Wave velocity dispersion and attenuation in media exhibiting

Wave velocity dispersion and attenuation in media exhibiting internal oscillations Marcel Frehner University of Vienna, Austria Holger Steeb Ruhr-University Bochum, Germany Stefan M. Schmalholz ETH Zurich, Switzerland, now at University of Lausanne, Switzerland

[0] Introduction Medium-internal oscillations Frequency [Hz] • Resonance of heterogeneities (resonant scattering) Nature (1996) • Oscillations on the pore-level Chouet, Mt. Redoubt, Alaska Continuous narrow-band • Oscillations in cracks seismic signal and volcanic tremor (Stoneley guided waves) Holzner et al. , 2009 • Oscillations in volcanic conduits Frehner & Schmalholz, 2010 Frehner & Schmalholz, Time [s] 2008 EGU – May 4 th 2010 marcel. frehner@univie. ac. at 2

[0] Introduction Overview GENERAL MODEL [1] [2] [3] [4] Model description 1 Bubble size distribution Comparison with laboratory data EXTENSION OF BIOT’S POROELASTIC THEORY [5] Model description [6] Dispersion and Attenuation EGU – May 4 th 2010 marcel. frehner@univie. ac. at 3

[1] Model description Interaction between oscillations and waves • A basic model – Acoustic medium (only one type of wave) – 1 D (no geometrical spreading, no redirection) Straight-forward analogy: Water with oscillating gas bubbles EGU – May 4 th 2010 marcel. frehner@univie. ac. at 4

[1] Model description Mathematical description • Assumption: Gas bubbles oscillate with a given resonance frequency • m Different gas bubble sizes can occur • Oscillations are coupled with acoustic medium through momentum interaction terms for j=1. . m: EGU – May 4 th 2010 marcel. frehner@univie. ac. at 5

[1] Model description Solution strategy • Analyze monochromatic wave • Leads to the generalized eigenvalue problem • Solution is the dispersion relation, from which the phase velocity c and attenuation factor a can be calculated: EGU – May 4 th 2010 marcel. frehner@univie. ac. at 6

[2] 1 Bubble size: Same bubble size, different gas volume fractions Dispersion EGU – May 4 th 2010 Attenuation marcel. frehner@univie. ac. at 7

[3] Bubble size distribution Dispersion Attenuation EGU – May 4 th 2010 marcel. frehner@univie. ac. at 8

[4] Comparison with laboratory data Commander & Prosperetti, J. Ac. Soc. , 1989 r = 0. 994 mm, fg = 0. 0377% Dispersion EGU – May 4 th 2010 Attenuation marcel. frehner@univie. ac. at 9

[5] Model description Poro-elastic theories Oil Water Full saturation • Continuous wetting phase • No non-wetting phase Biot theory (2 -phase model, Biot, 1962) Partial saturation • Continuous non-wetting phase • Continuous wetting phase 3 -phase model (e. g. , Steeb et al. , 2008) Residual saturation • Continuous non-wetting phase • Discontinuous wetting phase Effective 2 -phase model EGU – May 4 th 2010 marcel. frehner@univie. ac. at 10

[5] Model description Rheology of wetting phase • No propagating wave in discontinuous wetting phase • But relative displacement (elastic) between (viscous) wetting phase and solid skeleton damped oscillator rheology of wetting phase Hilpert et al. , 2000; Hilpert, 2007 Beresnev, 2006; Frehner et al. , 2009 EGU – May 4 th 2010 marcel. frehner@univie. ac. at 11

[5] Model description Residual saturation model • 2 -phase model (solid, non-wetting phase): Biot type • Partial balances of momentum for solid & non-wetting phase • Oscillator equation for discontinuous wetting phase Extended 2 -phase (Biot-type) model Oscillation and momentum interaction terms EGU – May 4 th 2010 marcel. frehner@univie. ac. at 12

[1] Model description Solution strategy • Analyze monochromatic wave • Leads to the generalized eigenvalue problem • Solution is the dispersion relation, from which the phase velocity c and attenuation factor a can be calculated: • 2 propagating compressional waves, 1 shear wave EGU – May 4 th 2010 marcel. frehner@univie. ac. at 13

[6] Dispersion and Attenuation Eigenvalue analysis for 2 oscillations Attenuation Phase velocity Fast P-wave EGU – May 4 th 2010 marcel. frehner@univie. ac. at 14

[6] Dispersion and Attenuation Eigenvalue analysis for 2 oscillations Slow P-wave & S-wave Phase velocity Attenuation Phase velocity S-wave Slow P-wave EGU – May 4 th 2010 marcel. frehner@univie. ac. at 15

Summary / Conclusions General model for interaction between wave propagation and oscillation Strong dispersion around resonance frequency Attenuation leads to frequency gap Distribution-function of resonance frequencies can be implemented Extended 2 -phase (Biot-type) model Valid for residual saturation in porous media Continous non-wetting phase and solid are treated with Biot’s equations Discontinuous wetting phase oscillates and is coupled to Biot’s equations through momentum interaction terms Strong dispersion and attenuation around resonance frequency EGU – May 4 th 2010 marcel. frehner@univie. ac. at 16

Thank you Frehner M. , Steeb H. & Schmalholz S. M. , 2010: Wave velocity dispersion and attenuation in media exhibiting internal oscillations, in Wave Propagation in Materials for Modern Applications, In-Tech Education and Publishing, ISBN 978 -953 -7619 -65 -7 Steeb H. , Frehner M. & Schmalholz S. M. , 2010: Waves in residual-saturated porous media, in Mechanics of Generalized Continua: One Hundred Years after the Cosserats, Springer Verlag, ISBN 978 -1 -4419 -5694 -1 EGU – May 4 th 2010 marcel. frehner@univie. ac. at 17