Paper 2 of this three-part series starts with a discussion of the question, Under what conditions is the aquifer inverse problem well-posed? After defining the terms uniqueness, identifiability, and stability, theoretical considerations and synthetic examples are used to demonstrate that ill-posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized. At the same time, the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases. Several test problems in the recent literature, which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem, are shown to be ill-posed; a question is thus raised about the validity of these conclusions in the general case. Various conjugate gradient algorithms, coupled with the adjoint state finite element method for computing the gradient of the estimation criterion, and with Newton's method for determining the optimum step size downgradient, are compared. A marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge.