可互溶流體於Hele-Shaw Cell 下界面不穩定現象─注入及拉升流場

Abstract

黏滯度指狀物是指於平行薄板或多孔性材料間，以低黏滯度流體驅動高黏度流體時，兩流體間介面形成代表流場不穩定的指狀物型態，在多種工業製程中，多相流流場界面的不穩定性，嚴重影響產品品質及生產效率，常見案例為原油開採時以水性溶劑為驅動流體注入多孔性岩層推動更為黏稠原油時，卻因指狀物型態出現，而使水性溶劑穿透原油，降低開採效率。另平行薄板間高度改變形成的徑向拉升流場，也因可運用於黏著與潤滑分析，成為另一重要研究議題，本論文中運用高精確模擬(highly accurate simulation)之數值方法，分別以一致形(monotonic)與非一致形(nonmonotonic)黏滯度剖面之可互溶流體，探討徑向注入微小間隙的兩平行板間(即Hele-Shaw Cell)與間隙隨時間增大之Hele-Shaw Cell 徑向流場之界面演變，論文內容包含兩大 部分： 第一部份於注入流場進行大量系統化的數值模擬，針對不同對流／擴散比(Peclet 值)與黏滯度剖面參數討論，首先除以過去學者慣用之指數型(下凹曲線) 一致形黏滯度剖面進行研究外，另外定義線性及反指數型(上凸曲線)一致性黏滯度剖面與非一致形黏滯度剖面進行比較，結果顯示黏滯度對比固定時，各種黏滯度剖面對注入流場之穩定性無顯著影響，但如非一致性黏滯度剖面與上凸黏滯度剖面交錯，將激化流體介面間的不穩定性。另經由系統化改變非一致形黏滯度分布各參數值，觀察其對界面指狀化圖形之影響，除觀察到多種有趣之介面型態，諸如產生於一致形黏滯度分布所無法觀察到的成對雙渦旋流場及逆指狀物結構等現象，最後對非一致性黏滯度剖面各參數對整體注入流場穩定性影響進行討論。 第二部分首先探討不同拉升函數與初始擾動對流場穩定性之影響，相對於定量注入可互溶流體於徑向Hele-Shaw Cell 流場會產生指狀物尾端開叉與分枝等多變流場現象，平行板間隙隨時間成指數化關係變化的拉升流場形成更錯綜複雜的流場現象，近期研究指出經由調整平板間隙與時間函數關係可控制流場指狀物的形態，本研究除得到指數拉升方式可較線性拉升方式獲得更不穩定之流場，亦歸納出高Péclet值與高黏滯度對比會增加指狀物長度，另對初始條件與擾動設定對流場穩定度影響於指數函數拉升時影響較大，線性函數拉升則幾乎沒有影響。進而討論不同黏滯度剖面之可互溶流體於指數拉升流場之界面 型態，影響情況較注入流場顯著。接著採與第一部份相同作法，探討不同黏滯度剖面之可互溶流體於拉升流場之影響，發現黏滯度剖面對流場穩定性影響程度於拉升流場較注入流場大，且不穩定性均遵循下凹曲線>直線>上凸曲線順序，最後對非一致性黏滯度剖面各參數對拉升流場穩定性影響進行討論。

Viscous fingering is an interfacial fluid flow instability that occurs when less viscous fluid displaces another more viscous one in a Hele-Shaw cell or porous media, leading to the formation of finger-like pattern at the interface of both fluids. The interfacial evolution of multiphase flows will severely impact on the quality of production and efficiency in a variety of practical application of industrial process. Most frequent example of this instability is that of oil recovery for which viscous fingering takes place when an aqueous solution displaces more viscous oil in underground reservoirs, leading to the formation of nontrivial fingerlike structure and reduce the efficiency of the displacement process. Another particularly interesting variation of the classic radial flow is the investigation of fingering instabilities in Hele-Shaw cells presenting variable gap spacing. This is also a very important issue in many industrial areas including adhesion, lubrication, and colloidal hydrodynamics. In this dissertation, we carried out the highly accurate simulation to investigate the interfacial evolution in two scenarios－radial injection-driven miscible flow and lifting radial Hele-Shaw flow, both with the monotonic and nonmonotonic viscosity profile. So, the thesis consists of two parts: Part 1 focus on radial injection-driven miscible flow in a Hele-Shaw cell and covers three major topics. To begin with, we perform numerical experiments in a wide range to study the dispersion relation on both the Péclet number and the parameters of the viscosity profile. A monotonic viscosity-concentration relation of exponential type (concave) by other scholars is assumed, and a linear and reverse (convex) monotonic viscosity profiles and nonmonotonic one are also discussed. Results of this study show that as the overall viscosity contrast held constant, nonmonotonic viscosity profile lead to a more stable flow than that of monotonic one, and there are no significant differences in different viscosity profiles. However, if the nonmonotonic viscosity profile crosses the convex monotonic viscosity profile, the nonmonotonic feature enhances the prominence of interfacial instability. Then, a great variety of morphological behaviors is systematically introduced. In general, the nonmonotonic feature enhances the prominence of interfacial instability. Formation of dual vortex pairs and “reverse fingering”, where the fingers spread farther in the backward than in the forward direction are observed, which are not present in monotonic viscosity profile. Finally, we have carried out a parameter study to understand the effects of nonmonotonicity on the stability of the injection flow. In part 2, discussions start with the investigation of the influence of lifting scenario and the perturbation set. Contrast to the injection-driven miscible flow in radial Hele-Shaw cells which leads to the formation of morphing flow phenomenon of finger tip-splitting and side-branch events are plentiful if the injection rate is constant with time. More complicated flow are present for time-dependent gap flow which results in different kinds of patterns, and leads to intricate morphologies if the cell’s gap width grows exponentially with time. Recent studies show that the growing of intricate patterns due to lifting can be ntrolled by properly adjusting the time-dependent gap width. Moreover, we found the exponential lifting case will cause the flow more unstable than the variant lifting situation. We also deduce higher Péclet number and viscous contrast (A in monotonic viscosity profile and μm in nonmonotonic one) demonstrate more vigorous fingering. The sensitivity of the system to changes in the initial conditions and perturbation set is also discussed. Next, the effects of four viscosity profiles as stated in part 1 have been investigated. Unlike injection flow,the stability of three monotonic viscosity profiles are always in the series of concave, linear and convex. However, as injection flow, if the nonmonotonic viscosity profile crosses the convex curve will enhances the prominence of interfacial instability. Finally, we have carried out a parameter study to understand the effects of nonmonotonicity viscosity profile on the stability of the lifting flow.