Geoscience Reference
In-Depth Information
1 At low frequencies, only a small fraction of the coun-
terions of the diffuse layer are dragged by the flow of
the pore water, and therefore, Q 0
V
of the pore space by the flow of the pore water (contribu-
tion of advective nature). If the Darcy velocity associated
with the poromechanical contribution is written as w w ,
the second contribution to the current density is given by
Q V . An expres-
sion to compute Q 0
V from the low-frequency permea-
bility, k 0 , is discussed further later.
2 At high frequencies, all the charge density existing in
the pores is uniformly dragged along the pore water
flow, and therefore, the charge density Q V is also equal
to the volumetric charge density of the diffuse layer.
An expression to compute Q V from the cation
exchange capacity is discussed further later.
Depending on the size of the electrical double layer with
respect to the size of the pores, two cases can be considered:
1 In the thick double layer approximation, Q V
Q V ω
w w
J m =
i
ω
3 23
The mechanical contribution to the filtration displace-
ment is given by the generalized Darcy
'
s law derived
previously:
k ω
η w
w w =
2
i
ω
p w ω
ρ w s w u
3 24
Q 0
V
(all the counterions of the diffuse layer are dragged
by the flow whatever the frequency), and therefore,
The total current density is therefore given by the sum
of the conductive and advective contributions, which
yields the following generalized Ohm
'
s law:
Q V
s w
Q V ω
, s w
3 19
Q V ω
η w
E + k ω
σ ω
2
J =
p w
ω
ρ w s w u
3 25
2 In the thin double layer approximation (see Chapter 1),
one can expect Q V
Q 0
The two constitutive equations for the generalized
Ohm
V .Therefore,wehave
'
s and Darcy
'
s laws are written as two coupled
equations:
Q V ω
Q 0
V s w
, s w
1
i
ωτ
k s w
3 20
σ
L ω , s w
s law, Equation (3.17), into the
Darcy equation, Equation (3.9), yields the following form
of Darcy
Introducing Coulomb
'
J
E
=
k ω
, s w
η w
2
i
ω
w w
− ∇
p w
ω
ρ w s w u
L ω
, s w
'
s law:
3 26
Q V ω
η w
k ω
η w
ρ w s w u + k ω
where the coefficient L (
2
ω
) is defined as
i
ω
w w =
p w ω
E
3 21
, s w Q V ω
, s w = k ω
, s w
L ω
3 27
η w
This equation shows the influence of three forcing
terms on the Darcy velocity: (i) the pore fluid pressure
gradient, (ii) the displacement of the solid framework,
and (iii) the electrical field through electroosmosis.
Now, we turn to investigate the macroscopic electrical
current density, J . The first contribution to J is the con-
duction current density given by Ohm
The generalized streaming potential coupling coeffi-
cient is defined by the following equations in the quasi-
static limit of the Maxwell equations:
× E =0=
E = −∇ ψ
3 28
'
s law:
, s w = ψ
L ω , s w
σ ω
C p ω
=
3 29
σ ω
J e =
E
3 22
p w j =0 ; u =0
, s w
k ω , s w Q V ω , s w
η w σ ω
(
where the conductivity
) denotes the complex
conductivity discussed in Chapter 1 and section 2.3.1.2
of Chapter 2.
The second contribution to the total current density
corresponds to the advective drag of the excess of charge
σ
ω
C p ω
, s w =
3 30
, s w
More explicitly, in the thin double layer approximation,
the streaming potential coupling coefficient is given by
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