The wellbore flow-rate solution of the constant-head test and the drawdown solution of the constant-flux test are generally employed to analyze measuring data for estimating the aquifer properties. This thesis develops mathematical models to describe the hydraulic head distribution for the constant-head test and the drawdown distribution for constant-flux test at a finite-domain confined aquifer. The hydraulic head solutions and the drawdown solutions in Laplace domain for skin zone and formation zone are derived using the Laplace transforms for both tests. In addition, the solution of wellbore flow rate for the constant-head test is derived based on the hydraulic head solution and Darcy’ law. The time-domain results of wellbore flow rate for the constant-head test and the drawdown for the constant-flux test are evaluated by the modified Crump algorithm. The results show that the dimensionless wellbore flow-rate solution and drawdown solution for a finite-domain aquifer are significantly different from the one of an infinite-domain aquifer at late pumping times.
For the constant-head test, the infinite-domain solution will underestimate the flow rate at the wellbore in a finite-domain aquifer when time is fairly large. The effect of finite boundary on the flow rate appears to be considerable less for a positive skin.
On the other hand, the infinite-domain solution for the constant-flux test may
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-overestimate the drawdown for a finite-domain aquifer at large time. The analysis for the effect of skin type on the drawdown shows that the drawdown is significant in the skin zone and insignificant in the formation zone for a positive skin.
For both the constant head test and constant-flux test, the infinite-domain solution can be used to determine the drawdown distribution, wellbore flow rate, or aquifer parameters if coupled with an optimization algorithm when the time is less than the boundary-effect time criterion. The infinite-domain solutions are generally in simpler forms and much easier to evaluate than the finite-domain solutions.
Therefore, time criteria provide a good reference for adopting the infinite-domain solution to calculate the drawdown or wellbore flow rate of finite-domain aquifers.
This thesis also derives large-time solutions for the constant head test and constant-flux test based on the relationship of small Laplace-domain variable p versus large time-domain variable t. It is found that the large-time solutions in finite-domain aquifers are equal to the steady-state solutions obtained from the Tauberian theorem. In addition, the large-time solutions can reduce to Thiem equation if neglecting the skin zone for finite-domain aquifers and approach infinity as the time goes infinitely large for infinite-domain aquifers.
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-APPENDIXES
Appendix A: Derivation of Equations (8) and (9) Appendix B: Derivation of Equations (32) and (33)
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-Appendix A: Derivation of Equations (8) and (9)
The solutions for the hydraulic head distribution in the skin zone and formation zone are derived by taking the Laplace Transform to the governing equation of Equations (1) and (2), initial condition (Equations (3) ), and boundary conditions (Equations (4), (5), (6), and (7) (Carslaw and Jaeger, 1959, p332) The results are
1 1 0( 1 ) 2 0( 1 )
h =C I q r +C K q r (A1)
2 1 0( 2 ) 2 0( 2 )
h =D I q r +D K q r (A2) The constant coefficients in Eqs. (A1) and (A2) can be solved with the boundary conditions of Equations (4) and (5) and continuity requirements of Equations (6) and (7) as
Consequently, the hydraulic head solutions in the skin zone and formation zone can be
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-obtained by substituting the constants of Equations (A3) - (A6) into Equations (A1) and (A2) as Equations (8) and (9), respectively.
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-Appendix B Derivation of Equations (32) and (33)
The solutions for the hydraulic head distribution in the skin zone and formation zone are derived by taking the Laplace Transform to the governing equation of Equations (1) and (2), initial condition (Equation (3)), and boundary conditions Equations (5), (6), (7) and (29) (Carslaw and Jaeger, 1959, p332). The results can be Accordingly, the drawdown solution can be written as
1 1 0( 1 ) 2 0( 1 )
s =C I q r +C K q r (B3)
2 1 0( 2 ) 2 0( 2 )
s =D I q r +D K q r (B4) The constant coefficients in Equations (B3) and (B4) can be solved with the boundary conditions of Equations (5) and (29) and continuity requirements, i.e., Equations (6) and (7). as
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-The drawdown solution in the skin zone and the formation zone can therefore be obtained by substituting Equations (B5) - (B8) into Equations (B3) and (B4) as Equations (32) and (33).
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-REFERENCE
Abramowitz, M., and I. A. Stegun (1970), Handbook of Mathematical Functions, Dover, New York.
Carslaw, H. S. and J. C. Jaeger (1959), Conduction of Heat in Solids, 2nd ed., Oxford University Press, London.
Chen C. S. (1984), A reinvestigation of the analytical solution for drawdown distributions in a finite confined aquifer, Water Resour.Res., 20(10), 1466-1468.
Crump K. S. (1976), Numerical inversion of Laplace transforms using a Fourier series approximation, J Assoc Comput Mach., 23(1), 89-96.
Cooper, H. H., and C. E. Jacob. (1946), A generalized graphical method for evaluating formation constants andsummerizing well field history, Trans. Am. Geophys.
Union, 27,526-534
de Hoog F. R., Knight J. H, and Stokes A. N. (1982), An improved method for numerical inversion of Laplace transforms, Soc Indus-trial Appl. Mathe. J. Sci.
Stat. Comput., 3(3), 357-366.
International Mathematics and Statistics Library (1987), Inc. IMSL user’s manual, 2, IMSL, Inc., Houston.
Peng, H.Y., H.D. Yeh, and S.Y. Yang (2002), Improved numerical evaluation for the radial groundwater flow equation, Adv. Water Resour., 25, 663-675,.
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-Sneddon, I. N. (1972), The Use of Integral Transforms, McGraw-Hill, New York.
Stehfest, H. (1970), Numerical inversion of Laplace transforms, Commun. ACM, 13(1), 47-49,.
Wang, C. T., and H. D. Yeh (2008), Obtaining the steady-state drawdown solutions of constant-head and constant-flux tests, Hydrol. Process., 22, 3456-3461, doi:10.1002/hyp.6950.
Yang, S. Y., H. D Yeh (2002), Solution for flow rates across the wellbore in a two-zone confined aquifer, J.Hydraulic Eng., 128, 175-183.
Yeh, H.D., S.Y. Yang, and H.Y. Peng (2003), A new closed-form solution for a radial two-layer drawdown equation for groundwater under constant-flux pumping in a finite-radius well, Adv. in Water Resour., 26, 747-757.
Yeh, H. D., and C. T. Wang (2007), Large-time solutions for groundwater flow problems using the relationship of small p versus large t, Water Resour. Res., 43(6), W06502, doi:10.1029/2006WR005472.
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-Table 1 List of Equations (26), (28), and two existing solutions for constant-head test
Solutions Outer boundary
Large-time solution Steady-state solution R is finite Thiem equation Thiem equation skin
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-Table 2 List of the large-time and steady-state drawdown solutions for the constant-flux test
Solutions Outer boundary
Large-time Steady-state R is finite Equations (43) and (44) Equations (43) and (44)
R is infinite Equations (47) and (48) No solution
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-Figure1 Schematic diagram for a constant-head test at a finite diameter well in a finite-domain confined aquifer.
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-Figure 2 Schematic diagram of the pumping test in a finite-domain confined aquifer.
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Figure 3 Dimensionless flow rate versus dimensionless pumping time for ρ1 =3 at ρ
= 1 and α = 0.1, 1 and 10. The solid line presents the flow-rate solution of infinite-domain aquifers and the dash line present the flow-rate solution of finite-domain aquifers.
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Figure 4 Comparison of the drawdown of a finite-domain aquifer to the drawdown of an infinite-domain aquifer for ρ1 = 3 at ρ = 1 and α = 0.1, 1, 5, and 10. The solid line presents the infinite-domain drawdown solution and the dash line represent the finite-domain drawdown solution.
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-Figure 5 Comparison of the drawdown of a finite-domain aquifer to the drawdown of an infinite-domain aquifer for ρ1 = 10 at ρ = 1 and α = 0.1, 1, 5, and 10. The solid line presents the infinite-domain drawdown solution and the dash line present the finite-domain drawdown solution.
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-Figure 6 Comparison of the drawdown of a finite-domain aquifer to the drawdown of an infinite-domain aquifer for ρ1 = 3 at ρ = 10 and α =0.1, 1, 5, and 10. The solid line presents the infinite-domain drawdown solution and the dash line present the finite-domain drawdown solution.
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Figure 7 Comparison of the drawdown of a finite-domain aquifer to the drawdown of an infinite-domain aquifer for ρ1 = 10 at ρ = 10 and α = 0.1, 1, 5, and 10. The solid line presents the infinite-domain drawdown solution and the dash line present the finite-domain drawdown solution.
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Figure 8 Time criterion τc versus dimensionless boundary distance ρR for various values of ρ, ρ1, and α.