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

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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.

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Data Availability

Some or all data that support the findings of this study are available from the corresponding author upon reasonable request.

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Funding

This research work was supported by the National Natural Science Foundation of China, 51978633, Xue-Liang Chen, 51878242, Hai-Long Wang, National Key Research and Development Program, 2017YFC1500205-03, 2019YFC1509403, Xue-Liang Chen, Natural Science Foundation of Hebei Province, E2020404007, Jie Dong, Basic Scientific Research of Provincial Colleges and Universities in Hebei Province, 2022QNJS15, Bo Zhang.

Author information

Authors and Affiliations

  1. Institute of Geophysics, China Earthquake Administration, Beijing, 100081, China

    Bo Zhang, Xueliang Chen, Lijun Qiu & Zhiwei Ji

  2. School of Civil Engineering, Hebei University of Architecture, Zhangjiakou, 075000, China

    Bo Zhang, Lijun Qiu, Jie Dong, Zhanxue Zhou & Lingyu Liang

Authors
  1. Bo Zhang
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  2. Xueliang Chen
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  3. Lijun Qiu
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  4. Jie Dong
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  5. Zhanxue Zhou
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  7. Lingyu Liang
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Contributions

BZ: Idea, Equation derivation, Data analysis, Writing—Original Draft, Review & Editing. XC: Review, Funding acquisition. LQ: Idea, Data collection and analysis. JD: Funding acquisition. ZZ: Participate in the review. ZJ: Participate in the review. LL: Participate in the review.

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Correspondence to Lijun Qiu.

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Appendix

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.

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Zhang, B., Chen, X., Qiu, L. et al. Characteristics of Elastic Wave Propagation in Fluid-Saturated Porous Media Based on the Model of Soil Mechanics. Pure Appl. Geophys. 180, 2309–2326 (2023). https://doi.org/10.1007/s00024-023-03269-z

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