本文應用孔彈性力學理論(poroelasticity)建立應力-應變關係，結合Lo et al. (2005)所建立的模式，將可變形孔隙介質二相流模式經包含互制行為的偏微分方程式推導成頻率方程式，並解出彈性波在含有兩種流體之孔隙介質中有三種不同速度的膨脹波存在。在此模式中將考慮固體與兩個流體間相互運動所造成慣性互制阻力和黏滯互制阻力的情況下分析彈性波在孔隙介質中之波傳特性，分別討論慣性互制與黏滯互制的影響。
依模式解出來的三種膨脹波之速度大小順序分別稱為P1、P2和P3波以進一步討論，為了增加這些孔隙介質的波傳現象在計量上的了解，採用哥倫比亞未壓密細砂質土壤並依兩種不可混合流體的系統(air-water or oil-water system)模擬實際孔隙介質問題(如：地下水層和油層)。且使用低頻的兩種震盪頻率(50和200 Hz)模擬之，將延伸Lo et al. 模式對三種膨脹波的傳波特性作進一步探討。
透過孔隙空間彎曲度因子(tortuosity)改變慣性互制效應，與Lo et al. (2005)模式比較，經數值模擬，評估慣性互制在孔隙介質中不同比例的流體飽和度下之波傳與衰減的結果顯示，慣性互制不影響P1波；慣性互制影響P2、P3波之波傳波速與能量衰減甚小，雖不能從物理機制上完全排除，但是在工程上小於工程容許範圍5%，因此在低頻範圍下可沿用Lo et al. (2005)模式的理論彎曲度因子值。
在另外一方面，為了評估孔隙介質中含有兩種不可混合流體所造成的黏滯(黏性)互制行為，使用由達西定律所引伸出來的模式。此模式比達西定律多了另一控制部份，因孔隙存在兩種流體，建立出孔隙介質中流體與固體相所造成的黏性阻力和兩種非混合流體間的黏性阻力之互制關係。透過界面互制因子改變黏性互制效應，模擬在孔隙介質中不同比例的流體飽和度下之波傳與衰減的結果顯示，不影響P1波之波速，但影響其衰退係數；黏性互制對P2波在兩種流體系統中呈現不同趨勢，在空氣和水系統中，當水飽和度約高於0.8時，P2波之波傳與衰減趨勢明顯受到影響，界面互制因子值越小時，P2波之波速越大，P2波之衰退係數越小，而在油和水系統中影響趨勢較不明顯。關於P3波，在空氣和水系統中，當水佔孔隙的比例較多時，P3波之波速隨著界面互制因子值越小而越小，P3波之衰退係數隨著界面互制因子值越小而越大。我們研究結果顯示黏性互制(包含流體間黏性阻力的考慮)對三種P波的作用不能以一概括之，也不能完全忽略流體間黏性阻力的影響。另外，兩種流體系統中流體黏滯比值和密度或飽和度比例的不同，甚至流體間相對滲透性的不同，皆有可能造成黏性互制參數的改變，影響波傳趨勢。

A lot of attention has been paid for acoustic wave propagation through porous media containing muti-phase fluids on the application of poroelasticity in recent years. We considered the coupling of inertial drag and viscous drag in this discussion. To apply Lo et al. model (2005), it is explicative to the behaviors of elastic waves in an elastic porous medium permeated by two immiscible, compressible and viscous fluids. The characteristics of dilatational waves for a homogeneous porous medium analysed through linear Stress-Strain relation and a general set of coupled partial differential equations derived from continuum mechanics of mixtures to describe these phenomena. The existence of three different motive modes of dilatational wave was solved to explain by the dispersion equations in a matrix form. These wave modes were called by P1, P2 and P3 in the magnitude of wave speed order, so as to talk about them.
Nowadays, it has been known from previous issues concerning wave propagation for an two-fluid system in unconsolidated porous media that P1 mode, which results from in-phase motions of the solid framework and the two fluids, moves with a speed equal to the square root of the ratio of an effective bulk modulus to an effective density of the fluid-containing porous medium, regardless of excitation frequency. The nature of the pore-fluids saturation and excitation frequency exert different measurable influence on the attenuation of the P1 wave and the speed and attenuation of the two diffusive modes (P2 and P3). The P2 wave results from out-of-phase motions of the solid framework and the fluids. After that, the lowest velocity of three dilatational waves propagating is P3 which arise from the presence of a second fluid in the pore space. The speed and attenuation of the two diffusive modes (P2 and P3) were associated with an effective dynamic shear viscosity of the pore fluids. In three dilatational waves there are different from the others on complex physical mechanisms, which probably concealed certain of the coupling relations, such as inertial and viscous forces.
Following this research of three waves, we took material parameter for Columbia Fine Sandy Loam Saturated with either an air-water or oil-water mixtures. The behaviors of dilatation wave were examined by the relative change of fluid saturation and wave excitation frequencies (50 and 200 Hz). The numerical simulations were accessible to assess the effects of inertial coupling and viscous coupling on account of the speed and attenuation of three waves. An assessment of inertial coupling in this fixed condition controlled was closely associated with tortuosity factor as compared with Lo et al. model (2005), besides it depended on volume fraction and material density of two phases. In conclusion, inertial coupling is neglected for low range of excitation frequencies. On the other hand, the suppression of viscous coupling involve generalized relative permeabilites or mobilities due to transport equations as a result of stretching Darcy’s law. Recently transport equations developed for two-phase flow through porous media have another term that has been included to account properly for interfacial coupling between the two flowing phases. Fluids have not only different material viscosity but also different material permeability. The porous media is saturated by fluids according to a relative proportion. Therefore, viscous coupling parameters were associated with relative saturation of fluid phase and generalized relative permeabilites and material viscosities of the fluid. We use generalized relative permeabilites to correct relative permeabilites of the wetting and nonwetting fluids in the theoretical pore size distridution model of Mualem (1976) by The interfacial coupling parameter. The interfacial coupling parameter controls the amount of viscous coupling. The effect of viscous coupling was not involved in the speed of P1 wave, but took part in the attenuation of P1 wave. The result showed different effect of viscous relations was dissimilar greatly in the speed and attenuation of P2 or P3 wave. Furthermore, our numerical results demonstrated that the viscous coupling effects for immiscible phase flowing in porous media are not implicated to ignore practically.

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