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where
p
is the pore fluid pressure and
L
is the length of
the cylindrical core. For a granular medium with a uni-
modal particle size distribution, an expression to deter-
mine the length scale
Watanabe and Katagishi, 2006, and references therein).
Gorelik (2004) used dimensional analysis to demonstrate
that the effect of the Reynolds number corresponds to a
multiplication of the Helmholtz
Λ
is given by (Revil, 2002)
Smoluchowski equation
by an unspecified function of the Reynolds number. In this
chapter, we look for an explicit (quantitative) relationship
between the streaming potential coupling coefficient and
the Reynolds number. At the scale of a representative
elementary volume, the current density is given by
-
d
0
2
mF
Λ
=
2 222
−
1
From Equations (2.221) and (2.222), the Reynolds
number is the solution of the following equation:
+
Q
0
J
=
−
σ
∇
ψ
V
w
,
2 229
2
f
g
2
m
αη
d
0
FF
ρ
h
L
kQ
0
V
η
f
∇
Re
2
+Re
−
=0
2 223
2
f
1
3
−
J
=
−
σ
∇
ψ
−
p
,
2 230
The positive root of Equation (2.223) is
where
k
is the apparent permeability defined earlier. The
streaming potential coupling coefficient can be related to
the excess charge of the diffuse layer per unit pore vol-
ume,
Q
0
Re =
1
2
1+
c
−
1 ,
2 224
V
,by
C
0
=
k
0
Q
0
. Equation (2.229) expresses
the fact that the source current density is equal to the
excess of charge of the pore fluid
Q
0
η
f
σ
V
d
0
FF
c
=
βρ
f
η
p
L
,
2 225
f
1
3
−
V
times the Darcy
velocity
w
. As the seepage velocity is influenced by the
increase of the Reynolds number (for Re >0.1), the
Reynolds number also influences the value of the stream-
ing potential coupling coefficient. Following Equations
(2.227) and (2.230), the streaming potential coupling
coefficient is related to the Reynolds number by
2.25 × 10
−
3
is a numerical constant (determined
fromthe constants givenearlier). Equation(2.224) is anew
equation that has a strongpractical value since it canbeeas-
ily used to determine the Reynolds number in a porous
material from the knowledge of the pressure gradient.
In the present case, the macroscopic body force corre-
sponds to the electrostatic force associated with the
excess of electrical charge per unit pore volume. There-
fore, the generalized Darcy equation, Equation (2.218),
can be written as
where
β
≈
C
C
0
=
1
1+Re
,
2 231
C
C
0
=1,
lim
Re 0
2 232
k
η
f
∇
Q
0
V
w
=
−
p
−
∇
ψ
,
2 226
where
C
0
is the streaming potential coupling coefficient in
viscous laminar flow conditions and
C
is themeasured cou-
pling coefficient. A comparison between Equations (2.227)
and (2.232) with experimental data obtained by Bolève
et al. (2007) is shown inFigure 2.14.We see a clear decrease
of thestreamingpotential couplingcoefficientwith theRey-
nolds number that is well reproduced by the model.
where
k
is an apparent permeability that is related to the
Reynolds number by
k
k
0
=
1
1+Re
,
2 227
k
k
0
=1,
lim
Re 0
2 228
2.4 Conclusions
where
k
0
is the permeability in viscous laminar flow
conditions.
The influence of inertial flow upon electrokinetic cou-
pling has been the subject of very few publications (see
In this chapter, we have developed a complete theory of
wave propagation in poroelastic media. In the first case,
the porous material is considered to be saturated with a
viscoelastic fluid able to sustain shear stresses. In this case,
