以孔頸單元系集模型推估濕潤相流體未飽和水力傳導係數之研究

Abstract

以水力等價孔頸單元系集模型為基礎，本研究建立一套能考量流體接觸角(contact angle)與界面流變性質(rheology)差異對流動行為影響的濕潤相流體未飽和水力傳導係數(hydraulic conductivity)推估方法，包含三個步驟： 1. 利用兩不互混流體置換機制毛細壓力門檻值，針對接觸角 之濕潤相流體，設計一組涵蓋各種置換行為的原型孔頸單元組。 2. 數值模擬不同毛細壓力下各原型孔頸單元內的靜態流體界面，並且數值求解濕潤相流體層流流場後，套配無因次流體界面面積平方根得傳導度函數。 3. 以與水力等價孔頸單元系集模型為依據，將原型孔頸單元組合為近似模型，採用逐步增減毛細壓力並累加各孔頸單元的濕潤相流體傳導度為水力傳導係數的方式，推估濕潤相流體未飽和水力傳導係數。 孔頸單元雖提供頸管與孔體分別描述狹長孔隙與大孔隙而更貼近真實土壤孔隙，卻因幾何上存在奇異點(singular point)而難以解析幾何線型描述流體界面。藉由Surface Evolver應用軟體，研究首先以增減毛細壓力與改變濕潤相流體接觸角 的方式，數值分析孔頸單元內各種置換機制觸發之臨界狀況的靜態流體界面並檢討置換機制毛細壓力門檻值的適用性。研究發現孔頸單元的「界面活塞型運動」置換機制需細分為：(1)界面從單元外進入頸管，與(2)界面由孔體進入頸管兩類。後者由於正立方孔體於垂直空間對角線上提供連續的漸變斷面，因此毛細壓力門檻值遠低於假設無限長直管的數值；而「攫斷」置換機制依觸發方式亦分為：(1)因為頸管內流體界面相接而觸發，與(2)因孔體內流體界面界面曲率不穩定而觸發。 針對接觸角 濕潤相流體於孔頸單元內的充滿單元、無頸管滿管組態(None Full Throat)、單頸管滿管組態(1 Full Throat)以及雙頸管滿管組態(2 Full Throats)四種流動空間，研究設定流體界面為無剪應力邊界條件，利用CFD-ACE套裝應用軟體數值求解濕潤相流體流場，並且以入口端斷面與出口端斷面的流場雷諾數小於0.1為層流判斷指標，計算濕潤相流體傳導度。研究發現單頸管滿管組態下的濕潤相流體傳導度與無因次流體界面面積平方根的函數關係，依照孔頸單元頸管尺寸比值分為指數函數、冪次函數與指數型半變異數函數等三類，而無頸管滿管組態的濕潤相流體傳導度則為無因次流體界面面積平方根的指數函數。此外，排退歷程的濕潤相流體傳導度高於汲取歷程，此乃因相同毛細壓力下的孔頸單元流體組態在排退與汲取歷程不同所導致。 研究以具有和水相同黏滯性、密度、表面張力但接觸角為 的虛擬流體為濕潤相流體，藉由已知傳導度變化的原型孔頸單元組，採用「線性規劃求取原型孔頸單元發生機率」與「依孔頸單元頸管尺寸比值分類與重組」兩種方式組成水力等價孔頸單元系集模型之近似模型，並推估濕潤相流體未飽和水力傳導係數。結果顯示，兩種方法均可反映濕潤相流體未飽和水力傳導係數的磁滯現象(hysteresis phenomena)。然而，前者因線性規劃得到的頸管尺寸發生機率與等價孔頸單元系集模型差異過大，導致高估未飽和水力傳導係數；後者除了可以定性描述未飽和水力傳導係數因非濕潤相流體入陷而降低的現象外，亦具有定量分析的可能。

Based on the framework of hydraulic equivalent Unit-Pore-Throat Ensemble Model (UPTEM), this study proposed an approach to qualitatively analyze the wetting phase fluid unsaturated hydraulic conductivity for a two-phase flow system. In order to take the contact angle and the rheology of fluid interface into consideration, the method is composed of the following three issues: 1. According to the capillary pressure thresholds, design a set of proto-Pore-Throat-Units (proto-PTUs) comprised all the possible mechanisms of fluid displacement within a pore-throat unit (PTU). 2. Given several capillary pressures, numerically simulate the static equilibrium capillary surfaces within each proto-PTU and analyze the Laminar flow of the wetting phase fluid to calculate the conductance. 3. Based on the hydraulic equivalent UPTEM, ensemble the proto-PTUs to analyze the unsaturated hydraulic conductivity. Although the PTU provides the elements of cube and equilateral triangular prism to depict the characteristics of the real soil pore structure better, the static equilibrium capillary surface cannot be described with the analytical geometry shapes due to the geometrical singular points. The shape of static equilibrium capillary surfaces within a PTU can be simulated numerically with the aid of Surface Evolver. The Surface Evolver is an interactive program that minimizes the energy of a surface subject to constraints, such as prescribed fluid volume and contact angle on the walls. According to the Surface Evolver graphical output, this study verified the critical conditions and the capillary pressure thresholds of 2 immiscible fluids displacement mechanisms within a PTU. The results revealed that the threshold of piston-type motion occurred at the intersection of cubic pore and equilateral triangular prism should be modified. Moreover, there should be another displacement mechanism, which snaps off the non-wetting phase in the pore body due to the thickening of wetting phase fluid. For contact angle , the study designed 25 proto-PTUs and applied the no-stress boundary condition on the fluid interface to simulate the flow field. With the criteria of Reynolds number less than 0.1, the Laminar flow field of wetting phase fluid are analyzed and the variation of conductance are fitted. Results show that the function of conductance and dimensionless area of capillary surface is the exponential type for the none-full-throat configuration; meanwhile, it could be the exponential type, power type or exponential semi-variogram type for the one-full-throat configuration depends on the throat size ratio. Furthermore, the conductance in drainage process is larger than the one in imbibition process for a given capillary pressure due to the change of the fluid configuration. To analyze the unsaturated hydraulic conductivity of a virtual wetting phase fluid based on the hydraulic equivalent UPTEM, which is deducted from the water-air soil column infiltration experiment, the study ensemble the 25 proto-PTUs by two ways and both can qualitatively depict the hysteresis phenomena. The first way employed the linear programming scheme to calculate the probability of each proto-PTU. Because the relation between probability and throat size deviated from the one of hydraulic equivalent UPTEM significantly, it overestimated the expected total pore volume, and therefore, the unsaturated hydraulic conductivity. On the other hand, the second way categorized the hydraulic equivalent UPTEM into 25 proto-PTUs based on the aspect ratio and the throat size ratio. It successfully revealed the reduction of unsaturated hydraulic conductivity due to non-wetting phase trapped during the imbibition process and simulated the scanning loops well.