Is the forward problem of ground water hydrology always well posed?

Complex aquifer systems are often modeled with quasi-three-dimensional models, which consider two-dimensional horizontal flow in the aquifers and one-dimensional vertical flow through aquitards. When the aquifer system consists of a phreatic aquifer and one or more semiconfined aquifers connected by aquitards, the discrete model consists of a nonlinear system of algebraic equations, because the transmissivity of the phreatic aquifer depends on the phreatic head. If the water extraction is very high, the phreatic aquifer can be depleted and the equations of the model must be modified accordingly. There are not simple and general criteria to state if the phreatic aquifer is depleted before solving the system of equations. Therefore, the iterative procedures (e.g., relaxation methods), used to find the solution to the forward problem, must handle these particular conditions and can suffer several problems of convergence. These problems can be caused by the choice of the initial head values or of the relaxation coefficient of the iterative algorithms; however, they can also be caused by the nonexistence or nonuniqueness of the solution to the system of nonlinear equations. The study of existence and uniqueness of the general problem is very difficult and, therefore, we consider a simplified problem, for which the discrete model can be handled analytically. The results of the numerical experiments show that the solution to the forward problem can be nonunique. Only for some cases it is possible to invoke physical arguments to eliminate tentative solutions.