Page 126 - Physical Principles of Sedimentary Basin Analysis

108 Heat flow

∂T
φ i i c i + φ i i c i v i ·∇T −∇ · (λ∇T ) = S (6.11)
∂t
i i
where the velocity of component i is the vector v i .
We will now look at temperature equation (6.11) for a porous medium, which is a
medium of two components: solid matrix and pore fluid. The coefficient before the term
∂T/∂t is the weighted mean of i c i for each component of the medium, using the volume
fractions φ i as normalized weights. The weighted mean of f c f and s c s is written as

b c b = φ f c f + (1 − φ) s c s (6.12)
where b and c b are the bulk density and the bulk specific heat capacity, respectively.
Be aware that the notation b c b hides the fact that we do not know each of these factors
separately, only their product.
The same averaging is carried out for the coefficient of the first-order term ∇T , assuming
that v i = v s for all components except the fluid (i = f ). It is done by replacing the fluid
velocity v f by the Darcy flux v D and the solid velocity v s . Recall from equation (2.31) that
the Darcy velocity is a volume flux measured relative to the velocity of the porous medium,

v D = φ · (v f − v s ) or φv f = v D + φv s . (6.13)
The coefficient before the ∇T -term is therefore

φ f c f v f + (1 − φ) s c s v s = f c f v D + b c b v s (6.14)
and the temperature equation (6.11) becomes
DT
b c b + f c f v D ·∇T −∇ · (λ∇T ) = S (6.15)
dt
using the coefficients (6.12) and (6.14). Notice that time differentiation is written as a
material derivative with respect to the velocity of the (solid) sediment matrix.
The full temperature equation (6.15) is rarely needed, and most problems can be treated
with a simplified version. The most common simplifications apply to problems in 1D with
no source term for heat generation (S = 0) and where there is no heat convection (v D = 0).
Furthermore, we rarely have to deal with problems where the solid material is moving. The
material derivative therefore becomes the normal partial derivative, and the temperature
equation simplifies to
2
∂T λ
∂ T
− = 0 (6.16)
∂t b c b ∂z 2
in the vertical direction. The only parameter left in the temperature equation is the thermal
diffusivity κ = λ/( b c b ). Furthermore, we can make a dimensionless version of the equa-
tion if there is a characteristic length l 0 and a characteristic temperature T 0 in the problem.
Introducing the dimensionless quantities
z T t
ˆ
ˆ
ˆ z = , T = and t = (6.17)
l 0 T 0 t 0