指形流為一種時常發生於土壤或岩層中的流動型態，兩種不同黏度的流體界面上會產生不穩定的現象，因其形狀與手指相似，故稱之為指形流。本文使用異質多孔介質為實驗模型模擬岩層中低黏度流體驅替高黏度流體的現象。實驗使用水平矩形Hele-Shaw cell系統為載台，並利用微機電製程的光刻技術於兩平板間設置兩種不同粒徑大小之顆粒。實驗觀察低黏度流體(空氣)驅替孔隙中高黏度流體(甘油水溶液)的過程中，探討其壓力的變化，並進行定量的分析。實驗設計四種不同粒徑之光刻顆粒、四種不同濃度的甘油水溶液，分析不同的空氣注入流率對於不同濃度的甘油水溶液在四種多孔介質中之結果。
本文提出低黏度流體(空氣)於多孔介質中壓力梯度的半經驗擬合式，此公式搭配先前學者提出高黏度流體(甘油水溶液)壓力梯度的公式能更清楚描述多孔介質內黏性指形發生後的動態壓力變化。實驗也發現異質孔徑界面上之壓差會隨著黏度與速度的提高而降低，此界面壓差亦能以半經驗公式來描述。

Finger flow often occurs in the soil or rock structure. The fluid interface of two fluid with different viscosities will evolve into an unstable phenomenon. Because the interfacial shape is similar to fingers, so it is called a finger flow. In this paper, heterogeneous porous media is used as an experimental model to investigate the phenomenon for a low-viscosity fluid displaces a high-viscosity fluids in rock formations. The experiment apparatus is a horizontal rectangular Hele-Shaw cell system, and lithography technology of MEMS was used to set two porous media of different particle sizes. In the process of the low-viscosity fluid (air) displacing the high-viscosity fluid (glycerin aqueous solution) through the pores, the pressure changes were investigated and quantitatively analyzed. The experiments were conducted with four different sizes of particles and four different concentrations of glycerin aqueous solution to analyze the results of different air injection flow rates.
The paper presents a semi-empirical fit of the pressure gradient of a low viscosity fluid (air) in a porous medium. This formula combine with previous studies in literature can be used to more clearly describe the dynamic pressure change after the occurrence of viscous finger in porous media. The experiment also found that the pressure difference at the interface of the heterogeneous pore size decreases with the increase of viscosity and velocity. The pressure difference at this interface can also be described by a semi-empirical formula.
Keywords: porous medium, heterogeneous pore-scale boundary, fingering flow.

Al-Housseiny, T. T., Tsai, P. A., & Stone, H. A. (2012). Control of interfacial instabilities using flow geometry. Nature Physics, 8(10), p. 747.
Burdine N.T., Magnolia Petroleum Co., Dallas. (1953). Relative Permeability Calculations
From Pore Size Distribution Data. Petroleum Transactions, AIME, T.P. 3519.,Vol. 198
Dawe, R. A., Caruana, A., & Grattoni, C. A. (2011). Microscale visual study of end effects at permeability discontinuities. Transport in porous media, 86(2), pp. 601-616.
Dawson, H. E., & Roberts, P. V. (1997). Influence of viscous, gravitational, and capillary forces on DNAPL saturation. Groundwater, 35.2, pp. 261-269.
Ferrari, A., & Lunati, I. (2013). Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy. Advances in water resources, 57, pp. 19-31.
Ferrari, A., Jimenez‐Martinez, J., Borgne, T. L., Méheust, Y., & Lunati, I. (2014). Challenges in modeling unstable two‐phase flow experiments in porous micromodels. Water Resources Research, 51(3), pp. 1381-1400.
Hill, S. (1952). Channeling in packed columns. Chemical Engineering Science, 1(6), pp. 247-253.
Harris Sajjad Rabbani, Dani Or, Ying Liu, Ching-Yao Laic, Nancy B. Lu, Sujit S. Datta, Howard A. Stone,and Nima Shokri. (2018). Suppressing viscous fingering in structured porous media. PNAS Latest Articles.
Islam, A., Chevalier, S., Salem I. B., Bernabe Y., Juanes R., Sassi, M. (2014). "Characterization of the crossover from capillary invasion to viscous fingering to fracturing during drainage in a vertical 2D porous medium." International Journal of Multiphase Flow, Vol. 58, pp. 279-291.
Jankov, M., Løvoll, G., Knudsen, H. A., Måløy, K. J., Planet, R., Toussaint, R., & Flekkøy, E. G. (2010). Effects of pressure oscillations on drainage in an elastic porous medium. Transport in porous media, 84(3), pp. 569-585.
Lenormand & Zarcone, C. (1984). Role of roughness and edges during imbition in square capillaries. Society of Petroleum Engineers, 13264, pp.16-19
Lenormand, R., Touboul, E., & Zarcone, C. (1988). Numerical models and experiments on immiscible displacements in porous media. Journal of fluid mechanics, 189, pp. 165-187.
Lake, L. W. (1989). Enhanced Oil Recovery. Englewood Cliffs, N.J: Prentice Hall.
Lin, K. Y., & Liu, S. J. (2009). The influence of processing parameters on fingering formation in fluid‐assisted injection‐molded disks. Polymer Engineering & Science, 49(11), pp. 2257-2263.
Løvoll, G., Jankov, M., Måløy, K. J., Toussaint, R., Schmittbuhl, J., Schäfer, G., & Méheust, Y. (2010). Influence of viscous fingering on dynamic saturation–pressure curves in porous media. Transport in porous media, 86(1), pp. 305-324.
Løvoll, G., Méheust, Y., Toussaint, R., Schmittbuhl, J., & Måløy, K. J. (2004). Growth activity during fingering in a porous Hele-Shaw cell. Physical Review E, 70(2), p. 026301.
Miner, C.S., & Dalton, N.N. (1953). Physical Properties of Glycerine and Its Solution. American Chemical Society Monograph, 117
Mukherjee, P. P., Kang, Q., & Wang, C. Y. (2011). Pore-scale modeling of two-phase transport in polymer electrolyte fuel cells—progress and perspective. Energy & Environmental Science, 4(2), pp. 346-369.
Nordbotten, J. M., Celia, M. A. and Bachu, S. (2005). "Injection and storage of CO2 in deep saline aquifers: Analytical solution for CO2 plume evolution during injection," Transport in Porous media, Vol. 58, pp. 339- 360.
Paterson, L. (1981). Radial fingering in a Hele Shaw cell. Journal of Fluid Mechanics, 113, pp. 513-529.
Pennell, K. D., Jin, M., Abriola, L. M., & Pope, G. A. (1994). Surfactant enhanced remediation of soil columns contaminated by residual tetrachloroethylene. Journal of Contaminant Hydrology, 16(1), pp. 35-53.
Pihler-Puzović, D., Illien, P., Heil, M., & Juel, A. (2012). Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Physical review letters, 108(7), p. 074502.
Rabbani Harris Sajjad., Or Dani., Liu Ying., Lai Ching-Yao., Lu Nancy B., Datta Sujit S., Stone Howard A., & Shokri Nima., (2018). Suppressing viscous fingering in structured porous media. PNAS May 8, 2018, 115 (19), 4833-4838
Schlu ̈ter S., Berg S., Ru ̈cker M.,Armstrong R.T., Vogel H.-J., Hilfer R., & Wildenschild D. (2015). Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media. Water Resour. Res., 52, pp. 2194–2205
Saffman, P. G., & Taylor, G. (1958). The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences(Vol. 245,No. 1242), pp. 312-329.

Takeshi Tsuji., Fei Jiang., Kenneth T. Christensen. (2016). Characterization of immiscible fluid displacement processes with various capillary numbers and viscosity ratios in 3D natural sandstone. Advances in Water Resources 95, pp. 3 – 15
Toussaint, R., Løvoll, G., Méheust, Y., Måløy, K. J., & Schmittbuhl, J. (2005). Influence of pore-scale disorder on viscous fingering during drainage. EPL (Europhysics Letters), 71(4), p. 583.
Toussaint, R., Måløy, K. J., Méheust, Y., Løvoll, G., Jankov, M., Schäfer, G., & Schmittbuhl, J. (2012). Two-phase flow: Structure, upscaling, and consequences for macroscopic transport properties. Vadose Zone Journal, 3.
Weitz, D. A., Stokes, J. P., Ball, R. C., Kushnick, A. P., (1987). Dynamic capillary pressure in porous media origin of the viscous-fingering length scale. Physical Review Letters, 59, 2967-2970.
林再興. (2004年4月). 未來的石油資源-油砂. 科學發展(436), 66-69.
林鴻諭. (2016). 利用異質孔徑界面增強多孔介質內流體驅替效果之研究. 國立中央大學碩士論文.
邱致衡. (2018). 異質多孔介質指形流之實驗. 國立中央大學碩士論文.