Page 160 - Physical Principles of Sedimentary Basin Analysis

142 Heat flow

where the Pe-number is the only parameter left. The Pe-number measures convective heat
transport relative to conductive heat transport, which we see by writing the Pe-number as
follows:
Convective heat transport f c f v D (T 2 − T 1 )
Pe = = . (6.151)
Conductive heat transport λ(T 2 − T 1 )/l 0
The numerator is the heat flux from a Darcy flux v D with a temperature difference T 2 − T 1 .
(Recall that v D is a volume flux – it is the volume of fluid per time through a unit area.) The
denominator is the heat flux resulting from the conductivity λ multiplied by the thermal
gradient (T 2 − T 1 )/l 0 . The stationary (time independent) version of equation (6.150)is
studied first:
dT ˆ d T
2 ˆ
Pe − = 0 (6.152)
dˆz dˆz 2
ˆ
where the boundary conditions in dimensionless form are T (ˆz=0) = 1 and T (ˆz=1) = 0.
ˆ
Equation (6.152) is straightforward to integrate two times and the stationary temperature
solution is
Pe ˆz
ˆ
T (ˆz) = c 1 e + c 2 . (6.153)
This solution is also easily verified by insertion into the stationary equation (6.152).
The two boundary conditions specify the two integration constants, and the stationary
temperature solution becomes

e Pe ˆz − e Pe
ˆ
T (ˆz) = . (6.154)
1 − e Pe
The (dimensionless) stationary temperature solution (6.154) is plotted in Figure 6.18 for
Pe =−10, −1, −0.1, 0, 0.1, 1 and 10. The temperature regime |Pe|
1 gives a geotherm
that is nearly a straight line and it is therefore conduction-dominated. The other regime,
|Pe|
1, has a geotherm that departs from the conduction-dominated straight line because
it is convection-dominated. Notice that the sign of the Pe-number reflects the direction of
the Darcy flux. The Pe-number is positive for flow upwards and negative for flow down-
wards. The temperature at the base is brought towards the top for Pe = 10, and the
temperature at the top is brought towards the base for Pe =−10. The regime |Pe|≈ 1
is where convection and conduction are roughly equally important. The scaling relations
give the temperature solution with units

z 2 − z

T (z) = T 1 + (T 2 − T 1 ) T ˆ . (6.155)
z 2 − z 1
Deviations from a near linear trend in subsurface temperature data are often explained by
groundwater flow. The Pe-number is useful because it tells us to what degree heat convec-
tion is important. It also gives an estimate of the vertical Darcy flux once it is estimated,
for example by matching the curve (6.154) against temperature observations. Furthermore,
the Darcy flux combined with pressure measurements allows for permeability estimations.