一維地下水傳輸方程式考慮與不考慮一階衰變及在第三邊界條件下之解析解

Abstract

溶質傳輸方程式常被用於描述污染物在地下水系統中的遷移及宿命，近年來，此方程式在幾種不同邊界條件下的解析解已經被解出。當地下水的傳流及延散都重要，且應同時考慮時，混合型邊界條件（或稱為第三邊界條件、柯西邊界條件）經常被使用。當污染物有化學、生物或放射性的衰變時，最常使用的就是一階衰變反應式。然而，在第三邊界條件下，考慮一階衰變所得到的解析解，當衰變速率常數（μ）趨近於零時，似乎出現奇異點。其實，在衰變速率常數趨近零時，考慮一階衰變所得到的解可化簡成未考慮一階衰變時所得的解。本論文的重點，即在利用數學的定理來證明及詳細推導此性質；此外，在第三邊界條件下與考慮一階衰變時，利用擾動法也可求得一個近似解，推導過程及此近似解的適用性，也於論文中探討。

The groundwater transport equation, or called advective-dispersion equation, is commonly used to describe the movement and fate of solutes in the groundwater flow system. Analytical solutions for the equation under various types of boundary condition have been developed in recent decades. Mixed-type boundary condition, also known as the third-type or Cauchy boundary condition, appropriates circumstances when both the dispersive and advective transports are about equally important. A first-order reaction term is often included in the groundwater transport equation if the solute is capable of chemical, biochemical, or radioactive decaying. The solution of the equation with the decay term and under the third-type boundary condition seems to become singular when the decay rate constant (□) approaches zero. In fact, these two solutions can be shown to be continuous and asymptotically reduces to the one of without the decay term even form as . Detailed mathematical derivations with employing the l’Hospital and Leibnitz’s rules are given to prove this assertion. Besides, an asymptotic solution for the case with the decay term and under the third-type boundary condition is solved by the perturbation method. The validity of this asymptotic solution is also discussed.