We consider the behaviour of the interface (free boundary) between fresh and salt water in a porous medium (a reservoir). The salt water is below the interface (with respect to the direction of gravity) and is stagnant. The fresh water is above the interface and moves towards the wells which are present in the reservoir. We give a description of the corresponding flow problem leading to a weak variational formulation involving a parameter Q which is related to the strength of the wells. We show that Q is a critical parameter in the following sense: there exists Qcr > 0 such that for Q <Qcr a smooth interface exists which is monotone with respect to Q. For Q = Qcr, a free boundary with one or more singularities (cusps) will occur at a positive distance from the wells. The global analysis for the problem (existence, uniqueness, monotonicity) is given here for two and three dimensional flow situations. The local cusp analysis is two-dimensional, and will be discussed in Part II.