Numerical analysis of surface waves on

Cooperation with: C. Lai (European Centre for Training and Research in Earthquake Engineering, Pavia, Italy), R. Lancellotta, S. Foti (Politecnico di Torino, Italy)

Description: The project is the continuation of the theoretical research on surface waves on the basis of the model by Wilmanski [5]. Here, the dispersion relation for surface waves on an impermeable boundary of a fully saturated poroelastic medium is investigated numerically in the whole range of frequencies. To this aim the linear simplified model of a two-component poroelastic medium is used. Similarly to the classical Biot model, it is a continuum mechanical model but it is much simpler due to the lack of coupling of stresses. The general dispersion relation has been ascertained, and results of an earlier work [6] for the high and low frequency approximations have been recalled. The main part of the project concerns numerical aspects: first we indicate the applied numerical procedure and then we illustrate the numerical results for the normalized velocities and attenuations of the Rayleigh and Stoneley waves, [1, 2]. It is known that surface modes of propagation in linear models result from the combination of bulk modes. Physically, this means that at any point of the boundary classical longitudinal and shear waves combine into the Rayleigh wave which must be slower than both bulk waves. The presence of the second longitudinal bulk wave P2 yields the existence of the second surface mode--the Stoneley wave which should be slower than the P2 wave--the slowest of bulk waves. Both quantities, velocities and attenuations, are shown for different values of the bulk permeability coefficient $\pi$ , in different ranges of frequencies. A decay of the Rayleigh wave velocity, mentioned in [3], has been confirmed in the range of small frequencies in spite of the lack of static coupling between components. Moreover, we compare the behavior of the two types of surface waves with the behavior of two bulk waves: P1 and P2.

Here, we only show a part of the project, namely the governing equations, the boundary conditions, and the numerical results for a chosen value of the permeability coefficient:

Within the linear model of a two-component poroelastic saturated medium the process is described by the macroscopic fields $\rho^{{F}}_{}$ $\left(\vphantom{ \mathbf{x,}t}\right.$ x, t $\left.\vphantom{ \mathbf{x,}t}\right)$ - partial mass density of the fluid, v^F $\left(\vphantom{ \mathbf{x,}t}\right.$ x, t $\left.\vphantom{ \mathbf{x,}t}\right)$ - velocity of the fluid, v^S $\left(\vphantom{ \mathbf{x,}t}\right.$ x, t $\left.\vphantom{ \mathbf{x,}t}\right)$ - velocity of the skeleton, e^S $\left(\vphantom{ \mathbf{x,}t}\right.$ x, t $\left.\vphantom{ \mathbf{x,}t}\right)$ - symmetric tensor of small deformations of the skeleton and the porosity n. These fields satisfy the following set of linear equations

		$\displaystyle {\frac{{\partial \rho ^{F}}}{{\partial t}}}$ + $\displaystyle \rho_{{0}}^{{F}}$ div v^F = 0, $\displaystyle \left\vert\vphantom{ \tfrac{\rho ^{F}-\rho _{0}^{F}}{\rho _{0}^{F}}% }\right.$ $\displaystyle {\tfrac{{\rho ^{F}-\rho _{0}^{F}}}{{\rho _{0}^{F}}% }}$ $\displaystyle \left.\vphantom{ \tfrac{\rho ^{F}-\rho _{0}^{F}}{\rho _{0}^{F}}% }\right\vert$ $\displaystyle \ll$ 1,
		$\displaystyle \rho_{{0}}^{{F}}$ $\displaystyle {\frac{{\partial \mathbf{v}^{F}}}{{\partial t}}}$ + $\displaystyle \kappa$ grad $\displaystyle \rho^{{F}}_{}$ + $\displaystyle \beta$ grad $\displaystyle \left(\vphantom{ n-n_{E}}\right.$ n - n_E $\displaystyle \left.\vphantom{ n-n_{E}}\right)$ + $\displaystyle \hat{{p}}$ = 0, $\displaystyle \hat{{p}}$ : = $\displaystyle \pi$ $\displaystyle \left(\vphantom{ \mathbf{v}^{F}-\mathbf{v}^{S}}\right.$ v^F - v^S $\displaystyle \left.\vphantom{ \mathbf{v}^{F}-\mathbf{v}^{S}}\right)$ ,
		$\displaystyle \rho_{{0}}^{{S}}$ $\displaystyle {\frac{{\partial \mathbf{v}^{S}}}{{\partial t}}}$ - div $\displaystyle \left(\vphantom{ \lambda ^{S}\left( {\rm tr} \mathbf{e}^{S}\r... ...) \mathbf{1}+2\mu \mathbf{e}^{S}+\beta \left( n-n_{E}\right) \mathbf{1}}\right.$ $\displaystyle \lambda^{{S}}_{}$ $\displaystyle \left(\vphantom{ {\rm tr} \mathbf{e}^{S}}\right.$ tr e^S $\displaystyle \left.\vphantom{ {\rm tr} \mathbf{e}^{S}}\right)$ 1 +2 $\displaystyle \mu$ e^S + $\displaystyle \beta$ $\displaystyle \left(\vphantom{ n-n_{E}}\right.$ n - n_E $\displaystyle \left.\vphantom{ n-n_{E}}\right)$ 1 $\displaystyle \left.\vphantom{ \lambda ^{S}\left( {\rm tr} \mathbf{e}^{S}\r... ...) \mathbf{1}+2\mu \mathbf{e}^{S}+\beta \left( n-n_{E}\right) \mathbf{1}}\right)$ - $\displaystyle \hat{{p}% }$ = 0,
		$\displaystyle {\frac{{\partial \mathbf{e}^{S}}}{{\partial t}}}$ = sym grad v^S, $\displaystyle \left\Vert\vphantom{ \mathbf{e}^{S}}\right.$ e^S $\displaystyle \left.\vphantom{ \mathbf{e}^{S}}\right\Vert$ $\displaystyle \ll$ 1, n_E : = n₀ $\displaystyle \left(\vphantom{ 1+\delta \mathrm{tr }\mathbf{e}^{S}}\right.$ 1 + $\displaystyle \delta$ tr e^S $\displaystyle \left.\vphantom{ 1+\delta \mathrm{tr }\mathbf{e}^{S}}\right)$ ,
		$\displaystyle {\frac{{\partial \left( n-n_{E}\right) }}{{\partial t}}}$ + $\displaystyle \Phi$ div $\displaystyle \left(\vphantom{ \mathbf{v}^{F}-\mathbf{v}^{S}}\right.$ v^F - v^S $\displaystyle \left.\vphantom{ \mathbf{v}^{F}-\mathbf{v}^{S}}\right)$ + $\displaystyle {\frac{{n-n_{E}}}{{\tau }}}$ = 0, $\displaystyle \left\vert\vphantom{ \tfrac{n-n_{0}}{n_{0}}}\right.$ $\displaystyle {\tfrac{{n-n_{0}}}{{n_{0}}}}$ $\displaystyle \left.\vphantom{ \tfrac{n-n_{0}}{n_{0}}}\right\vert$ $\displaystyle \ll$ 1.

Here $\rho_{{0}}^{{F}}$ , $\rho_{{0}}^{{S}}$ , n₀ denote constant reference values of partial mass densities, and porosity, respectively, and $\kappa$ , $\lambda^{{S}}_{}$ , $\mu^{{S}}_{}$ , $\beta$ , $\pi$ , $\tau$ , $\delta$ , $\Phi$ are constant material parameters. The first one describes the macroscopic compressibility of the fluid component, the next two are macroscopic elastic constants of the skeleton, $\beta$ is the coupling constant, $\pi$ is the coefficient of bulk permeability, $\tau$ is the relaxation time, and $\delta$ , $\Phi$ describe equilibrium and nonequilibrium changes of porosity, respectively. For the purpose of this work we assume $\beta$ = 0.

In order to determine surface waves in a saturated poroelastic medium we need conditions for z = 0. In the general case of a boundary between a saturated porous material and a fluid the boundary conditions were formulated by Deresiewicz & Skalak. We quote them here in a slightly modified form and for an impermeable boundary

		$\displaystyle \left.\vphantom{ T_{13}}\right.$ T₁₃ $\displaystyle \left.\vphantom{ T_{13}}\right\vert_{{z=0}}^{}$ $\displaystyle \equiv$ $\displaystyle \left.\vphantom{ T_{13}^{S}}\right.$ T₁₃^S $\displaystyle \left.\vphantom{ T_{13}^{S}}\right\vert_{{z=0}}^{}$ = $\displaystyle \mu^{{S}}_{}$ $\displaystyle \left.\vphantom{ \left( \frac{\partial u_{1}^{S}}{\partial z}+\frac{% \partial u_{3}^{S}}{\partial x}\right) }\right.$ $\displaystyle \left(\vphantom{ \frac{\partial u_{1}^{S}}{\partial z}+\frac{% \partial u_{3}^{S}}{\partial x}}\right.$ $\displaystyle {\frac{{\partial u_{1}^{S}}}{{\partial z}}}$ + $\displaystyle {\frac{{% \partial u_{3}^{S}}}{{\partial x}}}$ $\displaystyle \left.\vphantom{ \frac{\partial u_{1}^{S}}{\partial z}+\frac{% \partial u_{3}^{S}}{\partial x}}\right)$ $\displaystyle \left.\vphantom{ \left( \frac{\partial u_{1}^{S}}{\partial z}+\frac{% \partial u_{3}^{S}}{\partial x}\right) }\right\vert_{{z=0}}^{}$ = 0, $\displaystyle \left.\vphantom{ \frac{\partial }{\partial t}\left( u_{3}^{F}-u_{3}^{S}\right) }\right.$ $\displaystyle {\frac{{\partial}}{{\partial t}}}$ $\displaystyle \left(\vphantom{ u_{3}^{F}-u_{3}^{S}}\right.$ u₃^F - u₃^S $\displaystyle \left.\vphantom{ u_{3}^{F}-u_{3}^{S}}\right)$ $\displaystyle \left.\vphantom{ \frac{\partial }{\partial t}\left( u_{3}^{F}-u_{3}^{S}\right) }\right\vert_{{z=0}}^{}$ = 0,

		$\displaystyle \left.\vphantom{ T_{33}}\right.$ T₃₃ $\displaystyle \left.\vphantom{ T_{33}}\right\vert_{{z=0}}^{}$ $\displaystyle \equiv$ $\displaystyle \left.\vphantom{ (T_{33}^{S}-p^{F})}\right.$ (T₃₃^S - p^F) $\displaystyle \left.\vphantom{ (T_{33}^{S}-p^{F})}\right\vert_{{z=0}}^{}$ = $\displaystyle \left.\vphantom{ c_{P1}^{2}\rho _{0}^{S}\left( \frac{\partial u_{... ...al u_{1}^{S}}{\partial x}-\kappa \left( \rho ^{F}-\rho _{0}^{F}\right) }\right.$ c_P1² $\displaystyle \rho_{{0}}^{{S}}$ $\displaystyle \left(\vphantom{ \frac{\partial u_{1}^{S}}{% \partial x}+\frac{\partial u_{3}^{S}}{\partial z}}\right.$ $\displaystyle {\frac{{\partial u_{1}^{S}}}{{% \partial x}}}$ + $\displaystyle {\frac{{\partial u_{3}^{S}}}{{\partial z}}}$ $\displaystyle \left.\vphantom{ \frac{\partial u_{1}^{S}}{% \partial x}+\frac{\partial u_{3}^{S}}{\partial z}}\right)$ -2c_S² $\displaystyle \rho_{{0}}^{{S}}$ $\displaystyle {\frac{{\partial u_{1}^{S}}}{{\partial x}}}$ - $\displaystyle \kappa$ $\displaystyle \left(\vphantom{ \rho ^{F}-\rho _{0}^{F}}\right.$ $\displaystyle \rho^{{F}}_{}$ - $\displaystyle \rho_{{0}}^{{F}}$ $\displaystyle \left.\vphantom{ \rho ^{F}-\rho _{0}^{F}}\right)$ $\displaystyle \left.\vphantom{ c_{P1}^{2}\rho _{0}^{S}\left( \frac{\partial u_{... ...partial x}-\kappa \left( \rho ^{F}-\rho _{0}^{F}\right) }\right\vert_{{z=0}}^{}$ = 0,

**Fig.:** Numerical results for normalized velocities and attenuations of Rayleigh and Stoneley waves, each for a small frequency range (left) and a large frequency range (right) for the permeability coefficient: $\pi$ = 10⁷ ${\frac{{\mathrm{kg}}}{{\mathrm{m}^{3}\mathrm{s}}}}$
$\ProjektEPSbildNocap{\textwidth}{fg7_IUTAM.ps}$

·The velocity of propagation of this wave lies in the interval determined by the limits $\omega$ $\rightarrow$ 0 and $\omega$ $\rightarrow$ $\infty$ . The high frequency limit is approx. 4.7 % higher than the low frequency limit. The wave is always slower than the S wave. As a function of $\omega$ it possesses an inflection point and it is slightly nonmonotonous.

·This nonmonotonicity appears in the range of small frequencies. The velocity possesses in this range a minimum whose size is very small. Interestingly, the minimum value remains constant for the different values of $\pi$ . This means that the decay is not driven by the diffusion. Such a behavior is also observed within Biot's model.

·The attenuation of this wave grows from zero for $\omega$ = 0 to infinity as $\omega$ $\rightarrow$ $\infty$ . In the range of large frequencies it is linear (a constant positive quality factor). This means that it is a leaky wave.

·The velocity of this wave grows monotonically from the zero value for $\omega$ = 0 to a finite limit which is slightly smaller than the velocity of the P2 wave. The growth of the velocity of this wave in the range of low frequencies is much steeper than the one of Rayleigh waves similarly to the growth of the P2 velocity.

·Both the velocity and attenuation of the Stoneley wave approach zero as $\sqrt{{\omega }}$ .

·The attenuation of the Stoneley wave grows monotonically to a finite limit for $\omega$ $\rightarrow$ $\infty$ (zero quality factor). It is slightly smaller than the attenuation of P2 waves. Consequently, in contrast to the claims in the literature, the Stoneley wave is attenuated.

Numerical analysis of surface waves on impermeable boundaries of a poroelastic medium