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Analytical solution for Non-Darcian effect on transient confined-unconfined flow in a confined aquifer
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Abstract: This paper presents a new analytical solution to investigate the mechanism of transient confined-unconfined flow in a confined aquifer induced by pumping with a large rate during mine drainage. The study focuses on understanding the impact of non-Darcian effect on flow towards a fully penetrated pumping well. The nonlinear relationship between specific discharge and the hydraulic gradient is described using Izbash's equation. A novel approximate method is developed to linearize the mathematical model, and the solution is derived using the Boltzmann transform. The proposed solution is validated by comparing it with previous works. The findings indicate that increased non-Darcian index, quasi-hydraulic conductivity, and specific storage have negatively affect the development of the unconfined region and aquifer drawdown, as greater turbulence flow accelerates recharge to the pumping well. Drawdown is found to be sensitive to the non-Darcian index, quasi-hydraulic conductivity, while it is unaffected by specific yield and specific storage. The conclusions provide valuable insights for mine drainage and the application of geological and hydrological conditions.
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Key words:
- Non-Darcian flow /
- Izbash equation /
- Boltzmann transform /
- Sensitivity analysis
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Figure 1. A schematic diagram of the transient confined-unconfined flow towards a fully penetrating well in a confined aquifer.
Figure 2. Change of groundwater storage during the confined-unconfined conversion (after Hu and Chen, 2008).
Figure 3. Comparison of time-Drawdown curves with Xiao's model
Notes: Under the condition of $ Q=0.026\;{\mathrm{m}}^{3}/\mathrm{s} $, ${K}_{r}= $$ 0.0000695\;\mathrm{m}/\mathrm{s}$, ${S}_{s}=0.000002\;{\mathrm{m}}^{-1}$, $ {S}_{y}=0.3 $, $ b=30\;\mathrm{m} $, $ {h}_{0}=36\;\mathrm{m} $, $ r=10\;\mathrm{m} $ and $ n=1 $ in semi-log scales
Figure 4. Time-Drawdown behavior for different index
$ n $ Notes: With $ Q=0.04\;{\mathrm{m}}^{3}/\mathrm{s} $, ${K}_{r}=0.0000278\;\mathrm{m}/\mathrm{s}$, ${S}_{s}= 0.000002\; $$ {\mathrm{m}}^{-1}$, $ b=30\;\mathrm{m} $, $ {h}_{0}=36\;\mathrm{m} $ and $ r=10\;\mathrm{m} $ in semi-log scales.
Figure 5. Time-R value curves for different
$ {S}_{s} $ valueNotes: With $ Q=0.04\;{\mathrm{m}}^{3}/\mathrm{s} $, ${K}_{r}=0.00000348\;\mathrm{m}/\mathrm{s}$, $ {S}_{y}=0.3 $, $ b= $$ 30\;\mathrm{m} $, $ {h}_{0}=36\;\mathrm{m} $, $ n=1.4 $ and $ r=10\;\mathrm{m} $ in semi-log scales
Figure 6. Time-Drawdown curves for different
$ {S}_{s} $ valueNotes: With $Q=0.04\;{\mathrm{m}}^{3}/\mathrm{s}$, ${K}_{r}=0.00000348\;\mathrm{m}/\mathrm{s}$, $ {S}_{y}= 0.3 $, b = 30 m, $ {h}_{0}=36\;\mathrm{m} $, $ n=1.4 $ and $ r=10\;\mathrm{m} $ in semi-log scales
Figure 7. a)
$ R $ -time curves and b) time-drawdown curves for different$ {K}_{r} $ valueNotes: With $ Q=0.04\;{\mathrm{m}}^{3}/\mathrm{s} $, $ {S}_{y}=0.3 $, $ {S}_{s}=0.000002\;{\mathrm{m}}^{-1} $, b = 30 m, $ {h}_{0}=36\;\mathrm{m} $, $ n=1.4 $ and $ r=10\;\mathrm{m} $ in semi-log scales
Figure 8. Sensitivity analysis of aquifer parameters
Notes: With $ Q=0.04\;{\mathrm{m}}^{3}/\mathrm{s} $, ${S}_{s}=0.000002\;{\mathrm{m}}^{-1}$, $ {S}_{y}=0.3 $, b=30 m, $ {h}_{0}=36\;\mathrm{m} $, $ r=10\;\mathrm{m} $, $ n=1.4 $ and ${K}_{r}= 0.000003475\;\mathrm{m}/\mathrm{s}$ in semi-log scales
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