Page 207 - Physical Principles of Sedimentary Basin Analysis

6.21 Conservation of energy once more 189

specific entropy, which is defined as ds = de + pdν in thermodynamics. The reason for
introducing the specific entropy is that thermodynamics gives

c p α
ds = dT − dp (6.321)
T
where c p is the heat capacity at constant pressure, T is the absolute temperature in kelvin
and α is the thermal expansibility (see Notes 6.15, 6.16 and 6.17). The temperature
equation for a single phase medium is therefore
DT Dp ∂v i
c p − αT =∇ · (λ∇T ) + S + σ ij (6.322)
dt dt ∂x j
which includes a term for the rate of change of pressure. This term can normally be ignored
as justified by Exercise 6.33.
The derivation above generalizes to a multicomponent system, like a porous rock satu-
rated by a fluid, by expressing the internal energy (E), the kinetic energy (K) and the work
done on the system (W) as sums over each component. The derivation above is then car-
ried out for each component, where the continuity equation and Newton’s second law are
used separately for each component to simplify the expression. The only difference is that
the density is now replaced by φ k k , where φ k and k are the volume fraction and the
density of each component, respectively. The term Q =−λ∇T for heat conduction is not
written as a sum over each component, because it is a bulk property. The heat generation S
is also treated as a bulk property, but it could have been divided between each component
as S = φ k S k . The multicomponent version of the single component equation (6.322)is
then

DT ∂v i
φ k k c k −∇ · (λ∇T ) = S + σ ij . (6.323)
dt ∂x j
k k k k
The subscript k on the material derivative says that the velocity of component k is used.
This is exactly the same temperature equation as equation (6.11), except for the last term
on the right-hand side that involves the work done on the system per unit time for each
component. The temperature equation for a fluid saturated porous rock is
DT ∂v i
b c b + f c f v D ·∇T −∇ · (λ∇T ) = S + σ ij (6.324)
dt ∂x j
s
by introducing the bulk properties from equation (6.12) and the Darcy flux from equa-
tion (6.14). (The subscript b denotes the bulk and the subscript f the fluid.) The material

derivative of equation (6.324) is with respect to the solid velocity. The term (σ ∂v i /∂x j ) s
ij
represents the work per unit time in the solid phase, and the corresponding term in the fluid
phase is ignored.
Note 6.15 Entropy can be taken as a function of temperature and pressure, s = s(T, p),
and the increment ds is
∂s ∂s

ds(T, p) = dT + dp. (6.325)
∂T ∂p
p T