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Water table variations between drains have been investigated by various researchers in response to transient recharge. Recent studies have shown the importance of incorporating the effect of evapotranspiration (ET) in the design of subsurface drainage systems. In arid and semi-arid regions, ET plays a crucial role in lowering the water table resulting in increased drain spacing. In this paper, a numerical solution of two-dimensional free surface flow to ditch drains is presented in presence of transient recharge and depth-dependent ET from land surface for an aquifer with sloping impermeable base. The midpoint water table variations obtained from the proposed solution compare well with experimental results as well as already existing mathematical solution. When ET from the land surface is taken into account in combination with recharge, the model results can provide accurate and reliable estimates of water table fluctuation under complex situations, which are highly related to the hydrology of waterlogged and saline soils. NotationsL = Length of aquifer [L] B = Width of aquifer [L] b = A constant that depends on soil type [T 1 ] C = T / X E(h) = ET as a function of water table height [LT 1 ] E 0 = Constant ET [LT 1 ] f = Drainable porosity [L 3 /L 3 ] h = Variable water table height [L] h 0 = Initial water table height [L] H = Dimensionless water table height [L/L] H n l,m = Dimensionless water table elevation at 'l' th and 'm' th nodes corresponding to 'n' th time level K = Hydraulic conductivity [LT 1 ] l, m, n = Subscripts denoting the variables in space and time 780 S. SINGH AND C. S. JAISWAL R(t) = Transient rate of recharge [LT 1 ] R 0 + R 1 = Initial rate of transient recharge [LT 1 ] R 1 = Final rate of transient recharge [LT 1 ] r = Decay constant for recharge rate [T 1 ] t = Time [T] T = Dimensionless time V n l,m = Value of dimensionless head at 'l' th and 'm' th nodes corresponding to 'n' th time level x,y = Coordinate axes [L] X, Y = Dimensionless space coordinates α = Slope of impermeable aquifer base = Increment operator θ = Coefficient used to discriminate the finite-difference scheme

Experiments were conducted on a vertical Hele-Shaw model to study the effect of slope of an impermeable layer on flow profiles and flow rates in an unconfined aquifer. Experimental results were compared with the solutions of Pavlovsky (1930) and Childs (1971) for nonuniform seepage on a small sloping impermeable bed. These studies showed that the solution of Pavlovsky may be used for the prediction of the flow profile downslope up to 30% slope and upslope up to 15% slope. Pavlovsky's equations also predicted flow rates and normal depths satisfactorily up to 30% slope. Childs' equations also predicted similar results. None of these equations predicted the flow rate on negative slope satisfactorily. The flow of groundwater over an impermeable horizontal bed was analyzed by Dupuit [1863]. Boussinesq [1904] analyzed the unsteady state flow over a sloping bed with a small slope and having a small thickness of flow as compared to the lateral extent of aquifers. He assumed that the inertial forces are negligible and that the horizontal component of the velocity does not change with the height. As reported by Hart [1962] and Aravin and Numerov [1965], Pavlovsky analyzed the problem of slowly varying seepage through an unconfined aquifer resting on a sloping impermeable bed, and he arrived at three different possible solutions for different cases. He assumed that the streamlines of flow have very small curvature, have useful cross section of the flow approximate to planes, and are almost parallel to the bed. The head gradient was evaluated as dy/dl, where y is the height of intersection with the water table and I is the distance measured along the bed.

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An analytical solution for two-dimensional water table variation in an aquifer basin with inclined impermeable base has been proposed incorporating the effect of depth-dependent evapotranspiration (ET) from land surface. The proposed analytical solution has been obtained by devising a sound mathematical transformation, which transforms a complex partial differential equation into the simplest form. The results obtained from the proposed solution are in good agreement with the already existing mathematical solutions. The results of the proposed solution have been illustrated to study the variation of water table in the 2-D aquifer. Since the water table profiles are obtained lower by consideration of ET, the solution will result in wider drain spacing and, thus, economy in the drainage design. The proposed analytical solution can be used as an important tool for reliable prediction of water table variation in the salinity affected croplands of arid and semi-arid regions where the ET rate is very high. NotationsB Width of the aquifer [L] b A constant that depends on soil type [T −1 ] D Average depth of flow [L] E(h) ET as a function of water table height [LT −1 ] E 0 Constant ET [LT −1 ] f Drainable porosity [L 3 /L 3 ] h Variable water table height [L] h 0 Initial water table height [L] K Hydraulic conductivity [LT −1 ] L Length of the aquifer [L]