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once averaged,
ρ
can be replaced by the effective charge
passing per unit surface area of a cross section of the
porous material per unit time).
Expression for the electrical conductivity
density:
σ
can be
N
Q V
obtained by upscaling the local Nernst
Planck equation.
We assume that the wet oil carries a net charge per unit
volume to compensate the fixed charge density at the
surface of the minerals. As discussed in Chapter 1, the
total current density J (in Am 2 ) of the porous material
is given by J =
-
Q V =
q i C i ,
2 32
i =1
where C i is the phase average of the concentration of
species i in the pore space and q i the charge (in C) of
the ionic species i ( q i = 0 for neutral species) and Q V
denotes the total charge density per unit pore volume
of the diffuse layer. The effective charge density has been
introduced in Chapter 1 and its derivation will not be
repeated here.
The boundary value problem for the fluid flow is
expressed on the surface of an averaging cylinder (see
Pride, 1994)by
E + Q 0
V w . In this equation, the last term
corresponds to the source current density given by the
product between an effective charge density and the
Darcy velocity. This equation differs from the more con-
ventional form that uses the product of a cross-coupling
coefficient with the gradient of fluid pressure and an iner-
tial coupling term. The present approach has fewer para-
meters than the classical approach because the charge
density, Q 0
σ
V , can be determined from the low-frequency
permeability, k 0 , alone (see Figure 1.9).
At low frequency, the electrical field, E , is related to
the electrical potential,
η f
2 v + i
ωρ f v =
p f
F f ,
2 33
ψ
,by E =
−∇ ψ
, which satisfies
v =0,
2 34
× E = 0. The low-frequency coupling coefficient
is
given by
v =0on S ,
2 35
Q V k 0
η f σ
= ψ
z
p f
i
ωρ f u s , z = L
C 0 = lim
ω
0 C
ω
=
2 38
p f =
2 36
p f
0, z =0
J =0
where S represents the solid
pore fluid interface and z is
the unit vector normal to the disk face of the representa-
tive elementary volume. This boundary value problem is
exactly the same as the one given by Pride (1994) for a
poroelastic porous body saturated by a Newtonian fluid,
except that
-
s laws appear there-
fore as cross-coupled constitutive equations
The generalized Darcy
'
s and Ohm
'
k
ω
η f
C
ω σ
i
ω
w
η f . We can therefore
easily generalize the results obtained by Pride (1994) to a
poroelastic material saturated by a viscoelastic solvent.
This yields the following modified Darcy equation:
η f should be replaced by
=
2
J
ω σσ 1+ c
ω
η f σ
C
2 39
k
ω
−∇
p f +
ρ f ω
2 u s + F f
,
E
w = k
ω
η f
2 u s + F f
ω
−∇
p f + ρ f ω
ω σ
i
C
E ,
2 37
where F f
is the applied body force acting on the
fluid phase.
The normalized dynamic permeability is given by
where w =
u s is the phase average filtration dis-
placement of the fluid phase relative to the mineral
framework,
φ
u f
ϕ
is the interconnected porosity, k (
ω
) is the
ωτ m c
k
k 0
1
i
k
ω
=
,
2 40
dynamic permeability, C (
) is the dynamic coupling coef-
ficient described in the following text, and
ω
m c
1
i
ω ω
c 1
i
ωτ
is the electri-
cal conductivity of the porous material. Note that the
time derivative of the averaged filtration displacement
w
σ
2
where k 0 =
Λ
8 F (Johnson, 1986) is the permeability
at
low frequencies
ω
Min
ω
c ,
ω
m ,
ω
m =1
τ
m , and
corresponds to the Darcy velocity, that is,
the flux density of water (i.e., the volume of water
ω
=
ϕ
v
ω
ω
c =
η f k 0
ρ f F . Both F and
Λ
are two textural parameters
defined by (Johnson, 1986)
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