1. Acoustic Spectroscopy Simulation An 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. 1-D string Cylindrical Cavity Arbitrary Cavity Models for DARS

4. Sample Coating Resonator Circular DARS A Radially Layered Model for 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 Fluid Layer: 2x1 matrices Non-permeable Layer (solid) : 4x2 matrices Permeable Layer (porous): 6x3 matrices are unknown coefficients to be determined by boundary conditions jth layer are general solutions of wave equations

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. Pressure in Empty Cavity of the First Mode Radius (m) Vp (m/s) Density (kg/m 3 ) f (1) (Hz)f (2) (Hz)f (3) (Hz) 0.698410001000.131831.172655.43 Cavity Parameters (Zero displacement on cavity wall) Test Examples

15. First 3 Resonance Frequencies of an Empty Cavity

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

17. Sample Type Thickness of elastic coating layer (mm) Vs (m/s) Permeability (mDarcy) Porosity (%) f (1) (Hz) f (2) (Hz) f (3) (Hz) Acoustic---- 1012.381868.392726.53 Elastic-1650--1011.851866.872723.74 Poroelastic-1650370211010.621864.312719.84 Poroelastic-1650600211010.271863.592719.037 Poroelastic-16501370211009.831861.672716.17 Poroelastic-16506000211009.681859.952709.14 Poroelastic516501370211011.731866.502723.05 Poroelastic116501370211011.691866.402722.89 Poroelastic0.116501370211011.691866.382722.84 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. V p (m/s)V s (m/s)  (kg/m 3 ) f (1) (Hz)f (2) (Hz)f (3) (Hz) Berea 26561650 21011011.85 1866.87 2723.74 Boise 28371658 23091012.20 1867.88 2725.59 Chalk 30191611 17861012.15 1867.73 2725.33 Coal 2045840 1130 1010.061861.622714.06 Granite 51402720 26301012.96 1870.08 2729.61 Sandstone 20531205 19821010.95 1864.22 2718.83 Aluminum 64003100 27001013.06 1870.37 2730.13 Simulation for seven elastic samples

22.  Given (GPa) -1  Estimated (GPa) -1 Error (%) Berea0.13900.1322-4.9 Boise0.098800.09790-0.9 Chalk0.099030.10304.0 Coal0.27300.398213 Granite0.022970.023020.2 Sandstone0.2214 0 Aluminum0.01316 0 Compressibility estimated with the perturbation formula

23. Comparison with Diffusion Model for Porous Samples

24. Berea Perm (mDarcy)  (%)  m -Given (GPa) -1  e1 -Diffusion (GPa) -1  e2 –DARS (GPa) -1 Elastic--0.13900.1322-- Porous370210.13900.2505450.254153-1.4% Porous600210.13900.2853550.288843-1.2% Porous1370210.13900.3180420.332504-4.3% Porous6000210.13900.3282750.347402-5.5% Parameter estimation of Berea samples using different methods (Same porosity but different permeability) Biot model, Diffusion Model, Slow Wave

25. Estimation of Vp and Vs with DARS V p (m/s) Error (%) V s (m/s) Error (%)  (kg/m 3 ) Error (%) Berea 2568-3.31492-9.6 22406.6 Boise 2830-0.261630-1.7 23562.0 Chalk 31434.1181312 233931 Coal 230613135962 1.780 58 Granite 5123-0.332682-1.4 26550.96 Sandstone 2053012050 19820 Aluminum 6400031000 27000 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.