彈性波於未飽和彈性孔隙介質中傳遞及衰退的特性是目前水文地質方面重要的研究課題。在工程應用上，孔彈性力學理論可以精確地分析兩種非混合流體在孔隙介質中受彈性波之影響及變化。然而，在孔彈性力學理論中，其慣性阻力項、黏滯阻力項及應力項皆為互制(coupling)且為影響彈性波傳遞及衰退程度因子之一，因此在理論模式中無法直接求得邊界值問題之解析解。本研究利用有限差分法對孔彈性力學理論之偏微分方程式中各互制項進行離散化並數值模擬分析彈性波在未飽和孔隙介質中之傳波特性，進而觀察慣性項對孔彈性理論中之固體及兩項非混合孔隙流體的影響。
本文主要探討彈性波在拘限含水層傳遞時之動力行為的變化，經由數值模擬後可以獲得總應力及兩個孔隙非混合流體之壓力值。假設在振盪頻率為1 Hz、10 Hz、100 Hz的情況下，頻率越大，隨著距離傳遞之彈性波衰退程度也越大；比較不同起始孔隙流體飽和度的條件下(例如：非潤濕流體飽和度 =0.3、0.6、0.7、0.8、0.9)，其衰退程度隨著 飽和度增大而越大；然慣性項除了對特定範圍（震盪頻率8Hz~80Hz， =0.8~0.84）之近邊界處有明顯之影響外，對其他系統造成之效應較不顯著。

The study on the behavior of elastic wave propagation and attenuation through a fluid-containing porous medium is a central topic in hydrogeophysics. It has been demonstrated recently that the behavior can be well described by the theory of poroelasticity precisely. However, due to the fact that the model equations are decoupled in inertial and viscous coupling terms so that it is impossible to obtain exact analytical solutions for boundary problems, except in few special cases. In the current study, we utilize the finite difference method to disperse the governing partial difference equations and then numerically examine the characteristics of wave propagation and attenuation to gain physical insight on the influence of inertial effect on the pore pressure of two immiscible fluids and the stress of solid.
As an illustrative example, a semi-infinite Columbia Fine Sandy Loam containing two immiscible fluids (water-oil) subject to a stress-excitation boundary on the left side and an impermeable boundary on the up and down sides is investigated at seismic frequencies (1Hz, 10Hz, and 100Hz). Our numerical results show that the decay degree of the stress and pore fluid pressures increase with an increase in travel distance of elastic waves, excitation frequencies, and relative saturation of the non-wetting fluid. Most importantly, it is noted that as relative saturation of the non-wetting fluid takes the value from 0.8 to 0.84 and excitation frequency ranges 8Hz to 80Hhz, the total stress in the dynamic model (with inertial terms) yield a cyclic response while this attribute can not be observed in the diffusive model (inertial terms neglected).

【1】Alterman Z, and Karal F. C., Propagation of elastic waves in layered media by finite difference methods, Vol. 58, lussue: 1, pp. 367-398, 1968
【2】B. Gurevich and S.L. Lopatnikov, Velocity and Attenuation of elastic waves in finely layered porous rocks, Geophys. J. Int. 121. 993-947,1995.
【3】Biot, M. A., Theory of propagation of elastic waves in a fluid saturated porous solid, I. Low-frequency range, Journal of the Acoustic Society of America, Vol. 28, no.2,pp. 168-178, 1956a.
【4】Biot, M. A., Theory of propagation of elastic waves in a fluid saturated porous solid, II. Higher-frequency range, Journal of the Acoustic Society of America, Vol. 28, no.2,pp. 179-178191, 1956b.
【5】Berryman, J. G., Confirmation of Biot’s theory, Applied Physics Letters, Vol. 37, no. 4, pp. 382-384,1980.
【6】Berryman, J. G., Effective conductivity by fluid analogy for a porous insulator filled with a conductor, Physical Review B, Vol. 27, no. 12, pp. 7789-7792, 1983.
【7】Berryman, J. G., Thigpen, L., and Chin, R. C. Y., Bulk elastic wave propagation in partially saturated porous solids, Journal of the Acoustical Society of America, Vol.84, no. 1,pp. 360-373, 1988.
【8】Brutsaert, W., the propagation of elastic waves in unconsolidated unsaturated granular mediums, Journal of Geophysical Research, Vol. 69 no.2, pp.243-257,1964.
【9】Brutsaert, W., and J. M. Luthin, The velocity of Sound in Soils near the surface as a function of the moisture content, Journal of Geophysical Research, Vol. 69, no. 4, pp. 643-652, 1964.
【10】Chao-Lung, Yeh An Assessment of Characteristics of Elastic Wave Propagation and Attenuation through Saturated and Unsaturated Soils, Hydraulics & Ocean Engineering of NCKU, 2005

【11】David Tsiklauri and Igor Beresnev, Enhancement in the dynamic response of a viscoelastic fluid flowing through a longitudinally vibrating tube, Physical Review E, vol. 63 046304, 2001
【12】De-hua Han, A. Nur, and Dale Morgan, Effects of porosity and clay content on wave velocities in sandstones, Geophysics, Vol. 51. no. pp.2093-2107, 1986.
【13】Ferenc Szekely, Three-dimensional mesh resolution control in finite difference groundwater flow models through boxed spatial zooming, Journal of Hydrology 351, 261-267, 2008.
【14】Gray, W. G., General conservation equations for multi-phase systems: 4. Constitutive theory including phase change. Advances in water resources, Vol. 6, pp. 130-140, 1983
【15】Grag, S. K., and A. H., Compressional wave propagation in liquid and/or gas saturated elastic porous media , Journal of Applied Physics, Vol. 60, no. 9, pp.3045-3055, 1986
【16】Jhe-Wei Li, An Assessment of the Effect of Capillary Pressure Changes on Dilatational Wave Propagation and Attenuation through Unsaturated Poroelastic Media, Hydraulics & Ocean Engineering of NCKU, 2008
【17】Lo, W.-C., G. Sposito, E. Majer, Immiscible two-phase fluid flows in deformable porous media, Advances in Water Resources, Vol. 25(8-12), pp. 1105-1117, 2002.
【18】Lo, W.-C., G. Sposito, E. Majer, Wave propagation through elastic porous media containing two immiscible fluids, Water Resources Research, Vol. 41, no. 2, pp. W02025, 2005.
【19】Lo, W.-C., G. Sposito, E. Majer, Analytical decoupling of poroelasticity equations for acoustic-wave propagation and attenuation in a porous medium containing two immiscible fluids, Journal of Engineering Mathematics, Vol. 64, no. 2, pp. 219-235, 2009.
【20】O. L. Kuznetsov, E. M. Simkin, G. V. Chilingar, M. V. Gorfunkel, J. O. Robertson, JR., Seismic techniques of enhanced oil recovery: experimental and field results, Energy sources, 24:877-889, 2002
【21】O. L. Kuznetsov, E. M. Simkin, G. V. Chilingar, S. A. Katz, Improved oil recovery by application of vibro-energy to waterflooded sandstones, Journal of petroleum science and engineering 19 191-200,1998
【22】O. C. Zienkiewwicz, Dynamic behaviour of saturated porous media; the generalized biot formulation and its numerical solution, International journal for numerical and analytical methods in geomechanics. Vol. 8. 71-96, 1984
【23】R. Lenormand, C. Zarcone and A. Sarr, Mechanisms of the displacement of one fluid by another in a network of capillary ducts, J. Fluid Mech. Vol. 135, pp. 337-353, 1983
【24】Ren-Jun Xu, A numerical study on elastic wave propagation and attenuation through an unsaturated porous medium induced by a harmonic excitation, Hydraulics & Ocean Engineering of NCKU, 2009
【25】Santos. J. E., Corbero, J. M. And Douglas, J., Static and dynamic behavior of a porous solid saturated by a two-phase fluid, Journal of the acoustical society of America, Vol. 87, no. 4, pp. 1428-1438, 1990a
【26】Santos. J. E., Douglas, J., Corbero, J., and Lovera, O. M., A model for wave propagation in a porous medium saturated by a two-phase fluid, Journal of the acoustical society of America, Vol. 87, no. 4, pp. 1439-1448, 1990b
【27】Santos. J. E., Ravazzoli, C. L., Gauzellino, P. M., Carcione, J. M., and Cavallini, F., Simulation of waves in poro-viscoelastic rocks saturated by immiscible fluids. Numerical evidence of a second slow wave, Journal of computational acoustics, Vol. 12 no. 1 , pp. 1-21, 2004
【28】Tuncay, K., and Corapcioglu M. Y., Body waves in poroelastic media saturated by two immiscible fluids, Journal of geophysical research, Vol. 101, no. B11, pp. 25149-25159, 1996.
【29】Tuncay, K., and Corapcioglu M. Y., Wave propagation in poroelastic media saturated by two fluids, Journal of applied Mechanics, Vol. 64, no. 2, pp. 313-320, 1997
【30】van Genuchten, M. T., A closed-form equation for predicting the hydraulicc conductivity of unsaturated soils, Soil Science Society of America Journal, Vol. 44, no. 5, pp. 892-898, 1980
【31】Whitaker, S., Diffusion and dispersion in porous media, A1ChE Journal, Vol. 13, no. 3, pp. 420-427, 1967
【32】Whitaker, S., The governing equations for immiscible, two-phase flow, Transport in Porous Media 1, pp. 105-125. 1986b.
【33】 Wen-Feng Lee , An Assessment of Effect of Inertial and Viscous Couplings on Elastic Wave Propagation through Porous Media Containing Two Immiscible Fluids, Hydraulics & Ocean Engineering of NCKU, 2006