With increasing scarcity in water resources around the world, policy-makers are increasingly looking to water trading arrangements to ensure that available surface water is used as efficiently as possible. However, traded water must be able to be transported from its source to its point of use. In many real-world applications this transportation occurs over a gravity-fed river network. The physical characteristics of this network (such as the shape of weirs) determines how quickly water flows from point to point, as well as setting upper and lower limits on the rate of flow. The choice of the path of injections and extractions which maximises overall welfare must take into account these physical characteristics. This paper characterises the profile of water injections and extractions, and the corresponding path of prices, that maximises overall economic welfare, subject to the hydrology of a stylised river network. We illustrate the outcomes of the model in simple water networks, and show how even relatively simple changes in supply and demand conditions can lead to dynamic variation in the profile of prices and injection/extraction across the river system. We show how this model can act as the foundation for a smart market for water trading arrangements.