地下水超抽所引起地層下陷的理論，大致上可分為地下水流非耦合理論及Biot流固耦合理論。非耦合理論將壓力水頭及沉陷量分開處理，將計算後的壓力水頭代入適當的土體變形模型求得沉陷量。其優點為計算快速，但土體變形模型較缺乏理論依據；Biot耦合理論完整的考慮三維壓力水頭與土體變形的相互影響，但計算費時。本文將利用微擾法(Fallou, Mei, and Lee (1992))分析Biot耦合方程式在層狀土壤多層含水層系統抽水引起沉陷情形。將分析簡化後的耦合微擾方程式與非耦合方程式比較說明。並利用Laplace變換及數值反演方法建立耦合微擾分析模型。
此模型適用於多層含水層系統抽水問題，並考慮重力效應及自由含水層的影響，將三維空間問題轉化為數個二維空間問題。在理論基礎下，將壓力水頭與沉陷量分開計算，明顯增加計算效率。其計算結果與採用Biot耦合理論的有限元素模型結果比對，說明了此耦合微擾分析模型的正確性及有效性。
最後，利用此模型討論多層含水層系統在不同情況下的行為。

The theories of predicting the ground subsidence caused by excessive pumping of groundwater can be classified into two types: uncoupled theory of groundwater flow and coupled theory of Biot. Uncoupled theory calculates the drawdown and the subsidence separately. The subsidence was obtained by using an appropriate model of soil deformation with the calculated drawdown. Although the calculation can be fast, the soil deformation model lacks a theoretical basis. In the coupled theory of Biot, the full three-dimensional drawdown and deformation fields of soil interact mutually, but the calculation can be very time-consuming. This research adopts the perturbation theory developed by Fallou et al. (1992), which analyzes the coupled equations of Biot for the subsidence of a two-aquifer system due to pumping. We will compare the perturbed coupled equations with the uncoupled equations, and use the Laplace transform and numerical Laplace inversion method to establish the analysis model based on the perturbed coupled equations.
This model is applicable to the pumping problem of a multi-aquifer system, and it also includes the effects of gravity and the phreatic aquifer. Using this model, the three-dimensional problem is decomposed into several two-dimensional problems. Based on the theory, the drawdown and the subsidence can be calculated separately. It increases the computational efficiency significantly. The calculated results have been validated by those of finite element model of Biot theory.
Finally, using this analysis model, we discuss the time histories of the drawdown and the subsidence of a multi-aquifer system in different situations.

Abate, J., & Whitt, W. A Unified Framework for Numerically Inverting Laplace Transforms. Informs Journal on Computing, 18(4), 408-421, 2006.
Ai, Z. Y., Cheng, Z. Y., & Han, J. State Space Solution to Three-Dimensional Consolidation of Multi-Layered Soils. International Journal of Engineering Science, 46(5), 486-498, 2008.
Ai, Z. Y., Wang, Q. S., & Wu, C. A New Method for Solving Biot’s Consolidation of a Finite Soil Layer in the Cylindrical Coordinate System. Acta Mechanica Sinica, 24(6), 691-697, 2008.
Ai, Z. Y., Zeng, W. Z., Cheng, Y. H., & Wu, C. Uncoupled State Space Solution to Layered Poroelastic Medium with Anisotropic Permeability and Compressible Pore Fluid. Frontiers of Architecture and Civil Engineering in China, 5(2), 171-179, 2011.
Bear, J. (1979). Hydraulics of Groundwater: McGraw-Hill International Book Co.
Bear, J., & Corapcioglu, M. Y. Mathematical-Model for Regional Land Subsidence Due to Pumping .1. Integrated Aquifer Subsidence Equations Based on Vertical Displacement Only. Water Resources Research, 17(4), 937-946, 1981a.
Bear, J., & Corapcioglu, M. Y. Mathematical Model for Regional Land Subsidence Due to Pumping: 2. Integrated Aquifer Subsidence Equations for Vertical and Horizontal Displacements. Water Resources Research, 17(4), 947-958, 1981b.
Biot, M. A. General Theory of Three Dimensional Consolidation. Journal of Applied Physics, 12(2), 155-164, 1941.
Boulton, N. S. The Influence of Delayed Drainage on Data from Pumping Tests in Unconfined Aquifers. Journal of Hydrology, 19(2), 157-169, 1973.
Carbognin, L., Cecconi, G., & Ardone , V. Investigation to Safe Guard the Environment of the Venice Lagoon(Italy) against the Effects of Land Elevation Loss. Proceeding of the Sixth International Symposium on Land Subsidence(24-29 September), 2, 309-324, 2000.
Chen, C. S., & Stone, W. D. Asymptotic Calculation of Laplace Inverse in Analytical Solutions of Groundwater Problems. Water Resources Research, 29(1), 207-209, 1993.
Cheng, A. H. D., & Morohunfola, O. K. Multilayered Leaky Aquifer Systems .1. Pumping Well Solutions. Water Resources Research, 29(8), 2787-2800, 1993a.
Cheng, A. H. D., & Morohunfola, O. K. Multilayered Leaky Aquifer Systems .2. Boundary-Element Solutions. Water Resources Research, 29(8), 2801-2811, 1993b.
Cheng, A. H. D., & Ou, K. S. An Efficient Laplace Transform Solution for Multiaquifer Systems. Water Resources Research, 25(4), 742-748, 1989.
Chorley, D. W., & Frind, E. O. An Iterative Quasi-Three-Dimensional Finite Element Model for Heterogeneous Multiaquifer Systems. Water Resources Research, 14(5), 943-952, 1978.
Corapcioglu, M. Y., & Bear, J. A Mathematical-Model for Regional Land Subsidence Due to Pumping .3. Integrated Equations for a Phreatic Aquifer. Water Resources Research, 19(4), 895-908, 1983.
Detournay, E., & Cheng, A. H. D. Poroelastic Response of a Borehole in a Non-Hydrostatic Stress-Field. International Journal of Rock Mechanics and Mining Sciences, 25(3), 171-182, 1988.
Fallou, S. N., Mei, C. C., & Lee, C. K. Subsidence Due to Pumping from Layered Soil - a Perturbation-Theory. International Journal for Numerical and Analytical Methods in Geomechanics, 16(3), 157-187, 1992.
Gambolati, G., & Freeze, R. A. Mathematical Simulation of the Subsidence of Venice: 1. Theory. Water Resources Research, 9(3), 721-733, 1973.
Gambolati, G., Sartoretto, F., & Uliana, F. A Conjugate-Gradient Finite-Element Model of Flow for Large Multiaquifer Systems. Water Resources Research, 22(7), 1003-1015, 1986.
Gutierrez, M. S., & Lewis, R. W. Coupling of Fluid Flow and Deformation in Underground Formations. Journal of Engineering Mechanics-Asce, 128(7), 779-787, 2002.
Hantush, M. S. Modification of the Theory of Leaky Aquifers. Journal of Geophysical Research, 65(5), 1634-1634, 1960.
Hassanzadeh, H., & Pooladi-Darvish, M. Comparison of Different Numerical Laplace Inversion Methods for Engineering Applications. Applied Mathematics and Computation, 189(2), 1966-1981, 2007.
Herrera, I. Theory of Multiple Leaky Aquifers. Water Resources Research, 6(1), 185-193, 1970.
Herrera, I., & Figueroa V, G. E. A Correspondence Principle for the Theory of Leaky Aquifers. Water Resources Research, 5(4), 900-904, 1969.
Herrera, I., & Rodarte, L. Integrodifferential Equations for Systems of Leaky Aquifers and Applications: 1. The Nature of Approximate Theories. Water Resources Research, 9(4), 995-1005, 1973.
Herrera, I., & Yates, R. Integrodifferential Equations for Systems of Leaky Aquifers and Applications 3. A Numerical Method of Unlimited Applicability. Water Resources Research, 13(4), 725-732, 1977.
Hunt, B., & Scott, D. Extension of Hantush and Boulton Solutions. Journal of Hydrologic Engineering, 10(3), 223-236, 2005.
Hunt, B., & Scott, D. Flow to a Well in a Two-Aquifer System. Journal of Hydrologic Engineering, 12(2), 146-155, 2007.
Lee, C. K., Fallou, S. N., & Mei, C. C. Subsidence Due to Pumping from a Soil Stratum with a Soft Aquitard. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 339(1653), 193-230, 1992.
Lewis, R. W., & Schrefler, B. A Fully Coupled Consolidation Model of the Subsidence of Venice. Water Resources Research, 14(2), 223-230, 1978.
Liu, C. W., Lin, W. S., Shang, C., & Liu, S. H. The Effect of Clay Dehydration on Land Subsidence in the Yun-Lin Coastal Area, Taiwan. Environmental Geology, 40(4-5), 518-527, 2001.
Mathias, S. A., & Zimmerman, R. W. Laplace Transform Inversion for Late-Time Behavior of Groundwater Flow Problems. Water Resources Research, 39(10), 2003.
Mei, C. C. Gravity Effects in Consolidation of Layer of Soft Soil. Journal of Engineering Mechanics-Asce, 111(8), 1038-1047, 1985.
Neuman, S. P. Theory of Flow in Unconfined Aquifers Considering Delayed Response of the Water Table. Water Resources Research, 8(4), 1031-1045, 1972.
Neuman, S. P., Preller, C., & Narasimhan, T. N. Adaptive Explicit-Implicit Quasi Three-Dimensional Finite Element Model of Flow and Subsidence in Multiaquifer Systems. Water Resources Research, 18(5), 1551-1561, 1982.
Neuman, S. P., & Witherspoon, P. A. Applicability of Current Theories of Flow in Leaky Aquifers. Water Resources Research, 5(4), 817-829, 1969a.
Neuman, S. P., & Witherspoon, P. A. Theory of Flow in a Confined Two Aquifer System. Water Resources Research, 5(4), 803-816, 1969b.
Premchitt, J. A Technique in Using Integrodifferential Equations for Model Simulation of Multiaquifer Systems. Water Resources Research, 17(1), 162-168, 1981.
Rodarte, L. Theory of Multiple Leaky Aquifers: 1. The Integrodifferential and Differential Equations for Small and Large Values of Time. Water Resources Research, 12(2), 163-170, 1976.
Safai, N. M., & Pinder, G. F. Vertical and Horizontal Land Deformation in a Desaturating Porous Medium. Advances in Water Resources, 2(0), 19-25, 1979.
Safai, N. M., & Pinder, G. F. Vertical and Horizontal Land Deformation Due to Fluid Withdrawal. International Journal for Numerical and Analytical Methods in Geomechanics, 4(2), 131-142, 1980.
Schapery, R. A. Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis. Proc. U.S. Natl. Congr. Appl. Mech. 4th, 2, 1075-1085, 1962.
Stehfest, H. Algorithm 368: Numerical Inversion of Laplace Transforms [D5]. Commun. ACM, 13(1), 47-49, 1970.
Theis, C. V. The Relation between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground Water Storage. Transactions-American Geophysical Union, 16, 519-524, 1935.
Tsai, T. L., Chang, K. C., & Huang, L. H. Body Force Effect on Consolidation of Porous Elastic Media Due to Pumping. Journal of the Chinese Institute of Engineers, 29(1), 75-82, 2006.
Tseng, C. M., Tsai, T. L., & Huang, L. H. Effects of Body Force on Transient Poroelastic Consolidation Due to Groundwater Pumping. Environmental Geology, 54(7), 1507-1516, 2008.
Verruijt, A. Displacement Functions in the Theory of Consolidation or in Thermoelasticity. Zeitschrift für angewandte Mathematik und Physik ZAMP, 22(5), 891-898, 1971.
Wang, J., & Fang, S. State Space Solution of Non-Axisymmetric Biot Consolidation Problem for Multilayered Porous Media. International Journal of Engineering Science, 41(15), 1799-1813, 2003.
林瑞琦，一維耦合地層下陷模式之建立，國立台灣大學碩士論文，2001。
柳志錫，複雜含水地層之抽水沉陷行為，國立交通大學博士論文，2004。
洪以璇，多層含水層抽水引起壓密問題之耦合分析，國立成功大學碩士論文，2012。