Page 159 - Physical Principles of Sedimentary Basin Analysis

6.11 Forced convective heat transfer 141

and we are looking for a criterion that can decide when the fluid flow is sufficiently strong
to influence the temperature. The starting point is temperature equation (6.15), which
combines conductive and convective heat flow. In the vertical direction the temperature
equation (6.15) simplifies to
∂T ∂T ∂ ∂T
b c b + f c f v D − λ = 0, (6.146)
∂t ∂z ∂z ∂z

where b and f are the average bulk density and the fluid density, respectively, v D is the
Darcy flux and λ is the heat conductivity. The second term (with the Darcy flux v D ) repre-
sents convective heat transport and the third term (with the heat conductivity λ) represents
conductive heat flow. The rock is assumed to be at rest, (v s = 0), and the material deriva-
tive in equation (6.15) is replaced by a (normal) partial derivative in equation (6.146). The
relative importance of convective heat flow relative to conductive heat flow is best analyzed
by making a dimensionless version of the equation. The temperature equation (6.146)is
now rewritten as
2
2 ˆ
l c b b ∂T ˆ l 0 f c f v D ∂T ˆ ∂ T
0
+ − = 0 (6.147)
λ ∂t λ ∂ ˆz ∂ ˆz 2
ˆ
using the dimensionless temperature T = (T − T 1 )/(T 2 − T 1 ) and dimensionless vertical
position ˆz = (z 2 − z)/(z 2 − z 1 ), where the length of the system is l 0 = z 2 − z 1 .The
temperature is scaled using the difference between the highest temperature T 2 at the deepest
point along the aquifer and the surface temperature T 1 . The vertical coordinate is scaled
with the distance from the surface down to the low-permeable base (see Figure 6.17).
Notice that a hat above the symbol denotes a dimensionless quantity, and that the ˆz-axis is
pointing upwards. A positive Darcy flux is therefore upwards along the ˆz-axis. We see that
both the dimensionless T and ˆz variables are numbers between 0 and 1 (where ˆz = 0atthe
ˆ
base and ˆz = 1 at the top of the system). The coefficient of the time derivative is identified
as the characteristic time for the process

2
l c b b
0
t 0 = . (6.148)
λ
We will see in the next section that the temperature solution is close to a stationary solution
for t
t 0 , and t 0 therefore characterizes the time span of the thermal transients. The other
coefficient (before the convective) term is the dimensionless Peclet number

l 0 f c f v D
Pe = . (6.149)
λ
The dimensionless time is ˆ t = t/t 0 and the dimensionless version of temperature
equation (6.147) becomes

2 ˆ
∂T ˆ ∂T ˆ ∂ T
+ Pe − = 0 (6.150)
∂ ˆ t ∂ ˆz ∂ ˆz 2