Geoscience Reference
In-Depth Information
Dynamic coupling coeficient
10 -6
C 0
k 0 ρ w F
Berea sandstone
τ
=
C
p *(
ω
) ≈
;
k
η w
i ωτ k
1
10 -7
+
+ + + + + + +
+ + + + + + +
10 -8
Measurements
Model
0.012 S m -1
0.048 S m -1
0.095 S m -1
0.18 S m -1
0.32 S m -1
+
10 -9
10 4
10 5
10 6
1000
Frequency cHz
Figure 2.13 Dynamic streaming potential coupling coefficient (the values reported at 1 kHz are the static values). The data are from
Zhu and Toksöz (2013) for the same Berea sandstone (porosity 0.23, permeability 450 mD, NaCl). The relaxation is due to the transition
between the viscous laminar flow regime at low frequency and the inertial laminar flow regime at high frequencies. Below 10 kHz,
the streaming potential coupling coefficient can be considered independent of frequency.
Another way to look at the dependency of the stream-
ing potential coupling coefficient with the inertial effect
is to plot this coupling coefficient as a function of the
Reynolds number. Teng and Zhao (2000) recently
derived a generalized Darcy equation by volume aver-
aging the local Navier
Re = ρ f w R
η f
2 219
In a porous material, the radius of the capillary should
be replaced by a corresponding length scale of the porous
material. The Reynolds number is defined by
Stokes momentum equation over
a representative elementary volume of a porous material,
given by
-
Re = ρ f w
Λ
,
2 220
η f
ρ f d w
dt
+ 1+Re
k 0
η f w = −∇
p + F ,
2 218
where
Λ
is a characteristic length of the flow (for capillaries
Λ
= R where R is the radius of the capillary). If we replacew
by the Darcy equation (neglecting the electroosmotic
contribution), we can use the following approximation
to get an estimate of the Reynolds number:
where F is amacroscopic body force and Re is the Reynolds
number, a key dimensionless number that expresses the
ratio of inertial to viscous forces in the Navier
Stokes equa-
tion (Batchelor, 1972). For a capillary of radius R , U being
the strength of the seepage velocity, the Reynolds number
is then defined by (e.g., Batchelor, 1972)
-
ρ f k 0 Λ
p
L ,
Re =
2 221
η
f 1+Re
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