# Modeling of seismoelectric effects in the low frequency regime for field survey applications, American Geophysical Union/Advancing Earth And Space Sciences

Arjeta Heta, Hélène Barucq, Julien Diaz, Christina Morency

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Arjeta Heta, Hélène Barucq, Julien Diaz, Christina Morency. Modeling of seismoelectric effects in the low frequency regime for field survey applications, American Geophysical Union/Advancing Earth And Space Sciences. AGU 2023 - American Geophysical Union / Advancing Earth And Space Sciences, Dec 2023, San Francisco (CA), United States. 2023. hal-04394440

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SEE4GEO

# Modeling of seismoelectric effects in the low frequency regime for field survey applications, American Geophysical Union/Advancing Earth And Space Sciences

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UNIVERSITÉ
DE PAU ET DES
PAYS DE L'ADOUR

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Arjeta Heta¹,², Hélène Barucq¹,², Julien Diaz¹,², Christina Morency³

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Inria Bordeaux Sud-Ouest, Pau, France

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Université de Pau et des Pays de l'Adour, Pau, France

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Lawrence Livermore National Laboratory, Livermore, CA, USA

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arjeta.heta@inria.fr

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helene.barucq@inria.fr

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julien.diaz@inria.fr

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morency1@llnl.gov

## Introduction

A conductive porous medium excited by a seismic wave is the scene of complex physical phenomena known as seismo-electric effects. Understanding them opens new prospects for imaging geological objects invisible to seismic imaging. Their modeling began with the equations proposed by Pride in the 90s, which are constructed by coupling Maxwell's equations with Biot's wave equations. The coupling terms are written in terms of a frequency-dependent operator, which poses a major difficulty in the time domain, as it is an integral Fourier operator whose discretization is not trivial, especially as it must also consider the co-existence of waves propagating a thousand times faster than seismic waves.

## Pride's equations for seismoelectric effects

Pride's equations Pride (1994) and Haartsen and Pride (1994) model SEE by coupling Maxwell's (electromagnetics) and Biot's (elastodynamics) equations. Here, we consider only the seismo-electric coupling.

$$ \begin{cases} i\omega\rho_a\dot{\mathbf{u}}_s + i\omega\rho_f\dot{\mathbf{w}} = \nabla \cdot \tau + f_u \\ i\omega p = -M\nabla \cdot \dot{\mathbf{w}} - M\alpha : \epsilon(\dot{\mathbf{u}}_s) \end{cases} \quad \begin{array}{l} \text{Poroelastic} \\ \text{constitutive laws} \end{array} $$

$$ \begin{cases} i\omega\tau + i\omega\alpha p = \mathbf{C} : \epsilon(\dot{\mathbf{u}}_s) \\ i\omega\delta_0\mathbf{E} = \nabla \times \mathbf{H} - \mathbf{J} + \mathbf{f}_C \\ i\omega\mu_0\mathbf{H} = -\nabla \times \mathbf{E} \end{cases} \quad \begin{array}{l} \text{Dynamical} \\ \text{equations} \end{array} $$

$$ \begin{cases} i\omega\rho_f\dot{\mathbf{u}}_s + i\omega\rho_{dyn}(\omega)\dot{\mathbf{w}} = -\nabla\mathbf{p} + \underbrace{i\omega\rho_{dyn}(\omega)L(\omega)\mathbf{E}}_{\text{Transport}} + f_\omega \\ \mathbf{J} = \sigma\mathbf{E} - L(\omega)(\nabla p + i\omega\rho_f\dot{\mathbf{u}}_s) \end{cases} $$

Pride's equations are solved in time in Morency and Tromp (2008) and Morency (2020):

$$ \begin{cases} \rho\partial_t^2 \mathbf{u}_s + \rho_f\partial_t^2 \mathbf{w} = \nabla \cdot \mathbf{T} + \mathbf{F} \\ \eta k^{-1} \partial_t \mathbf{w} + \frac{\rho_f c}{\phi} \partial_t^2 \mathbf{w} + \rho_f \partial_t^2 \mathbf{u}_s = \nabla \cdot \mathbf{T}_f + \mathbf{F} \end{cases} \quad \begin{array}{l} \text{Biot's} \\ \text{equations} \end{array} $$

$$ \mathbf{T} = (B\nabla \cdot \mathbf{u}_s + C\nabla \cdot \mathbf{w})\mathbf{I} + 2\mu_{fr}\mathbf{D}_s \\ \mathbf{T}_f = -p_f\mathbf{I} = (C\nabla \cdot \mathbf{u}_s + M\nabla \cdot \mathbf{w})\mathbf{I} $$

$$ \epsilon\partial_t^2\mathbf{E} = \nabla \cdot (\partial\tilde{\mathbf{H}}) - \sigma\partial_t\mathbf{E} - \partial_t\mathbf{j}_s \\ \partial_t\tilde{\mathbf{H}} = \tilde{\mu}^{-1} : \nabla\mathbf{E} $$

$$ \mathbf{j}_s = \underbrace{Q_v k \eta^{-1}}_{L_0} (\nabla \cdot \mathbf{T}_f - \rho_f \partial_t^2 \mathbf{u}_s + \mathbf{F}) \quad \begin{array}{l} \text{Maxwell's} \\ \text{equations} \end{array} $$

## Modeling of the problem

<table><caption>Table 1 – Unknowns</caption><tr><td>u<sub>s</sub></td><td>solid displacement</td></tr><tr><td>w</td><td>fluid displacement</td></tr><tr><td>p</td><td>fluid pressure</td></tr><tr><td>τ</td><td>stress tensor</td></tr><tr><td>E</td><td>electric density field</td></tr><tr><td>H</td><td>magnetic intensity field</td></tr><tr><td>J</td><td>electric current density</td></tr><tr><td>T</td><td>total stress</td></tr><tr><td>T<sub>f</sub></td><td>fluid stress</td></tr></table>

<table><caption>Table 2 – Porous sub-soil physical parameters</caption><tr><td>ρ<sub>a</sub>, ρ<sub>f</sub>, ρ<sub>dyn</sub></td><td>averaged, fluid, dynamic densities</td></tr><tr><td>α</td><td>effective stress tensor</td></tr><tr><td>f<sub>u</sub>, f<sub>w</sub>, f<sub>C</sub></td><td>exterior forces</td></tr><tr><td>L(ω)</td><td>coupling coefficient</td></tr><tr><td>C</td><td>stiffness tensor</td></tr><tr><td>ε(u)</td><td>strain tensor</td></tr><tr><td>M</td><td>fluid-solid coupling modulus</td></tr><tr><td>δ<sub>0</sub>, μ<sub>0</sub></td><td>electric factor and permeability</td></tr><tr><td>σ</td><td>conductivity</td></tr></table>

<table><caption>Table 3 – Porous sub-soil physical parameters</caption><tr><td>μ<sub>fr</sub></td><td>frame shear modulus</td></tr><tr><td>F</td><td>macroscopic source</td></tr><tr><td>D<sub>s</sub></td><td>solid strain deviator</td></tr><tr><td>ρ</td><td>bulk density</td></tr><tr><td>η</td><td>fluid viscosity</td></tr><tr><td>k</td><td>fluid permeability</td></tr><tr><td>B,C,M</td><td>Biot coefficients</td></tr><tr><td>c</td><td>Tortuosity</td></tr></table>

## Frequency dependent coupling

Electrokinetic (Maxwell-Biot coupling) coefficient $L(\omega)$:

$$ L(\omega) = L_0 \left(1 - i \frac{m \omega}{4 \omega_t}\right)^{-1/2} \quad \text{with} \quad L_0 = \frac{\phi \epsilon_0 e_f \zeta^p}{\alpha \infty \eta} \underbrace{\left(1 - 2 \frac{d^l}{\lambda}\right)}_{\approx 1} $$

In time domain, Pride's equations are solved with $L(\omega) \approx L_0$. This is a low-frequency approximation that only applies to field-type simulations. We show that, at frequencies encountered in laboratory experiments (>20kHz), the effect of frequency on coupling is too significant to be neglected. Frequency-dependent coupling is very difficult to account for in time domain as $L(t)$ is a Fourier integral operator global in time and space (inverse Fourier):

$$ LE(\vec{x}, t) = L_0 \int_{\mathbb{R}} (1 - i \frac{\omega}{\omega_t})^{-1/2} \hat{E}(\vec{x}, \omega) e^{i\omega t} d\omega \quad (1) $$

Simplified frequency dependent coupling operators introduced in Heta, Barucq, and Diaz (2022) can be used. In polynomial or rational form, the equivalent time domain equations are simple to obtain. For $\omega/\omega_t$ small enough,

$$ L(\omega) \approx L_n(\omega) = L_0 \prod_{j=1}^{n} \left( 1 - i c_j^{(n)} \frac{m\omega}{4\omega_t} \right) / \left( 1 - i b_j^{(n)} \frac{m\omega}{4\omega_t} \right) $$

where

$$ b_j^{(n)} = \cos \left( \frac{j\pi}{2n+1} \right)^2, \quad c_j^{(n)} = \sin \left( \frac{j\pi}{2n+1} \right)^2, \quad a_j^{(n)} = \frac{2c_j^{(n)}}{2n+1} $$

## RESULTS & DISCUSSION

It has been shown in Heta, Barucq, and Diaz (2022) that $L(\omega) \approx L_0$ is a good approximation at low frequencies as a function of cut-off frequency $\omega_t$. For comparison with the time-domain equations using $L_0$, we use $L(\omega) = L_0$. The frequency-domain equations are solved using the Hybridizable Discontinuous Galerkin method with the solver hou10ni(Figure 2).

The use of a fictitious permittivity $\epsilon \approx 7.965 \cdot 10^{-4} \text{Fm}^{-1}$ to address the diffusive nature of the EM signal has been introduced with success in e.g. Maaø (2007) and Mittet (2010) to carry out time-domain simulations valid on a field scale. The time-domain equations are solved using the Spectral Element Method with the solver SPECFEM(Figure 3).

Using a discrete inverse Fourier transform, seismograms are computed from a frequency sweep. These results are compared to the solutions computed directly in time domain.

<figure><img src="image_12.png" /><figcaption>Figure 2 - hou10ni seismogram results using real and fictitious permittivity.</figcaption></figure>

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## Observations

* Results with true and with fictitious permittivity are very similar in frequency domain, consistent with time domain findings.
* SPECFEM was run with 150000 time steps. hou10ni results were computed on fine meshes but for 400 distinct frequencies up to 400Hz.
* In frequency domain, only the source and reflected wave are visible in $E_y$. In time domain, the converted wave is visible in domain 2 (arrives at $t \approx 180\mu s$).
* Despite the higher resolution of the time domain computations, arrival times coincide with the two methods.

## Conclusion

A use of a fictitious permittivity allows to compute solutions in time but it modifies the behaviour of the converted wave. The comparisons between our solvers, one in time(SPECFEM) and the other one in frequency(Hou10ni), show a quite good consistency regarding the field V and the co-seismic wave. We have a perfect arrival time but the amplitudes do not match.

## References

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* HAARTSEN; PRIDE. Modeling of coupled electroseismic wave propagation from point sources in layered media. In: **SEG Technical Program Expanded Abstracts 1994.** [S.I.]: Society of Exploration Geophysicists, 1994. p. 1155–1158.
* HETA, A.; BARUCQ, H.; DIAZ, J. Approximate Coupling Terms in Seismoelectric Theory: from Frequency Domain to Time Domain. In: **AGU22-AGU Fall Meeting 2022.** [S.I.: s.n.], 2022.
* MAAØ, F. A. Fast finite-difference time-domain modeling for marine-subsurface electromagnetic problems. **Geophysics**, Society of Exploration Geophysicists, v. 72, n. 2, a19–a23, 2007.
* MITTET, R. High-order finite-difference simulations of marine CSEM surveys using a correspondence principle for wave and diffusion fields. **Geophysics**, Society of Exploration Geophysicists, v. 75, n. 1, f33–f50, 2010.
* MORENCY, C. Electromagnetic wave propagation based upon spectral-element methodology in dispersive and attenuating media. **Geophysical Journal International**, Oxford University Press, v. 220, n. 2, p. 951–966, 2020.
* MORENCY, C.; TROMP, J. Spectral-element simulations of wave propagation in porous media. **Geophysical Journal International**, Blackwell Publishing Ltd Oxford, UK, v. 175, n. 1, p. 301–345, 2008.
* PRIDE. Governing equations for the coupled electromagnetics and acoustics of porous media. **Physical Review B**, APS, v. 50, n. 21, p. 15678, 1994.

## Acknowledgments

This research was funded by the Geothermica european co-fund initiative, project SEE4GEO, including LLNL under Contract DE-AC52-07NA27344, and the E2S UPPA project.

SEE4GEO

GEOTHERMICA

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