Jiawei Liu, Wen-An Yong, Stability analysis associated with Biot/squirt models for revolution propagation in saturated porous news, Geophysical Journal Global.
This work is focused on the Biot/squirt (BISQ) models for revolution propagation in saturated porous news. We reveal that the models allow exponentially exploding solutions, as time would go to infinity, once the characteristic squirt-flow coefficient is negative or features a non-zero imaginary component. We additionally reveal that the coefficient that is squirt-flow have non-zero imaginary components for a few experimental parameters or even for low angular frequencies. Since the models are linear, the presence of such exploding solutions suggests uncertainty associated with the BISQ models. This outcome, for the time that is first offers a theoretical description associated with well-known empirical observation that BISQ model just isn’t dependable ( perhaps maybe perhaps not in line with Gassmann’s formula) at low frequencies. It calls for a reconsideration regarding the trusted BISQ concept. Having said that, we show that the 3-D isotropic BISQ model is stable once the squirt-flow coefficient is good. In specific, the initial Biot model is unconditionally stable in which the squirt-flow coefficient is 1.
The propagation of seismic waves in rocks with fluids happens to be being among the most research that is active in geoscience. Different theories (Biot 1956a; Dvorkin & Nur 1993; Mavko et al. 2009; MГјller et al. 2010) concentrate on relating attenuation and dispersion of seismic waves in planet materials to real properties of this stones and liquids. Attenuation means the exponential decay of revolution amplitude with distance and dispersion is just a variation of propagation velocity with frequency. The Biot and squirt-flow mechanisms are believed to be the most important ones (Dvorkin & Nur 1993; Yang & Zhang 2002) among the various mechanisms related to attenuation and dispersion. They’ve offered as rigorous and formal fundamentals to review wave that is acoustic in saturated porous media.
In Biot’s theory (Biot 1956a, b), pore liquids are forced to be involved in the solid’s motion because of viscosity and inertia. The Biot system is described through macroscopic stone properties and based on the macroscopic flow. The mechanism that is squirt-flow linked to the squirting of this pore fluid away from cracks because they are deformed by the compressional waves. It really is pertaining to microscopic rock properties and on the basis of the neighborhood fluid movement (Dvorkin & Nur 1993). Plainly, the two mechanisms happen simultaneously and influence the procedure for seismic power attenuation and propagation.
Discover that the models could be changed to a method of first-order partial differential equations (see Section 3) and therefore hold the kind of hyperbolic leisure systems. According to this, our company is led because of the concept of hyperbolic leisure systems developed in Yong ( 1992, 2001) and conduct a security analysis for the BISQ model and its particular 3-D extension that is isotropic. Our outcomes reveal that the models enable spatially bounded but time exponentially exploding solutions once the characteristic squirt-flow coefficient is negative or includes an imaginary part that is non-zero. We also reveal that the coefficient that is squirt-flow includes a non-zero imaginary part at low angular frequencies. More over, by numerically examining some experimental information parameters from Dvorkin & Nur ( 1993), King et al. ( 2000), Parra ( 2000) and Yang et al. ( 2011), we come across it is usual and practical when it comes to squirt-flow coefficient to have non-zero imaginary parts. The existence of such exploding solutions indicates instability of the widely used models because the models are linear. This appears the time that is first mention uncertainty of this BISQ models. Moreover, we show that, if the coefficient that is squirt-flow good, the 3-D isotropic model satisfies a structural security condition for hyperbolic leisure systems and so is stable. In specific, the initial Biot model is unconditionally stable in which the squirt-flow coefficient is 1.
Our outcomes offer reasonable explanations for a few problematic simulations that are numerical when you look at the literary works. In Parra ( 2000) for a transversely isotropic medium, it absolutely was remarked that the velocity dispersion and attenuation predicted by the BISQ model aren’t in keeping with experimental data during the low frequency that is angular. For the snapshot of seismic revolution areas, Yang et al. ( 2002) unearthed that it really is difficult to gain the sluggish P-waves because of the BISQ model utilizing the tiny regularity and the sluggish P-waves happen just after increasing the regularity. These experimental and numerical findings claim that the BISQ model just isn’t dependable ( maybe maybe not in keeping with Gassmann’s formula) at zero regularity limitation (Chapman et al. 2002) and concur perfectly with your theoretical end in Theorem 2.3: the BISQ model is unstable in the event that angular regularity f is little but non-zero. The numerical results in Yang & Wang ( 2002) indicate that the slow P-waves only occur when the fluid viscosity is negligible on the other hand. We observe that the squirt-flow coefficient is genuine if the viscosity vanishes. These also coincide perfectly with this theoretical outcomes: the BISQ model is stable once the coefficient that is squirt-flow good.
The paper is arranged the following. In part 2, we reveal the presence of time-exponentially exploding answers to the 1-D BISQ model as soon as the squirt-flow coefficient is negative or features a non-zero imaginary component. Right right Here we additionally analyse the squirt-flow coefficient at low frequencies and numerically examine some experimental parameters. Area 3 is specialized in the 3-D isotropic hinge expansion, which can be transformed to a method of first-order partial differential equations because of the coefficient matrices given in Appendix. It really is demonstrated that the uncertainty outcome is legitimate for the 3-D model and the good squirt-flow coefficient guarantees the structural security regarding the expansion. The final area summarizes our primary outcomes.