Water table fluctuation in aquifers overlying a semi-impervious layer due to transient recharge from a circular basin
Introduction
Artificial recharge of unconfined aquifers by means of infiltrating or recharge basins is applied to augment groundwater resources. Solutions to the problem of water table fluctuation underneath recharge basins are essential in order to evaluate artificial recharge schemes, which depend on the variation of the infiltration rate, and furthermore, to evaluate water management projects.
Many researchers have investigated groundwater flow in unconfined aquifers owing to the recharge through infiltrating basins and have derived analytical expressions for the rise and decline of groundwater mound.
Baumann, 1952, Glover, 1960, Hantush, 1967 obtained linearised solutions of the Boussinesq equation. They considered groundwater flow in homogeneous, isotropic, unconfined aquifers receiving constant recharge from infiltrating basins that percolates vertically downwards until it reaches the water table. They assume that recharge rate is small in comparison with hydraulic conductivity and it is almost completely refracted in the direction of the slope of the water table. They also assume that the maximum rise of the water table is small relative to the initial water table height above the base of the aquifer.
Hunt (1971) solved the Laplace equation subject to a linearised boundary condition on the free surface to take into account the vertical component of the velocity vector near the free surface in the recharge zone.
Rai et al., 1994, Rai and Singh, 1995, Rai et al., 1998 among others, considered recharge rates exponentially decreasing with time to obtain analytical solutions of the linearised Boussinesq equation to predict the decline of the groundwater mound due to the decrease of recharge as a result of the clogging of the soil pores at the bottom of the basin (Bear, 1979).
Manglik et al., 1997, Manglik and Rai, 1998, Rai and Manglik, 1999, Manglik et al., 2004, Rai et al., 2006, developed analytical solutions of two-dimensional linearised Boussinesq equation to describe water table fluctuations for different recharge operations from infiltrating basins. Since in practice the basins are operated intermittently they approximated the time varying recharge by series of line segments of different length and slopes.
In all of the above-mentioned works it is assumed that the unconfined aquifer is resting on an impervious layer. However, the term “impervious” is a relative term. Commonly, in analysing groundwater systems, a low permeability layer is considered to act as an impervious barrier to flow. In many cases (Zlotnik, 2004), a significant amount of water can be transported vertically downward or upward by leakage through the low permeability layer, between hydraulically connected adjacent aquifers. It is also assumed that the pattern of horizontal flow, based on Dupuit–Forchheimer assumptions, is not seriously affected by the flow through the low permeability base.
In addition, since the recharge rate is the rate at which water replenishes the water table zone, it may be determined by the variation of the infiltration rate through the bottom of the recharge basin, but in that case the upward or downward leakage through the semi-impervious layer should be evaluated. In addition, estimates of net recharge to the unconfined aquifer will be incorrect if they are based solely on the infiltration rates determined by loss from the recharge basin, without accounting for leakage through the semi-impervious base.
The aim of this paper is to study the influence of both the semi-impervious base of the aquifer and the transient recharge rate on the development of groundwater mound underneath a circular infiltration basin. For that purpose a new analytical solution of linearised form of Boussinesq equation is presented for the fluctuation of the water table in unconfined aquifers. It is assumed that the unconfined aquifer is overlying a semi-impervious layer and the vertical downward leakage depends on the water table height. Leakage occurs over the large area where the rising of the water table takes place and this solution show that is important even when the semi-impervious layer has extremely low hydraulic conductivity. Furthermore, recharge rate is considered to decrease exponentially with time during a single cycle of recharge. In addition a recharge scheme is considered consisting of alternate cycles of varying recharge and dry periods. Various recharge cycles, each one followed by a dry period is essentially a “real life” condition. Dry cycles are commonly used in recharge basin operations to provide the necessary time for maintenance of the basin by scraping, amongst others, its bottom in an attempt to improve its infiltrability. In this paper, a case of two cycle recharge is approximated by small piecewise continuous elements, each of which may be represented in general by Nth degree polynomials. Therefore, few non-linear elements may represent accurately the variation of replenishment almost without additional computational effort. The analytical solutions were compared with the results obtained by applying the explicit Mac Cormack finite difference computational scheme which was used for the numerical solution of the non-linear form of the governing equation in order to validate the applied linearisation.
Section snippets
Governing equations
Fig. 1 illustrates the groundwater recharge cone in an unconfined aquifer, induced from a circular infiltration basin at the land surface. In analyzing the groundwater flow system the following simplifying assumptions are made: (1) the unconfined aquifer is homogeneous, isotropic, resting on a semi-impervious layer and is characterised by constant hydrogeological parameters; (2) the maximum rise of the water table is small relative to the initial water table height, h0; (3) the flow in the
Analytical solution
To solve the boundary value problem described by Eqs. (13), (14a), (14b), (14c), the finite Hankel transform (Sneddon, 1995) of the displacement F(ξ, τ), is introducedwhere J0(x) denotes the Bessel function of the first kind and zero order of argument x, and μi are the positive roots of the transcendental equation J0(μi) = 0, i = 0, 1, 2, …
Multiplying both sides of Eq. (13) by ξJ0(μiξ) and integrating with respect to ξ from 0 to 1, yields
Results
The analytical solution for the water table height variation, in response to constant recharge rate, given by Eq. (31), was compared with numerical solutions of the non-linear equation (1), in order to establish the validity of the applied linearisation.
The Mac Cormack finite difference explicit computational scheme was used to obtain the numerical solution of Eq. (1) (Mac Cormack, 1969). It is a two-step (predictor–corrector) scheme, of second-order accuracy, conditionally stable and
Conclusions
New analytical solutions of a linearised Boussinesq equation are presented in this study, developed for the problem of water table fluctuation in an unconfined aquifer underlain by semi-impervious layer in response to time varying recharge. Numerical solution of the non-linear form of the governing equation was used to validate the applied linearisation. It was found that proper adjustment of the weighted mean of the depth of saturation can further improve the accuracy of the solutions of the
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