Characteristics of Elastic Wave Propagation in Fluid-Saturated Porous Media Based on the Model of Soil Mechanics

Abstract

In the present study, elastic body wave propagation in fluid-saturated porous media is investigated, and the analytical model is solved in terms of the model of soil mechanics. The potential function expressions of elastic body waves in fluid-saturated porous media with different permeability (i.e., finite, zero, and infinity) are derived by using potential functions of solid- and liquid-phase displacements. Then, the elastic wave dispersion equations are obtained by using the complex forms of the plane harmonic wave. Furthermore, in the case of finite, zero, and infinite permeability, the plane wave solutions of the elastic waves are derived and the analytical formulations of propagation velocities and attenuation coefficients are calculated. Finally, numerical examples are carried out to analyze the dispersion and attenuation characteristics of three waves (fast compressional wave P₁, low compressional wave P₂, and shear wave S) in two-phase media. In addition, the effects of various properties (i.e., dynamic permeability coefficient, dissipation coefficient, porosity, and Poisson ratio) on the propagation characteristics of three waves are studied numerically. This study is of value to the wave propagation phenomenon as well as other related disciplines.

Appendix

The variable symbols in Appendix are consistent with the text. The process of modeling is as follows.

The dynamic equilibrium equations for solid and liquid phases are given by

$$\sum\limits_{j} {\frac{{\partial \sigma^{\prime}_{ij} }}{{\partial x_{j} }}} + (1 - n)p_{f} + R_{i} = \rho_{1} \ddot{u}_{i}$$

(18)

$$(1 - n)\frac{{\partial p_{f} }}{{\partial x_{i} }} - R_{i} = \rho_{2} \ddot{U}_{i}$$

(19)

Here, $\sigma^{\prime}_{ij}$(= $\lambda \delta_{ij} \varepsilon$ + $2\mu \varepsilon_{ij}$) is the effective stress of the soil skeleton, in which $\delta_{ij}$ is the Kronecker delta, and the subscripts i (= x, y, z) and j (= x, y, z) represent directions. The strain components of solid skeleton $\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i} )$. The bulk strain components of solid skeleton $\varepsilon$ = $div{\varvec{u}}$. $\ddot{u}_{i}$ and $\ddot{U}_{i}$ are the acceleration components of solid and fluid phases, respectively. $R_{i}$ is the force component of the liquid on the solid phase that can be obtained by

$$R_{i} = \frac{{n^{2} \rho_{w} g}}{K}\frac{\partial }{\partial t}(U_{i} - u_{i} ) = b_{i} (\dot{U}_{i} - \dot{u}_{i} )$$

(20)

where g is the gravitational acceleration. K is the permeability coefficient satisfying Darcy's law. $u_{i}$ and $U_{i}$ are the displacement components of solid and fluid phases. $b_{i}$ is the dissipation coefficient in the i-direction, and $b_{i}$ = $b$ = ${{n^{2} } \mathord{\left/ {\vphantom {{n^{2} } k}} \right. \kern-0pt} k}$ in the isotropic medium. k (= ${K \mathord{\left/ {\vphantom {K {\rho_{w} g}}} \right. \kern-0pt} {\rho_{w} g}}$) is the dynamic permeability coefficient. $\dot{u}_{i}$ and $\dot{U}_{i}$ are the velocity components of solid and fluid phases.

The mass–conservation laws of solid and fluid phases are as follows:

$$\left\{ \begin{array}{l} \frac{{\partial ((1 - n)\rho_{s} )}}{\partial t} + \nabla \cdot ((1 - n)\rho_{s} \dot{\varvec{u}}) = 0 \hfill \\ \frac{{\partial (n\rho_{w} )}}{\partial t} + \nabla \cdot (n\rho_{w} \dot{\varvec{U}}) = 0 \hfill \\ \end{array} \right.$$

(21)

The equations of state in two-phase media are given by

$$\left\{ \begin{array}{l} \frac{{d\rho_{s} }}{{\rho_{s} }} = - \frac{{d(\Theta^{s} /(1 - n))}}{{E_{s} }} \hfill \\ \frac{{d\rho_{w} }}{{\rho_{w} }} = - \frac{{dp_{f} }}{{E_{w} }} \hfill \\ \end{array} \right.$$

(22)

where $\Theta^{s}$(= ${{\sum {\sigma^{\prime}_{ii} } } \mathord{\left/ {\vphantom {{\sum {\sigma^{\prime}_{ii} } } 3}} \right. \kern-0pt} 3} + (1 - n)p_{f}$) is the mean normal stress of the solid phase.

Add the two formulas of Eq. (21) and substitute Eq. (22) into Eq. (21). The compressibility of solid particles is assumed to infinity, then the mass continuity equation of fluid-saturated porous media is written as

$$(1 - n)\nabla \cdot \dot{\varvec{u}} + n\nabla \cdot \dot{\varvec{U}} - \frac{n}{{E_{w} }}\frac{{\partial p_{f} }}{\partial t} = 0$$

(23)

Equations (18), (19), and (23) are the basic formulas for the model of soil mechanics.