兩種可混合或不可混合的流體在隨機多孔介質中的變化

Abstract

這是一個三年期的計劃。我們想要探討的是兩種不同的可混合或不可混合 的流體在隨機多孔介質中的變化問題。它主要的應用是在瞭解污染源(如地底下 的化學廢水或儲存在地底下的核能電廠的廢棄物等) 在地下的擴散情形，及地 下的碳氫化合物抽取的問題。 地底下的縫隙結構與地質性質是十分複雜的，而流體在地底下的運動的方 向、快慢、time-scale 等卻和它們息息相關。在數學的modeling 上，縫隙結構 與地質性質都是以permeability field 來表示，然而經由測量我們往往只能得到 有限個點的permeability field 的值。因此在應用上一個數學模式的正確與否完 全取決於我們如何用數學式子將有限的地質結構的資料與流體的運動的關係描 述出來。這對瞭解地下污染源的分佈情形是十分重要的。這也對應於我們是否 能瞭解流體在隨機多孔介質中的變化問題。之前的計劃是討論二相流在破裂多 孔介質中的宏觀模式，破裂多孔介質是一種特殊的有週期性的多孔介質，在這 種介質中流體會呈現出兩種不同的time scale 的現象。現在想考慮的隨機多孔介 質則包含了多個time scale 的現象。我們希望借助之前的經驗來幫助瞭解更實際 一點的物理現象。

This is a three-year project. We propose to study two-component miscible or immiscible displacement in random porous media. The benefits from this research are the understanding the distribution of underground contaminants (for example, underground waste from chemical factories, radioactive waste from nuclear plants) and the enhanced oil recovery process. It is known that the geometrical structure and geological properties of porous media are complex, and they have strong effects on fluid flow, for example, flow direction, speed, and time-scale etc.. In mathematical modeling, those properties are represented by permeability field (random functions of position). In practice, experiments provide the data of the permeability field only at a finite set of points of the porous media. So an effective mathematical model strongly depends on how to relate the finite data with fluid flow in the media. From mathematical point of view, this corresponds to whether we are able to understand fluid flow in random porous media. In previous projects, we study the two-phase flows in fractured porous media, a special type of porous media with periodic structure. Fluid flows in these media may show two different time scales. Now fluid flows in random porous media may have many time-scales due to many levels of heterogeneities. We plane to study this more realistic problems based on the previous experiences in this project.