S-wave propagation across material discontinuities in poroelasticity

The shear motion in Newtonian fluids, i.e., the fluid vorticity, represents an intrinsic loss mechanism governed by a diffusion equation. Its description involves the trace-free part of the fluid viscous stress tensor. This part is missing in the Biot theory of poroelasticity. As a result, the fluid vorticity is not captured, and only one S-wave is predicted. The missing fluid vorticity has implications for the propagation of S-waves across discontinuities. This becomes most apparent in the problem of S-wave propagation across the welded contact of an elastic solid with a porous medium. At such a contact, the no-slip condition between the elastic solid and the constituent parts of the porous medium, the solid-frame, and the pore-fluid, must hold. This requirement translates into a vanishing relative motion of the fluid with respect to the solid-frame, i.e., filtration field, at the contact. Nevertheless, our analysis shows that for the Biot theory, in the low-frequency regime, a non-zero, although insignificantly small filtration field exists at the contact. But, more importantly, the filtration field is noticeable when the transition to the high-frequency regime occurs. This constitutes a disagreement with the requirement of a no-slip boundary condition and renders the prediction unphysical. This shortcoming is circumvented by including the fluid viscous stress tensor into the poroelastic constitutive relations, as stipulated by the de la Cruz-Spanos poroelasticity theory. Then, a second S-wave is predicted which manifests as the fluid vorticity at macroscale. This process is distinct from the fast S-wave, the other predicted S-wave akin to the Biot S-wave. We find that the generation of this process at the contact induces a filtration field equal and opposite to that associated with the fast S-wave. Therefore, the no-slip condition is satisfied, and the S-wave reflection/transmission across a discontinuity becomes physically meaningful.

Predicting acoustic performance of high surface area particle stacks with a poro-elastic model

Because of the high sound absorption they offer at low frequencies, there is a growing interest in high surface area particles and how they might be applied in noise control. Therefore, a model that can accurately predict the acoustic behavior of this type of materials will be useful in relevant applications. A poro-elastic model based on a combination of Biot theory and an existing rigid model of granular activated carbon (GAC) is introduced in the current work. The input parameters for this model consist of a certain number of properties that are known by measurement, and a set of values obtained by matching the model prediction with acoustic measurements. Measured absorption coefficients and surface impedance of stacks of several types of different activated carbon particles are shown in this paper. A fitting procedure that determines the unknown parameters is also described. It is shown that the model is able to predict the acoustic behavior of the particle stacks, and especially to capture the frame resonances at low frequencies, thus, validating the proposed model. Beyond the activated carbon used in the present tests, it is reasonable to generalize this model to stacks of other high surface area particles.

Modeling and Vibration Analysis of a Porous Rotational Shell Based on Biot Theory

Combining the Biot theory and classical elastic theory for thin shells, a new dynamic model of a thin fluid-saturated porous rotational shell is proposed. First-order ordinary differential control equations of the porous rotational shell are derived in the frequency domain. These equations are then solved by using the precise element method. The accuracy of this model has been verified by comparing with a vibration experiment. Moreover, the comparisons between the present model and two equivalent property models are carried out. Because the present approach considers the fluid-solid coupling effect and makes no assumptions for the fluid displacements, it is more accurate in the high-frequency range. Lastly, the dynamic characteristics of porous rotational shells are demonstrated by the proposed method.

Comparison of Fluid Pressure Wave between Biot Theory and Storativity Equation

Compressibilities of pore fluid and rock skeleton affect pressure profile and flow velocity of fluid in aquifers. Storativity equation is often used to characterize such effects. The equation suffers from a disadvantage that at infinite large frequency, the predicted velocity of fluid pressure wave is infinitely large, which is unrealistic because any physical processes need certain amounts of time. In this paper, Biot theory is employed to investigate the problem. It is shown that the key equations of Biot theory can be simplified to storativity equation, based on low-frequency assumption. Using Berea sandstone as an example, we compare phase velocity and the quality factor between Biot theory and storativity equation. The results reveal that Biot theory is more accurate in yielding a bounded wave velocity. At frequency lower than 100 kHz, Biot theory yields a wave velocity 8 percent higher than storativity equation does. Apparent permeability measured by fluid pressure wave (such as Oscillatory Hydraulic Tomography) may be 14 percent higher than real permeability measured by steady flow experiments. If skeleton is rigid, Biot theory at very high frequencies or with very high permeabilities will yield the same velocity as sound wave in pure water. The findings help us for better understanding of the physical processes of pore fluid and the limitations of storativity equation.

Q values and wave inhomogeneity parameters of reflected inhomogeneous P and S waves at the free surface of an effective Biot solid

SUMMARY In this study, new methods are developed to estimate the dissipation factors, inhomogeneity parameters and phase velocities of the reflected waves at the free surface of a poro-viscoelastic solid in which the seismic wave propagation is described by effective Biot theory. The Christoffel equations of an effective Biot medium are solved for a general harmonic plane wave and the three complex velocities obtained corresponding to the shear wave (SV), fast-P wave and slow-P wave, together with their polarizations. Based on the complex form of the energy balance equation in an effective Biot material, expressions are derived for the energy ratios at the free surface. Moreover, the equations for the inhomogeneity parameters are derived as functions of the complex slowness or the unit polarization vectors. Based on the implicit and the explicit dissipation factor expressions, two methods are developed to obtain the dissipation factors, the inhomogeneity parameters and the phase velocities of mode-converted waves. These methods are illustrated by numerical examples which show that the dissipation factors, inhomogeneity parameters and phase velocities of reflected waves can strongly depend on the incidence angle (also reflected angle), the incident wave inhomogeneity parameter and the wave frequency. Ignoring these dependencies and using dissipation factors only valid for homogeneous waves can cause discrepancies in computed phase velocities and dissipation factors for interface generated (reflected/transmitted) inhomogeneous waves.