Page 54 - Physical Principles of Sedimentary Basin Analysis

36 Properties of porous media

which is equal to the average heat conductivity function (2.103) at the two porosities φ = 0
and φ = φ 2 has c given by

1
c = 1 − exp φ 2 ln(λ f /λ s ) . (2.110)
φ 2
(b) What is c when λ f = 0.64 W/Km and λ s = 2.5 W/Km?
(c) How much does a change φ in the porosity change the linear average heat
conductivity?

Exercise 2.25 Plot the following three functions of φ in the interval from 0 to 1 for a = 1
and b = 0.1:

f a (φ) = φa + (1 − φ)b arithmetic
φ (1−φ)
f g (φ) = a b geometric
1
f h (φ) = harmonic. (2.111)
(φ/a) + ((1 − φ)/b)
Which average is largest, in between and least?
Exercise 2.26 The temperature-dependent heat conductivity may be represented as

1

λ(T ) = (2.112)
A + BT
where T is the temperature in kelvin. An example is the data provided by Seipold (1995,

1998). Show that (2.112) can be rewritten in the form of (2.104) where λ 0 = 1/(A + BT 0 ),
c 0 = B/(A + BT 0 ) and T 0 = 273 C.
◦
Exercise 2.27 This exercise shows how in situ measurements of sediment heat conductiv-
ities can be used to estimate a porosity–depth function. Let heat conductivity as a function
of porosity be linearly approximated as
λ(φ) = λ s (1 − cφ) (2.113)

where c = 0.988. (See Exercise 2.24.) Assume that the Athy porosity–depth function also
can be linearly approximated,

φ(z) = φ 0 exp(−z/z 0 ) ≈ φ 0 (1 − z/z 0 ), (2.114)
which is a reasonable approximation for shallow depths compared to z 0 .
(a) Show that a linear representation of in-situ measurements of heat conductivity as a
function of depth
λ(z) = λ 0 + az (2.115)
gives the following parameters in Athy’s porosity function:

1
λ 0 1
φ 0 = 1 − and z 0 = λ s − λ 0 ). (2.116)
c λ s a