# Transient-state Analytical Solution for Arbitrarily-Located Multiwells in Triangular-Shaped Unconfined Aquifer

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## Abstract

This work presents an analytical solution for the linearized Boussinesq equation describing the nature of well hydraulics in equilateral triangular-shaped unconfined aquifer. This homogeneous, isotropic, fully-saturated porous media is hydraulically connected to three surrounding streams of constant-head. The solution enables modeling aquifer response to a system of arbitrarily-located, fully-penetrating multiwells (injection, extraction or combination of both), each characterized by stepwise time-varying rate. First, a fundamental solution is provided for multiwell-induced head distribution in an infinite aquifer domain. Image well theory is then efficiently implemented to create an equivalent flow field for the intended domain. Spatiotemporal head distribution is obtained in the form of fivefold series involving exponential integrals. Expressions are also derived to quantify stream depletion rates caused by a single pumping well, under both transient and steady-state conditions. As a hypothetical example, an aquifer remediation scheme is planned by combining two extraction wells with an injection one. The computed head profiles reveal strictly close agreement with numerical results obtained by finite element method. Sensitivity map for stream depletion rate is also discussed. The present results are found to exactly reproduce those available for the wedge-shaped domain, under certain geometric constraint. Finally, the solution is extended to the case of hemi-equilateral triangular-shaped aquifer with or without an impervious boundary line.

## Keywords

Multiwell hydraulics Analytical solution Triangular aquifer Image wells Stream depletion rate Linearized Boussinesq equation## Notes

## References

- Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 9. Dover, New YorkGoogle Scholar
- Asadi-Aghbolaghi M, Chuang MH, Yeh HD (2012) Groundwater response to tidal fluctuation in a sloping leaky aquifer system. Appl Math Model 36(10):4750–4759CrossRefGoogle Scholar
- Asadi-Aghbolaghi M, Rakhshandehroo GR, Kompani-Zare M (2011) Analytical solutions for the capture zone of a pumping well near a stream. Hydrogeol J 19:1161–1168CrossRefGoogle Scholar
- Asadi-Aghbolaghi M, Seyyedian H (2010) An analytical solution for groundwater flow to a vertical well in a triangle-shaped aquifer. J Hydrol 393(3):341–348CrossRefGoogle Scholar
- Bresciani E, Kang PK, Lee S (2019) Theoretical analysis of groundwater flow patterns near stagnation points. Water Resour Res 55(2):1624–1650CrossRefGoogle Scholar
- Butler JJ, Zhan X, Zlotnik VA (2007) Pumping-induced drawdown and stream depletion in a leaky aquifer system. Groundwater 45(2):178–186CrossRefGoogle Scholar
- Chang YC, Yeh HD (2007) Analytical solution for groundwater flow in an anisotropic sloping aquifer with arbitrarily located multiwells. J Hydrol 347(1):143–152CrossRefGoogle Scholar
- Erdélyi A (1954) Tables of integral transforms (Vol. I). McGraw-Hill, New YorkGoogle Scholar
- Gander W, Gautschi W (2000) Adaptive quadrature—revisited. BIT Numer Math 40(1):84–101CrossRefGoogle Scholar
- Hunt B (1999) Unsteady stream depletion from ground water pumping. Groundwater 37(1):98–102CrossRefGoogle Scholar
- Kacimov AR, Kayumov IR, Al-Maktoumi A (2016) Rainfall induced groundwater mound in wedge-shaped promontories: the Strack–Chernyshov model revisited. Adv Water Resour 97:110–119CrossRefGoogle Scholar
- Kacimov AR, Maklakov DV, Kayumov IR, Al-Futaisi A (2017) Free surface flow in a microfluidic corner and in an unconfined aquifer with accretion: the Signorini and Saint-Venant analytical techniques revisited. Transp Porous Media 116(1):115–142Google Scholar
- Lin CC, Chang YC, Yeh HD (2018) Analysis of groundwater flow and stream depletion in the L-shaped fluvial aquifer. Hydrol Earth Syst Sci 22(4):2359–2375CrossRefGoogle Scholar
- Mahdavi A (2019) Response of triangular-shaped leaky aquifers to rainfall-induced groundwater recharge: an analytical study. Water Resour Manag 33(6):2153–2173CrossRefGoogle Scholar
- Mahdavi A, Seyyedian H (2013) Transient-state analytical solution for groundwater recharge in triangular-shaped aquifers using the concept of expanded domain. Water Resour Manag 27(8):2785–2806CrossRefGoogle Scholar
- Mahdavi A, Seyyedian H (2014) Steady-state groundwater recharge in trapezoidal-shaped aquifers: a semi-analytical approach based on variational calculus. J Hydrol 512:457–462CrossRefGoogle Scholar
- Rai SN, Manglik A (2012) An analytical solution of Boussinesq equation to predict water table fluctuations due to time varying recharge and withdrawal from multiple basins, wells and leakage sites. Water Resour Manag 26(1):243–252CrossRefGoogle Scholar
- Samani N, Zarei-Doudeji S (2012) Capture zone of a multi-well system in confined and unconfined wedge-shaped aquifers. Adv Water Resour 39:71–84CrossRefGoogle Scholar
- Simpson MJ, Jazaei F, Clement TP (2013) How long does it take for aquifer recharge or aquifer discharge processes to reach steady state? J Hydrol 501:241–248CrossRefGoogle Scholar
- Verhoest NE, Troch PA (2000) Some analytical solutions of the linearized Boussinesq equation with recharge for a sloping aquifer. Water Resour Res 36(3):793–800CrossRefGoogle Scholar
- Watanabe K (2014) Integral transform techniques for Green’s function. Springer, New YorkCrossRefGoogle Scholar
- Yeh HD, Chang YC (2006) New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 29(3):471–480CrossRefGoogle Scholar
- Yeh HD, Chang YC, Zlotnik VA (2008) Stream depletion rate and volume from groundwater pumping in wedge-shape aquifers. J Hydrol 349(3):501–511CrossRefGoogle Scholar
- Yeo IW, Lee KK (2003) Analytical solution for arbitrarily located multiwells in an anisotropic homogeneous confined aquifer. Water Resour Res 39:1–5CrossRefGoogle Scholar
- Zlotnik VA, Kacimov A, Al-Maktoumi A (2017) Estimating groundwater mounding in sloping aquifers for managed aquifer recharge. Groundwater 55(6):797–810CrossRefGoogle Scholar
- Zlotnik VA, Tartakovsky DM (2008) Stream depletion by groundwater pumping in leaky aquifers. J Hydrol Eng 13(2):43–50CrossRefGoogle Scholar