A semianalytical solution for the Boussinesq equation with nonhomogeneous constant boundary conditions

The Boussinesq groundwater equation is widely used in hydrology to predict streamflow from an unconfined aquifer and derive the aquifer's saturated hydraulic conductivity and drainable porosity, and to predict water table height in drainage engineering. In this work, we solve this equation in an unconfined horizontal aquifer for nonhomogeneous boundary conditions for the water table height. The solution is found in the form of a Taylor series that has a finite radius of convergence, which is different for each initial condition. We also present an expression for the flux boundary condition at the origin as a function of the depth of the adjoining stream that automatically satisfies the boundary condition at infinity, and thus eliminates the need for a trial‐and‐error approach for the solution, which is accurate to 10−7. In order to obtain an approximation for the water table height in the region where the series solution diverges, first we computed a diagonal Padé approximation from the series coefficients, which converges in a larger interval than the series, and then we matched it with a new asymptotic approximation for large values of the independent variable. We found that the proposed matched solution is better suited to cases where the water head at the origin is close to the initial water head in the aquifer. Key Points An analytical solution of the Boussinesq equation is obtained Expressions are derived for practical applications The new expressions may be applied in drainage engineering and Hydrology