Unsteady seepage flow over sloping beds in response to multiple localized recharge
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Abstract
New generalized solutions of linearized Boussinesq equation are derived to approximate the dynamic behavior of subsurface seepage flow induced by multiple localized timevarying recharges over sloping ditch–drain aquifer system. The mathematical model is based on extended Dupuit–Forchheimer assumption and treats the spatial location of recharge basins as additional parameter. Closed form analytic expressions for spatiotemporal variations in water head distribution and discharge rate into the drains are obtained by solving the governing flow equation using eigenvalue–eigenfunction method. Downward and zerosloping aquifers are treated as special cases of main results. A numerical example is used for illustration of combined effects of various parameters such as spatial coordinates of the recharge basin, aquifer’s bed slope, and recharge rate on the dynamic profiles of phreatic surface.
Keywords
Boussinesq equation Recharge Slopes Ditch–drain Groundwater moundList of symbols
 D
Average saturated depth of the aquifer [L]
 H
Dimensionless water head height in ith zone
 H^{*}(X)
Dimensionless steadystate water head height
 h(x, t)
Water head height measured from sloping bed [L]
 h_{0}
Initial water level in the drains [L]
 ħ (x, t)
Variable water head height measured from horizontal datum [L]
 K
Hydraulic conductivity [L/T]
 L
Lateral extant of the unconfined aquifer [L]
 m
Number of recharge basins in the domain []
 N_{i}, N_{f}
Initial and final recharge rate per unit area of the aquifer [LT^{−1}]
 N_{ki}, N_{kf}
Initial and final recharge rate in the kth basin [LT^{−1}]
 Q
Dimensionless flow rate
 Q(X = 0, τ)
Dimensionless flow rate into left drain
 Q(X = 1, τ)
Dimensionless flow rate into right drain
 Q_{0}^{*}
Dimensionless steadystate flow rate into the left drain
 Q_{1}^{*}
Dimensionless steadystate flow rate into the right drain
 q
Flow rate per unit area of the aquifer [L^{2} T^{−1}]
 R(x, t)
Source term in the domain [LT^{−1}]
 S
Specific yield []
 t
Time [T]
 X
Dimensionless spatial coordinate
 x
Horizontal xaxis [L]
 x_{i}, x_{i}′
Spatial coordinates of the end points of the recharge basin [L]
 x_{k}, x_{k}′
Spatial coordinates of the kth recharge basin [L]
 β
Sloping angle measured in radian
 α
Dimensionless value of sloping angle β
 λ
Recharge rate controlling parameter [T^{−1}]
 λ_{k}
Recharge rate controlling parameter in the kth basin [T^{−1}]
 τ
Dimensionless time
Introduction
Managing sustainability of limited water resources is a challenging task in the face of everincreasing demand for water and the geotechnical problems associated with overuse of available water. Proper resource management techniques based on precise information of groundwater response to recharge and withdrawal activities can help in improving sustainability and efficiency of an aquifer. In this context, mathematical and numerical models have emerged as effective tools to determine what is currently happening in an aquifer and what will happen as a result of variation in hydrologic and hydraulic parameters.
In areas where overdevelopment has depleted groundwater resources, artificial recharge is often used for augmentation of subsurface water reservoir and to enhance the sustainable yield of the region. Analytic models are extensively used for assessing the temporal development of groundwater system in response to controlled activities such as localized replenishment and pumping from wells. Evolution and stabilization of the free surface in unconfined semiinfinite aquifers subjected to rectangular recharge at uniform rate was first analyzed by Glover (1960) and Hantush (1967). Since then, numerous mathematical models have been developed to predict the dynamic behavior of groundwater in response to the constant or periodically applied recharge in an unconfined aquifer (Hunt 1971; Marino 1974; Rao and Sarma 1981; Manglik et al. 1997; Rai and Manglik 1999). Numerical model developed by Zomorodi (1991) highlighted the effects of unsaturated zone on the recharge rate and the effect of intransit water in reducing the fillable pore space. The calculation presented in his study demonstrated that estimation of the recharge rate by a constant value can severely misjudge the actual results. Rastogi and Pandey (1998) simulated numerically the phreatic head distribution in response to constant recharge from recharge basins of different geometry but of equal areas. They showed that the mound height underneath a basin decreased when the perimeter of the recharge basin increased. Manglik et al. (1997) proposed a new method for simulating the discontinuous cycles of recharge operation by a sequence of line segments of varying lengths and slopes. Rai et al. (2006) and Rai and Manglik (2012) applied this scheme to predict the water head distribution in unconfined aquifers for multirecharge and pumping operations. Although, these studies provide useful insight into the groundwater flow system; the upland watershed hydrology concerning subsurface drainage over hillslope cannot be satisfactorily explained with these results.
The movement of groundwater depends considerably on the stratigraphical and structural units of the region. In hillslope terrains, the flow of groundwater inherits certain unique features that are seldom seen in horizontal strata. Most importantly, aquifers in hillslope regions are underlain by sloping impervious beds. In such cases, approximation of groundwater flow based on the assumption that the streamlines are nearly parallel to the sloping bed (Dupuit–Forchheimer assumption) yields more accurate results. Childs (1971) formulated the unsteady groundwater flow in skewed coordinate system (along and perpendicular to the sloping base) by a nonlinear Boussinesq equation. Chapman (1980) converted this approximation in rectangular coordinate system. Analytic solutions of linearized Boussinesq equation under varying hydrologic conditions are presented by a number of investigators (Brutsaert 1994; Verhoest and Troach 2000; Upadhyaya and Chauhan 2001; Bansal and Das 2009, 2011; Bansal 2012). Using finite Fourier transform, effects of localized recharge from single basin on the evolution of groundwater mound in downward sloping aquifers of rectangular shape were analyzed by Ram and Chauhan (1987), Singh et al. (1991) and Ramana et al. (1995). By considering varying hydraulic conductivities along x and yaxes, Chang and Yeh (2007) analyzed twodimensional transient groundwater flow in a sloping aquifer due to timevarying recharge from single recharge basin and extraction from multiwells. Closed form analytic expressions for spatiotemporal variations in water head over semiinfinite sloping bed due to localized recharge is presented by Bansal and Das (2010) and Bansal (2013). The aforementioned studies highlight the importance of bed slope in the determination of transient profiles of phreatic surface; however, the results are either exceedingly complicated or suffer from slow convergence.
The aim of the current study is to develop new analytic model for prediction of groundwater mound in a sloping ditch–drain aquifer system due to multiple localized recharge. The study uses a simple approach to develop analytical expressions for water head distribution in the aquifer due to uniform percolation from a single recharge basin. The solution is then extended to multiple basins of varying recharge rate which are arbitrarily located in the model domain. Unlike previous works based on Laplace transform technique, the current study treats the entire aquifer as a single zone system. Closed form analytic expressions for water head distribution in the aquifer and discharge rate into the ditches are obtained by solving the linearized Boussinesq equation with eigenvalue–eigenfunction method. Some special cases such as no slope and uniform localized recharge are derived as limiting cases of the analytic results. Combined effects of bed slope, spatial coordinate of recharge basin, and the recharge rate parameter on the transient profiles of groundwater mound are illustrated with the help of a numerical example.
Development of model for single recharge basin
Extension of the model for multirecharge basins
Discharge rate into drains
Numerical solution of the nonlinear flow equation
Discussions of results
To demonstrate the combined effects of bed slopes and timevarying localized recharges, a numerical example is considered with K = 2.5 m/d, S = 0.25, h _{0} = 5 m, and L = 200 m. Recharge schemes are applied through two basins R_{1} and R_{2} extending in the ranges of 25 ≤ x ≤ 50 m and 150 ≤ x ≤ 175 m, respectively. In the current example, it is assumed that the recharge schemes applied in both R_{1} and R_{2} are the same. The initial recharge rate is 4 mm/h which finally drops to 2 mm/h by the exponential function 2 + 2 e^{−0.2t }. Three different values of sloping angle are considered, namely, β = 10°, 0°, and −10°. Analytic values of water head are computed using Eq. (35) in which the mean saturated depth D of the aquifer is successively approximated using an iterative formula D = (h _{0} + h _{ t })/2, where h _{ t } is the water table height at time t at the end of which D is approximated. From computational viewpoint, implementation of Eq. (35) is simple and straightforward. Numerical experiments with various values of aquifer parameters indicate that the infinite sequence in the righthand side of Eq. (35) converges very fast to a final value. We have taken first 50 terms of the sequence to represent the sum of the whole infinite series. Performance of analytic solution is tested against numerical solution of the nonlinear Boussinesq equation at 200 equally spaced grid points of length 1 m for t = 2, 5, 10, 20, and 50 days. It is observed that the two solutions are in excellent agreement during the initial stages (t = 2, 5, 10, and 20 days). As time increases (e.g., 50 days), analytic solutions marginally underestimate the numerical solutions. Numerical experiments do not reveal any definite interplay between the relative percentage difference (RPD) and the aquifer parameters. However, a clear observation is that the RPD is least in the proximity of aquifer–drain interfaces, and decreases as λ decreases.
Comparison of water head height under the midpoint of R_{1} (x = 37.5 m) due to constant (λ = ∞) and timevarying (λ = 0.2 d^{−1}) recharge
t (days)  β = 0°  β = 10°  β = −10°  

λ = ∞  λ = 0.2  λ = ∞  λ = 0.2  λ = ∞  λ = 0.2  
2  5.3101  5.5602  5.3097  5.5593  5.3097  5.5593 
5  5.6003  5.9516  5.5912  5.9354  5.5911  5.9353 
10  5.9318  6.264  5.8963  6.2089  5.8946  6.206 
20  6.3504  6.5618  6.2653  6.4551  6.2367  6.4111 
50  7.0092  7.0976  7.0857  7.2167  6.6005  6.6371 
Comparison of water head height under the midpoint of R_{2} (x = 162.5 m) due to constant (λ = ∞) and timevarying (λ = 0.2 d^{−1}) recharge
t (days)  β = 0°  β = 10°  β = −10°  

λ = ∞  λ = 0.2  λ = ∞  λ = 0.2  λ = ∞  λ = 0.2  
2  5.3101  5.5602  5.3097  5.5593  5.3097  5.5593 
5  5.6003  5.9516  5.5911  5.9353  5.5912  5.9354 
10  5.9318  6.264  5.8946  6.206  5.8963  6.2089 
20  6.3504  6.5618  6.2367  6.4111  6.2653  6.4551 
50  7.0092  7.0976  6.6005  6.6371  7.0857  7.2167 
Conclusions

Unlike horizontal aquifers, the groundwater mound in sloping aquifers is asymmetric and its peak drifts along the bed slope.

Given two basins of equal dimensions and subjected to same recharge rate, the mound height beneath the upslopelocated basin is lesser than that of the down slopelocated basin.

The bed slope accelerates the process of stabilization of water head profiles and significantly controls the mound height.

The prediction of spatiotemporal variations of water head distribution can be erroneous if the recharge rate is approximated by a constant value.
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