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There are extensive laboratory measurements of elastic moduli dispersion of porous materials, yet the mechanisms for the dispersion are not fully understood. Related analytical solutions are missing in the literature, potentially due to complex mathematical derivations. In this paper, we use Biot's theory of poroelastodynamics (PED) and present the first analytical solution for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. This is done by introducing Helmholtz decomposition, two scalar potentials, and two vector potentials to decouple the original governing partial differential equations. The methods of matrix diagonalization and separation of variables are then used to solve the decoupled equations. The elastodynamics (ED) solution is also presented. We demonstrate the coupled responses of displacement, pore pressure, and stress and interpret systematically the mechanisms for the elastic moduli dispersion. Furthermore, we investigate comprehensively the effects of loading frequency, material's poromechanical characteristics, sample size, and boundary conditions on the elastic moduli dispersion. Finally, excellent matches are found between the analytical solution and published measurements of the dynamic Young's moduli and Poisson's ratios of a water-saturated clastic sediment rock and a glycerin-saturated limestone. The analytical solution captures the materials’ elastic moduli dispersion at both seismic and sonic frequencies. This work should be of interest to scientists and engineers who work on dynamics, vibration, and poromechanics. Introduction Operations, such as the generalization of seismic waves by an explosion, ultrasound imaging, dynamic test on a fluid-saturated specimen, hydraulic fracturing, etc., are usually accompanied by a sudden force and wave behavior. Wave propagation in a fluid-saturated porous medium can generate pore pressure gradients, between the peaks and troughs of the wave, and therefore induce fluid flow in the porous medium [1,2]. It is well known that the relative motion between pore fluid and porous matrix results in viscous friction and energy dissipation, causing wave velocities and elastic moduli dispersion [3], i.e., frequency-dependency. The wave velocity dispersion and elastic moduli dispersion are of particular interest to geophysics and rock mechanics. Many laboratory experiments have been done to explore the effects of pore fluid on the wave and elastic moduli dispersion of porous materials. Four methods are commonly used, i.e., the pulse transmission approach using a “pitch and catch” method [4], the differential acoustic resonance spectroscopy measuring the change of resonant frequencies of a cavity with and without a test sample [5], the resonant bar technique oscillating a thin rod of material at a resonant frequency [6,7], and the forced deformation test using axial vibration and the stress-strain method [8], [9], [10], [11], [12], [13]. Scientists have applied these methods to many rocks and found that the dispersion is impacted by many factors, including fluid viscosity, saturation, confining pressure, effective stress, pore fluid distribution, and anisotropy [14], [15], [16], [17], [18], [19]. Measurements on Fontainebleau sandstones [20] show that a high value of fluid viscosity leads to the material in an undrained state even at low frequencies from 0.004 Hz to 0.4 Hz. The dynamic bulk moduli of Savonnières limestones at 1 Hz and 10 Hz exhibit nonmonotonic variations as the saturation increases [21]. Experiments on Mancos shale and Pierre shale reveal that the sensitivity of the dynamic stiffness of shale samples to the confining pressure varies between seismic and ultrasonic frequencies. Measurements of the elastic moduli dispersion of a tight sandstone indicate that the dispersion decreases as the effective pressure increases [22]. At the same saturation, the elastic moduli dispersion for an Indiana limestone varies with saturation methods, indicating the impact of fluid distribution on the dispersion [23]. The phenomenon of dispersion is also reported in many fields, such as the Kankakee limestone formation [24], the Utsira formation [25], and the North slope of Alaska [26]. In laboratory measurements, cylindrical samples are usually preferable. Analytical simulations for cylindrical specimens subject to various laboratory setups can be found in the literature [27], [28], [29], [30]. These simulations greatly help researchers understand the porous sample's poromechanical responses, including pore pressure, displacement, and stress. Nevertheless, the simulations are quasi-static, assuming that the summation of all forces over a mass of any size is equal to zero. In a forced deformation test, a cyclic load is applied and generates elastic waves in the sample. As a result, the inertial effect, or Newton's second law, becomes significant and needs to be considered. The theory of elastodynamics [31] utilizes Hooke's law and Newton's second law and well interprets the responses of stress and deformation during the elastic wave propagation in a solid elastic medium. For a fluid-saturated porous medium, Biot's theory of poroelastodynamics [2,32] has been a popular one describing the fluid-solid interactions through generalized Hooke's law, Newton's second law, and dynamic Darcy's law. Pride and Berryman [33] generated Biot's single porosity theory to dual-porosity dual-permeability poroelastodynamics to further account for the existence of fractures in a fluid-saturated porous medium. Analytical or numerical solutions based on the theories of dynamics [2,31,33] can be found for problems with a variety of configurations, including a one-dimensional column [34,35], a half-space [36], an infinite three-dimensional space [37], [38], [39], [40], [41], a borehole [42], a circular cylinder embedded in an elastic medium [43], a two-dimensional square [44], [45], [46], [47], [48], an anisotropic porous plate [49], and inclusion in an unbounded domain [50]. These solutions focus on the poromechanical responses, including pore pressure, stress, and displacement, without investigating elastic moduli dispersion. Analytical and numerical methods for the study of elastic moduli dispersion can be found in other publications. Using the squirt flow model [51,52], Adelinet et al. [53] calculated bulk modulus dispersion in a cracked-porous rock and showed that the dispersion can increase from 10% to 37% as the crack fraction raises from 4% to 12%. Seyfaddini et al. [49] proposed two numerical methods to increase the accuracy in computing wave dispersion in an anisotropic poroelastic plate. Nejadsadeghi and Misra [54] showed that higher-order inertia significantly impacts wave dispersion in an infinite 1D material with a granular microstructure. Using Biot's poroelastodynamics [2], Guo et al. [55] showed that P-wave anisotropy is highly impacted by the angle of intersected fractures. Other publications show that the dispersion could be impacted by many factors, such as the existence of finite initial strain and stress [43,56], lateral inertia [57], and the alignment of fractures [58,59]. A thorough review is out of the scope of the paper. Analytical poroelastodynamics modeling of the pore pressure, displacement, and stress of a cylindrical fluid-saturated porous cylinder subject to a dynamic test is missing in the literature, which is potentially due to the complex mathematical derivations. The response of stress and pore pressure fields to the loading frequency and how they impact the dynamic elastic moduli are not fully understood. As we know, materials’ dynamic elastic moduli are usually significantly different from the static ones. The study of the difference between the dynamic and the static elastic moduli has been a historical and challenging topic [3,60]. A key to studying the difference is to investigate the factors impacting materials’ dynamic elastic moduli. Publications have shown that there are many factors contributing to the difference, e.g., the magnitude of the strain amplitude, the frequency of the dynamic measurements, the presence of pore fluid and natural fractures, and material anisotropy [61], [62], [63]. In this paper, we will show that the dynamic elastic moduli are also controlled by other factors, such as Biot's coefficient, permeability, and sample size. In this work, we use Biot's theory of poroelastodynamics to derive the analytical expressions of displacement, pore pressure, and stress of a fluid-saturated porous cylinder subject to a dynamic forced deformation test. We then use analytical expressions of stress and displacement to calculate the cylindrical sample's dynamic Young's modulus and Poisson's ratio. A Ruhr sandstone specimen is chosen as an illustration example to demonstrate the analytical solutions. At low frequencies when the rock's permeability is high enough to equilibrate the pressure gradients, the pore pressure is almost uniform across the sample. At high frequencies, pore pressure becomes non-uniform. The locations where the pore pressure reaches the minimal and maximal are highly frequency dependent. The dynamic Young's modulus and Poisson's ratio reach their local minimum and maximum at the sample's resonance frequencies. At 20 kHz, Biot's coefficient increases from 0.5 to 1, the dynamic Young's modulus decreases then increases while the Poisson's ratio continues to decrease. Elastic moduli are also sensitive to the dimension of the rock sample. 3D plots show that neither the dynamic Young's modulus nor the dynamic Poisson's ratio is monotonic to the sample's radius or length. We also show that the analytical simulation matches well with published laboratory measurements on a clastic sediment rock sample from the North Sea and a limestone from a Dogger outcrop of Paris Basin. Section snippets Dynamic forced deformation test In this section, we present the schematics of a dynamic forced deformation test and the related boundary conditions. As illustrated in Fig. 1, an isotropic fluid-saturated cylindrical sample, with radius R and height h , is under a harmonic excitation of the form uz ( t ) = U z 0sin ω t , where ω is the angular frequency, t is the time, uz is the vertical displacement, and U z 0 is the amplitude of the vertical displacement at the bottom surface of the sample. The top surface of the sample is fixed. Both Governing equations for the poroelastodynamics modeling In this section, we represent the governing equations for the theory of poroelastodynamics originally presented by Biot in 1956 [2,32]. This theory couples the generalized Hooke's law, Newton's second law, dynamic Darcy's law, and pore fluid mass balance equations to describe the phenomenon of elastic wave propagation in a fluid-saturated porous material. The original equations were expressed in the spatial and time domains. Because harmonic loading is applied to the rock sample studied in this Analytical solutions The analytical solution to Eqs. (4) and (5) for a cylindrical geometry is presented in this section. To derive the analytical solution, we first apply Helmholtz decomposition to decompose the displacement fields into divergence- and curl-free components. Then we introduce two scalar potentials and two vector potentials to decouple Eqs. (4) and (5) into one partial differential equation, one algebraic equation, and two coupled partial differential equations. Eventually, we use the methods of Demonstration example In this section, an example is presented to demonstrate the poromechanical responses, including pore pressure, displacement, stress, and force, of a water-saturated cylindrical Ruhr sandstone. Furthermore, we interpret systematically the mechanisms of elastic moduli dispersion. The effects of loading frequency, boundary conditions, permeability, Biot's coefficient, and sample size on the elastic moduli dispersion are investigated. The poromechanical and physical parameters for a water-saturated Simulate published laboratory measurements In this section, we use the analytical simulation to match published laboratory measurements of the dynamic Young's modulus and Poisson's ratio of a fluid-saturated clastic sediment rock, i.e., X #2, from the North Sea [11] and a limestone from a Dogger outcrop of Paris Basin [70]. Using Archimedes’ principle and Boyle's law, the total porosity, i.e., the difference between the bulk and grain volumes, of the clastic sediment rock is measured to be 29.1%. The sample is fully saturated with water Discussions The newly derived solution for isotropic porous cylinders has several applications. Firstly, it can help researchers obtain a better understanding of the mechanisms of the poromechanical responses and the elastic moduli dispersion of fluid-saturated isotropic materials. Secondly, in the oil and gas industry, the knowledge of the poromechanical responses is critical in optimizing the loading frequency in the ultrasound wellbore stimulation. Thirdly, it can be used to estimate dynamic Conclusions In this paper, we presented the first analytical solution of pore pressure, stress, force, displacement, and elastic moduli dispersion for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. We used Helmholtz decomposition and introduced two scalar potentials and two vector potentials to decouple the governing equations for Biot's theory of poroelastodynamics. The decoupled equations were then solved by the methods of matrix diagonalization and separation of Acknowledgments The author would like to thank Prof. Younane Abousleiman, Dr. Houzhu Zhang, and Prof. Marian Wiercigroch for their fruitful discussions and comments improving this paper. References (75) G. Han et al. Uncoupled poroelastic and intrinsic viscoelastic dissipation in cartilage J. Mech. Behav. Biomed. Mater. (2018) B.K. Connizzo et al. Tendon exhibits complex poroelastic behavior at the nanoscale as revealed by high-frequency AFM-based rheology J. Biomech. (2017) A.D. Cheng Material coefficients of anisotropic poroelasticity Int. J. Rock Mech. Min. Sci. (1997) M. Aleyaasin et al. Wave dispersion and attenuation in viscoelastic polymeric bars: analysing the effect of lateral inertia Int. J. Mech. Sci. (2010) X. Guo et al. Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with initial stresses Int. J. Mech. Sci. (2016) N. Nejadsadeghi et al. Role of higher-order inertia in modulating elastic wave dispersion in materials with granular microstructure Int. J. Mech. Sci. (2020) M. Adelinet et al. Dispersion of elastic moduli in a porous-cracked rock: theoretical predictions for squirt-flow Tectonophysics (2011) Z. Liu et al. The method of fundamental solutions for the elastic wave scattering in a double-porosity dual-permeability medium Appl. Math. Model. (2021) F. Seyfaddini et al. Semi-analytical IGA-based computation of wave dispersion in fluid-coupled anisotropic poroelastic plates Int. J. Mech. Sci. (2021) C. Liu et al. Poroelastodynamic responses of a dual-porosity dual-permeability material under harmonic loading Partial Differ. Equ. Appl. Math. (2021) X. Su et al. The poroviscoelastodynamic solution to Mandel's problem J. Sound Vib. (2022) A. Mehrabian et al. Mandel's problem reloaded J. Sound Vib. (2021) S. Akbarov et al. The influence of the finite initial strains on the axisymmetric wave dispersion in a circular cylinder embedded in a compressible elastic medium Int. J. Mech. Sci. (2010) S.K. Hoang et al. Poroviscoelasticity of transversely isotropic cylinders under laboratory loading conditions Mech. Res. Commun. (2010) M. Lee et al. Unique problems associated with seismic analysis of partially gas-saturated unconsolidated sediments Mar. Pet. Geol. (2009) X. Guo et al. Dispersion relations of elastic waves in three-dimensional cubical piezoelectric phononic crystal with initial stresses and mechanically and dielectrically imperfect interfaces Appl. Math. Model. (2019) S. Manna et al. Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer Appl. Math. Model. (2020) Y. Ma et al. A semi-analytical method for the dispersion analysis of orthotropic composite plates with periodically attached acoustic black hole resonators Appl. Math Model. (2022) P. Kumar et al. Analytical study on shear wave propagation in anisotropic dry sandy spherical layered structure Appl. Math. Model. (2022) J. Frenkel On the theory of seismic and seismoelectric phenomena in a moist soil J. Eng. Mech. (2005) M.A. Biot Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range J. Acoust. Soc. Am. (1956) T.M. Müller et al. Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks-a review Geophysics (2010) F. Birch The velocity of compressional waves in rocks to 10 kilobars: 1 J. Geophys. Res. (1960) S.X. Wang et al. Differential acoustic resonance spectroscopy for the acoustic measurement of small and irregular samples in the low frequency range J. Geophys. Res. Solid Earth (2012) S. Nakagawa Split Hopkinson resonant bar test for sonic-frequency acoustic velocity and attenuation measurements of small, isotropic geological samples Rev. Sci. Instrum. (2011) N. Lucet et al. Sonic properties of rocks under confining pressure using the resonant bar technique J. Acoust. Soc. Am. (1991) J.W. Spencer Stress relaxations at low frequencies in fluid-saturated rocks: attenuation and modulus dispersion J. Geophys. Res. Solid Earth (1981) M.L. Batzle et al. Fluid mobility and frequency-dependent seismic velocity-direct measurements Geophysics (2006) N. Tisato et al. Measurements of seismic attenuation and transient fluid pressure in partially saturated Berea sandstone: evidence of fluid flow on the mesoscopic scale Geophys. J. Int. (2013) R. Hofmann, Frequency dependent elastic and anelastic properties of clastic rocks,... S. Lozovyi et al. From static to dynamic stiffness of shales: frequency and stress dependence Rock Mech. Rock Eng. (2019) T. Cadoret et al. Influence of frequency and fluid distribution on elastic wave velocities in partially saturated limestones J. Geophys. Res. Solid Earth (1995) S. Chapman et al. Seismic attenuation in partially saturated Berea sandstone submitted to a range of confining pressures J. Geophys. Res. Solid Earth (2016) L. Pimienta et al. Dispersions and attenuations in a fully saturated sandstone: experimental evidence for fluid flows at different scales Lead. Edge (2016) L. Pimienta et al. Bulk modulus dispersion and attenuation in sandstones Geophysics (2015) V. Mikhaltsevitch et al. Laboratory measurements of the effect of fluid saturation on elastic properties of carbonates at seismic frequencies Geophys. Prospect. (2016) H. Yin et al. Pressure and fluid effect on frequency-dependent elastic moduli in fully saturated tight sandstone J. Geophys. Res. 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