# Improved Solutions to the Linearized Boussinesq Equation with Temporally Varied Rainfall Recharge for a Sloping Aquifer

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Conceptual and Mathematical Models

^{2}/T), K is the hydraulic conductivity (L/T), ${H}_{w}$ is the elevation of the groundwater table measured perpendicularly to the underlying impermeable layer (L), $\theta $ is the inclined angle of the aquifer bottom (-), and r = r(t) is the rainfall recharge rate (L/T).

#### 2.2. Present Improved Solutions

## 3. Results and Discussions

#### 3.1. Comparison of Analytical and Numerical Solutions

#### 3.2. Comparison of Unsteady State and Quasi-Steady State

^{−5}m for the groundwater level and 10

^{−3}m

^{2}/day for the outflow.

## 4. Conclusions

- According to the error analysis, in the case of a constant recharge rate for a sloping aquifer, the results of the proposed solution are better than the results proposed by Verhoest and Troch [7] after comparing with the numerical solutions; therefore, the present analytical solution appears to be more feasible than that proposed in a previous study.
- The proposed solutions reach the convergence criteria faster than the solutions of Verhoest and Troch [7], thus saving computation time.
- The present solution can be directly applied to unsteady recharge rate cases without the requirement of the quasi-steady state method which was employed in the study of Verhoest and Troch [7].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Paniconi, C.; Wood, E. A detailed model for simulation of catchment scale subsurface hydrological processes. Water Resour. Res.
**1993**, 29, 1601–1620. [Google Scholar] [CrossRef] - Brutsaert, W. The unit response of groundwater outflow from a hillslope. Water Resour. Res.
**1994**, 30, 2759–2763. [Google Scholar] [CrossRef] - Chapman, T.G. Recharge-induced groundwater flow over a plane sloping bed: Solutions for steady and transient flow using physical and numerical models. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef] - Berne, A.; Uijlenhoet, R.; Troch, P.A. Similarity analysis of subsurface flow response of hillslopes with complex geometry. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef] - Troch, P.A.; Paniconi, C.; van Loon, E.E. The hillslope-storage Boussinesq model for subsurface flow and variable source ares along complex hillslopes: 1. Formulation and characteristic response. Water Resour. Res.
**2003**, 39, 1316. [Google Scholar] [CrossRef] - Dralle, D.N.; Boisramѐ, G.F.S.; Thompson, A.E. Spatially variable water table recharge and the hillslope hydrologic response: Analytical solutions to the linearized hillslope Boussinesq equation. Water Resour. Res.
**2014**, 50, 8515–8530. [Google Scholar] [CrossRef] - Verhoest, E.C.; Troch, P.A. Some analytical solution of the linearized Boussinesq equation with recharge for a sloping aquifer. Water Resour. Res.
**2000**, 36, 793–800. [Google Scholar] [CrossRef] - Arfken, G.B.; Weber, H.J. Mathematical Methods for Physicists; Academic Press Inc.: San Diego, CA, USA, 1995. [Google Scholar]
- Zissis, T.S.; Teloglou, I.S.; Terzidis, G.A. Response of a sloping aquifer to constant replenishment and to stream varying water level. J. Hydrol.
**2000**, 243, 180–191. [Google Scholar] [CrossRef] - Bansal, R.K.; Das, S.K. Response of an unconfined sloping aquifer to constant recharge and seepage from the stream of varying water level. Water Resour. Manag.
**2010**, 25, 893–911. [Google Scholar] [CrossRef] - Kazezyılmaz-Alhan, C.M. An improved solution for diffusion waves to overland flow. Appl. Math Model.
**2012**, 36, 1465–1472. [Google Scholar] [CrossRef] - Chapman, T.G. Modeling groundwater flow over sloping beds. Water Resour. Res.
**1980**, 16, 1114–1118. [Google Scholar] [CrossRef] - Childs, E.C. Drainage of groundwater resting on a sloping bed. Water Resour. Res.
**1971**, 7, 1256–1263. [Google Scholar] [CrossRef] - Özisik, M.N. Boundary Value Problems of Heat Conduction; Dover Publications Inc.: New York, NY, USA, 1968. [Google Scholar]
- Swanson, R.C.; Turkel, E. On central-difference and upwind schemes. J. Comput. Phys.
**1992**, 101, 292–306. [Google Scholar] [CrossRef] - Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys.
**1988**, 77, 439–471. [Google Scholar] [CrossRef] - Koussis, A.D. A linear conceptuals ubsurfaces torm flow model. Water Resour. Res.
**1992**, 28, 1047–1052. [Google Scholar] [CrossRef]

**Figure 2.**Spatial variation in the groundwater table under constant recharge for (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 1 day).

**Figure 3.**Spatial variation in the groundwater table under constant recharge for (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 3 days).

**Figure 4.**Spatial variation in the groundwater table under constant recharge for (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 5 days).

**Figure 5.**Relative percentage difference (RPD) between analytical and numerical solutions. (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 1 day).

**Figure 6.**RPD between analytical and numerical solutions. (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 3 days).

**Figure 7.**RPD between analytical and numerical solutions. (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$ (t = 5 days).

**Figure 8.**Transient variation of outflow under constant recharge for (

**a**) $\theta =2\xb0$ (

**b**) $\theta =6\xb0$.

**Figure 9.**Variation in the outflow corresponding to varying recharge rates, as illustrated in the study of Verhoest and Troch [7].

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**MDPI and ACS Style**

Wu, M.-C.; Hsieh, P.-C.
Improved Solutions to the Linearized Boussinesq Equation with Temporally Varied Rainfall Recharge for a Sloping Aquifer. *Water* **2019**, *11*, 826.
https://doi.org/10.3390/w11040826

**AMA Style**

Wu M-C, Hsieh P-C.
Improved Solutions to the Linearized Boussinesq Equation with Temporally Varied Rainfall Recharge for a Sloping Aquifer. *Water*. 2019; 11(4):826.
https://doi.org/10.3390/w11040826

**Chicago/Turabian Style**

Wu, Ming-Chang, and Ping-Cheng Hsieh.
2019. "Improved Solutions to the Linearized Boussinesq Equation with Temporally Varied Rainfall Recharge for a Sloping Aquifer" *Water* 11, no. 4: 826.
https://doi.org/10.3390/w11040826