Perturbation solutions of the Boussinesq equation for horizontal flow in finite and semi-infinite aquifers

Groundwater flow in an unconfined aquifer can be described by the Boussinesq equation. The Boussinesq equation is nonlinear and approximate solutions exist only for specific flow conditions. In the present work, the nonlinear equation is tackled analytically via a refinement of its linearized solution in a perturbation series framework. The perturbation series solution applies for any time-dependent variation of the stream level, and they are evaluated for various flow recharge and drainage conditions. Particular results for a sudden change and a piecewise linear variation are presented for both finite and semi-infinite aquifers. The approximate solutions can simulate the original nonlinear behavior with high accuracy due to the introduction of the representative depth parameter alpha that is evaluated in an optimal fashion for any given flow condition. The derivation of the optimal alpha parameter has also allowed the generalization and improvement of the classical series results pertaining to the abrupt rise or drop of the stream level in a semi-infinite aquifer. In the case of a gradual variation of the stream level in a finite aquifer domain, the linear solution with a time-varying estimate of alpha is of sufficient accuracy for practical purposes. Explicit algebraic expressions are presented for the exchange flow rates and the associated flow volumes that are of prime interest in stream-aquifer interaction studies.

Automated summary

The study explores the analytical solution of the nonlinear Boussinesq equation using a perturbation series framework, providing approximate solutions for various groundwater flow conditions. By introducing the representative depth parameter alpha, these approximate solutions can accurately simulate the original nonlinear behavior and evaluate different variations in stream level.