Generalized viscoelastic wave equation

Abstract

S U M M A R YThis paper presents a generalized wave equation which unifies viscoelastic and pure elastic cases into a single wave equation. In the generalized wave equation, the degree of viscoelasticity varies between zero and unity, and is defined by a controlling parameter. When this viscoelastic controlling parameter equals to 0, the viscous property vanishes and the generalized wave equation becomes a pure elastic wave equation. When this viscoelastic controlling parameter equals to 1, it is the Stokes equation made up of a stack of pure elastic and Newtonian viscous models. Given this generalized wave equation, an analytical solution is derived explicitly in terms of the attenuation and the velocity dispersion. It is proved that, for any given value of the viscoelastic controlling parameter, the attenuation component of this generalized wave equation perfectly satisfies the power laws of frequency. Since the power laws are the fundamental characteristics in physical observations, this generalized wave equation can well represent seismic wave propagation through subsurface media.Key words: Non-linear differential equation; Seismic attenuation; Wave propagation. I N T RO D U C T I O NFor seismic wave simulation, there are two basic types of wave equations: pure elastic wave equation and viscoelastic wave equation. This paper proposes to unify these two types of wave equations into a single wave equation.The seismic property of subsurface media should generally be considered to be a superposition of Hook's pure elastic model and a viscoelastic model. Customarily, the Kelvin-Voigt model assumes the seismic property to be a stack of the elastic model and a Newtonian viscous model. The latter is a linear viscosity in which the stress-strain relationship is presented by a first-order temporal derivative. Thus, the Kelvin-Voigt model is given as the following:where σ (t) is the time-dependent tensile stress, ε(t) is the corresponding strain and E 0 and E 1 are two parameters of the model. In eq.(1), E 0 ε is the elastic term and E 1 dε/dt is the linear viscous term. This leads to a well-known wave equation called the Stokes equation. However, subsurface media behave in a much generalized viscoelastic manner. To reflect this characteristic, a stress-strain relationship can be described by a fractional order, instead of the first order, temporal derivative. Then, the superposition model can be expressed aswhere β is the fractional order of the time differential. The last term E 1 d β ε/dt β uses a compact form of fractional derivative to describe the frequency and time dependency of a viscoelastic system (Scott-Blair 1947).The mathematical-physics history of this empirical method, using a fractional derivative to represent the viscoelastic property, begins with Nutting (1921) who reported the observation that the stress relaxation could be modelled by fractional powers of time. Such a model is in sharp contrast to the traditional view that stress relaxation is best described by decaying exponentials. Gemant (1936) f...

Nonlinear fractional waves at elastic interfaces

“…This fractional wave equation has a simple physical interpretation in terms of the frequency-dependent penetration depth of the surface wave into the liquid subphase. Our derivation complements previous approaches where fractional wave equations were obtained by invoking response functions with fractional exponents [45][46][47][48][49], and constitutes the first derivation of a fractional wave equation from first physical principles. Therefore, on a fundamental level, our theory sheds light on how fractional wave behavior emerges from the viscous coupling of an interface to the embedding bulk medium.…”

“…non-integer, time derivative. Although linear fractional wave equations have been amply described in the literature [43][44][45][46][47][48][49], until now no derivation of such an equation based on physical first principles had been available. In a second step, we also include nonlinear effects in the wave amplitude by accounting for the nonlinear interfacial compressibility.…”

Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method (FRDTM)

“…Recently, fractional differential equations have been successfully applied to describe (model) real world problems. For instance, the generalized wave equation, which contains fractional derivatives with respect to time in addition to the second-order temporal and spatial derivatives, was used to model the viscoelastic case and the pure elastic case in a single equation (Wang, 2016). The time fractional Boussinesq-type equations can be used to describe small oscillations of nonlinear beams, long waves over an even slope, shallow-water waves, shallow fluid layers, and nonlinear atomic chains; the time-fractional ) 1 , 1 , 2 ( B -type equations can be used to study optical solitons in the two cycle regime, density waves in traffic flow of two kinds of vehicles, and surface acoustic soliton in a system supporting love waves; the time fractional Klein-Gordon-type equations can be applied to study complex group velocity and energy transport in absorbing media, short waves in nonlinear dispersive models, propagation of dislocations within crystals (Abu .…”

Vectorial Iterative Fractional Laplace Transform Method for the Analytic Solutions of Fractional Cauchy-Riemann Systems Partial Differential Equations

“…Recently, fractional differential equations have been successfully applied to describe (model) real-world problems. For instance, the generalized wave equation, which contains fractional derivatives with respect to time in addition to the second-order temporal and spatial derivatives, was used to model the viscoelastic case and the pure elastic case in a single equation [42]. The time fractional Boussinesq-type equations can be used to describe small oscillations of nonlinear beams, long waves over an even slope, shallow-water waves, shallow fluid layers, and nonlinear atomic chains; the time-fractional ) 1 , 1 , 2 ( B -type equations can be used to study optical solitons in the two-cycle regime, density waves in traffic flow of two kinds of vehicles, and surface acoustic soliton in a system supporting love waves; the time fractional Klein-Gordon-type equations can be applied to study complex group velocity and energy transport in absorbing media, short waves in nonlinear dispersive models, propagation of dislocations within crystals as cited in [43].…”