André Revil

Associate Professor at the Colorado School of Mines

Directeur de Recherche at the National Centre for Scientific Research (CNRS)

Abderrahim Jardani

Associate Professor at the University of Rouen

Paul Sava

Associate Professor of Geophysics at the Colorado School of Mines

Allan Haas

Senior Engineering Geophysicist

The edition first published 2015 © 2015 by John Wiley & Sons, Ltd.

When a seismic wave passes through a porous rock, it causes movement of both the rock frame and the pore fluid. Normally their movements are not at same rate. Relative movement between the water and solid disturbs the electrical double layer which in turn produces electric charge movement. Eventually the variation in electric charge can be detected as a seismoelectric signal. Therefore. velocity and frequency of seismoelectric signals depend directly on the seismic wave (Dong, 1990)

To mathematically derive expressions of seismoelectric effect propagation, an important assumption is made that seismoelectric signals “propagate” at the same velocity as its inducing seismic wave. This assumption is validated through its similarity to headlight traces observed on a highway. A travelling car and its taillight represent the seismic wave and the seismoelectric signal it generates, respectively. As light speed is obviously faster than car speed, we can observe the movement of a car by tracking its tailing light in a dark night. Similarly, electric current transmits much faster than seismic waves in rocks. The transit time of electric current from the affected rock to a detector on the ground is omitted. By honoring our assumption, one can plot the propagation of seismoelectric effect the same way as of its inducing seismic wave.

In the case where two (one pair of) electrodes were placed inside of the seismic field, the streaming potential is measured during the interval while the seismic wave travels from the one electrode to the other. After time zero, as seismic waves penetrate into deeper layers of subsurface along with wave front, the seismoelectric signal measured corresponds to response at a certain time or depth. Apply the Helmholtz equation to the whole body effect, and express the results in terms of stress in the rock, yields

E(t)= [ω2Δρ(p1(t)-p2(t))εζ]/[4πση]

where E(t) is streaming potential, ω is the frequency of the seismic wave, Δρ is the density difference between the rock and the water, P1 and P2 are rock stresses of the seismic wave at electrodes 1 and 2, ε is the permittivity of dielectric of the fluid, ζ is the zeta potential, η is the viscosity and σ is the fluid conductivity.

Pride(1994) uncovered the relationships between electromagnetic and acoustic signal in porous media. However, direct application of the equation in real data is usually difficult. According to previous lab experiments (Dong, 1990), it can be safely assumed that in seismic frequency range (1) fluid movement behavior is similar to that of laminar flow, and (2) water molecule is incompressible. Then we made simplifications to the Pride equations and derived the two-dimensional seismoelectric signal propagation equation (Dong, 2002).

Two dimensional seismoelectric signal propagation equations in accordance with the stress-strain relation (Voigt's solid formula) can then be developed from the Helmholtz equation:

E(x12)=(ω2Δρ∈ζ/(4πησ))[∈xx(e1)+∈zz(e1)+∂∈xx(e1)/∂t + ∂∈ zz(e1)/∂t−∈ xx(e2)−∈zz(e2) −∂∈xx(e2)/∂t−∂∈ zz(e2)/∂t]

E(z12)=(ω2Δρ∈ζ/(4πησ))[∈zz(e1)+∈xx(e1)+∂∈zz(e1)/∂t + ∂∈ xx(e1)/∂t−∈ zz(e2)−∈xx(e2) −∂∈zz(e2)/∂t−∂∈ xx(e2)/∂t]

In which E(x12) is the streaming potential along the x axis between electrodes 1 and 2 and E(z12) is the streaming potential along the z axis between the same electrodes. In addition, ∈xx (e1) is the displacement in the x direction along the x axis at electrode 1, ∈xx (e2) is the displacement in the x direction along the x axis at electrode 2, ∈zz (e1) is the displacement in the z direction along the z axis at electrode 1 and ∈zz (e2) is the displacement in the z direction along the z axis at electrode 2. Partial derivatives of the displacement with respect to time are included, as are the same variables used in Equation 1. Use of these equations allows us to analyze a streaming potential signal in respect to time and arrive at the motions of the seismic wave which generated the streaming potential seismoelectric signal. Note that these equations are given in the two dimensional version, but the equivalent sets of equations for three dimensions can obviously be derived. It is also possible to reduce them back to the one dimensional versions and use that for some surveying.

The relation between the ζ streaming potential and the radius of a tube (after Parkhomenko 1971)

Electrokinetic coupling coefficient as a function of permeability with fluid resistivity is 200 Ωm. Empty square are measured values for limestone. Stars are computed value by Jouniaux and slope of straight line are proportional to K