In a river claims problem, agents are ordered linearly, and they have both an initial water endowment as well as a claim to the total water resource. We provide characterizations of two solutions to this problem, using Composition properties which have particularly relevant interpretations for the river claims problem. Specifically, these properties relate to situations where river flow is uncertain or highly variable, possibly due to climate change impacts. The only solution that satisfies all Composition properties is the ‘Harmon rule’ induced by the Harmon Doctrine, which says that agents are free to use any water available on their territory, without concern for downstream impacts. The other solution that we assess is the ‘No-harm rule’, an extreme interpretation of the “no-harm” principle from international water law, which implies that water is allocated with priority to downstream needs. In addition to characterizing both solutions, we show their relation to priority rules and to sequential sharing rules, and we extend our analysis to general river systems.