Upstream, towards the source, up the stream, against the current, upstream from, higher up the river, towards the headwaters, towards the mountains, against the tide, counterflow. These are all synonyms that can be used instead of "up river". Each of these phrases implies movement against the flow of the river towards its origin. These phrases can be useful in mapping out river journeys, describing geographies and describing water currents. Whether you're a river conservationist or a nature enthusiast, incorporating these synonyms into your vocabulary will help you communicate more effectively and concisely.