Application to the vadose zone - The Seismoelectric Method - page 228

Assuming that vertical changes in parameter A are
negligible compared to changes in the volumetric water
content, Equation (7.8) describes a linear relationship,
with slope A , between the magnitude of the seismoelec-
tric conversions,

∇ φ t = C ∇ pt , 7 3

where C is the streaming potential coupling coefficient
described in detail in Chapter 3. This assumes that
induced flow is laminar, that all pore radii are much lar-
ger than the thickness of electrical double layer at the
fluid

δ s ek t , and volumetric water content

changes, δθ , with depth.

solid interface, and that surface conductivity is
negligible compared to that of the bulk fluid. We use
the simplified relationship discussed in Chapter 3:

-

7.2.5 Discussion
In the present case, we have no reason to expect signif-
icant spatial changes to occur in any of the fluid or inter-
facial properties. The range of porosities (0.33 to 0.36;
West & Truss, 2006) is smaller than the range of volumet-
ric water contents (~0.15 to 0.33; see Figure 7.9b) and, as
such, supports a linear relationship according to
Equation (7.7). Notwithstanding, the spatial variation
of the porosity, albeit small, could explain some of the
variability in Figure 7.10. The inverse relationship
between the seismoelectric conversions and water con-
tent changes (Figure 7.10) must imply that Q 0

Cs w = Cs w =1 s w , 7 4

Q V k
η w σ

Cs w =1 = − 7 5

In these equations, Q V (expressed in C m − 3 ) denotes
the effective charge density of the diffuse layer per unit
pore volume that is dragged by the flow of the pore
water, k the permeability (in m 2 ), the electrical conduc-

tivity of the porous material (in S m − 1 ), and

σ

V is positive

η w the
dynamic viscosity of the pore water (in Pa s). Combining
Equations (7.3) and (7.4), and by replacing s w with the
volumetric water content,

(negative zeta potential, ) because all other parameters

affecting A in Equation (7.7) are positive. This is not
surprising, however, since the zeta potential is indeed
negative in most naturally occurring environments (see
Revil et al., 1999a, b and Chapter 1). In the present case,
A has a value of ~0.33 (Figure 7.10).
Haines and Pride (2006) demonstrated that layers as
thin as 1/20th of the dominant seismic wavelength can
be responsible for observable seismoelectric conversions.
In this case study, the mean seismic velocity in the vadose
zone was 811 m s − 1 , and the dominant seismic frequency
was 61 Hz, implying that layers as thin as ~0.6 m should
generate detectable seismoelectric conversions. Vadose
zone volumetric water content typically varies on this
scale (Figures 7.5 and 7.9b), effectively representing ver-
tically successive thin layers, each producing a strong
electrical dipole. The inference of a temporally continu-
ous seismoelectric conversion (Figure 7.9a), produced
at the surface by the superimposed effect of these dipoles,
is therefore consistent with Haines and Pride (2006).
Since Figure 7.10 effectively represents a calibration of
seismoelectric conversions for volumetric water content,
Equation (7.7) could thus be used to either complete the
volumetric water content-depth profile between 11.25 m
depth and the water table at 14.5 m or to estimate volu-
metric water content-depth profiles at some distance
from this study site where no borehole control is availa-
ble. Alternatively, if the pressure pulse ( P 0 ( t )) was meas-
ured in field surveys and the zeta potential (

ζ

θ , where s w = θ / ϕ ( ϕ denotes

porosity), we have

Q 0

V k
η w σϕ

p 0 t 1 ∇ ψ t = Cs w ∇ pt = −

r θ

Ht , 7 6

assuming the equation

1
r Ht ,

∇ pt = p 0 t 7 7

corresponds to the differential fluid pressure caused by
the propagating seismic wave, where p 0 ( t ) is the pressure
pulse generated by the hammer impact at the ground
surface, r is radial distance from the shot point, and H ( t )
denotes the Heaviside (step) function. We can substitute
s ek ( t )= ( t ) r as the sp heric ally c orrect ed seismoelectric

conversion and assign s ek t and p 0 t , respectively, as
the stacked seismoel ect ric conversion signal and stacked
pressure pulse. If

∇ φ

are the peak-to-peak
positive or negative change in seismoelectric conversion
and volumetric water content, respectively, as consid-
ered in Figure 7.10, we obtain

∇ s ek t and ∇ θ

δ s ek t = − A δθ , 7 8

Q 0

V k
η w σϕ

A ≡ p 0 t 7 9

ζ )or Q V could