Geoscience Reference
In-Depth Information
CHAPTER 3
Seismoelectric theory in partially
saturated conditions
Now that we have developed a complete theory in the
saturated case (both with the poroelastic and acoustic
approximations), we consider in this chapter some
extensions to two cases. The first extension concerns
the unsaturated case for which the material is partially
saturated in water (considered to be the wetting phase
for the solid grains) but the nonwetting phase (air) is at
the atmospheric pressure. The second extension corre-
sponds to the case where there are two immiscible
Newtonian fluids present in the pore space. In this case,
we need to account explicitly for the capillary pressure.
Finally, we reinvestigate the acoustic approximation
extending its validity in partially saturated conditions.
We end the chapter by comparing some of the predictions
of our model with available experimental data.
an excess of (positive) charges in the pore space of the
porous material in the electrical double layer. As dis-
cussed in Chapter 1, the electrical double layer coating
thesurfaceofthegrainsismadeoftwolayers:(1)alayer
of (counter) ions sorbed onto the mineral surface and
(2) a diffuse layer where the electrostatic (Coulomb)
force prevails. In the following, we will use the subscript
a
to describe the properties of air (the nonwetting
phase), and subscripts
w
and
s
will be used to describe
the properties of thewater and solid phases, respectively.
Water is assumed to be thewetting phase. The term skel-
eton will be used to describe the assemblage of grains
alone without the two fluid phases in the pore network.
Another set of assumptions used later pertains to
the capillary pressure curve. Hysteretic behavior will
be neglected, and therefore, the porous material will be
characterized by a unique set of hydraulic functions
(for instance, the Brooks & Corey, 1964). We will work
also with the Richards model, which makes the assump-
tion that the air pressure is constant and equal to the
atmospheric pressure. This implies in turn that the air
phase is infinitely mobile and connected to the atmos-
phere. This represents a simplification of the problem that
makes the analysis more tractable. The capillary pressure
p
c
(in Pa) is defined as the pressure of the nonwetting
phase minus the pressure of the wetting phase (Bear &
Verruijt, 1987):
3.1 Extension to the unsaturated case
3.1.1 Generalized constitutive equations
In this section, we develop a new theory of the seismo-
electric effect in unsaturated porous material. This
theory can be used to understand the electromagnetic
signals associated with seismic wave propagation in
saturated and unsaturated porous media. We consider
in the following text an isotropic representative
elementary volume of a porous material with connected
pores. The surfaces of the minerals in contact with the
pore water are negatively charged. We consider therefore
p
c
=
p
a
−
p
w
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