孔彈性理論在工程的應用中已經被廣泛地認為可以精確地分析含兩種非混合流體孔隙介質內複雜力學機制的有效解析方法。然而，孔彈性理論其偏微分方程式中所含之慣性阻力項、黏滯阻力項和應力項均為互相耦合 (coupled)。因此，除非在特定情況，並無法求出其邊界值問題的封閉 (closed-form) 之穩態解析解。本研究證明，未飽和孔隙介質的孔彈性理論之偏微分方程式可以被成功地去耦合成為三個赫姆霍茲 (Helmholtz) 方程式，在物理上意指在未飽和孔隙介質中存在有三種不同膨脹波。我們所導出之赫姆霍茲方程式其正向座標為複數值且在頻率域內可以表示成由固體體積應變量和兩個孔隙流體之線性體增量所組成的三種不同線性組合，亦可表示成由總正向應力和兩種孔隙流體壓力所組成之三種不同線性組合。由數值計算結果亦顯示，在含兩種非混合流體之孔隙介質中，由赫姆霍茲方程式所表示的三種膨脹波的波速和衰減係數與之前未去耦合的頻率方程式解出之數值結果完全相同。如此，在不同邊界條件下，未飽和孔隙介質膨脹波的傳遞和衰減行為，可以成功地由赫姆霍茲方程式精確模擬分析出。

Poroelastic theory has become an accurate approach to analyzing the complex mechanical behavior of an elastic porous medium containing two immiscible fluids in many subsurface engineering applications. However, the resulting partial differential equations in the theory take on a coupled form in the terms relevant to inertial drag, viscous damping, and applied stress. Except in special cases, it is difficult to obtain closed-form, steady-state analytical solutions for boundary-value problems.
In the present study, we demonstrate that these partial differential equations can be decoupled into three Helmholtz equations featuring complex-valued, frequency-dependent normal coordinates for dilatational wave motions, physically corresponding to three independent modes of dilatational waves. The normal coordinates in turn can be expressed as three different linear combinations of the solid dilatation and the linearized increment of fluid content for each pore fluid in the frequency domain, or as three different linear combinations of total dilatational stress and two pore fluid pressures, representations which are applicable respectively to strain-controlled and stress-prescribed boundary conditions.
Numerical simulation shows that the phase speed and attenuation coefficient of the three dilatational waves calculated from the Helmholtz equations we derived are exactly the same as those obtained previously form the dispersion relations of the coupled partial differential equations. Thus, dilatational wave motions in unsaturated porous media subject to suitable boundary conditions can be accurately modeled by our Helmholtz equations analytically.

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