Poroelastodynamics

Abstract

In the preceding chapters we have been dealing with poroelastic theories and problems under the assumptions similar to elastostatics; that is, at any instant of a loading, the poroelastic body is at a state of static equilibrium. In other words, for a body of any size, finite or infinitesimal, the summation of all forces, including surface and body forces, must equal to zero, $\sum \vec{F} = 0$, such that there is no acceleration created by the imbalance of forces. This, however, does not mean that there is no motion. One of the characteristics of poroelastic body is that its deformation is time-dependent, giving the impression of a creeping-like motion, even if the applied load is constant in time. This transient behavior is the consequence of a fluid phase. Fluid has no shear strength to resist shear deformation, but has a viscosity that resists the rate of shear deformation. Hence the force equilibrium of a fluid can be accompanied by motion. So even without considering the acceleration caused by force imbalance, the poroelastic body is not exactly static, and the poroelastic theory presented in the preceding chapters can be called a quasi-static theory. When a force is rapidly applied, such as by an explosion in the air, by the impact of a solid body, or due to the slippage of a fault, the inertial effect, that is, the right hand side of Newton’s second law of motion $\vec{F} = m\vec{a}$, cannot be neglected. A dynamic theory should be introduced. When the inertial effect is considered in a continuum body that is compressible, a wave phenomenon results. Particularly, the stress caused by the applied force is not instantly felt throughout the body—it has a finite speed of propagation. Sound propagation in the air as a wave phenomenon was recognized by philosophers and scientists as early as Aristotle (384–322 BC), and then by Galileo Galilei (1564–1642) (Imelda and Subramaniam, Phys Educ 42(2):173–179, 2007). In fact, Aristotle already recognized that sound is a longitudinal wave when he wrote “(the air) is set in motion …by contraction or expansion or compression” (Barnes J (ed), The complete works of Aristotle. Revised Oxford translation, vol 1. Princeton University Press, Princeton, 1984, 1256pp). This is indicative of the definition of a longitudinal wave, in which the particle motion is parallel to the wave propagation direction. In a solid, elastic medium, there exist two types of waves: in addition to the longitudinal, or compressional wave, there also exists a transverse wave, also called a shear wave, in which the particle motion is perpendicular to the wave propagation direction. These two waves propagate at different wave speed. The historical development of elastic wave theory, or elastodynamics, is well summarized in Love (A treatise on the mathematical theory of elasticity, vol 1. Cambridge University Press, Cambridge/New York, 1892, 354pp). In this chapter we are interested in the wave propagation in porous medium, or the theory of poroelastodynamics. In such medium, a third wave, called the second compressional wave, is observed, due to the existence of two phases, a solid and a fluid. The reasoning and theoretical demonstration of such waves were first presented by Yackov Frenkel (J Phys USSR 13(4):230–241, 1944. Republished, J Eng Mech ASCE 131(9):879–887, 2005) (see Sect. F.13 for a biography). Frenkel’s work was motivated by the field observation of Ivanov (Doklady Akademii Nauk SSSR 24(1):42–45, 1939; Izvestiya Akademii Nauk SSSR, Ser. Geogr. Geofiz 5:699–727, 1940), who discovered the so-called seismoelectric effect of the second kind (E-effect) generated by underground explosion—when a seismic wave is generated by an explosion, electric potential differences can be observed between electrodes situated at different distances from the source of the waves. Based on the continuum mechanics theory, Frenkel demonstrated that in a fluid infiltrated isotropic porous medium, in addition to a longitudinal and a shear wave, there existed a second longitudinal wave characterized by the out-of-phase movement between solid and fluid. He then showed that in the presence of electrolytes in liquids, electric current was generated due to the relative movement between the phases. The alternating directions of the electrical current in turn generate an electromagnetic wave. However, as quoted in the prologue of the chapter, after the proclamation of the discovery of a second wave, Frenkel did not further pursue its characteristics. Twelve years later, citing Frenkel’s original contribution, Biot (J Acoust Soc Am 28(2):168–178, 1956; J Acoust Soc Am 28(2):179–191, 1956) re-derived the theory of wave propagation in porous medium. Biot not only demonstrated the existence of the waves, but also presented the wave speeds. Particularly, it was shown that the second compressional wave is highly dissipative, and propagates at a much lower speed than the first compressional wave; hence they are respectively called the slow wave and the fast wave. In the higher frequency range, Biot also introduced a physical model of capillary flow in parallel plates or tubes, to account for the viscous-inertial attenuation. He also discovered a characteristic frequency at which the attenuation reaches its maximum. Biot’s model became immensely popular and the second wave has largely been referred to as the Biot second wave.

We shall not write down the expressions for its roots and shall only remark that …one of them corresponds to waves with a very small damping, and the other to waves with a very large damping. The waves of the second kind are thus really non-existent.

—Yacov Frenkel (1944)