CHAPTER 3 NUMERICAL RESULTS
3.5 Trapezoid Aquifer
The present solutions can also be used to predict the flow induced by pumping in a trapezoid aquifer with 45 degree at the acute angle as shown in Figure 7. Following two conditions must be met if employing the present solution to predict the flow in a trapezoid aquifer. They are the aquifer must be isotropic and homogeneous and the hydraulic heads at two constant-head boundaries have to be equal. A dashed line can be drawn between two re-entrant angles as shown in Figure 7b and an imaginary pumping well should be placed at the location which is symmetric to the pumping well with respect to the dashed line. Thus, the dashed line can be considered as an impermeable boundary for the trapezoid aquifer.
Figure 7b demonstrates the transient drawdown distribution at t= 2 days for the pumping well located at (180 m, 76 m) and (76 m, 180 m) with the rate of 40 m /3 day. The
conductivities K and x Ky are 2×10−2 m /s and specific storage S is s 1×10−5 m for −1 the whole region, and the hydraulic heads at the boundaries (x= 200 m and y=200 m) are equal to 30 m. The aquifer thickness is 20 m. As a result, a symmetric pattern of the hydraulic head distribution can be observed in the figure. The hydraulic head distribution in the right trapezoid aquifer with a pumping well shown in Figure 7a can therefore be simulated as a part of hydraulic head distribution in Figure 7b. To our knowledge, the semi-analytical solution describing the flow induced by pumping in trapezoid aquifers has never before been presented. The solution derived herein for an L-shaped aquifer can accordingly be applied to solve the flow problems in trapezoid aquifers.
CHAPTER 4 CONCLUDING REMARKS
A new semi-analytical model has been developed to analyze the hydraulic head distributions induce by pumping in a heterogeneous and anisotropic aquifer with an L-shaped domain. Method of domain decomposition is used to divide the aquifer into two sub-regions with different hydraulic conductivities. Steady-state solution is first derived and used as the initial condition before pumping in the L-shaped aquifer system. The Laplace-domain solution of the model is derived using the Fourier finite cosine transform and the Laplace transform. The Stehfest algorithm is then adopted to evaluate the time-domain results. The effects of anisotropy, heterogeneity, and boundary conditions due to pumping are investigated.
The present solutions can describe the steady-state and transient distribution of hydraulic head induced by pumping in L-shaped aquifers. Following conclusions can be drawn from this study:
1. The steady-state hydraulic head distributions indicate that the flow in region 1 is mainly in x-direction and in region 2 is in y-direction.
2. The anisotropic ratio has a significant effect on the flow pattern. The flow velocity in x-direction increases with anisotropic ratio.
3. The developed solution can be used to assess the effects of heterogeneity and multi-well locations on the head distribution. In addition, it can also be used to solve the flow problems in trapezoid aquifers.
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APPENDIX A: STEADY STATE SOLUTIONS FOR AN L-SHAPED AQUIFER WITHOUT PUMPING
In accordance with the dimensionless variables defined in section 2.2, Eqs. (1) and (2) can be written, respectively, as
∑= The dimensionless boundary conditions for region 1 can be expressed as:
* 0
Similarly, the dimensionless boundary conditions for region 2 are
* 0
1
The continuity conditions of hydraulic head and flux in dimensionless form are, respectively,
)
Without the pumping, the steady-state solution for groundwater flow can be solved after removing all the right-hand side terms in Eqs. (A1) and (A2). Multiplying Eq. (A1) by
)
∫
The general solutions of Eqs. (A12) and (A13) are
* the following relation can be obtained
m
φ lead to Eqs. (12) and (13). The Fourier finite cosine inversions for regions 1 and 2 are 2
∑
∞ and the continuity conditions Eqs. (A10) and (A11) are given in Eqs. (17) and (18), respectively.APPENDIX B: TRANSIENT STATE SOLUTIONS
Multiplying Eq. (A1) by cos(λix*) and integrating it from x*=0 to x* =1 in region 1 with boundary conditions Eqs. (A3) and (A4), Eq. (A1) can be transformed as
∑
=Then, taking Laplace transforms with respect to t of Eqs. (B1) and (B2) with the initial conditions Eqs. (12) and (13), respectively, results in
∑
∑
The general solutions of Eqs. (B3) and (B4) can be expressed as ) are, respectively, from
*
boundary conditions Eqs. (A5) and (A9), the following relations can be obtained
Substituting Eqs. (B11) and (B12) into Eqs. (B5) and (B6), respectively, the inversions of
* the continuity conditions Eqs. (A10) and (A11) are presented as Eqs. (28) and (29), respectively.
Table 1. Notations.
i
Table 2. The constant-head boundaries and the anisotropic ratio of the hydraulic conductivities for cases (a) – (c)
Case
constant-head
boundary (m) hydraulic conductivity (m/s)
anisotropic ratio of hydraulic conductivity
a represents the anisotropic ratio of the hydraulic conductivity in region 1
κ2
b represents the anisotropic ratio of the hydraulic conductivity in region 2
Figure 1. Map of an aquifer in Hsinchu, Taiwan. The study area is shaded. (modified from Hydrological year book of Taiwan, 2010, Part II-River stage and discharge, and Taiwan Active Fault Information System)
Figure 2. An L-shaped alluvial aquifer with two sub-regions.
(a)
(b)
(c)
Figure 3. Steady-state hydraulic head distribution in the aquifer with anisotropic ratio of (a) 1, (b) 2, (c) 0.5.
(a) t=2 hrs
(b) t=2 days
(c) t =7 days
Figure 4. Hydraulic head distributions in aquifers with Kx =Ky = 520 m/day at time (a) 2 hrs, (b) 2 days, (c) 7 days.
(a)
(b)
Figure 5. Transient hydraulic head distributions induce by pumping at t=7 days in (a) heterogeneous aquifer and (b) homogeneous aquifer.
Region 1(0≤x≤900m,0≤ y≤300m): coarse sand Region 2(0≤x≤300m,300m≤ y≤600m): fine sand
Figure 6. Hydraulic head distributions induced by two injection wells and one pumping well at t =1 day.
(a)
(b)
Figure 7. (a) A trapezoid aquifer (b) The head distribution induced by two pumping wells located at (76 m, 180 m) and (180 m, 76 m) at t =2 days.