Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers

[1] An approximate nonlinear solution of the one-dimensional Boussinesq equation is presented using the traveling wave approach. The solution pertains to a semi-infinite phreatic aquifer with a uniform water table that is subject to a time-varying water level at the stream-aquifer boundary. The advantage of the traveling wave method is in its versatility in handling transient boundary conditions while preserving the inherent nonlinearity of the physical phenomenon. The nonlinear solution is of a simple logarithmic form and describes the position of the water table as a function of time. It yields the exact solution for the special case of uniform water level rise at the boundary. Algebraic expressions that quantify the main flow processes are derived from the basic solution. These include the stream-aquifer exchange flow rates, bank storage and depletion, front position and propagation speed, and an improved working relationship for aquifer parameter estimation. A comparison with two exact solutions and numerical solutions of the Boussinesq equation validates the accuracy of the approximation and highlights the limitation of the method in specific flow conditions. The traveling wave model performs best for sharp front movements and monotonic water table profiles and provides excellent estimates of the flow rate and volume at the inlet boundary. The accuracy of the solution deteriorates for fluctuating inlet conditions and worsens in cases when there is a sharp reversal of flow conditions.