Introduction to the basic concepts - The Seismoelectric Method - page 30

achieve notational brevity. The strain tensor is related to
its components by

pressure P , and we consider no change in the fluid pres-
sure p =0. In this case, the bulk deformation is given by

P
K

= ε ij x i x j 1 121

ε

ε kk P ,0 ≡∇ u P ,0 = − 1 122

where x i ( i = 1, 2, 3) denote the basis vectors of the Car-
tesian framework of reference ( x i x j =

where the drained bulk modulus of the porous material
is defined by

δ ij
denotes the Kronecker delta) and a b represents the
tensorial product between vectors a and b .
We first consider two states, E1 and E2, that will be
combined soon to determine the effective pressure in
a porous material. In state E1, we apply a confining

δ ij where

1
K =

1
V

∂V
∂

− 1 123

P p =0, T

and T denotes temperature. In state E2, we apply a
confining stress P , and we imposed a fluid pressure equal
to the confining stress p = P . In this second state, the
deformation of the material is controlled by the stiffness
of the solid phase (not by the stiffness of the skeleton).
Therefore, we have

X 3

X 1

O

P
K S

ε kk P = p , p ≡∇ u P = p , p = − 1 124

X 2

T 31

where K S , the drained bulk modulus of the solid phase
(e.g., silica), is defined by

T 11

1
K S = −

1
V

∂ V

∂

1 125

P p = P , T

T 21

Now, we can describe the general bulk deformation of
a porous material in state E as the linear superposition of
the two states E1 and E2 as shown in Figure 1.26. This
can be written as

Figure 1.25 Definition of the stress tensor components on a cube
of an elastic material. Each component defines a force on one of
the face of the cube in a certain direction. These forces are
imposed from the external world.

P

P - p

P=p

Solid

Solid Solid

p

=

+

p

p = 0

E ( P , p ) E 1 ( P - p , 0) E 2 ( P = p, p )

Figure 1.26 Application of the superposition principle used to determine the effective pressure law in a linear poroelastic material. The
porous material is composed of a solid phase with an interconnected pore space, which is also connected to the external world. In the
general case, we impose a confining pressure P to the porous material, and there is an internal pressure corresponding to the pore fluid
pressure p . The general case can be considered as the superposition of two states E1 and E2. In the first state, we apply a confining
pressure P - p , and therefore is no internal pressure in the pores. In state E2, we apply a confining pressure that is equal to the fluid

pressure.