# On the well-posedness of Pride system for modeling seismo-electric effects wave propagation in conductive porous media

Arjeta Heta, Hélène Barucq, Rabia Djellouli, Julien Diaz

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Arjeta Heta, Hélène Barucq, Rabia Djellouli, Julien Diaz. On the well-posedness of Pride system for modeling seismo-electric effects wave propagation in conductive porous media. WAVES 2024 - 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation, WAVES 2024, Jun 2024, Berlin, Germany. hal-04902504

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Submitted on 20 Jan 2025

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# On the well-posedness of Pride system for modeling seismo-electric effects wave propagation in conductive porous media

Arjeta Heta<fnref n="1" />,*, Rabia Djellouli<fnref n="2" />, Hélène Barucq<fnref n="3" />, Julien Diaz<fnref n="3" />

## Abstract

A conductive porous medium excited by a seismic wave is the scene of seismo-electric effects. Understanding them opens new prospects for imaging geological objects invisible to seismic imaging. This phenomenon has been modeled by Pride in the 90s, by a strong (two-way) coupling of Maxwell's (electromagnetics) with Biot's (porous elasticity) equations. A simplified system considering only Maxwell-to-Biot (electroseismic) or Biot-to-Maxwell (seismo-electric) coupling is often considered. In this context, it is enough to show existence and uniqueness of the solutions to the separate systems.

We analyze the well-posedness of the boundary value problem resulting from strong Maxwell-Biot coupling. We consider the low-frequency regime which yields a constant coupling term and seek solutions in time domain. The main idea to prove the existence and uniqueness of the solution is to reformulate the considered Maxwell-Biot system such that the fixed-point theorem can be employed. Such a proof requires the derivation of various a priori estimates on both elastic and electromagnetic fields, that appear to be of general interest. Constraints on the physical parameters are respected for a quite general class of physical media.

**Keywords:** Maxwell system, Biot system, Pride system, well-posedness, seismo-electric

## 1 Mathematical model

We adopt a three-dimensional Pride's system [1], which couples Biot's and Maxwell's systems using a current source depending on the Biot solutions. We denote by $\Omega$ the computational domain and by $(0, T)$ the time domain of the propagation, with $T > 0$. In addition, the defi-

nitions of the physical parameters of the system are reported in Table 1.

### 1.1 Biot's system

The displacement vector relative to the fluid phase $u$ and the characteristic vector of the relative fluid/solid movement $w$ satisfy in $\Omega \times (0, T)$ the following system:

$$ 
\left\{ 
\begin{aligned}
\rho \partial_t^2 u + \rho_f \partial_t^2 w &= \nabla \cdot T + f_u \\
\frac{\rho_f c}{\phi} \partial_t^2 w + \eta k^{-1} \partial_t w + \rho_f \partial_t^2 u &= \nabla \cdot T_f + \eta k^{-1} L_0 E + f_w
\end{aligned}
\right. 
$$

where $f_u, f_w$ are two prescribed source terms, and the tensors $T, T_f$, and $D_s$ are defined as follows:

$$ 
\left\{ 
\begin{aligned}
\nabla \cdot T_f &= C \nabla(\operatorname{div} u) + M \nabla(\operatorname{div} w) \\
\nabla \cdot T &= B \nabla(\operatorname{div} u) + C \nabla(\operatorname{div} w) + 2\mu_{fr} \nabla \cdot D_s \\
2\mu_{fr} \nabla \cdot D_s &= \mu_{fr} \nabla \cdot \epsilon(u) - 2\mu_{fr}/3 \nabla \operatorname{div} u
\end{aligned}
\right. 
$$

### 1.2 Maxwell's system

The electric field $E$ (resp. the magnetic field $H$) satisfy in $\Omega \times (0, T)$ the following system:

$$ 
\left\{ 
\begin{aligned}
\epsilon \partial_t E &= \operatorname{curl} H - \sigma E - j_s + f_E \\
\partial_t H &= -\mu^{-1} \operatorname{curl} E + f_H
\end{aligned}
\right. 
$$

where $f_E, f_H$ are two prescribed source terms and the current density $j_s$ is given by:

$$ j_s = L_0 \left( \frac{\rho_f c}{\phi} \partial_t^2 w + \eta k^{-1} \partial_t w - \eta k^{-1} L_0 E \right) $$

### 1.3 Boundary and initial condition

The wall boundary conditions correspond to homogeneous Dirichlet in speed, solid displacements, and electric field:

$$ 
\left\{ 
\begin{array}{ll}
E \times n = 0 & \text{on } \partial\Omega; \quad t \ge 0 \\
w \cdot \mathbf{n} = 0 & \text{on } \partial\Omega; \quad t \ge 0 \\
u = 0 & \text{on } \partial\Omega; \quad t \ge 0
\end{array}
\right. 
$$

<footnote n="1">¹Université de Pau et des Pays de l'Adour, France</footnote>

<footnote n="2">²Departement of Mathematics & IRIS, California State University Northridge, USA</footnote>

<footnote n="3">³EPI Makutu, Inria Bordeaux Sud-Ouest, France</footnote>

<footnote>*Email: arjeta.heta@inria.fr</footnote>
The initial conditions are homogeneous:

$$ egin{cases} E(0, x) = H(0, x) = 0 & \text{on } \Omega \\ w(0, x) = u(0, x) = 0 & \text{on } \Omega \\ \partial_t w(0, x) = \partial_t u(0, x) = 0 & \text{on } \Omega \end{cases} $$

## 2 Mathematical results

The Biot and Maxwell system are coupled via the electrical term $L_0E$ and the density $j_s$. We must point out that Pride's system was originally posed in the frequency domain. This involves a frequency dependent coupling parameter $L(\omega)$ and permeability $k(\omega)$ which do not translate directly into integro-differential time operators (not rational functions). The system presented here corresponds to a 0-th order approximation of $L$ and $k$ corresponding to a low frequency regime. Furthermore, well-posedness in time domain has been shown for the Biot and Maxwell systems when uncoupled [2], [3]. Note that existence and uniqueness of the solution of the Pride system with a "weak" coupling form of the Biot system with Maxwell system (a one-way coupling system from Maxwell to Biot) has been established in [3]. In this paper, the Pride system is formed of two-way coupling which can be viewed as a "strong" coupling. Next, we announce the main result of this paper. To this end, we introduce the following spaces:

$$ \mathcal{H}_M = C^0(0, T; H_{\text{curl}} \times H_{\text{curl}}) \cap C^2(0, T; L^2(\Omega) \times L^2(\Omega)) $$

$$ \mathcal{H}_B = C^0(0, T; \{(u, w) \in H^1(\Omega) \times H_{\text{div}}(\Omega), \\ \nabla \text{div } u \in L^2(\Omega), \nabla \text{div } w \in L^2(\Omega)\}) \cap C^1(0, T; H^1(\Omega) \times H_{\text{div}}(\Omega)) \\ \cap C^2(0, T; L^2(\Omega) \times L^2(\Omega)) $$

We then have,

**Theorem** If $f_u, f_w, f_E, f_H \in C^1(0, T; H^1(\Omega) \times H_{\text{div}}(\Omega) \times L^2(\Omega) \times L^2(\Omega))$, then the initial boundary value problem defined by Pride's equations admits a unique solution $(u, w, E, H) \in \mathcal{H}_B \times \mathcal{H}_M$.

### 2.1 Idea of the proof

The proof employs the fixed point theorem defined on Banach space. This requires reformulating the Pride system as a fixed point

problem by constructing a mapping between the sources and the fields in appropriate Banach spaces. Then, we derive stability estimates on the field vectors $(u, w, E, H)$. Last, we prove that the constructed mapping is a contraction and therefore conclude the proof of the theorem.

<table><caption>Table 1: Physical parameters</caption><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>ρ</td><td>effective density</td></tr><tr><td>ρ<sub>f</sub></td><td>fluid density</td></tr><tr><td>c</td><td>tortuosity</td></tr><tr><td>φ</td><td>porosity</td></tr><tr><td>η</td><td>viscosity</td></tr><tr><td>μ<sub>fr</sub></td><td>shear modulus</td></tr><tr><td>B, C, M</td><td>Biot coefficients</td></tr><tr><td>k</td><td>permeability</td></tr><tr><td>L<sub>0</sub></td><td>coupling coefficient</td></tr><tr><td>ε</td><td>permittivity</td></tr><tr><td>σ</td><td>conductivity</td></tr><tr><td>μ</td><td>magnetic permeability</td></tr></tbody></table>

## References

[1] S. Pride, Governing equations for the coupled electromagnetics and acoustics of porous media, *Physical Review B* **50-21** (1994), pp. 15678.

[2] H. Barucq, Well-posedness and exponential stability of Maxwell-like systems coupled with strongly absorbing layers, *Journal de mathématiques pures et appliquées* **87-3** (2007), pp. 253-273.

[3] E. Bonnetier and F. Triki and Q. Xue, An inverse problem for an electroseismic model describing the coupling phenomenon of electromagnetic and seismic waves, *Inverse Problems* **35-4** (2019), pp. 045002.

[4] D. R. Smart, Fixed point theorems, *Cup Archive* (1980).