Page 125 - Physical Principles of Sedimentary Basin Analysis

6.1 The temperature equation 107

where the first term q is the heat flux and the second term is heat transport by convection,
and v i is the velocity of component i. Fourier’s law gives the heat flux

dT
q =−λ (6.4)
dx
as the product of the temperature gradient and the heat conductivity λ (see Section 2.12).
Notice that the bulk heat conductivity is an average heat conductivity for all components in
the system. The heat conductivity is often isotropic, but it can also be non-isotropic just like
the permeability. The minus sign in Fourier’s law is needed because heat flows from “high”
to “low” temperatures. Energy conservation E 1 = E 2 requires that expressions (6.1)
and (6.2) are equal, and we get

∂ ∂ ∂ ∂T
φ i i e i + φ i i v i e i − λ = S (6.5)
∂t ∂x ∂x ∂x
i i
when both sides are divided by t x.The t- and x-differentiations in equation (6.5)are
carried out inside the parentheses and terms cancel because of mass conservation of each
component
∂ ∂
(φ i i ) + (φ i i v i ) = 0 (6.6)
∂t ∂x
(see section 3.18). Equation (6.5) then simplifies to

∂e i ∂e i
∂ ∂T
φ i i + φ i i v i − λ = S. (6.7)
∂t ∂x ∂x ∂x
i i
The specific internal energy e i is a function of temperature and volume. Assuming that the
volume is constant gives that
∂e i ∂T ∂e i ∂T
= c i and = c i (6.8)
∂t ∂t ∂x ∂x
where

∂e i
c i = (6.9)
∂T V
is the specific heat capacity at constant volume for component i. The rate of change of
internal energy (6.8) gives that expression (6.7) for energy conversion becomes

∂T ∂T ∂ ∂T
φ i i c i + φ i i c i v i − λ = S. (6.10)
∂t ∂x ∂x ∂x
i i
Although the temperature equation is derived in 1D it is straightforward to generalize it to
3D. All we have to do is to replace ∂/∂x with the ∇-operator, and the temperature equation
becomes