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We experimentally validate a relatively recent electrokinetic formulation of the streaming potential (SP) coefficient as developed by Pride (1994). The start of our investigation focuses on the streaming potential coefficient, which gives rise to the coupling of mechanical and electromagnetic fields. It is found that the theoretical amplitude values of this dynamic SP coefficient are in good agreement with the normalized experimental results over a wide frequency range, assuming no frequency dependence of the bulk conductivity. By adopting the full set of electrokinetic equations, a full-waveform wave propagation model is formulated. We compare the model predictions, neglecting the interface response and modeling only the coseismic fields, with laboratory measurements of a seismic wave of frequency 500 kHz that generates electromagnetic signals. Agreement is observed between measurement and electrokinetic theory regarding the coseismic electric field. The governing equations are subsequently adopted to study the applicability of seismoelectric interferometry. It is shown that seismic sources at a single boundary location are sufficient to retrieve the 1D seismoelectric responses, both for the coseismic and interface components, in a layered model.

The first observation of coupling between electromagnetic and mechanical effects (also known as electroosmosis, which is one of the electrokinetic effects) dates back to the beginning of the 19th century. In 1809, Reuss [

The electrokinetic effect works as follows. In a fully fluid-saturated porous medium, a charged nanolayer at the solid-liquid interface is present (see Figure

Electric double-layer according to the Stern model. The inner and outer Helmholtz planes are indicated as IH and OH, respectively. The slipping plane is denoted by S and its charge is characterized by the

Gouy [

In 1936, Thompson [

Regarding wave modeling, Neev and Yeatts [

The governing equations of Pride describe coupled seismic and electromagnetic wave propagation effects. A schematic description of the coseismic and interface response effects is given in Figure

Schematic of a “standard” geometry for a seismoelectric survey (modified from Haines [

However, in seismoelectric surveys, the interface response is known to be very weak, that is, the response suffers from a very low signal-to-noise ratio. Therefore, the sources in classical seismoelectric surveys need to be strong. This is not always possible and therefore it is beneficial to be able to replace those strong sources by receivers: the principle of interferometry. In addition, by doing interferometry, stacking inherently takes place with a possible improvement of the signal-to-noise ratio as a result [

The foundations of the principle of interferometry were lain in 1968 by Claerbout who showed that by using the autocorrelation of the 1D transmission response of a horizontally layered medium (bounded by a free surface), the reflection response of this medium can be obtained [

Although the individual constituents of Pride’s model (i.e., Biot’s theory and Maxwell’s theory) have been experimentally validated, the dynamic

It is shown that measurements of both the dynamic

The governing equations for seismoelectric and electroseismic wave propagation in a fluid-saturated porous medium are derived from the compilation of Biot’s theory [

Expressing the expanded Biot equations, for the solid as well as the fluid and adopting an

We experimentally validate ^{3}, a viscosity of

Sample properties.

Property | Symbol | Value | Unit |
---|---|---|---|

Permeability^{a} | |||

Shape factor^{b} | 1.75 | [—] | |

Porosity^{c} | 0.093 | [—] | |

Tortuosity^{d} | 1.8 | [—] | |

Debye length^{e} | [m] | ||

Weighted pore volume-to-surface ratio^{f} | [m] |

^{
a}The permeability is measured directly. ^{b, d}The shape factor and the tortuosity are derived from an independent dynamic head experiment [^{c}The porosity is computed from [^{e, f}The Debye length and the characteristic pore size are computed from theory (see [

Schematic of the dynamic Darcy cell with borosilicate sample and Monel disks (modified from [

Exploded view of the capillary core.

The 50 Hz electromagnetic frequency radiating from the equipment is suppressed by shielding the setup and its wires (therefore use has been made of shielded twisted cable pairs). To reduce uncorrelated noise the data are averaged multiple times.

In Figures

Amplitude of the normalized dynamic permeability. Theory of Johnson et al. [

Phase value of the dynamic permeability. Theory of Johnson et al. [

The measurements of the normalized dynamic SP coefficient (normalized to the measured value at 11 Hz, where

Amplitude of the normalized dynamic SP coefficient. Theory of Pride [

Phase values of the dynamic SP coefficient. Theory of Pride [

The difference between measurement and theory in the high-frequency range can be caused by the possibility of the system to function as a capacitor [

Electrokinetic theory in isotropic, homogeneous, and fluid-saturated poroelastic media predicts the existence of a fast and a slow

Using (

The longitudinal fluid-solid ratio, which describes the fluid-to-solid displacement amplitude ratio, is derived from the first row in (

We now model coseismic electric potentials generated within a porous medium due to a fast

(Modified from [

Property | Symbol | Value | Unit |
---|---|---|---|

Bulk modulus skeleton grains^{a} | [Pa] | ||

Bulk modulus (pore) fluid^{b} | [Pa] | ||

Bulk modulus framework of grains^{c} | [Pa] | ||

Shear modulus framework of grains^{c} | [Pa] | ||

Pore fluid viscosity^{b} | [kg/(m s)] | ||

Pore fluid density^{b} | 1000 | [kg/ | |

Solid density^{c} | 2570 | [kg/ | |

Weighted pore volume-to-surface ratio^{d} | [m] | ||

Porosity of the porous medium^{c} | 0.52 | [—] | |

Permeability^{c} | |||

Tortuosity^{c} | 1.7 | [—] | |

Sample width^{e} | [m] | ||

Relative permittivity of the (pore) fluid^{b} | 80.1 | [—] | |

Relative permittivity of the solid^{b} | 4 | [—] | |

Fluid magnetic permeability (= | [H/m] | ||

(Pore) fluid conductivity^{e} | [S/m] | ||

Zeta-potential^{f} | [V] |

^{
a}[^{b}[^{c}see N5B in [^{d}[^{e}measured values, and ^{f}see [

(Modified from [

For the geometry of Figure

(Modified from [

Schakel et al. [

By comparing the model for the first coseismic response (CS1 in Figure

The model for the second coseismic response (CS2 in Figure

Model of filtered coseismic responses in time (a) and frequency domain (c) and coseismic measurements in time (b) and frequency domain (d). Pulse abbreviations are: coseismic response 1 (CS1) and coseismic response 2 (CS2).

It is possible to model all interface responses and coseismic effects of Figure

Considering the combined character of seismo-electromagnetic waves it can be very beneficial to use them for a wide range of applications. (The application for oil-field exploration has already been shown by Thompson et al. [

Interferometry makes use of the cross-correlation of responses at different receivers in order to obtain the Green’s function of the field response between these stations. In other words, it is the deterministic response from one station to the other.

Figure

Schematic seismoelectric interferometry setting. Cross-correlation of electric (

A second well-known problem in these conventional seismoelectric surveys is the very low signal-to-noise ratio. By doing interferometry, stacking inherently takes place with a possible improvement of the signal-to-noise ratio as a result. After deriving the system of equations for coupled seismic and electromagnetic waves in saturated porous media [

Following Wapenaar and Fokkema [

Next, considering the Fourier transform of an impulsive source acting at time

Starting from the general interferometric Green’s function representation (

We can distinguish two terms in this integral representation. The first term on the right-hand side represents correlations of recorded responses of sources on the boundary of the domain of reciprocity, whereas the second term on the righthand-side represents correlations of recorded responses of sources throughout the reciprocity domain.

As shown by de Ridder et al. [

Looking at expression (

Due to the fact that wave energy is dissipated during wave propagation, the domain sources are necessary to account for these losses. However, these sources are not likely to exist in reality or cannot be rewritten for practical applications and therefore we would like to be able to ignore their contributions.

As is already shown in three examples by de Ridder et al. [

In the following section, we will increase the complexity of the numerical configuration by adding an extra layer to the system, to investigate the Green’s function retrieval for a 1D, three-layered system bounded by a free-surface. In other words, we will look at the applicability of the interferometric seismoelectric Green’s function representation (

Overview of the relevant medium parameters for the 1D seismoelectric interferometry model.

Property | Unit | Value medium A | Value medium B | Dimension |
---|---|---|---|---|

Porosity | 0.4 | 0.2 | [—] | |

Pore fluid density | [kg/ | |||

Solid density | [kg/ | |||

Shear modulus framework of grains | [Pa] | |||

Pore fluid viscosity | [kg/(m s)] | |||

Static permeability | [ | |||

Static electrokinetic coupling | [ | |||

Tortuosity | 3.0 | 3.0 | [—] | |

Relative perm. of the (pore) fluid | 80 | 80 | [—] | |

Relative perm. of the solid | 4 | 4 | [—] | |

Bulk electric conductivity | [S/m] |

We consider a three-layered 1D medium bounded by a vacuum half-space. The top and bottom layer consist of medium parameters belonging to medium A and the sandwiched layer has the properties of medium B (see Table

The geometry of the 1D numerical experiment. Positions

Figure

Separated contributions of the domain integral and the boundary points to the retrieved Green’s functions. In other words, it shows the relative contributions of the two right-hand side terms in (

Several events can be recognized in Figure

The obtained correlation gather of the domain integral for a three-layered medium bounded by a vacuum. The scale is taken as the logarithm of the absolute value of the amplitude. Summing this correlation gather panel yields the total contribution of the domain integral as shown in Figure

Comparison between the exact Green’s function

Comparison between the exact Green’s function

Looking at Figure

This is visible in Figure

As is visible, the correlation gather of this relatively simple 1D example already shows a great complexity of events. It contains lots of multiple arrivals and free-surface ghosts. Therefore, distinguishing all the different events is quite a task. Looking at the different events, some contributions are so-called non-stationary. That means that this contribution of a certain source position to a certain event shifts in time as a function of the source position [

The numerical 1D SH-TE example presented here has shown that the presence of seismic sources only is sufficient to retrieve an accurate seismoelectric response. This means effectively that both seismic and electromagnetic signals are registered at different receivers (without the need of explicit electromagnetic sources) and that by cross-correlating these registered signals, the accurate seismoelectric Green’s function (less than 10% amplitude difference) is retrieved. In addition, it has been shown that the electromagnetic boundary source contribution to the Green’s function retrieval in the positive time window is negligible. However, the numerical example presented here is of course far from resembling a real Earth setting. Nevertheless, recent seismic interferometry studies performed on real data have shown that, for example, by using seismic noise sources (e.g. from microseisms),

It was shown that the computed amplitude and phase for the dynamic permeability correlate well with the normalized measurements, whereas for the dynamic SP coefficient, only the normalized amplitude correlates well with the predictions of the theory. This difference could be due to a capacitor effect of the set-up. To prevent the capacitor effect, using insulating plates and electrodes perforated in them may be a solution. In addition, this difference could be related to a slight frequency-dependence of the bulk conductivity. Using independent impedance measurements of the sample could also improve the results. A full-waveform seismoelectric model in a layered geometry was obtained from the solution of a mechanical boundary value problem and the electric-solid ratio of the fast

Substituting plane wave expressions into the poroelastic boundary conditions (

Substituting plane wave expressions into the poroelastic boundary conditions (

The research was performed at Delft University of Technology. The research was partly funded as Shell-FOM (Fundamental Research on Matter) projects within the research programs “The physics of fluids and sound propagation” and “Innovative physics for oil and gas.” The authors would like to thank Kees Wapenaar for useful comments, as well as the editor and two anonymous reviewers for constructive remarks and suggestions to improve this journal paper.