Analytical Modeling of Elastic Moduli Dispersion and Poromechanical Responses of a Dual-Porosity Dual-Permeability Porous Cylinder Under Dynamic Forced Deformation Test

Abstract

The elastic moduli dispersion of materials is a topic of great interest in many engineering practices. Such dispersion is regularly measured on laboratory-preferred cylindrical samples of various natures, from solid dry to single-porosity and dual-porosity dual-permeability fluid-saturated materials. Analytical modeling of these laboratory tests is desired as it is effective and time-efficient in the interpretation of the experimental results and the characterization of the poromechanical properties of the material. To this end, for the first time, this paper presents analytical modeling of the elastic moduli dispersion and the poromechanical behaviors of a dual-porosity dual-permeability fluid-saturated cylinder under a dynamic forced deformation test. Our dual-porosity dual-permeability poroelastodynamics solution can be easily reduced to the elastodynamics and the single-porosity poroelastodynamics ones. We demonstrate the capabilities of our analytical solution by modeling the dynamic elastic moduli and poromechanical responses of a naturally fractured rock sample. Our results show that spatial distributions of poromechanical quantities, such as pore pressure and strain, are mostly uniform at low frequencies and become increasingly nonuniform at high frequencies. Additionally, the dynamic Young’s modulus and Poisson’s ratio are highly dependent on Biot’s and Skempton’s coefficients, the sample’s dimension, and the loading frequency. The dispersion trends of these dynamic elastic moduli vary as the loading frequency approaches the material's resonant and anti-resonant frequencies. Finally, we use our analytical solution to successfully match laboratory data from dynamic forced deformation tests on three clastic sediment rock samples from the North Sea and a shale sample from Mont Terri.

Highlights

• The first analytical poroelastodynamics solution for a dual-porosity dual-permeability fluid-saturated porous cylinder.

• Comprehensively interpretation of the poromechanical responses and the mechanisms of the dispersion.

• Excellent matches between the analytical solution with laboratory measurements.

Appendices

Appendix 1

1.1 Definitions of Parameters

$$\begin{array}{c}\left[\begin{array}{c}{a}_{11}\\ {a}_{12}\\ \begin{array}{c}{a}_{13}\\ {a}_{21}\\ \begin{array}{c}{a}_{22}\\ {a}_{23}\end{array}\end{array}\end{array}\right]=\frac{1}{{\overline{M} }_{11}{\overline{M} }_{22}-{\overline{M} }_{12}^{2}}\\ \times \left[\begin{array}{c}{\overline{M} }_{11}{\overline{M} }_{12}\left({\overline{\alpha }}_{1}{\overline{M} }_{12}-{\overline{\alpha }}_{2}{\overline{M} }_{22}\right)\\ {\overline{M} }_{11}{\overline{M} }_{12}^{2}\\ \begin{array}{c}-{\overline{M} }_{11}{\overline{M} }_{12}{\overline{M} }_{22}\\ {\overline{M} }_{12}{\overline{M} }_{22}\left(-{\overline{\alpha }}_{1}{\overline{M} }_{11}+{\overline{\alpha }}_{2}{\overline{M} }_{12}\right)\\ \begin{array}{c}-{\overline{M} }_{11}{\overline{M} }_{12}{\overline{M} }_{22}\\ {\overline{M} }_{12}^{2}{\overline{M} }_{22}\end{array}\end{array}\end{array}\right]\end{array}$$

(38)

$$\begin{array}{c}\left[\begin{array}{c}\frac{1}{{\overline{M} }_{11}}\\ \frac{1}{{\overline{M} }_{12}}\\ \frac{1}{{\overline{M} }_{22}}\end{array}\right]=\left[\begin{array}{c}\frac{1}{{M}_{11}}-\frac{\gamma }{i\omega }\\ \frac{1}{{M}_{12}}+\frac{\gamma }{i\omega }\\ \frac{1}{{M}_{22}}-\frac{\gamma }{i\omega }\end{array}\right]\end{array}$$

(39)

$$\begin{array}{c}\gamma ={\gamma }_{0}\sqrt{1-\frac{i\omega }{{\omega }_{r}}}\end{array}$$

(40)

$$\begin{array}{c}{\rho }_{23}=\frac{{\rho }_{f}}{2}\left[\begin{array}{c}\left(\tau -1\right)\phi -\left({\tau }_{1}-1\right){\mathrm{v}}_{1}{\phi }_{1}\\ -\left({\tau }_{2}-1\right){\mathrm{v}}_{2}{\phi }_{2}\end{array}\right]\end{array}$$

(41)

$$\begin{array}{c}{v}_{s}=\sqrt{\frac{G}{\rho +{\frac{2{b}_{12}-{b}_{11}-{b}_{22}}{{b}_{11}{b}_{22}-{b}_{12}^{2}}\rho }_{f}^{2}}}\end{array}$$

(42)

$$\begin{array}{c}\left[\begin{array}{c}{b}_{11}\\ {b}_{12}\\ {b}_{22}\end{array}\right]=\left[\begin{array}{c}\frac{{\tau }_{1}{\rho }_{f}}{{\mathrm{v}}_{1}{\phi }_{1}}+\frac{i}{\omega {\kappa }_{11}}\\ \frac{{\rho }_{23}}{{\mathrm{v}}_{1}{\mathrm{v}}_{2}{\phi }_{1}{\phi }_{2}}\\ \frac{{\tau }_{2}{\rho }_{f}}{{\mathrm{v}}_{2}{\phi }_{2}}+\frac{i}{\omega {\kappa }_{22}}\end{array}\right]\end{array}$$

(43)

$$\begin{array}{c}M=\left[\begin{array}{ccc}\left(\begin{array}{c}\lambda +2G\\ -{\overline{\alpha }}_{1}{a}_{11}\\ -{\overline{\alpha }}_{2}{a}_{21}\end{array}\right)& \left(\begin{array}{c}-{\overline{\alpha }}_{1}{a}_{12}\\ -{\overline{\alpha }}_{2}{a}_{22}\end{array}\right)& \left(\begin{array}{c}{-\overline{\alpha }}_{1}{a}_{13}\\ -{\overline{\alpha }}_{2}{a}_{23}\end{array}\right)\\ {a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\end{array}\right]\end{array}$$

(44)

$$\begin{array}{c}N=\left[\begin{array}{ccc}-{\omega }^{2}\rho & -{\omega }^{2}{\rho }_{f}& -{\omega }^{2}{\rho }_{f}\\ {\omega }^{2}{\rho }_{f}& \left(\begin{array}{c}{\omega }^{2}\frac{{\tau }_{1}{\rho }_{f}}{{\mathrm{v}}_{1}{\phi }_{1}}\\ +\frac{i\omega }{{\kappa }_{11}}\end{array}\right)& {\omega }^{2}\frac{{\rho }_{23}}{{\mathrm{v}}_{1}{\mathrm{v}}_{2}{\phi }_{1}{\phi }_{2}}\\ {\omega }^{2}{\rho }_{f}& {\omega }^{2}\frac{{\rho }_{23}}{{\mathrm{v}}_{1}{\mathrm{v}}_{2}{\phi }_{1}{\phi }_{2}}& \left(\begin{array}{c}{\omega }^{2}\frac{{\tau }_{2}{\rho }_{f}}{{\mathrm{v}}_{2}{\phi }_{2}}\\ +\frac{i\omega }{{\kappa }_{22}}\end{array}\right)\end{array}\right]\end{array}$$

(45)

where $\tau \left(=\phi \frac{{\mathrm{v}}_{2}{\phi }_{1}+\left(3-{\mathrm{v}}_{2}\right){\tau }_{1}}{\left(3-2{\mathrm{v}}_{2}\right){\phi }_{1}+2{\mathrm{v}}_{2}{\tau }_{1}}\right)$ and $\phi \left(={\mathrm{v}}_{1}{\phi }_{1}+{\mathrm{v}}_{2}{\phi }_{2}\right)$ are the average tortuosity and porosity, respectively, $\gamma$ is the inter-porosity fluid exchange coefficient, ${\gamma }_{0}$ is the low-frequency limit, ${\omega }_{r}$ is the relaxation frequency.

At high frequencies, the fluid viscosity needs to be multiplied by a viscosity correction factor. For example, Biot’s parallel plates flow model (Biot 1956b; Cheng 2016) can be used to simulate the frequency-dependent viscosity, i.e., $\mu =-{\mu }_{0}\frac{\sqrt{i}{\kappa }_{u}\mathrm{tan}\sqrt{i}{\kappa }_{u}}{3(1-\mathrm{tan}\sqrt{i}{\kappa }_{u}/\sqrt{i}{\kappa }_{u})}$, where ${\kappa }_{u}={\delta }_{p}\sqrt{\frac{k{\rho }_{f}\omega }{\phi {\mu }_{0}}}$ and ${\delta }_{p}$ is a factor dependent on pore geometry.

Appendix 2

2.1 Exact Expressions of Displacements, Pore Pressures, and Stresses

Substitution of Eqs. (17, 20, 26, 27) into Eqs. (28–30) provide the solutions of displacements as follows

$$\begin{array}{c}{u}_{r}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{1m}{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)\mathrm{cos}\frac{n\pi }{h}z\\ -{C}_{m}{n}_{1m}\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)\\ -{\sum }_{n=1}^{\infty }{A}_{n}\frac{n\pi }{h}{J}_{1}\left(r{\xi }_{n}\right)\mathrm{cos}\frac{n\pi }{h}z\end{array}$$

(46)

$$\begin{array}{c}{w}_{1r}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{2m}{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)\mathrm{cos}\frac{n\pi }{h}z\\ -{C}_{m}{n}_{2m}\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)\\ -\frac{{\rho }_{f}\left({b}_{12}-{b}_{22}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}{\sum }_{n=1}^{\infty }{A}_{n}\frac{n\pi }{h}{J}_{1}\left(r{\xi }_{n}\right)\mathrm{cos}\frac{n\pi }{h}z\end{array}$$

(47)

$$\begin{array}{c}{w}_{2r}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{3m}{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)\mathrm{cos}\frac{n\pi }{h}z\\ -{C}_{m}{n}_{3m}\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)\\ -\frac{{\rho }_{f}\left({b}_{12}-{b}_{11}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}{\sum }_{n=1}^{\infty }{A}_{n}\frac{n\pi }{h}{J}_{1}\left(r{\xi }_{n}\right)\mathrm{cos}\frac{n\pi }{h}z\end{array}$$

(48)

$$\begin{array}{c}{u}_{z}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{1m}{J}_{0}\left(r{\eta }_{mn}\right)\frac{n\pi }{h}\mathrm{sin}\frac{n\pi }{h}z\\ +{n}_{1m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}z\\ +{\sum }_{n=1}^{\infty }{A}_{n}\left[\begin{array}{c}\frac{{J}_{1}\left(r{\xi }_{n}\right)}{r}\\ +{\xi }_{n}\frac{{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)}{2}\end{array}\right]\mathrm{sin}\frac{n\pi }{h}z\end{array}$$

(49)

$$\begin{array}{c}{w}_{1z}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{2m}\frac{n\pi }{h}{J}_{0}\left(r{\eta }_{mn}\right)\mathrm{sin}\frac{n\pi }{h}z\\ +{n}_{2m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}z\\ +\frac{{\rho }_{f}\left({b}_{12}-{b}_{22}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}\\ \times {\sum }_{n=1}^{\infty }{A}_{n}\left[\begin{array}{c}\frac{{J}_{1}\left(r{\xi }_{n}\right)}{r}\\ +{\xi }_{n}\frac{{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)}{2}\end{array}\right]\mathrm{sin}\frac{n\pi }{h}z\end{array}$$

(50)

$$\begin{array}{c}{w}_{2z}=-{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{3m}\frac{n\pi }{h}{J}_{0}\left(r{\eta }_{mn}\right)\mathrm{sin}\frac{n\pi }{h}z\\ +{n}_{3m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}z\\ +\frac{{\rho }_{f}\left({b}_{12}-{b}_{11}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}\\ \times {\sum }_{n=1}^{\infty }{A}_{n}\left[\begin{array}{c}\frac{{J}_{1}\left(r{\xi }_{n}\right)}{r}\\ +{\xi }_{n}\frac{{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)}{2}\end{array}\right]\mathrm{sin}\frac{n\pi }{h}z\end{array}$$

(51)

Pore pressures and stresses are determined from Eqs. (7–12) as follows

$$\begin{array}{c}{p}_{1}=-{\sum }_{n=1}^{\infty }\begin{array}{c}{B}_{mn}{a}_{1k}{n}_{km}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}\frac{{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)}{r}\\ +\frac{{\eta }_{mn}^{2}\left[{J}_{0}\left(r{\eta }_{mn}\right)-{J}_{2}\left(r{\eta }_{mn}\right)\right]}{2}\\ +\frac{{n}^{2}{\pi }^{2}{J}_{0}\left(r{\eta }_{mn}\right)}{{h}^{2}}\end{array}\right\} \end{array}\\ +{C}_{m}{a}_{1k}{n}_{km}\left\{\begin{array}{c}\frac{{\lambda }_{m}\left[{J}_{0}\left(r\sqrt{-{\lambda }_{m}}\right)-{J}_{2}\left(r\sqrt{-{\lambda }_{m}}\right)\right]}{2}\\ -\frac{\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)}{r}\end{array}\right\}\\ +{a}_{1k}{n}_{km}{D}_{m}{\lambda }_{m}\mathrm{cosh}\sqrt{{\lambda }_{m}}z\end{array}$$

(52)

$$\begin{array}{c}{p}_{2}=-{\sum }_{n=1}^{\infty }\begin{array}{c}{B}_{mn}{a}_{2k}{n}_{km}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}\frac{{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)}{r}\\ +\frac{{\eta }_{mn}^{2}\left[{J}_{0}\left(r{\eta }_{mn}\right)-{J}_{2}\left(r{\eta }_{mn}\right)\right]}{2}\\ +\frac{{n}^{2}{\pi }^{2}{J}_{0}\left(r{\eta }_{mn}\right)}{{h}^{2}}\end{array}\right\} \end{array}\\ +{C}_{m}{a}_{2k}{n}_{km}\left\{\begin{array}{c}\frac{{\lambda }_{m}\left[{J}_{0}\left(r\sqrt{-{\lambda }_{m}}\right)-{J}_{2}\left(r\sqrt{-{\lambda }_{m}}\right)\right]}{2}\\ -\frac{\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)}{r}\end{array}\right\}\\ +{a}_{2k}{n}_{km}{D}_{m}{\lambda }_{m}\mathrm{cosh}\sqrt{{\lambda }_{m}}z\end{array}$$

(53)

$$\begin{array}{c}{\sigma }_{rz}=2G{\sum }_{n=1}^{\infty }{B}_{mn}{n}_{1m}{\eta }_{mn}\frac{n\pi }{h}{J}_{1}\left(r{\eta }_{mn}\right)\mathrm{sin}\frac{n\pi }{h}z\\ +G{\sum }_{n=1}^{\infty }\begin{array}{c}{A}_{n}\mathrm{sin}\frac{n\pi }{h}z\times \\ \left[\begin{array}{c}\left(\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-\frac{1}{{r}^{2}}-\frac{3{\xi }_{n}^{2}}{4}\right){J}_{1}\left(r{\xi }_{n}\right)\\ +{\xi }_{n}\frac{{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)}{2r}+\frac{{\xi }_{n}^{2}{J}_{3}\left(r{\xi }_{n}\right)}{4}\end{array}\right]\end{array}\end{array}$$

(54)

$$\begin{array}{c}{\sigma }_{rr}={C}_{m}\left\{\begin{array}{c}\left[{n}_{1m}\left(\lambda +2G\right)-{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right]{\lambda }_{m}\\ \times \frac{{J}_{0}\left(r\sqrt{-{\lambda }_{m}}\right)-{J}_{2}\left(r\sqrt{-{\lambda }_{m}}\right)}{2}\\ +\left({\overline{\alpha }}_{l}{a}_{lk}{n}_{km}-{n}_{1m}\lambda \right)\\ \frac{\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)}{r}\end{array}\right\}\\ -{\sum }_{n=1}^{\infty }\begin{array}{c}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}{B}_{mn}\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right)\times \\ \left[\frac{{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)}{r}+\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}{J}_{0}\left(r{\eta }_{mn}\right)\right]+\\ {B}_{mn}\left[{n}_{1m}\left(\lambda +2G\right)-{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right]{\eta }_{mn}^{2}\\ \times \frac{{J}_{0}\left(r{\eta }_{mn}\right)-{J}_{2}\left(r{\eta }_{mn}\right)}{2}\\ +{A}_{n}\frac{n\pi }{h}G{\xi }_{n}\left[{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)\right]\end{array}\right\}\end{array}\\ +(\lambda {n}_{1m}-{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}){D}_{m}{\lambda }_{m}\mathrm{cosh}\sqrt{{\lambda }_{m}}z\#\end{array}$$

(55)

$$\begin{array}{c}{\sigma }_{zz}={C}_{m}\left\{\begin{array}{c}\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right){\lambda }_{m}\times \\ \frac{{J}_{0}\left(r\sqrt{-{\lambda }_{m}}\right)-{J}_{2}\left(r\sqrt{-{\lambda }_{m}}\right)}{2}\\ +\left({\overline{\alpha }}_{l}{a}_{lk}{n}_{km}-{n}_{1m}\lambda \right)\\ \times \frac{\sqrt{-{\lambda }_{m}}{J}_{1}\left(r\sqrt{-{\lambda }_{m}}\right)}{r}\end{array}\right\}\\ -{\sum }_{n=1}^{\infty }\begin{array}{c}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}{B}_{mn}\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right)\times \\ \left[\begin{array}{c}\frac{{\eta }_{mn}{J}_{1}\left(r{\eta }_{mn}\right)}{r}\\ +\frac{{\eta }_{mn}^{2}\left[{J}_{0}\left(r{\eta }_{mn}\right)-{J}_{2}\left(r{\eta }_{mn}\right)\right]}{2}\end{array}\right]\\ +{B}_{mn}\left[\begin{array}{c}{n}_{1m}\left(\lambda +2G\right)\\ -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\end{array}\right]{J}_{0}\left(r{\eta }_{mn}\right)\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}\\ -{A}_{n}\frac{n\pi }{h}2G\left[\begin{array}{c}{\xi }_{n}\frac{{J}_{0}\left(r{\xi }_{n}\right)-{J}_{2}\left(r{\xi }_{n}\right)}{2}\\ +\frac{1}{r}{J}_{1}\left(r{\xi }_{n}\right)\end{array}\right]\end{array}\right\}\end{array}\\ \begin{array}{c}+\left[\begin{array}{c}\left(\lambda +2G\right){n}_{1m}\\ -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\end{array}\right]{D}_{m}{\lambda }_{m}\mathrm{cosh}\sqrt{{\lambda }_{m}}z\end{array}\#\end{array}$$

(56)

$$\begin{array}{c}F=-2\pi {C}_{m}\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right) \\ \times R\sqrt{-{\lambda }_{m}}{J}_{1}\left(R\sqrt{-{\lambda }_{m}}\right)\\ -2\pi {\sum }_{n=1}^{\infty }\begin{array}{c}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}{B}_{mn}\left(\begin{array}{c}{n}_{1m}\lambda \\ -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\end{array}\right)R{\eta }_{mn}{J}_{1}\left(R{\eta }_{mn}\right)\\ +{B}_{mn}\left[\begin{array}{c}{n}_{1m}\left(\lambda +2G\right)\\ -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\end{array}\right]\\ \times \frac{R{J}_{1}\left(R{\eta }_{mn}\right)}{{\eta }_{mn}}\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}\\ -{A}_{n}\frac{n\pi }{h}2GR{J}_{1}\left(R{\xi }_{n}\right)\end{array}\right\}\end{array}\\ \begin{array}{c}+\pi {R}^{2}\left[\begin{array}{c}\left(\lambda +2G\right){n}_{1m}\\ -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\end{array}\right]{D}_{m}{\lambda }_{m}\mathrm{cosh}\sqrt{{\lambda }_{m}}z\end{array}\#\end{array}$$

(57)

where Einstein summation convention has been applied to indices $l\left(=\mathrm{1,2}\right)$, $k\left(=\mathrm{1,2},3\right)$, and $m(=\mathrm{1,2},3)$.

Appendix 3

3.1 Determination of Solution Coefficients

Before applying the boundary conditions, we express the function of $\mathrm{cosh}\sqrt{{\lambda }_{m}}z$ in the cosine series as below

$$\begin{array}{c}\mathrm{cosh}\sqrt{{\lambda }_{m}}z={\gamma }_{m}+{\sum }_{n=1}^{\infty }{\gamma }_{mn}\mathrm{cos}\frac{n\pi z}{h}\end{array}$$

(58)

$$\begin{array}{c}where {\gamma }_{m}=\frac{Sinh\left(h\sqrt{{\lambda }_{m}}\right)}{h\sqrt{{\lambda }_{m}}}\end{array}$$

(59)

$$\begin{array}{c}{\gamma }_{mn}=\frac{2{\left(-1\right)}^{n}h\sqrt{{\lambda }_{m}}Sinh\left(h\sqrt{{\lambda }_{m}}\right)}{{n}^{2}{\pi }^{2}+{h}^{2}{\lambda }_{m}}\end{array}$$

(60)

Applying boundary conditions (1–3) into the corresponding solutions expressed in Appendix 2, we can obtain 10 equations for 10 unknowns, i.e., ${A}_{n}$, ${B}_{mn}$, ${C}_{m}$, and ${D}_{m}$, where $m=1, 2, 3$. These equations are listed as follows:

$$\begin{array}{c}2{B}_{mn}{n}_{1m}{\eta }_{mn}\frac{n\pi }{h}{J}_{1}\left(R{\eta }_{mn}\right)+{A}_{n}{\Omega }_{n}=0\end{array}$$

(61)

$$\begin{array}{c}{C}_{m}{n}_{2m}\sqrt{-{\lambda }_{m}}{J}_{1}\left(R\sqrt{-{\lambda }_{m}}\right)=0\end{array}$$

(62)

$$\begin{array}{c}{C}_{m}{n}_{3m}\sqrt{-{\lambda }_{m}}{J}_{1}\left(R\sqrt{-{\lambda }_{m}}\right)=0\end{array}$$

(63)

$$\begin{array}{c}{B}_{mn}{n}_{2m}{\eta }_{mn}{J}_{1}\left(R{\eta }_{mn}\right)+{A}_{n}{\Psi }_{n}=0\#\end{array}$$

(64)

$$\begin{array}{c}{B}_{mn}{n}_{3m}{\eta }_{mn}{J}_{1}\left(R{\eta }_{mn}\right)+{A}_{n}{\Delta }_{n}=0\end{array}$$

(65)

$$\begin{array}{c}{C}_{m}{\beta }_{m}+{D}_{m}{\chi }_{m}={P}_{c}\#\end{array}$$

(66)

$$\begin{array}{c}{B}_{mn}{\Phi }_{mn}+{A}_{n}{\Lambda }_{n}-{D}_{m}{\Psi }_{mn}=0\end{array}$$

(67)

$$\begin{array}{c}{n}_{1m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}h=-{U}_{z0}\#\end{array}$$

(68)

$$\begin{array}{c}{n}_{2m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}h=0\end{array}$$

(69)

$$\begin{array}{c}{n}_{3m}{D}_{m}\sqrt{{\lambda }_{m}}\mathrm{sinh}\sqrt{{\lambda }_{m}}h=0\end{array}$$

(70)

where the intermediate variables are defined by

$$\begin{array}{c}{\Omega }_{n}=\left(\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-\frac{1}{{R}^{2}}-\frac{3{\xi }_{n}^{2}}{4}\right){J}_{1}\left(R{\xi }_{n}\right)\\ \quad+{\xi }_{n}\frac{{J}_{0}\left(R{\xi }_{n}\right)-{J}_{2}\left(R{\xi }_{n}\right)}{2R}+\frac{{\xi }_{n}^{2}{J}_{3}\left(R{\xi }_{n}\right)}{4}\end{array}$$

(71)

$$\begin{array}{c}{\Theta }_{n}=\frac{{\rho }_{f}\left({b}_{12}-{b}_{22}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}\frac{n\pi }{h}{J}_{1}\left(R{\xi }_{n}\right)\end{array}$$

(72)

$$\begin{array}{c}{\Delta }_{n}=\frac{{\rho }_{f}\left({b}_{12}-{b}_{11}\right)}{{b}_{11}{b}_{22}-{b}_{12}^{2}}\frac{n\pi }{h}{J}_{1}\left(R{\xi }_{n}\right)\end{array}$$

(73)

$$\begin{array}{c}{\beta }_{m}=\left[{n}_{1m}\left(\lambda +2G\right)-{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right]{\lambda }_{m} \\ \times \frac{{J}_{0}\left(R\sqrt{-{\lambda }_{m}}\right)-{J}_{2}\left(R\sqrt{-{\lambda }_{m}}\right)}{2}\\ -\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right)\frac{\sqrt{-{\lambda }_{m}}{J}_{1}\left(R\sqrt{-{\lambda }_{m}}\right)}{R}\end{array}$$

(74)

$$\begin{array}{c}{\chi }_{m}=\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right){\lambda }_{m}{\gamma }_{m}\end{array}$$

(75)

$$\begin{array}{c}{\Phi }_{mn}=\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right)\\ \times \left[\frac{{\eta }_{mn}{J}_{1}\left(R{\eta }_{mn}\right)}{R}+\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}{J}_{0}\left(R{\eta }_{mn}\right)\right]\\ +\left[{n}_{1m}\left(\lambda +2G\right)-{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right]{\eta }_{mn}^{2}\\ \times \frac{{J}_{0}\left(R{\eta }_{mn}\right)-{J}_{2}\left(R{\eta }_{mn}\right)}{2}\end{array}$$

(76)

$$\begin{array}{c}{\Lambda }_{n}=\frac{n\pi }{h}G{\xi }_{n}\left[{J}_{0}\left(R{\xi }_{n}\right)-{J}_{2}\left(R{\xi }_{n}\right)\right]\end{array}$$

(77)

$$\begin{array}{c}{\Psi }_{mn}=\left({n}_{1m}\lambda -{\overline{\alpha }}_{l}{a}_{lk}{n}_{km}\right){\lambda }_{m}{\gamma }_{mn}\end{array}$$

(78)

Solving Eqs. (60, 61, 62, 63, 64, 65, 66, 67,68 and 69), we have.

$$\begin{array}{c}\left[\begin{array}{c}{D}_{1}\\ {D}_{2}\\ {D}_{3}\end{array}\right]={\left({X}_{ij}\right)}_{3\times 3}^{-1}\left[\begin{array}{c}-{U}_{z0}\\ 0\\ 0\end{array}\right]\end{array}$$

(79)

$$\begin{array}{c}\left[\begin{array}{c}{C}_{1}\\ {C}_{2}\\ {C}_{3}\end{array}\right]={\left({Y}_{ij}\right)}_{3\times 3}^{-1}\left[\begin{array}{c}-{D}_{m}{\chi }_{m}\\ 0\\ 0\end{array}\right]\end{array}$$

(80)

$$\begin{array}{c}\left[\begin{array}{c}{A}_{n}\\ {B}_{1n}\\ \begin{array}{c}{B}_{2n}\\ {B}_{3n}\end{array}\end{array}\right]={\left({Z}_{ij}\right)}_{4\times 4}^{-1}\left[\begin{array}{c}{D}_{m}{\Psi }_{mn}\\ 0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\right]\end{array}$$

(81)

$$\begin{array}{c}{X}_{ij}={n}_{ij}\sqrt{{\lambda }_{j}}\mathrm{sinh}\sqrt{{\lambda }_{j}}h, \quad i,\quad j=\mathrm{1,2},3\end{array}$$

(82)

$$\begin{array}{c}{Y}_{1j}={\beta }_{j},\quad j=\mathrm{1,2},3\end{array}$$

(83)

$$\begin{array}{c}{Y}_{ij}={n}_{ij}\sqrt{-{\lambda }_{j}}{J}_{1}\left(R\sqrt{-{\lambda }_{j}}\right), \quad i=\mathrm{2,3};\quad j=\mathrm{1,2},3\end{array}$$

(84)

$$\begin{array}{c}{Z}_{11}={\Lambda }_{n};\quad{Z}_{21}={\Omega }_{n};\quad{Z}_{31}={\Theta }_{n};\quad{Z}_{41}={\Delta }_{n}\end{array}$$

(85)

$$\begin{array}{c}{Z}_{1j}={\Phi }_{\left(j-1\right)n},\quad j=\mathrm{2,3},4\end{array}$$

(86)

$$\begin{array}{c}{Z}_{2j}=2{n}_{1(j-1)}{\eta }_{(j-1)n}\frac{n\pi }{h}{J}_{1}\left(R{\eta }_{(j-1)n}\right),\quad j=\mathrm{2,3},4\end{array}$$

(87)

$$\begin{array}{c}{Z}_{ij}={n}_{\left(i-1\right)\left(j-1\right)}{\eta }_{\left(j-1\right)n}{J}_{1}\left(R{\eta }_{\left(j-1\right)n}\right)\\ i=\mathrm{3,4};\quad j=\mathrm{2,3},4\end{array}$$

(88)

Appendix 4

4.1 Elastodynamics Solutions

The elastic solutions can be obtained by removing the pore pressure related terms from the poroelastic ones and are listed as follows.

$$\begin{array}{c}{u}_{r}=-{\sum }_{n=1}^{\infty }{\overline{B} }_{1n}{\overline{\eta }}_{1n}{J}_{1}\left(r{\overline{\eta }}_{1n}\right)\mathrm{cos}\frac{n\pi }{h}z\\ -{\overline{C} }_{1}\sqrt{-{\overline{\lambda }}_{1}}{J}_{1}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)\\ -{\sum }_{n=1}^{\infty }{\overline{A} }_{n}\frac{n\pi }{h}{J}_{1}\left(r{\overline{\xi }}_{n}\right)\mathrm{cos}\frac{n\pi }{h}z\end{array}$$

(89)

$$\begin{array}{c}{u}_{z}=-{\sum }_{n=1}^{\infty }{\overline{B} }_{1n}{J}_{0}\left(r{\overline{\eta }}_{1n}\right)\frac{n\pi }{h}\mathrm{sin}\frac{n\pi }{h}z\\ +{\overline{D} }_{1}\sqrt{{\overline{\lambda }}_{1}}\mathrm{sinh}\sqrt{{\overline{\lambda }}_{1}}z\\ +{\sum }_{n=1}^{\infty }{\overline{A} }_{n}\left[\begin{array}{c}\frac{{J}_{1}\left(r{\overline{\xi }}_{n}\right)}{r}\\ +{\overline{\xi }}_{n}\frac{{J}_{0}\left(r{\overline{\xi }}_{n}\right)-{J}_{2}\left(r{\overline{\xi }}_{n}\right)}{2}\end{array}\right]\mathrm{sin}\frac{n\pi }{h}z\end{array}$$

(90)

Pore pressures and stresses are determined from Eqs. (7–12) as follows

$$\begin{array}{c}{\sigma }_{rz}=2G{\sum }_{n=1}^{\infty }{\overline{B} }_{1n}{\overline{\eta }}_{1n}\frac{n\pi }{h}{J}_{1}\left(r{\overline{\eta }}_{1n}\right)\mathrm{sin}\frac{n\pi }{h}z\\ +G{\sum }_{n=1}^{\infty }{\overline{A} }_{n}\left[\begin{array}{c}\left(\begin{array}{c}\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-\frac{1}{{r}^{2}}\\ -\frac{3{\overline{\xi }}_{n}^{2}}{4}\end{array}\right){J}_{1}\left(r{\overline{\xi }}_{n}\right)\\ +{\overline{\xi }}_{n}\frac{{J}_{0}\left(r{\overline{\xi }}_{n}\right)-{J}_{2}\left(r{\overline{\xi }}_{n}\right)}{2r}\\ +\frac{{\overline{\xi }}_{n}^{2}{J}_{3}\left(r{\overline{\xi }}_{n}\right)}{4}\end{array}\right]\mathrm{sin}\frac{n\pi }{h}z\end{array}$$

(91)

$$\begin{array}{c}{\sigma }_{rr}={C}_{1}\left[\begin{array}{c}\left(\lambda +2G\right){\overline{\lambda }}_{1}\\ \times \frac{{J}_{0}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)-{J}_{2}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)}{2}\\ -\lambda \frac{\sqrt{-{\overline{\lambda }}_{1}}{J}_{1}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)}{r}\end{array}\right]\\ -{\sum }_{n=1}^{\infty }\begin{array}{c}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}{\overline{B} }_{1n}\lambda \left[\begin{array}{c}\frac{{\overline{\eta }}_{1n}{J}_{1}\left(r{\overline{\eta }}_{1n}\right)}{r}\\ +\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}{J}_{0}\left(r{\overline{\eta }}_{1n}\right)\end{array}\right]\\ +{\overline{B} }_{1n}\left(\lambda +2G\right){\overline{\eta }}_{1n}^{2}\\ \times \frac{{J}_{0}\left(r{\overline{\eta }}_{1n}\right)-{J}_{2}\left(r{\overline{\eta }}_{1n}\right)}{2}\\ +{\overline{A} }_{n}\frac{n\pi }{h}G{\overline{\xi }}_{n}\left[{J}_{0}\left(r{\overline{\xi }}_{n}\right)-{J}_{2}\left(r{\overline{\xi }}_{n}\right)\right]\end{array}\right\}\end{array}\\ +\lambda {\overline{D} }_{1}{\overline{\lambda }}_{1}\mathrm{cosh}\sqrt{{\overline{\lambda }}_{1}}z\end{array}$$

(92)

$$\begin{array}{c}{\sigma }_{zz}={\overline{C} }_{1}\lambda \left[\begin{array}{c}{\overline{\lambda }}_{1} \frac{{J}_{0}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)-{J}_{2}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)}{2}\\ -\frac{\sqrt{-{\overline{\lambda }}_{1}}{J}_{1}\left(r\sqrt{-{\overline{\lambda }}_{1}}\right)}{r}\end{array}\right]\\ -{\sum }_{n=1}^{\infty }\begin{array}{c}\mathrm{cos}\frac{n\pi }{h}z\times \\ \left\{\begin{array}{c}{\overline{B} }_{1n}\lambda \left[\begin{array}{c}\frac{{\overline{\eta }}_{1n}{J}_{1}\left(r{\overline{\eta }}_{1n}\right)}{r}\\ +\frac{{\overline{\eta }}_{1n}^{2}\left[{J}_{0}\left(r{\overline{\eta }}_{1n}\right)-{J}_{2}\left(r{\overline{\eta }}_{1n}\right)\right]}{2}\end{array}\right]\\ +{\overline{B} }_{1n}\left(\lambda +2G\right){J}_{0}\left(r{\overline{\eta }}_{1n}\right)\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}\\ -{\overline{A} }_{n}\frac{n\pi }{h}2G\left[\begin{array}{c}{\overline{\xi }}_{n}\frac{{J}_{0}\left(r{\overline{\xi }}_{n}\right)-{J}_{2}\left(r{\overline{\xi }}_{n}\right)}{2}\\ +\frac{1}{r}{J}_{1}\left(r{\overline{\xi }}_{n}\right)\end{array}\right]\end{array}\right\}\end{array}\\ \begin{array}{c}+\left(\lambda +2G\right){\overline{D} }_{1}{\overline{\lambda }}_{1}\mathrm{cosh}\sqrt{{\overline{\lambda }}_{1}}z\end{array}\end{array}$$

(93)

$$\begin{array}{c}F=-2\pi {\overline{C} }_{1}\lambda R\sqrt{-{\overline{\lambda }}_{1}}{J}_{1}\left(R\sqrt{-{\overline{\lambda }}_{1}}\right)\\ -2\pi {\sum }_{n=1}^{\infty }\left\{\begin{array}{c}{\overline{B} }_{1n}\lambda R{\overline{\eta }}_{1n}{J}_{1}\left(R{\overline{\eta }}_{1n}\right)\\ +{\overline{B} }_{1n}\left(\lambda +2G\right)\\ \times \frac{R{J}_{1}\left(R{\overline{\eta }}_{1n}\right)}{{\overline{\eta }}_{1n}}\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}\\ -{\overline{A} }_{n}\frac{n\pi }{h}2GR{J}_{1}\left(R{\overline{\xi }}_{n}\right)\end{array}\right\}\mathrm{cos}\frac{n\pi }{h}z\\ \begin{array}{c}+\pi {R}^{2}\left(\lambda +2G\right){\overline{D} }_{1}{\overline{\lambda }}_{1}\mathrm{cosh}\sqrt{{\overline{\lambda }}_{1}}z\end{array}\end{array}$$

(94)

where

$$\begin{array}{c}{\overline{\lambda }}_{1}=-\frac{\rho {\omega }^{2}}{\lambda +2G}\end{array}$$

(95)

$$\begin{array}{c}{\overline{\eta }}_{1n}=\sqrt{-\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-{\overline{\lambda }}_{1}}\end{array}$$

(96)

$$\begin{array}{c}{\overline{\xi }}_{n}=\sqrt{-\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-\frac{\rho {\omega }^{2}}{G}}\end{array}$$

(97)

$$\begin{array}{c}{\overline{D} }_{1}=-\frac{{U}_{z0}}{\sqrt{{\lambda }_{1}}\mathrm{sinh}\sqrt{{\lambda }_{1}}h}\end{array}$$

(98)

$$\begin{array}{c}{\overline{C} }_{1}=\frac{-{\overline{D} }_{1}{\overline{\chi }}_{1}}{{\overline{\beta }}_{1}}\end{array}$$

(99)

$$\begin{array}{c}\left[\begin{array}{c}{\overline{A} }_{n}\\ {\overline{B} }_{1n}\end{array}\right]={\left[\begin{array}{cc}{\overline{\Omega } }_{n}& 2{\overline{\eta }}_{1n}\frac{n\pi }{h}{J}_{1}\left(R{\overline{\eta }}_{1n}\right)\\ {\overline{\Lambda } }_{n}& {\overline{\Phi } }_{1n}\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ {\overline{D} }_{1}{\overline{\Psi } }_{1n}\end{array}\right]\end{array}$$

(100)

$$\begin{array}{c}{\overline{\beta }}_{1}=\left(\lambda +2G\right){\overline{\lambda }}_{1}\\ \times \frac{{J}_{0}\left(R\sqrt{-{\overline{\lambda }}_{1}}\right)-{J}_{2}\left(R\sqrt{-{\overline{\lambda }}_{1}}\right)}{2}\\ -\lambda \frac{\sqrt{-{\overline{\lambda }}_{1}}{J}_{1}\left(R\sqrt{-{\overline{\lambda }}_{1}}\right)}{R}\end{array}$$

(101)

$$\begin{array}{c}{\overline{\chi }}_{1}=\lambda {\overline{\lambda }}_{1}{\overline{\gamma }}_{1}\end{array}$$

(102)

$$\begin{array}{c}{\overline{\Phi } }_{1n}=\lambda \left[\frac{{\overline{\eta }}_{1n}{J}_{1}\left(R{\overline{\eta }}_{1n}\right)}{R}+\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}{J}_{0}\left(R{\overline{\eta }}_{1n}\right)\right]\\ \quad+\left(\lambda +2G\right){\overline{\eta }}_{1n}^{2}\frac{{J}_{0}\left(R{\overline{\eta }}_{1n}\right)-{J}_{2}\left(R{\overline{\eta }}_{1n}\right)}{2}\end{array}$$

(103)

$$\begin{array}{c}{\overline{\Omega } }_{n}=\left(\frac{{n}^{2}{\pi }^{2}}{{h}^{2}}-\frac{1}{{R}^{2}}-\frac{3{\overline{\xi }}_{n}^{2}}{4}\right){J}_{1}\left(R{\overline{\xi }}_{n}\right)\\ \quad+{\overline{\xi }}_{n}\frac{{J}_{0}\left(R{\overline{\xi }}_{n}\right)-{J}_{2}\left(R{\overline{\xi }}_{n}\right)}{2R}+\frac{{\overline{\xi }}_{n}^{2}{J}_{3}\left(R{\overline{\xi }}_{n}\right)}{4}\end{array}$$

(104)

$$\begin{array}{c}{\overline{\Lambda } }_{n}=\frac{n\pi }{h}G{\overline{\xi }}_{n}\left[{J}_{0}\left(R{\overline{\xi }}_{n}\right)-{J}_{2}\left(R{\overline{\xi }}_{n}\right)\right]\end{array}$$

(105)

$$\begin{array}{c}{\overline{\Psi } }_{1n}=\lambda {\overline{\lambda }}_{1}{\overline{\gamma }}_{1n}\end{array}$$

(106)

$$\begin{array}{c}{\overline{\gamma }}_{1}=\frac{Sinh\left(h\sqrt{{\overline{\lambda }}_{1}}\right)}{h\sqrt{{\overline{\lambda }}_{1}}}\end{array}$$

(107)

$$\begin{array}{c}{\overline{\gamma }}_{1n}=\frac{2{\left(-1\right)}^{n}h\sqrt{{\overline{\lambda }}_{1}}Sinh\left(h\sqrt{{\overline{\lambda }}_{1}}\right)}{{n}^{2}{\pi }^{2}+{h}^{2}{\overline{\lambda }}_{1}}\end{array}$$

(108)