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Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium

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Abstract

A linear viscoporoelastic model is developed to describe the problem of reflection and transmission of an obliquely incident plane P-wave at the interface between an elastic solid and an unsaturated poroelastic medium, in which the solid matrix is filled with two weakly coupled fluids (liquid and gas). The expressions for the amplitude reflection coefficients and the amplitude transmission coefficients are derived by using the potential method. The present derivation is subsequently applied to study the energy conversions among the incident, reflected, and transmitted wave modes. It is found that the reflection and transmission coefficients in the forms of amplitude ratios and energy ratios are functions of the incident angle, the liquid saturation, the frequency of the incident wave, and the elastic constants of the upper and lower media. Numerical results are presented graphically. The effects of the incident angle, the frequency, and the liquid saturation on the amplitude and the energy reflection and transmission coefficients are discussed. It is verified that in the transmission process, there is no energy dissipation at the interface.

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Abbreviations

α :

phases (α = S, L, G)

l, n :

direction vectors of waves

β :

various P-waves (β = 1, 2, 3)

θ 0 :

angle of the incident P-wave

F :

fluid phases (F = L, G)

H αu :

y-component of the vector potential function of the α phase

ρ α :

material density of the α phase

H e :

y-component of the vector potential function of the elastic solid

k rp :

wave number of the reflected P-wave

λ S, μ S :

Lamé constants of the solid skeleton of the porous medium

k rs :

wave number of the reflected S-wave

k tpβ :

wave number of the transmitted P β -wave

λ e, μ e :

Lamé constants of the elastic solid

k ts :

wave number of the transmitted Swave

n α :

volume fraction for the α phase

l rs :

direction vector of the reflected Swave

l ts :

direction vector of the transmitted Swave

ρ e :

material density of the elastic solid

u α :

displacement vector of the α phase

σ α :

stress tensor of the α phase

l ip :

direction vector of the incident Pwave

l rp :

direction vector of the reflected Pwave

l tpβ :

direction vector of the transmitted Pβ-wave

σ e :

stress tensor of the elastic solid

S r :

liquid saturation degree

c rs :

velocity of the reflected S-wave

c ts :

velocity of the transmitted S-wave

θ F :

material parameter representing the energy due to the variation of the F fluid volume

c ip :

velocity of the incident P-wave

c rp :

velocity of the reflected P-wave

c tpβ :

velocity of the transmitted P β -wave

η F :

viscosity of the F fluid phase

p c :

capillary pressure

A αtpβ :

amplitude of the transmitted P β -wave in the α phase

K α :

bulk modulus of the α phase

A ip :

amplitude of the incident P-wave

A rp :

amplitude of the reflected P-wave

k r F :

relative permeability of the F fluid phase

k :

intrinsic permeability of the porous medium

B αts :

amplitude of the transmitted S-waves in the α phase

α vg,m vg :

fitting parameters in the van Genuchten model

B rs :

amplitude of the reflected S-wave

B ts :

amplitude of the transmitted S-wave

ω :

angular frequency

k p :

wave number of the P-wave

p ez :

normal component of the Poynting energy vectors in the elastic solid

p uz :

normal component of the Poynting energy vectors in the unsaturated porous medium

k s :

wave number of the S-wave

c p :

phase velocity of the P-wave

F e :

time-average of the energy fluxes in the elastic solid

F u :

time-average of the energy fluxes in the unsaturated porous medium

c s :

phase velocity of the S-wave

ψ α :

scalar potential function for the α phase

R :

amplitude reflection and transmission coefficient

H α :

vector potential function for the α phase

E :

energy reflection and transmission coefficient

i:

\(\sqrt { - 1} \)

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Authors and Affiliations

  1. MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, Hangzhou, 310058, P. R. China

    Wei-yun Chen  (陈炜昀), Tang-dai Xia  (夏唐代) & Chao-jiao Zhai  (翟朝娇)

  2. Institute of Geotechnical Engineering, Zhejiang University, Hangzhou, 310058, P. R. China

    Wei-yun Chen  (陈炜昀), Tang-dai Xia  (夏唐代) & Chao-jiao Zhai  (翟朝娇)

  3. Institute of Geotechnics, TU Bergakademie Freiberg, Freiberg, 09599, Germany

    Wei Chen  (陈 伟)

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  1. Wei-yun Chen  (陈炜昀)
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Correspondence to Tang-dai Xia  (夏唐代).

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Chen, Wy., Xia, Td., Chen, W. et al. Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium. Appl. Math. Mech.-Engl. Ed. 33, 829–844 (2012). https://doi.org/10.1007/s10483-012-1589-6

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  • DOI: https://doi.org/10.1007/s10483-012-1589-6

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