破裂介質中的熱傳導方程式

Abstract

這是一個二年期的計劃。我們希望探討熱傳導在破裂多孔介質中的宏觀模 式。這有助於了解污染源(如化學工廠排放的廢水，核電廠的廢棄物等) 在地底 擴散時的平流與對流現象。地底下的縫隙結構與地質性質是十分複雜的，然而 流體在地底下的運動卻與它們息息相關。從微觀的方程式去了解流體的變化是 不切實際。如何利用數學式子正確的將地質結構與流體的運動的宏觀模式描述 出來，這是了解地下污染源的分佈情形的一個重要課題。破裂多孔介質是一種 特殊的地質結構，在這種介質中流體的運動往往呈現出兩種不同的time-scale 的現象。污染源擴散時平流與對流現象則常發生在污染源的附近或較遠處。熱 傳導現象對應到流體的運動中的平流與對流的問題，此現象在多孔介質中的可 混合與不可混合流體的運動中也會發生。換言之，了解熱傳導現象在破裂多孔 介質中的的數學模式可視為要進一步了解污染源在地底擴散情形的基礎。 之前的計劃討論了很多流速方程(一種橢圓形方程式)在破裂多孔介質中的 情形。現在主要考慮的是transport equation 的情形。若能結合這兩部份的 結果，相信多孔介質中的可混合與不可混合流體在破裂多孔介質中的運動問題 一定能有更進一步的了解。

This is a two-year project. We plan to find the macroscopic equations for heat conduction in fractured porous media. The benefit from this research is the understanding waste contaminant transport in soil. The contaminant transportation is strongly related to the geology of soil and is complicated. It is not practical to study the problem in microscopic level. To derive a proper mathematical model in macroscopic level for contaminant transport problem is an important subject. Fractured porous medium is a kind of special periodic porous medium with two different structures and flows in the medium may show two different time-scale phenomenon. For multiphase flows in porous media, the differential equations to model the transport problem are usually convection and diffusion equations as the equations to describe heat conduction problem. In other words, to understand the heat conduction in fractured porous media is a basic step to understand the waste contaminant transport in soil. Different from previous study which focusing on flow equations (elliptic equations) in fractured media, we now concentrate on transport problem (parabolic equations). Combining with previous results, we should have powerful tools to tackle multiphase flow problem in porous media.