Electrical disturbances associated with seismic sources - The Seismoelectric Method - page 138

500

0

1000

Offset (m)

01 02 03 04 05 06 07 08

Well

0

First layer L1
Second layer L2

400

Focal mechanism

S(650,700)

Figure 5.2 Sketch of the domain used for the
modeling. It consists of two layers named L1
(shallow layer) and L2 (deeper layer). A set of
seismic and electrical receivers (geophones
and electrodes) are located at the ground
surface (upper boundary of L1).

Receivers
Seismic source

1000

papers in seismoelectric modeling (see Chapter 2).
Therefore, the poroelastodynamic equations of motion
expressed in the frequency domain are given by

Table 5.1 Material properties of the two layers L1 and L2 used for
the forward model.

Parameter L1 L2 Units

2 S

ω

− ω ρ u + θ ω ∇ p = ∇ T+ F , 5 1

S m − 1 σ 0.01 0.05

C m − 3 4.2 1234.8

0
V

Q

u T T = λ ∇ u I + G ∇ u + ∇ , 5 2

kg m − 3 ρ 2650 2650

s

p
M +

kg m − 3 ρ f 1040 1000

2

∇ k ω ∇ p − ω ρ f u − f = − α ∇ u , 5 3

ϕ 0.20 0.15

—

K s 36.5 35.7 GPa

K 2.22 17.9 GPa

where I is the identity matrix, F corresponds to the body
(volumetric) force acting on the bulk porous medium, u
denotes the (averaged) displacement of the solid phase,
and f denotes the bulk force acting on the fluid phase.
These three equations correspond to (i) a macroscopic
momentum conservation equation for the solid phase,
(ii) a constitutive equation for the effective stress tensor
T , and (iii) a momentum conservation equation for the
pore fluid. In these equations, p is the average pressure in
the pore fluid phase, and the coefficient M is one of the
Biot moduli defined by M = K f K S K f 1

fr

K f 0.25 2.25 GPa

G 4.0 17.7 GPa

1×10 − 12 1×10 − 15 m 2 k 0

1×10 − 3 1×10 − 3 η Pa s f

m s − 1 c p 1925 4322

m s − 1 c s 1310 2721

elastic, and (iv) other attenuation mechanisms like
squirt-flow mechanisms are neglected. In the following,
we assume an e − i ω t time dependence for the mechanical
disturbances ( i being the pure imaginary number used in
Chapter 1, i 2 = − 1, ω is the angular frequency, and t is
time). We use the displacement of the solid phase and
the pore fluid pressure as unknowns instead of the solid
and fluid phase displacement vectors like inmost previous

− ϕ − K fr K S +

ϕ K S , where K S and K f denote the bulkmoduli of the solid

phase and fluid phase, respectively; K fr symbolizes the
bulk modulus of the solid frame (skeleton); and ϕ corre-
sponds to the connected porosity.
The modulus λ corresponds to the drained Lamé
coefficient defined by λ = K fr − 2 3 G where G = G fr