Page 134 - Physical Principles of Sedimentary Basin Analysis

116 Heat flow

span. One particular application is the estimation of geotherms in stable continental areas.
There are no direct measurements of continental geotherms, and the estimates have to be
based on surface (or near surface) observations. In particular observations of surface heat
flow, heat production in surface rocks and geophysical measurements of the thickness of
the crust are important. The heat production is negligible in mantle rocks, but important in
crustal rocks (see Table 6.2). It is therefore important to know where the crust ends and the
mantle begins.
The stationary temperature equation in the vertical direction is
d dT
λ(z) =−S(z) (6.49)
dz dz
when there are zero velocity terms. Notice that both the heat conductivity λ(z) and the
heat production term S(z) depend on the depth z. It is straightforward to solve the second-
order temperature equation (6.49) with two integrations. Integration two times brings two
integration constants into the solution, which are found by requiring the solution to fulfill
two boundary conditions. One boundary condition is a constant temperature at the surface.
The other boundary condition will either be the mantle heat flow at the base of the crust or
the temperature at the base of the lithosphere. The latter boundary condition is preferred
when there is some knowledge about the thickness of the lithosphere.
An integration of the temperature equation (6.49) from the depth z to the depth of the
crust z m gives the heat flux at any position in z in the crust:
dT z m
q(z) = λ(z) = S(u)du + q m (6.50)
dz z
where q m is the mantle heat flow into the base of the crust (at z = z m ). The heat production
is S(z) ≈ 0 in the lithospheric mantle (z > z m ) and its contribution to the heat flow is
negligible. The heat flow increases from the depth z m towards the surface because of the
integral over the heat source. The surface heat flow
z m

q s = S(u) du + q m (6.51)
0
is the mantle heat flow added to the contribution from heat production from the entire
thickness of the crust, and it is the maximal heat flow. We will later see that the heat
production in the crust is often the most important part of the surface heat flow.
The average heat production along a vertical through the crust is
1 z m
S av = S(u) du (6.52)
z m 0
and we have from equation (6.51) that this average is simply
1
S av = (q s − q m ) . (6.53)
z m
An estimate of the mantle heat flux (q m ) combined with observation of surface heat flux
(q s ) and the average thickness of the crust z m gives the average heat production in the crust.