Page 170 - Physical Principles of Sedimentary Basin Analysis

152 Heat flow

the conduction-dominated level. This turns out to imply that the two regimes of conductive
and convective heat flow are controlled by a different Peclet number than the one defined
in Section 6.11. We can see this by looking at the energy balance for a small element dz
along the fracture. The energy transported into the element minus the energy transported
out of the element during a time step dt is

∂T
E 1 = (c f f T v 0 w dt) z − (c f f T v 0 w dt) z+dz ≈−c f f v 0 w dz dt (6.188)
∂z
where c f is the heat capacity of the fluid, f is the fluid density, v 0 is the average fluid
velocity in the fracture and w is the width of the fracture. The energy E 1 has to be equal
to the energy E 2 lost through the vertical sides,
∂T
E 2 = 2λ dz dt (6.189)
∂x
where λ is the heat conductivity and the factor 2 accounts for the two sides of the fracture.
Energy conservation then requires that E 1 = E 2 , which implies that
∂T ∂T
Pe f = 2 (6.190)
∂z ∂x
where
c f f v 0 w
Pe f = (6.191)
λ
is the Peclet number for heat transport in the fracture. The Pe f -number defines two regimes,
one for conduction- and another for convection-dominated heat transport, in the same way
as the Pe-number from Section 6.11. We can see this by looking at an approximate tem-
perature solution for the two cases. The conduction-dominated regime has T ≈ 1 −ˆz, and
ˆ
ˆ
ˆ
x
therefore ∂T /∂ ˆ ≈ 0 and ∂T /∂ ˆz ≈ 1, and from equation (6.190) we get that Pe
1.
In the convection-dominated regime we can approximate the temperature along most of
the fracture by the temperature at the inlet, T ≈ 1, and we therefore have that ∂T /∂ ˆz ≈ 0
ˆ
ˆ
for most of the fracture. We will see below that ∂T /∂ ˆ ≈ 1 for the stationary tem-
x
ˆ
ˆ
perature field around a fracture when the temperature is T = 1 along the fracture. The
relation (6.190) therefore requires that Pe f
1 for the convection-dominated case.
The difference between the Pe-number for 1D heat flow and the Pe f -number for 2D heat
flow around fractures is the characteristic length.
The length of the system is the characteristic length for 1D heat flow, but for 2D heat
flow around the fracture it is the width of the fracture that enters the Peclet number as a
characteristic length.
The temperature solution in the regime Pe f
1 can be approximated by the con-
stant temperature of the inlet along the fracture. The surroundings of the fracture become
heated to this temperature. Solving the stationary temperature equation in 2D in the case
of constant heat conductivity
2
∇ T = 0 (6.192)