Seismoelectric theory in saturated porous media - The Seismoelectric Method - page 50

no change in the fluid pressure (e.g., considering an
empty or drained porous material). In this case, the bulk
deformation of the solid frame, the variation of the poros-
ity, and the deviators are

2 d s T D =2 d s T D

π − π 0 +2 d s π π , 2 70

e

T D

=2 d f T D 0 +2 d f π π 2 φ d f π 2 71

P
K fr ,

This yields

∇ u s P 0 = − 2 60

P p f

K fr −

− p f

K s ,

K fr
K s − ϕ

P
K fr ,

∇ u s P p f = − 2 72

Δϕ P 0 = − 1 − 2 61

−

K fr
K s − ϕ

P p f

K fr

Δϕ P p f = − 1 − , 2 73

2 d s T D 0 = T D

G fr ,

2 62

1
G fr T D

+ G s , 2 d s T D

π − π

= 2 74

G fr
G s − ϕ

1
G fr T D ,

e

T D 0 =1

2 ϕ d f − 2 63

G fr
G s − ϕ

1
G fr

e

T D T D 2 ϕ d f π =1 − − π , 2 75

where G fr and K fr are the shear and bulkmoduli of the dry
porous frame (in other words the shear and bulk moduli
of the skeleton of the material, both expressed in Pa).
In the second thought experiment, we apply a fluid
pressure p f everywhere throughout the pore space and,
simultaneously, a confining pressure P = p f to the exter-
nal surface of the sample. At the same time, we apply
the mean deviatoric stress π to the pore space and, simul-
taneously, a deviatoric stress T D = π to the external sur-
face of the sample. In this case, the bulk deformation of
the solid frame, the variation of the porosity, and the var-
iation of the deviators are given by

and therefore

P
K fr +

p f
K fr ,

∇ u s P p f = − α 2 76

P p f

K fr

−

Δϕ P p f = − α − ϕ , 2 77

1
G fr T D

− α G G fr ,

2 d s T D

π

= 2 78

1
G fr

e

T D T D p f

K s ,

2 ϕ d f π = α − ϕ − π , 2 79 G

∇ u s p f p f = − 2 64

where α =1 − K fr K s is the classical Biot coefficient

Δϕ p f p f =0, 2 65

( G fr G s is a second Biot shear coef-

ficient introduced by Revil and Jardani (2010).
The equation for the porosity variation

ϕ ≤ α ≤ 1) and α G =1 −

= G s ,

2 d s π π 2 66

P , p f can be
written as a function of the increment of the fluid content
∇

Δ ϕ

w P , p f using the continuity equation for the mass of
the pore fluid. This equation is given by

e

2 φ d f π π =0 2 67

Using the superposition principle, we add together the
results of the two thought experiments discussed. For
the general case, the bulk deformation of the solid frame,
the variation of the porosity, and the deviator are

+ K f p f + Δϕ ∇ w + ϕ ∇ u s =0, 2 80

P

K fr − K f p f +

− p f P

K fr − ϕα

p f
K fr ,

∇ w = α − ϕ ϕ 2 81

∇ u s P p f = ∇ u s P − p f 0 + ∇ u s p f p f , 2 68

P
K fr −

p f
K fr α

K fr
K f −

K fr
K s

∇ w = α + ϕ 2 82

Δϕ P p f = Δϕ P − p f 0 + Δϕ p f p f , 2 69