Forward and inverse modeling - The Seismoelectric Method - page 101

CHAPTER 4
Forward and inverse modeling

2 s

ω

In Chapters 2 to 3, we developed all the field equations
required to model the seismoelectric effect in saturated
and unsaturated conditions (including two-phase flow
conditions). We wish to use these field equations to
numerically forward-model the occurrence of seismo-
electric signals for various applications in earth sciences.
Therefore, we need to discuss how to numerically
implement the forward modeling of these equations with
the finite-element method. Then, as geophysicists, we
want to solve the inverse problem in order to determine
the information content of these seismoelectric signals
(e.g., White, 2005; White & Zhou, 2006). To accomplish
the solution of the inverse problem, we present in this
chapter both stochastic and deterministic algorithms to
invert seismoelectric signals in terms of material
properties or in terms of localizing boundaries between
geological formations.

u−θ ω ∇ p + ∇ ^ TðÞ =0 ω ρ ð 4 : 1 Þ

1
M p +

∇ ð k ω ∇ p Þ + θ ω ∇u =0 ð 4 : 2 Þ

where the bulkmodulus, M, is given by Equation (2.189),
k ω by Equation (2.193), ρ

s
ω

by Equation (2.195), and the

θ ω by Equation (2.202) and T denotes the
effective stress tensor (see Eqs. 2.199 and 2.200). The so-
called weak formulation can be obtained by multiplying
the governing Equations (4.1) and (4.2) by the admissible
variations of displacement of the solid phase and pore
water pressure fields, δ u and δ p. These equations are then
integrated on the volume, Ω , of the poroelastic material

coupling term

ð

δu d Ω =0

2 s

ω

ω ρ u−θ ω ∇ p + ∇ ^ TðÞ ð 4 : 3 Þ

Ω

ð

1
M p +

∇ ð k ω ∇ p Þ + θ ω ∇u

∂ pd Ω

=0 ð 4 : 4 Þ

Ω

4.1 Finite-element implementation

The integral of the each of these equations is then
rewritten using Green

4.1.1 Finite-element modeling
In water-saturated conditions, the poroelastic model
described in Chapter 2 (dynamic Biot - Frenkel theory)
can be expressed in the frequency domain under the
following equations for the displacement of the solid
phase, u , and the pore fluid pressure, p (see Eqs. 2.201
and 2.203):

' s formula (Attalla et al., 1998):

ð

2 ð

T ðÞε

s s

ω

δð d Ω−ω ρ uδ ud Ω

Ω

Ω

+ ð
Ω

ð

θ ω ∇ p δ ud Ω− T ðÞnδ ud Γ =0 ð 4 : 5 Þ

Γ