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Acoustic Spectroscopy Simulation An Exact Solution for Poroelastic Samples

Acoustic Spectroscopy Simulation An Exact Solution for Poroelastic Samples. Youli Quan November 13, 2006. Model Theory Applications (1) Verification of perturbation theory (2) Comparison with diffusion model for porous samples (3) Estimation of Vp & Vs with DARS. Models for DARS.

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Acoustic Spectroscopy Simulation An Exact Solution for Poroelastic Samples

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  1. Acoustic Spectroscopy SimulationAn Exact Solution for Poroelastic Samples Youli Quan November 13, 2006

  2. Model • Theory • Applications • (1) Verification of perturbation theory • (2) Comparison with diffusion model for porous samples • (3) Estimation of Vp &Vs with DARS

  3. Models for DARS Arbitrary Cavity Cylindrical Cavity 1-D string

  4. A Radially Layered Model for DARS Resonator Sample Coating Circular DARS

  5. Generalized Reflection and Transmission Method for Circular DARS Governing Equations H, G, C, M, … are poroelastic parameters

  6. Formal Solution in jth layer are general solutions of wave equations jth layer Fluid Layer: 2x1 matrices Non-permeable Layer (solid) : 4x2 matrices Permeable Layer (porous): 6x3 matrices are unknown coefficients to be determined by boundary conditions

  7. Boundary Conditions Three types of materials are considered: Fluid, Solid, and Porous Nine types of boundary conditions must be handled: Fluid - Fluid Fluid - Solid Fluid – Porous Solid - Fluid Solid - Solid Solid – Porous Porous - Fluid Porous - Solid Porous - Porous

  8. An Example: Fluid – Porous

  9. Ordinary Reflection and Transmission Coefficients

  10. They can be directly calculated from

  11. Generalized Reflection and Transmission Coefficients

  12. They can be iteratively calculated from with given initial condition at last layer for 1. Pressure = 0 2. Displacement = 0

  13. Normal Modes and Resonance Frequencies The normal modes are the non-trivial solutions of the source-free wave equation under given boundary conditions. The requirement of a non-trivial solution leads to the dispersion relation: Its solution, for a model m, gives the resonance frequency.

  14. Radius (m) Vp (m/s) Density (kg/m3) f(1)(Hz) f(2)(Hz) f(3)(Hz) 0.6 984 1000 1000.13 1831.17 2655.43 Test Examples Pressure in Empty Cavity of the First Mode Cavity Parameters (Zero displacement on cavity wall)

  15. First 3 Resonance Frequencies of an Empty Cavity

  16. A closer look of the first mode Q-value of the cavity is defined by the imaginary part of the frequency.

  17. Sample Type Thickness of elastic coating layer (mm) Vs (m/s) Permeability (mDarcy) Porosity (%) Acoustic - - - - 1012.38 1868.39 2726.53 Elastic - 1650 - - 1011.85 1866.87 2723.74 Poroelastic - 1650 370 21 1010.62 1864.31 2719.84 Poroelastic - 1650 600 21 1010.27 1863.59 2719.037 Poroelastic - 1650 1370 21 1009.83 1861.67 2716.17 f(1) (Hz) Poroelastic - 1650 6000 21 1009.68 1859.95 2709.14 f(2) Poroelastic 5 1650 1370 21 1011.73 1866.50 2723.05 (Hz) Poroelastic 1 1650 1370 21 1011.69 1866.40 2722.89 f(3) (Hz) Poroelastic 0.1 1650 1370 21 1011.69 1866.38 2722.84 Simulation results for 4 types of 7 samples (Berea)

  18. Resonance frequency changes vs. permeability (4 open porous samples)

  19. Applications • Verification of perturbation theory • Comparison with diffusion model for porous samples • Estimation of Vp & Vs with DARS

  20. Estimation of Compressibility Using Perturbation Theory

  21. Vp (m/s) Vs (m/s) r (kg/m3) f(1)(Hz) f(2)(Hz) f(3)(Hz) Berea 2656 1650 2101 1011.85 1866.87 2723.74 Boise 2837 1658 2309 1012.20 1867.88 2725.59 Chalk 3019 1611 1786 1012.15 1867.73 2725.33 Coal 2045 840 1130 1010.06 1861.62 2714.06 Granite 5140 2720 2630 1012.96 1870.08 2729.61 Sandstone 2053 1205 1982 1010.95 1864.22 2718.83 Aluminum 6400 3100 2700 1013.06 1870.37 2730.13 Simulation for seven elastic samples

  22. Compressibility estimated with the perturbation formula

  23. Comparison with Diffusion Model for Porous Samples

  24. Berea Perm (mDarcy) f (%) km -Given (GPa)-1 ke1 -Diffusion (GPa)-1 Elastic - - 0.1390 0.1322 - - Porous 370 21 0.1390 0.250545 0.254153 -1.4% Porous 600 21 0.1390 0.285355 0.288843 -1.2% Porous 1370 21 0.1390 0.318042 0.332504 -4.3% ke2 –DARS (GPa)-1 Porous 6000 21 0.1390 0.328275 0.347402 -5.5% Parameter estimation of Berea samples using different methods (Same porosity but different permeability) Biot model, Diffusion Model, Slow Wave

  25. Vp (m/s) Error (%) Vs (m/s) Error (%) r (kg/m3) Error (%) Berea 2568 -3.3 1492 -9.6 2240 6.6 Boise 2830 -0.26 1630 -1.7 2356 2.0 Chalk 3143 4.1 1813 12 2339 31 Coal 2306 13 1359 62 1.780 58 Granite 5123 -0.33 2682 -1.4 2655 0.96 Sandstone 2053 0 1205 0 1982 0 Aluminum 6400 0 3100 0 2700 0 Estimation of Vp and Vs with DARS i=1,2,3

  26. Remarks • This simulation tool can also be used for other studies, e.g., the empirical equations for Q-value estimation. • Boit model and the diffusion model are consistent in our case.

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