A two-step explicit solution of the Boussinesq equation for efficient simulation of unconfined aquifers in conjunctive-use models

Mathematical models that simultaneously simulate surface and groundwater components and their interaction are required to evaluate management alternatives in conjunctive-use systems. Complex systems and scenarios defined over long time periods often force the use of computationally efficient aquifer models. Thus the groundwater flow equation needs to be solved using explicit techniques such as influence functions or the eigenvalue technique. The eigenvalue technique provides a continuous-in-time solution using a state equation with important computational advantages. These methods are strictly applicable only to confined aquifers, which are modeled with a linear groundwater flow equation. However, many commonly exploited aquifers connected with the surface system are unconfined and should be simulated using the nonlinear Boussinesq equation. A two-step explicit solution of the unconfined groundwater flow problem is presented for aquifers with nearly horizontal bottom. It is based on a new approach to linearize the Boussinesq equation. Using a change of variable, it is possible to define an equation with a structure similar to the linear groundwater flow equation. The only difference is found in a term that depends on the solution. Approaching this term by means of a fictitious stress, we obtain a linear equation analogous to the confined groundwater flow equation. The boundary conditions can also be formulated as a function of this new variable with linear expressions. Therefore the unconfined groundwater flow problem can be solved by applying the superposition principle, and the solution can be obtained with a reduced computational cost using the eigenvalue technique.