SME
Geophysics
Heat flow
Use this for “Granite with a conductivity of 2.5 W/(m·K) warms 25 °C per km down. Find the heat flow, and the temperature 10 km down if the surface is at 10 °C.”
Chart
T(z) = 25z + 10
The vertical axis grows downward.
This is an example. Type your own values to start a new problem.
q (mW/m²)
=
k (W/(m·K))
×
G (°C/km)
T (°C)
=
T₀ (°C)
+
G (°C/km)
×
z (km)
Formulas
q
=
k
×
G
62.5 = 2.5 × 25
T
=
T₀
+
G
×
z
260 = 10 + 25 × 10
Assumptions
•
Steady conduction with no heat made in the layer, so the temperature climbs at one rate: T = T₀ + Gz.
•
Heat flows up, from hot rock to cold: the heat flow is the conductivity times the gradient (q = kG).
•
W/(m·K) times °C/km is mW/m², the unit heat flow is quoted in.
Step-by-step
WE KNOW
k = 2.5 W/(m·K), G = 25 °C/km, T₀ = 10 °C, z = 10 km
FIND
heat flow (q), temperature at depth (T)
Step 1 · Find heat flow (q)
Use q = k × G
Fourier’s law: the conductivity times how fast the temperature climbs with depth.
q
=
2.5
×
25
q
=
62.5
mW/
m
2
Step 2 · Find temperature at depth (T)
Use T = T₀ + G × z
Start
at
the
surface
temperature
and
add
G
degrees
for
each
kilometer
down.
T
=
10
+
25
×
10
T
=
10
+
250
T
=
260
°C
Check
Put the numbers back in. Both sides should match.
62.5 = 2.5 × 25 ✓
260 = 10 + 25 × 10 ✓
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Temperature with radiogenic heat
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More problem types
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