定水頭部分貫穿井之水井水力學模式為混合邊界值問題，因為在井的邊界上同時存在兩種邊界：在井篩上為定水頭邊界，井篩之外為不透水邊界。一般而言，混合邊界值問題無法直接由積分轉換求解。然而，本研究使用拉普拉斯轉換及有限傅利葉餘弦轉換求得定水頭部分貫穿井於受壓含水層之半解析解。在此方法中，將井篩不均勻地分割成許多小段，每一小段上皆維持相同的定水頭，但有不同的流通量。然後，將井篩上的定水頭邊界置換成對應的流通量邊界，即將混合邊界轉換成均勻邊界，利用積分轉換便可求解。最後，由井篩上的定水頭可求得井篩上的流通量變化，獲得混合邊界值問題的半解析解。由此半解析解可獲得當含水層厚度為無限大時的暫態解及穩態解。當井篩底部（或頂部）至含水層底部（或頂部）的距離大於100倍的井篩長度時，含水層厚度可視為無限大，地下水流可到達穩定狀態，抽水井流量是從無限遠的含水層底部所供應的。若含水層厚度為有限時，地下水流為暫態，抽水井的流量隨時間增加而衰減。部份貫穿效應在定水頭井周圍最大，隨著水平距離增加而減小，在1至2倍含水層厚度除以含水層異向性之平方根的距離上消失，視井篩位置而定。水平方向之水力導數及儲水係數可由全程貫穿井的比洩降獲得。當井篩由含水層頂部貫穿時，垂直方向之水力導數可由抽水井的比洩降求得。

An analytical approach using integral transform techniques is developed to deal with a well hydraulics model involving a mixed boundary of a flowing partially penetrating well, where constant drawdown is stipulated along the well screen and no-flux condition along the remaining unscreened part. The aquifer is confined and of finite thickness. First, the mixed boundary is changed into a homogeneous Neumann boundary by discretizing the well screen into a finite number of segments, each of which at constant drawdown is subject to unknown a priori well bore flux. Then, the Laplace and the finite Fourier transforms are used to solve this modified model. Finally, the prescribed constant drawdown condition is reinstated to uniquely determine the well bore flux function, and to restore the relation between the solution and the original model. The transient and the steady-state solutions for infinite aquifer thickness can be derived from the semi-analytical solution, complementing the currently available dual integral solution. If the distance from the edge of the well screen to the bottom/top of the aquifer is 100 times greater than the well screen length, aquifer thickness can be assumed infinite for times of practical significance, and groundwater flow can reach a steady-state condition, where the well will continuously supply water under a constant discharge. However, if aquifer thickness is smaller, the well discharge decreases with time. The partial penetration effect is most pronounced in the vicinity of the flowing well, decreases with increasing horizontal distance, and vanishes at distances larger than 1 to 2 times the aquifer thickness divided by the square root of aquifer anisotropy. The horizontal hydraulic conductivity and the specific storage coefficient can be determined from vertically averaged drawdown as measured by fully penetrating observation wells. The vertical hydraulic conductivity can be determined from the well discharge under two particular partial penetration conditions.

Abdul, A. S., A new pumping strategy for petroleum product recovery from contaminated hydrogeologic systems: Laboratory and field evaluations, Ground Water Monit. R., 12, 105-114, 1992.
Abramowitz, M., and I.A. Stegun, Handbook of Mathematical Functions, 1046 pp., Dover, Mineola, New York, 1970.
Agarwal, R. G., R. Al-Hussainy, and H. J. Ramey, Jr., An investigation of wellbore storage and skin effect in unsteady liquid flow: I. Analytical treatment, Soc. Pet. Eng. J., 279-290, 1970.
Akindunni, F. F., and R. W. Gillham, Unsaturated and saturated flow in response to pumping of an unconfined aquifer: Numerical investigation of delayed drainage, Ground Water, 30(6), 873-884, 1992.
Boulton, N. S., Unsteady radial flow to a pumped well allowing for delayed yield from storage, in Gen. Assem. Rome, Tome II, Int. Assoc. Sci. Hydrol. Publ., 37, 472-477, 1955.
Boulton, N. S., Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage, Proc. Inst. Civ. Eng., 26, 496-482, 1963.
Bowles, M. W., L. R. Bentley, B. Hoyne, and D. A. Thomas, In situ ground water remediation using the trench and gate system, Ground Water, 38(2), 172-181, 2000.
Butler, J. J., Jr., Pumping tests in nonuniform aquifers－the radially symmetric case, J. Hydrol., 101, 15-30, 1988.
Cassiani, G., Z. J., Kabala, and M. A. Medina Jr., Flowing partially penetrating well: solution to a mixed-type boundary value problem., Adv. Water Resour., 23, 59-68, 1999.
Cassiani, G., and Z. J., Kabala, Hydraulics of a partially penetrating well: solution to a mixed-type boundary value problem via dual integral equations, J. Hydrol., 211, 100-111, 1998.
Chang, C. C., and C. S. Chen, An integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin, Water Resour. Res., 38(6), 1071, doi:10.1029/2001WR001091, 2002a.
Chang, C. C., and C. S. Chen, Field experiment and data analysis of a constant-head injection test with skin effects in a low-transmissivity aquifer, TAO, 13(1), 15-38, 2002b.
Chang, C. C., and C. S. Chen, A flowing partially penetrating well in a finite-thickness aquifer: A mixed-type initial boundary value problem, J. Hydrol., 271, 101-118, 2003.
Chen, C. S., and C. C. Chang, Use of cumulative volume of constant-head injection test to estimate aquifer parameters with skin effects: field experiment and data analysis, Water Resour. Res., 38(5), 1056, doi:10.1029/2001WR000300, 2002.
Chen, C. S., and C. C. Chang, Well hydraulics theory and data analysis for the constant head test in an unconfined aquifer with skin effect, Water Resour. Res., 39(5), 1121, doi:10.1029/2002WR001516, 2003.
Chen, C. S., and W. D. Stone, Asymptotic calculation of Laplace inverse in analytical solutions of groundwater problems, Water Resour. Res., 29 (1), 207-209, 1993.
Chen, X. H., and J. F. Ayers, Aquifer properties determined from 2 analytical solutions, Ground Water, 36(5), 783-791, 1998.
Clegg, M. W., Some approximate solutions of radial flow problems associated with production at constant well pressure, Soc. Pet. Eng. J., 240, 31-42, 1967.
Chu, W. C., J. Garcia-Rivera, and R. Raghavan, Analysis of interference test data influenced by wellbore storage and skin at the flowing well, J. Pet. Technol., 32(1), 171-178, 1980.
Dagan, G., A note on packer, slug, and recovery tests in unconfined aquifers, Water Resour. Res., 14, 929-934, 1978.
Dougherty, D. E., and D. K. Babu, Flow to a partially penetrating well in a double-porosity reservoir, Water Resour. Res., 20(8), 1116-1122, 1984.
Earlougher, R. C., Jr., Advances in Well Test Analysis, Monograph Series, vol. 5, Society of Petroleum Engineers, Dallas, 1977.
Edwards, K. B., and L. C. Jones, Modeling pumping tests in weathered glacial till, J. Hydrol., 150(1), 41-60, 1993.
Ehlig-Economides, C. A., and H. J. Ramey Jr., Transient rate decline analysis for wells produced at constant pressure, Soc. Pet. Eng. J., 21(1), 98-104, 1981a.
Ehlig-Economides, C. A., and H. J. Ramey Jr., Pressure buildup for wells produced at a constant pressure, Soc. Pet. Eng. J., 21(1), 105-114, 1981b.
Fabrikant,V. I., Mixed Boundary Value Problems of Potential Theory and their Application in Engineering, pp.451, Kluwer Academic Publisher, Netherlands, 1991.
Fetkovich, M. J., Decline curve analysis using type curves, J. Pet. Technol., 32(6), 1065-1077, 1980.
Freeze, R. A., and D. B. McWhorter, A framework for assessing risk reduction due to DNAPL mass removal from low-permeability soils, Ground Water, 35(1), 111-123, 1997.
Gradshteyn, I. S., and I. M. Ryzhik, Table of Integrals, Series, and Products, 1160pp., Academic Press, INC., New York, 1980.
Gringarten, A.C., and H. J. Ramey, Jr., An approximate infinite conductivity solution for a partially penetrating line-source well, Soc. Pet. Eng. J., 259, 140-148, 1975.
Guppy, K. H., S. Kumar, and V. D. Kagawan, Pressure-transient analysis for fractured wells producing at constant pressure, SPE Formation Eval., March, 169-178, 1988.
Hantush, M. S., Aquifer tests on partially penetrating wells. J. Hydraul Div Proc ASCE87(HY5) 95-171, 1961.
Hantush, M. S., Hydraulics of wells, in Advances in Hydroscience, edited by V. T. Chow, vol. 1, Academic Press, New York, 1964.
Hiller, C. K., and B. S. Levy, Estimation of aquifer diffusivity from analysis of constant head pumping test data, Ground Water, 32(1), 47-52, 1994.
Huang, S. C., Unsteady-state heat conduction in semi-infinite regions with mixed-type boundary conditions, J. Heat Transfer, 107, 489-391, 1985.
Huang. S. C., and Y. P. Chang, Anisotropic heat conduction with mixed boundary conditions, J. Heat Transfer, 106, 646-648, 1984.
Hurst, W., J. D. Clark, and E. B. Brauer, The skin effect in producing wells, J. Pet. Technol., 246, 1483-1489, 1969.
Jacob, C. E. and S. W. Lohman, Nonsteady flow to a well of constant drawdown in extensive aquifers, Am. Geophys. Union Trans., 33, 559-569,1952.
Jargon, J. R., Effect of wellbore storage and wellbore damage at the active well on interference test analysis, J. Pet. Technol., 28, 851-858, 1976.
Javandel, I., Analytical solutions in subsurface fluid flow, Geol. Soc. Am. Spec. Paper, 189, 223-235,1982.
Jones, L., A comparison of pumping and slug tests for estimating the hydraulic conductivity of unweathered Wisconsin age till in Iowa, Ground Water, 31(6), 896-904, 1993.
Jones, L., T. Lemar, and C.-T. Tsai, Results of two pumping tests in Wisconsin age weathered till in Iowa, Ground Water, 30(4), 529-538, 1992.
Kruseman, G. P., and N. A. de Ridder, Analysis and Evaluation of Pumping Test Data, ILRI publication 47, 377 pp., 1990.
Mishra, S. and D. Guyonnet, Analysis of observation-well response during constant-head testing, Ground Water, 30(4), 523-528, 1992.
Moench, A. F., Specific yield as determined by type-curve analysis of aquifer-test data, Ground Water, 32(6), 949-957, 1994.
Moench, A. F., Combining the Neuman and Boulton models for flow to a well in an unconfined aquifer, Ground Water, 33(3), 378-384, 1995.
Moench, A. F., Flow to a well of finite diameter in a homogeneous, anisotropic water table aquifer, Water Resour. Res., 33(6), 1397-1407, 1997.
Mucha, I., and E. Paulikova, Pumping test using large-diameter production and observation wells, J. Hydrol., 89, 157-164, 1986.
Murdoch, L. C., and J. Franco, The analysis of constant drawdown wells using instantaneous source functions, Water Resour. Res., 30(1), 117-124, 1994.
Narasimhan, T. N., and M. Zhu, Transient flow of water to a well in an unconfined aquifer: Applicability of some conceptual models, Water Resour. Res., 29(1), 179-191, 1993.
Neuman, S. P., Theory of flow in unconfined aquifers considering delayed response of the water table, Water Resour. Res., 8(4), 1031-1044, 1972.
Neuman, S. P., Effects of partial penetration on flow in unconfined aquifers considering delayed aquifer response, Water Resour. Res., 10(2), 303-312, 1974.
Neuman, S. P., Analysis of pumping test data from anisotropic unconfined aquifers considering delayed gravity response, Water Resour. Res., 11(2), 329-342, 1975.
Noble, B., Methods Based on the Wiener-Hopf Technique. Pergamon Press, New York, 1958.
Novakowski, K. S., A composite analytical model for analysis of pumping tests affected by well bore storage and finite thickness skin, Water Resour. Res., 25(9), 1937-1946, 1989.
Novakowski, K. S., Interpretation of the transient flow rate obtained from constant-head tests conducted in situ in clays, Can. Geotechnical J., 30, 600-606, 1993.
Nwankwor, G. I., R. W. Gillham, G. van der Kamp, and F. F. Akindunni, Unsaturated and saturated flow in response to pumping of an unconfined aquifer : Field evidence of delayed drainage, Ground Water, 30(5), 690-700, 1992.
Olarewaju, J. S., and W. J. Lee, A comprehensive application of a composite reservoir model to pressure-transient analysis, SPE Reserv. Eng., 4(3), 325-331, 1989.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Frotran 77; The Art of Scientific Computing, 933pp., Cambridge University Press, New York, 1992.
Rice, J. B., Constant drawdown aquifer tests: an alternative to traditional constant rate tests, Ground Water Monit. R., 18(2), 76-78, 1998.
Ruud, N. C., and Z. J. Kabala, Response of a partially penetrating well in a heterogeneous aquifer: integrated well-face flux versus uniform well-face flux boundary conditions, J. Hydrol., 194, 76-94, 1997.
Schapery, R. A. Approximate methods of transform inversion for viscoelastic stress analysis, Proc., 4th U. S. National Congress of Applied Mechanics, 1075-1085, 1961.
Selim, M. S., and D. Kirkham, Screen Theory for Wells and Soil drainpipes, Water Resour. Res., 10(5), 1019-1030, 1974.
Shames, I. H., Mechanics of Fluids, McGraw-Hill, New York, 1962.
Sneddon, I. N., Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, and Wiley, New York, 1966.
Sneddon, I. N., The Use of Integral Transforms, 540pp., McGraw-Hill, New York,1972.
Stehfest, H., Numerical inversion of Laplace transforms, Commun. ACM, 13, 47-39, 1970.
Streltsova, T. D., Well Testing in Heterogeneous Formations, An Exxon Monograph, 423 pp., John Wily and Sons, New York, 1988.
Talbot, A., The accurate numerical inversion of Laplace transforms, J. Inst. Math. Applic., 23, 97-120, 1979.
Tavenas, F., P. Jean, P. Leblond, and S. Leroueil, The permeability of nature soft clays. Part II: Permeability characteristics, Can. Geotech. J., 20, 645-660, 1983.
Tavenas, F., M. Diene, and S. Leroueil, Analysis of the in situ constant-head permeability test in clays. Can. Geotech. J., 27, 305-314, 1990.
Uraiet, A. A., and R. Raghavan, Unsteady flow to a well producing at a constant pressure, J. Pet. Technol., 32(10), 1803-1812, 1980a.
Uraiet, A. A., and R. Raghavan, Pressure buildup analysis for a well produced at constant bottomhole pressure, J. Pet. Technol., 32(10), 1813-1824, 1980b.
van Everdingen, A. F., The skin effect and its influence on the productive capacity of a well, Trans. Am. Inst. Min. Metall. Pet. Eng., 198, 171-176, 1953.
van Everdingen, A. F., and W. Hurst, The application of the Laplace transformation to flow problems in reservoirs, Trans. Am. Inst. Min. Metall. Pet. Eng., 186, 305-324, 1949.
Wilkinson, W. B., Constant head in situ permeability tests in clay strata, Geotechnique, 18, 172-194, 1968.
Wylie, C. R., and L. C. Barrett, Advanced Engineering Mathematics, 1103 pp., McGraw-Hill, New York, 1982.