Page 127 - Physical Principles of Sedimentary Basin Analysis

6.2 Stationary 1D temperature solutions 109

2
where the characteristic time is t 0 = l /κ, gives the dimensionless temperature equation
0
∂T ˆ ∂ T
2 ˆ
− = 0. (6.18)
∂ ˆ t ∂ ˆz 2
We notice that the dimensionless temperature equation has no parameters, and it is there-
fore just one dimensionless solution of the problem. The dimensionless solution can be
converted to a solution for any specific case using the scaling relationships (6.17).

6.2 Stationary 1D temperature solutions
A simple version of the temperature equation is a time-independent equation with zero
velocity terms and no heat generation. The temperature equation is then (an elliptic
equation)
∇· (λ∇T ) = 0. (6.19)

A further simplification is to allow only heat flow in the vertical direction, which is a
good assumption when a sedimentary basin shows little lateral variation. The temperature
equation is then reduced to a simple 1D equation
d dT
λ(z) = 0 (6.20)
dz dz
where the heat conductivity is a function of the depth. This z dependence is typically due
to changing lithologies with depth, or from a changing porosity inside the same lithology.
The porosity has normally a decreasing trend with depth. The average heat conductivity,
therefore, increases with depth inside the same lithology, because the heat conductivity of
the rock matrix is higher than the heat conductivity of water. Temperature equation (6.20)
is solved by two integrations. The first integration recovers Fourier’s law

dT
λ(z) = q 0 (6.21)
dz
with a constant (z-independent) heat flux q 0 . Notice that the minus sign in Fourier’s law
is left out, because we want positive heat flow to be upwards, (although the z-axis points
downwards). The next integration yields the temperature
z
dz
T (z) = T 0 + q 0 (6.22)
0 λ(z)
where T 0 is the temperature at the surface, z = 0. The temperature in the case of constant
heat conductivity λ 0 is
q 0
T (z) = T 0 + z (6.23)
λ
where the temperature increases linearly with depth. The temperature as a function of the
depth is called a geotherm.