為了描述溶質在地表底下的傳輸行為，數學上常用移流-延散方程式描述控制移流和水力延散傳輸的物理機制。在延散傳輸的理論中，延散度是量測溶質分散的重要的參數。傳統數學模式預測溶質傳輸，多採用常數延散度。然而，現地研究指出延散度並非常數，而是會隨溶質傳輸的距離增加而改變，且在長距離時漸近常數。本研究中，考慮溶質傳輸問題發生於有限長度的孔隙介質中，並考慮延散度隨距離增加而趨近一常數。本研究以Laplace轉換級數方法解析漸近形式的尺度延散度之移流-延散方程式。發展之解析解與數值解進行濃度穿透曲線比較以檢驗其正確性，比較的結果顯示，在不同觀測位置的濃度穿透曲線，解析解與數值解十分吻合。然而，解析解於極限延散度小而特徵半展距大時，在小時間無法進行數值計算。此外，藉由分析解析解函數的數學行為，可了解無法進行數值計算的困難之處。

To describe solute transport in a subsurface porous medium, the advection-dispersion equation is widely used to mathematically describe the physical processes governing advective and hydrodynamic dispersive transport. In the theory of dispersive transport, dispersivity is an important parameter for the measurement of the spreading of solute. Classical mathematical models for predicting solute transport are based on advection-dispersion equation with space-invariant dispersivity. However, field study indicated that dispersivity is not constant but generally increases with solute transport distance, and becomes asymptotically constant at large distance. In this study, a solute transport problem in a finite porous medium where the dispersion process depends on distance and increases up to some constant limiting value is considered. The Laplace transform power series technique is applied to analytically solve the advection-dispersion equation with asymptotic distance-dependent dispersivity. The developed analytical solution is compared to the numerical solution to examine its accuracy. Results shows that the breakthrough curve at different observation points from the power series solution have good agreements with those from the numerical solution. However, the solution can not been numerically computed at the early time when the asymptotic dispersivity is small and the characteristic length is large. In addition, the mathematical behaviors of the developed solution functions are analyzed to address the difficulty in numerical computation.

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