We consider the problem of sharing water among agents located along a river, who have quasi-linear preferences over water and money. The benefit of consuming an amount of water is given by a continuous, concave benefit function. In this setting, a solution efficiently distributes water over the agents and wastes no money. Since we deal with concave benefit functions, it is not always possible to follow the usual approach and define a cooperative river game. Instead, we directly introduce axioms for solutions on the water allocation problem. Besides three basic axioms, we introduce two independence axioms to characterize the downstream incremental solution, introduced by Ambec and Sprumont (J Econ Theory 107:453-462, 2002), and a new solution, called the UTI incremental solution. Both solutions can be implemented by allocating the water optimally among the agents and monetary transfers between the agents. We also consider the particular case in which every agent has a satiation point, constant marginal benefit equal to one up to its satiation point and marginal benefit of zero thereafter. This boils down to a water claim problem, where each agent only has a nonnegative claim on water, but no benefit function is specified. In this case, both solutions can be implemented without monetary transfers. © 2013 Springer-Verlag Berlin Heidelberg.