Geoscience Reference
In-Depth Information
Normalised pressure ield (
t
= 219 ms)
Normalised electric potential (
t
= 219 ms)
200
200
5
5
100
100
0
0
0
0
−100
−100
−5
−5
−200
−200
Geophone
Electrode
−200
−100
0
100
200
−200
−100
0
100
200
Offset (m)
Offset (m)
Recorded pressure at geophone
Recorded potential at electrode
4
4
2
2
0
0
−2
−2
−4
−4
0
100
200
300
0
100
200
300
Time (ms)
Time (ms)
Figure 1.22
Same as Figure 1.18 for time 219.0 ms. The traveling P-wave is located in between the source and the position of
the recording geophone located with an electrode (reference electrode to infinity). We see no disturbances at the position
of the geophone and electrode. (See insert for color representation of the figure.)
1.5 Elements of poroelasticity
ε
ij
=
1
2
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
;
ð
1
:
118
Þ
In this section, we provide some of the basic tools
required to understand static and dynamic poroelasticity.
These elements will be used further in Chapter 2 to estab-
lish some general models for the seismoelectric theory.
;
ε
ij
=
1
2
u
i, j
+ u
j, i
ð
1
:
119
Þ
where we are using the engineering notations and
where the vector
u
refers to the displacement of the solid
phase. The (small) change in volume of the material is
described as
1.5.1 The effective stress law
We consider a porous material with a connected porosity
and a solid frame (the skeleton) that behaves elastically.
The material is homogeneous at the scale of the represen-
tative elementary volume and monomineralic. The stress
tensor applied to the porous material is written as
T
. The
component of the stress tensor corresponds to T
ij
, which
represents the force per unit surface area imposed from
outside in the ith-direction along a surface with normal
in the jth-direction (Figure 1.25). The strain (deforma-
tion) tensor is defined by
ε
ij
=
∇u
=
δ
V
ε
kk
=Tr
V
;
ð
1
:
120
Þ
where Tr refers to the trace of the tensor (the sum of the
diagonal elements) and
δ
V/V denotes a volume element
increment. In Equation (1.120), we have used Einstein
(summation) convention on repeated indices (Einstein,
1916). This convention implies summation over a set
of repeated indices in an equation with the goal to
