Electromagnetic/Acoustic Coupling in Partially Saturated Porous Rocks: An Extension of Pride’s Theory

Abstract

In this paper, a set of equations governing the electromagnetic/acoustic coupling in partially saturated porous rocks in the low-frequency regime is derived. The equations are obtained by volume averaging of fundamental electromagnetic and mechanical equations valid at the pore-scale, following the same procedure as the one developed in the seminal paper of S. Pride for porous media where the fluid electrolyte fully saturates the pore space. In the present approach, it is assumed that the porous rock is partially saturated with a wetting-fluid electrolyte (water) and a non-wetting fluid (air). We also assume that an electromagnetic/mechanical coupling exists at the water–solid and water–air contact surfaces through adsorbed excess charges balanced by mobile ions in the water. The proposed approach is valid at the low-frequency regime, where capillary pressure perturbations can be safely neglected. The governing equations thus derived are similar to the ones obtained by Pride with the main difference that the various coefficients, including the electrokinetic coupling coefficient and electric conductivity appearing in the transport equations, are functions of the water saturation and depend on electrical and topological properties of both electric double layers.

Appendix A Mechanical Constitutive Relations in the Case of Stagnant Non-Wetting Fluid

If the non-wetting fluid is assumed to be stagnant ($p_{nw}=0$, $\varvec{w_{nw}}=0$ and $p_{w}=-p_{c}$) then the volume average of the stress–strain relations are taken without the consideration of Eq. (60). Following Pride et al. (1992), taking into account that the wetting-fluid occupies a volume $s_{w}\phi$ we obtain

$$\begin{aligned}{} & {} \varvec{\tau _{B}}=(\tilde{K}_c \nabla \cdot \varvec{u_s} + \tilde{C} \nabla \cdot \varvec{w_{w}}) {\textbf {I}}+ G \left( \nabla \varvec{u_s} + \nabla \varvec{u_s}^T - \frac{2}{3} \nabla \cdot \varvec{u_s} {\textbf {I}}\right) , \ \ \ \ \ \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} p_c= \tilde{C} \nabla \cdot \varvec{u_s} + \tilde{M} \nabla \cdot \varvec{w_{w}}, \end{aligned}$$

(A2)

where

$$\begin{aligned}{} & {} \tilde{K}_{c} = \frac{K_{m} + s_{w} \phi K_{w} + \left[ (1-\phi )+2 s_w \phi \right] K_{s} \tilde{\Delta }}{1 + \Delta }, \end{aligned}$$

(A3)

$$\begin{aligned}{} & {} \tilde{C}=\frac{K_w + K_s \tilde{\Delta }}{1+\tilde{\Delta }}, \end{aligned}$$

(A4)

$$\begin{aligned}{} & {} \tilde{M}=\frac{1}{s_w\phi } \frac{K_w}{1+\tilde{\Delta }}. \end{aligned}$$

(A5)

In these expressions,

$$\begin{aligned} \tilde{\Delta } = \frac{K_w}{s_w \phi K_s^2 }\left[ (1-\phi ) K_s -K_m \right] . \end{aligned}$$

(A6)

From Eq. (A2), if the solid displacement is negligible, then taking the first time derivative we have $\dot{p}_{c}= \tilde{M} \nabla \cdot \varvec{\dot{w}_{w}}$.