Geoscience Reference
In-Depth Information
CHAPTER 2
Seismoelectric theory in saturated
porous media
We present in this chapter a complete theory for the
generation of seismoelectric effects in the quasistatic limit
of the Maxwell equations and for various types of rheo-
logical constitutive laws for the porous material and the
pore fluid. We start with a description of the poroelastic
wave propagation in a poroelastic material filled by a vis-
coelastic fluid that can sustain shear stresses (extended
Biot
than the classical Biot
-
Frenkel theory and provides inter-
esting mechanisms for some resonance of the fluid inside
the pore space of the material. At low frequencies, it has
been observed that electrokinetic phenomena in oil-wet
porous media are similar to those existing in water-
saturated porous materials (Yasufuku et al., 1977;
Alkafeef et al., 2001; Alkafeef & Smits, 2005). Various
authors have shown that the components of oil that
are responsible for wettability are also polar components
(Buckley & Liu, 1998; Alkafeef et al., 2006) and therefore
they are good solvents (Delgado et al., 2007). We will
show in this chapter that the extended Biot
-
Frenkel
symmetry for such a Maxwell fluid yields symmetric
constitutive relationships and the existence of
Frenkel theory). This represents the general case
of wave propagation that will be discussed in this chapter.
Then we present the equations describing the propaga-
tion of the seismic waves in a poroelastic material satu-
rated by a Newtonian fluid (classical Biot
-
Frenkel
theory) as a special case of the more general theory. In
this second case, we will describe the properties of the
most important sensitivity coefficient entering into the
coupled equations, the so-called streaming potential cou-
pling coefficient.
-
two
(compressional) P-waves and two shear (S-)waves.
2.1.1 Properties of the two phases
We start this section by describing the constitutive model
of the stress
2.1 Poroelastic medium filled with a
viscoelastic fluid
strain relationships for a viscoelastic fluid
such as a wet oil. A linear Maxwell fluid model consists
of a linear dashpot in series with a linear spring. In this
situation, the fluid behaves like a Newtonian viscous fluid
at low frequency (or for long time scales) and like a solid
at high frequencies (or short time scales). The transition
frequency is discussed in the following text. We denote
by
T
ij
(in Pa) and
e
ij
(dimensionless) the components of
the stress and deformation tensors
T
f
and
e
f
of this fluid
(indicated by the subscript f), respectively. Both
T
f
and
e
f
are symmetric second-order tensors. The stress
-
Traditionally, we consider the wave propagation inside
the framework of the classical Biot
Frenkel theory. In
this theory, the skeleton of the porous material is consid-
ered to be linear elastic while the pore fluid is Newtonian.
The present section provides a constitutive model of
the seismoelectric response of a charged porous medium
filled with a (Maxwell) viscoelastic solvent like a heavy
wet oil. We will show that this approach is more general
-
