裂隙含水層中進行水力試驗時，地下水流場之水流維度(flow dimension)可能不是一維(單向)、二維(徑向)或三維(球狀)情況，而是分數維度，須由碎形幾何的概念來討論。新竹尖石裂隙含水層場址進行雙封塞微水試驗(double packer slug test，DPT)，獲得具定常振盪週期和固定振幅比的振盪反應，與高滲透顆粒性含水層微水試驗的振盪反應有相同特徵，屬二維徑向流情況。又因DPT的初始壓差較小(<0.5 m)，不影響低滲透性岩體母質的水壓，故DPT資料分析簡化為單孔隙率(僅考慮裂隙中水壓變化)、二維徑向流、類穩態(quasi-stead-state)模式，成功地分析DPT資料。一般的DPT 並無觀測井水位記錄，本研究則在執行DPT時，於測試井鄰近的觀測井量測水位變化。發現觀測井水位亦呈現振盪反應，但振幅較小，週期較長。因地下水流入(出)測試井時，地下水流出(入)觀測井，故測試井與觀測井的水位振盪存有相位差。目前尚無合適模式可分析觀測井的水位振盪。未來將發展觀測井振盪水位資料之分析模式，以便與測試井資料分析對比，提升裂隙含水層水文地質的瞭解。

The dimension of the groundwater flow field associated with a hydraulic test in a fractured formation may not be one-dimensional (uniform)、two-dimensional (radial) or three-dimensional (spherical). Instead, the flow dimension may be fractional, representative of certain fractal geometry. The oscillatory response of double-packer slug tests in the fractured formation in Jianshi county have constant periods and constant amplitude ratio, satisfying the characteristics of a two-dimensional radial flow condition. Because the initial pressure water pressure in the fracture can be likely neglected without introducing significant error. As a result, data analysis of DPT can be simplified using a single porosity (fracture only)、two-dimensional radial flow and quasi-stead-state model. Successful results are obtained using this modeling approach. Moreover, water level measured at an observation well close to the tested well of the DPT exhibits oscillatory response with smaller amplitudes and shorter periods. There is a phase lag between the oscillatory response from the tested well and the observation well, indicating that when groundwater flowed into (out) the observation well, the groundwater flowed out (in) the test well. So far, there has no suitable model for analyzing the oscillation of water level in observation well, and thus the future study should focus on the development of a DPT model that can be used to analyze the oscillatory response from the observation well in a fractured formation.

[1] Muskat M., The flow of homogeneous fluids through porous media, New York：McGraw-Hill, 1937.
[2] Barenblatt, G. I., Yu. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rock ”, Prikladna matematika i mekhanika, 24(5), 852-864, 1960.
[3] Warren, J. E., and P. J. Root, “The behavior of naturally fractured reservoirs ”, Society of Petroleum Engineers journal, 3, 245-255, 1963.
[4] Kazemi, H., “Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution ”, Society of Petroleum Engineers journal, 9, 451-462, 1969.
[5] Dontsov, K. M., and V. T. Boyarchuck, “Effect of characteristics of a naturally fractured medium upon the shape of buildup curves ”, Izvestiya Vysshikh Uchebnykh Zavedenii, Neft Gaz, 1, 42-46, 1971.
[6] de Swaan O. A., “Analytic solutions for determining naturally fractured reservoir properties by well testing ”, Society of Petroleum Engineers journal, 16, 117-122, 1976.
[7] Boulton, N. S., and T. D. Streltsova, “Unsteady flow to a pumpws well in a fissured water-bearing formation ”, Journal of Hydrology, 35, 257-269, 1977.
[8] Boulton, N. S., and T. D. Streltsova, “Unsteady flow to a pumped well in a fissured aquifer with a free surface level maintained constant ”, Water Resources Research, 14, 527-532,1978.
[9] Najurieta, H. L., “A theory for pressure transient analysis in naturally fractured reservoirs”, Journal of Petroleum Technology, 32, 1241-1250, 1980.
[10] Serra, K., A. C. Reynolds, and R. Raghavan, “New pressure transient analysis methods for naturally fractured reservoirs ”, Journal of Petroleum Technology, 35, 1902-1914, 1983.
[11] Streltsova, T. D., “Well pressure behavior of a naturally fractured reservoir ”, Society of Petroleum Engineers journal, 23, 769-780, 1983.
[12] Gringarten, A. C., “Interpretation of tests in fissured and multilayered reservoirs with double-porosity behavior：Theory and practice ”, Journal of Petroleum Technology, 36, 549-564, 1984.
[13] Moench, A. F., “Double-porosity models for a fissured groundwater reservoir with fracture skin ”, Water Resources Research, 20, 831-846,1984.
[14] Streltsova, T. D., “Well testing in heterogeneous formations ”, John Wiley and Sons, New York, 1988.
[15] Barker, J.A. “A generalized radial flow model for hydraulic tests in fractured rock ”, Water Resources Research, 24(10), 1796-1804, 1988.
[16] Oliver Audouin and Jacques Bodin, “Cross-borehole slug test analysis in a fractured limestone aquifer ”, Journal of Hydrology, 384, 510-523, 2008.
[17] Butler Jr., J. J., and X. Zhan., “Hydraulic tests in highly permeable aquifers ”, Water Resources Research, 40, W12402. doi：10.1029/2003/WR002998, 2004.
[18] Zhan, X., and J. J. Butler Jr., “Mathematical derivations of semianalytical solutions for hydraulic tests in highly permeable aquifers”, Kansas geological survey open-file report 2003-60, Lawrence, 2003
[19] Hvorslev M. J., “Time lag and soil permeability in ground-water observations. US Army Corps of Engineers”, Waterways Experiment Station Bulletin No. 36, Mississippi, USA, 1951.
[20] Mathias, S. A., and A. P. Butler, “Shape factors for constant-head double-packer permeameters ”, Water Resources Research, 43, W06430. doi：0043-1397/07/2006WR005279, 2007.
[21] Moon, P., and D. E. Spencer, “Field Theory for Engineers ”, Van Nostrand Reinhold, New York, 1961.
[22] Ratnam, S., K. Soga, and R. W. Whittle, “Revisiting the Hvorslev’s intake factors using the finite element method”, Géotechnique ,51(7), 641-645, 2001.
[23] Mathias, S. A., and A. P. Butler, “An improvement on Hvorslev’s shape factors ”, Geotechnique, 56(10), 705-706, 2006.
[24] Widdowson, M. A., F. J. Molz, and J. G. Melville, “An analysis technique for multilevel and partially penetrating slug test data ”, Ground Water, 28(6), 937-945, 1990.
[25] Chen C. S., “An analytical data analysis method for oscillatory slug test ”, Ground Water , 44, 604-608, 2006.
[26] Van der Kamp, G., “Determining aquifer transmissivity by means of well response test: The underdamped case ”, Water Resources Research , 12(1), 71-77, 1976.
[27] Chen, C. S., “An analytical method of analyzing the oscillatory pressure head measured at any depth in a well casing”. Hydrological Processes, 22, 1119-1124, 2008.
[28]
[29] Zurbuchen, B. R., V. A. Zlotnik, and J. J. Butler Jr., “Dynamic interpretation of slug tests in highly permeable aquifers”. Water Resources Research, 38(3), 1025, 10.1029/2001/WR000354, 2002.
[30] Ostendorf, D. W., D. J. DeGroot, P. J. Dunaj, and J. Jakubowski, “A closed form slug test theory for high permeability aquifers”, Ground Water, 43(1), 87-101, 2005.
[31] 萬力，胡伏生，田開銘，「三段水壓試驗」，地球科學-中國地質大學學報，第20卷第4期，389~392頁，1995年7月。