Analytical solutions of unsteady drainage problems for soils subjected to variable recharge from a semi-confined aquifer

In drainage of agricultural lands, the upward vertical recharge from a semi-confined aquifer depends on the difference of the piezometric heads on the two sides of the semi-impermeable layer through which this recharge takes place. This means that the recharge through the semi-impermeable base depends on the unknown height of the unsteady water table. In the nonhomogeneous Boussinesq equation, which describes the drainage problems, the downward uniform rate of the recharge from rain or irrigation and the recharge from the semiconfined aquifer are expressed by two terms. By solving the Boussinesq equation expressions for the nondimensional height of the water table and the nondimensional discharge of the drains per unit drained area are obtained for three different initial conditions. Some known solutions are shown as special cases of the present solutions. Variation of nondimensional water table heights at half distance of the drain spacing and the nondimensional discharge of the drains with nondimensional time have been graphically illustrated with the help of synthetic examples.Notation B _s thickness of the semi-impervious layer L] - c hydraulic resistance of the semi-impervious layer T] - D depth of the drains from the base L] - d _e equivalent depth L] - h=h(x, t) height of the water table L] - h ₀ initial height of the water table L] - h _t water table height at mid-distance of drains att L] - h _j ,h _k water table height at mid-distance of drains at timej andfk, respectively L] - H ₀ piezometric head in the semi-confined aquifer L] - K hydraulic conductivity of the soil LT^–1] - K _s hydraulic conductivity of the semi-impervious layer LT^–1] - k ₀,k ₁,k ₂ nondimensional constants - L distance between the drains L] - q ₀ upward recharge per unit surface area through the semi-impervious layer LT^–1] - q _t discharge per unit drainable area of drains at timet LT^–1] - R,R ₀ recharge per unit surface area from rain or irrigation during the unsteady and steady-state, respectively, LT^–1] - S specific yield of the soil - t time of observation T] - x distance measured from the drain L] - leakage factor L] - nondimensional distance - nondimensional time