CHAPTER 2 METHEMATICAL MODEL
2.4 The Large-time Solution for the Constant-head Test
1 1
K ρ p /K0
(
ρ1 p)
if ρR →∞. The dimensionless flow-rate solution for an infinite domain aquifer in Laplace-domain given by Yang and Yeh (2002) was expressed asObviously, the flow-rate solution for an aquifer with an infinite-domain is in a simpler form and much easier to evaluate than the one with a finite-domain.
2.4 The Large-time Solution for the Constant-head Test 2.4.1 In a Finite Confined Aquifer
The large-time flow-rate solution at the wellbore can be evaluated from Equation (14) by utilizing the SPLT technique. The limiting forms of the Bessel functions for
small arguments used for computing Equation (25) are I x0( ) ~ 1/ (1),Γ
( )
~ 2( )
21 x x Γ
I , K x0( ) ~ ln( ),− x and ( ) ~ 1/K x1 x where ( )Γ x is the gamma
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-function (Abramowitz and Stegun 1970, p.375). Applying L’Hospital’s rule to Equation (14) with p approaching zero, the Laplace-domain wellbore flow rate for
small p gives
where the negative sign in Equation (25) expresses withdrawal at the test well.
Accordingly, the large-time wellbore flow-rate solution can be easily obtained by taking the inverse Laplace transform to Equation (25) as
1
Equation (26) is independent of time and naturally is a steady-state solution. By applying the Tauberian theory to Equation (14) (Sneddon, 1972), one can also obtain Equation (26). This result implies that the wellbore flow rate does approach steady state at large time condition for an aquifer with a finite-domain (Wang and Yeh, 2008).
In addition, Equation (26) can be simplified to the Thiem equation if the skin zone is absent (i.e., r1 equals rw).
2.4.2 In an Infinite Confined Aquifer
If an infinite extended boundary is considered, i.e., R→ ∞, Equation (14) therefore is equivalent to Equation 15. Again, the Laplace-domain wellbore flow rate for small p in an infinite confined aquifer can be obtained as
12 Equation (27) is, therefore, identical to the one of Yeh and Wang (2007, Equation 3).
The large-time wellbore flow-rate solution is obtained after taking the inverse Laplace transform to Equation (27) as (Yeh and Wang, 2007)
( ) ( )
where 0.5772...γ = is Euler’s constant and the Riemann Zeta function ξ(3) = 1.2020569032. The numerators of the right-hand side terms of Equation (28) are all constants and the denominators are a function of time sinceη =λt=T2
(
r1 r2)
2T2T1t/rw2S2 . The value of lnη tends to infinity as t approaches infinity; consequently, Equation (28) becomes zero. It means that the steady-state wellbore flow rate in an infinite confined aquifer is zero implying the constant head has reached to the infinite extended boundary.When neglecting the skin zone, Equation (28) can reduce to the solution of Yeh and Wang (2007, Equation 6). Table 1 provides a list for comparing Equations (26) and (28) with two existing solutions.
2.5 Drawdown Solution for Constant-flux Test
This section presents the mathematical model for the constant-flux test. The
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-assumptions for the constant-flux test are the same as those for the constant-head test except that there is a constant discharge rate, rather than a constant head, maintained at the wellbore through out the entire pumping test. Therefore, the mathematical model describing the constant-flux test is identical to the constant-head test except the boundary condition specified at the wellbore. The boundary condition specified for a constant pumping with the flow rate Q can be expressed as
1
Equations (6) and (7) representing the continuity requirements of the hydraulic head and the flux at the interface of the skin zone and the formation zone are also applicable to the constant-flux test. Figure 2 illustrates the schematic diagram of an aquifer for the constant-flux test.
2.6 The Drawdown Solution for the Constant-flux Test in a Finite Confined
Aquifer
The obtained hydraulic head solution are
( ) ( )
Appendix B lists the derivation for the drawdown solution for the constant-flux test in a finite confined aquifer. The drawdown can be represented by s = h0 –h, so the
14
-drawdown solution can be written as
( ) ( )
2.7 Dimensionless Drawdown Solution for the Constant-flux Test
The dimensionless drawdown is defined as sD=s(4πT2) Q. Equations (32) and (33) can then be respectively written as
( ) ( )
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These two solutions can also be inverted to time domain numerically by the modified Crump algorithm (de Hoog et al. 1982). The dimensionless Laplace-domain drawdown solutions for the aquifer with an infinite domain provided by Yeh et al.
(2003) can be expressed as
( ) ( )
2.8 The Large-time Drawdown Solution for the Constant-flux Test 2.8.1 In Finite Confined Aquifers
The drawdown solution at late time can be obtained by applying the SPLT technique to Equations (32) and (33). Based on the L’Hospital rule, the Laplace-domain solutions for drawdowns in skin zone and the formation zone when p is small can be obtained respectively as
1 1
The large-time drawdown solutions are then obtained by taking the inverse Laplace transform to Equations (41) and (42) as
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Equations (43) and (44) are independent of time and equal to the steady-state solution, which can also be obtained by applying the method of Tauberian theory (Sneddon 1972) to Equations (32) and (33). Wang and Yeh (2008) also showed that the drawdowns can reach steady state if aquifers have finite domain. In addition, Equations (43) and (44) can reduce to the Thiem equation if the skin zone is absent.
2.8.2 In Infinite Confined Aquifers
Equations (43) and (44) are the large-time solutions of Equations (32) and (33), respectively, for finite-domain confined aquifers. By applying the SPLT relationship to Equations (34) and (35), the Laplace-domain drawdown solutions for small p in skin zone and formation zone can be obtained respectively as
1 1
Finally, the large-time drawdown solutions in time domain can be obtained after employing the inverse Laplace transform as
1 1 2
17 Equation (48) can reduce to cooper and Jacob equation (1946) if neglecting the effect of skin zone. In addition, this large-time drawdown solution for an aquifer with infinite extended boundary is equal to the heat flow solution at late time presented by Carslaw and Jaeger (1959, p.339) if the limiting form of the Bessel function is
( ) ~ [ln( / 2)K x0 − x + instead ofγ] K x0( ) ~ ln( )− x .
Equations (45) and (46) are function of time. Equations (47) and (48) approach infinite when t reaches infinity due to the term ln t approaches infinity. This phenomenon indicates that the drawdown solution of the infinite extended boundary increases with time because the infinitely extended boundary can not provide enough groundwater to maintain the constant pumping rate at wellbore. In addition, Equations (47) and (48) decrease with increasing r indicating that the condition of zero drawdown at the outer boundary condition can be held. Table 2 provides a list of comparison between the large-time drawdown and steady state solutions in finitely and infinitely extended confined aquifers.
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-CHAPTER 3 RESULTS AND DISCUSSION
3.1 Comparison of Infinite-Domain Solution to Finite-Domain Solution for
Constant-head Test
3.1.1 Effects of skin zone and Finite Domain
Figure 3 illustrates that the curve of the wellbore flow-rate as a function of time at ρ = 1 for ρ1 = 3 and α = 0.1, 1, and 10. For α = 0.1 and 1, the finite-domain flow-rate matches with the infinite-domain flow-rate in small time (i.e., 10 < τ < 100). These results indicate that the infinite-domain flow-rate solution can approximate the finite-domain flow-rate solution if the test time is less than the time criterion. In other words, the boundary distance R has no effect on the wellbore flow-rate as the test time is shorter than the time criterion. However, these two solutions deviate for one another in the period of moderate time (100 < τ < 1000). The finite-domain flow-rate solution approaches its asymptotic limit, i.e., the steady state solution, when the test time is large while the infinite-domain flow-rate solution continuously declines with dimensionless time.
The dimensionless flow rate with a positive skin is smaller than that with a negative skin (Yang and Yeh, 2002) at a specific time. This result indicates that the effect of finite boundary on the flow-rate may be negligible for a high value of α (say, e.g., 10).
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-3.1.2 Time Criterion for Using the Infinite-Domain Solution
Figure 3 shows that the time criterion increases with ρR (= /R r ). At w ρ1 = 3 and
3.2Comparision of the Infinite-Domain Solution to the Finite-Domain Solution
for Constant-flux Test
3.2.1 Effects of Skin Type, Skin Thickness, and Finite-Domain
Figure 4 shows the curves of dimensionless drawdown at wellbore for aquifers with finite-domain and infinite-domain with ρ1 = 3 and α = 0.1, 1, 5, and 10. The drawdown curves for both aquifers match exactly at early pumping time (1 < τ < 10).
However, the curves gradually deviate from one another in the middle time period (10
< τ < 100) due to boundary effect. It indicates that the infinite-domain drawdown solution can approximate the finite-domain solution when the time is less than the time criterion. Finially, the finite-domain drawdown solution tends to an asymptotic limit, the steady state solution, while the infinite-domain drawdown continuously
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-increases with dimensionless time. These results reflect that the pumped water comes from the remote constant-head boundary for the finite-domain aquifer and is from the aquifer storage for the infinite-domain aquifer. Figure 4 also shows the effect of skin type on the drawdown distribution. The drawdown increases with α indicating a larger α has a larger drawdown value.
Figure 5 demonstrates the curves of dimensionless drawdown at wellbore for aquifers with finite-domain and infinite-domain with ρ1 = 10 and α = 0.1, 1, 5, and 10.
This figure has a similar pattern of temporal drawdown distribution to Figure 4 except that the time criterion relating to the boundary effect is delayed. Yeh et al. (2003) also mentioned that the thickness of the wellbore ρ1 may influence the magnitude of the dimensionless drawdown.
Figures 6 and 7 show the dimensionless drawdown curves for ρ1 = 3 and 10, respectively, with ρ = 10, α = 0.1, 1, 5, and 10. There two figures indicate that the effect of skin type on the drawdown in the formation zone contrasts to that in the skin.
The drawdown of the formation zone is less sensitive for a positive skin aquifer.
3.2.2 Time Criterion for Various Values of Skin Type, Skin Thickness and Radial
Distance
The time criterion increases with the dimensionless boundary distance ρR as illustrated in Figures 4 - 7. Figure 8 shows the time criterion versus ρR for varying
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-α , ρ1, and ρ. This figure indicates that the time criterion τc increases with α
(skin type) and ρ1 (skin thickness) but decreases with ρ (dimensionless radial distance from the center of the test well) if ρR is fixed. Additionally, τc also
increases with ρR if α is fixed.
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