×
A (m²)
×
i
Δh
Head drop
example · m
L
Flow length
example · m
q
Darcy flux
calculated · m/day
n
Porosity
example
v
Seepage velocity
calculated · m/day
t
Travel time
calculated · days
Formulas
i
=
Δh
L
0.005
=
2
400
Q
=
K
×
A
×
i
50 = 10 × 1,000 × 0.005
q
=
Q
A
0.05
=
50
1,000
v
=
q
n
0.2
=
0.05
0.25
t
=
L
v
2,000
=
400
0.2
Assumptions
•
Laminar flow through connected pores; water flows from high head to low.
•
Flow is proportional to how fast the head falls along the path; on a steady slope that gradient is Δh ÷ L.
•
The seepage velocity is faster than the Darcy flux q because water moves only through the pores.
Step-by-step
WE KNOW
K = 10 m/day, Δh = 2 m, L = 400 m, A = 1,000 m², n = 0.25
FIND
hydraulic gradient (i), discharge (Q), Darcy flux (q), seepage velocity (v), travel time (t)
Step 1 · Find hydraulic gradient (i)
Use i = Δh ÷ L
Divide the head drop by the length it falls over.
i
=
2
400
i
=
0.005
Step 2 · Find discharge (Q)
Use Q = K × A × i
Darcy’s law: the conductivity times the area times the gradient.
Q
=
10
×
1,000
×
0.005
Q
=
10,000
×
0.005
Q
=
50
m³/day
Step 3 · Find Darcy flux (q)
Use q = Q ÷ A
Divide the flow by the area it crosses.
q
=
50
1,000
q
=
0.05
m/day
Step 4 · Find seepage velocity (v)
Use v = q ÷ n
Divide the flux by the porosity: water moves only through the pores.
v
=
0.05
0.25
v
=
0.2
m/day
Step 5 · Find travel time (t)
Use t = L ÷ v
Divide the length by the seepage speed.
t
=
400
0.2
t
=
2,000
days
Check
Put the numbers back in. Both sides should match.
0.005
=
2
400
✓
50 = 10 × 1,000 × 0.005 ✓
0.05
=
50
1,000
✓
0.2
=
0.05
0.25
✓
2,000
=
400
0.2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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