Page 124 - Physical Principles of Sedimentary Basin Analysis

106 Heat flow

Heat flow

0 40 60 85 120 180 240 350
mW m –2
Figure 6.1. The map of the world heat flux shows that the spreading ridges have the highest heat
flow, and that oceanic plates that are formed at these ridges cool as they move away from the
ridges. The continents have heat flow values in the interval from 40 W m −2 to 80 W m −2 ,which
is partly from heat generation in the crust and partly from cooling of the Earth. The figure is taken
from International Heat Flow Commission (2008).

6.1 The temperature equation
A temperature equation is an expression for conservation of energy. We consider energy
conservation in a box of length x placed along the x-axis as shown in Figure 3.18.We
then have that the increase in the internal energy E 1 in the box is equal to the net inflow
of energy E 2 . A change in the internal energy in the box is

E 1 = φ i i e i A x (6.1)
i
where A is the section area, φ i is the volume fraction of each component (or phase) i, i is
the density of component i and e i is the internal energy of component i per unit mass. The
sum is over each component. A two-component system is typically a porous rock where
one volume fraction φ 1 = φ is the porosity and the other volume fraction φ 2 = 1−φ is the
solid matrix. The summation is written explicitly with the summation sign instead of using
the summation convention, because the summation is over three indices and not a pair of
two identical indices. Net heat influx in the box is

E 2 = (F(x) − F(x + x)) A t + SA x t (6.2)
where F(x) is the flux of energy per unit area at position x and S is the heat generation per
unit volume and unit time at the center of the box. The energy flux is

F = q + φ i i v i e i (6.3)
i