We present SurfRiver, a new visual transformation approach that flattens stream surfaces in 3D to rivers in 2D for comparative visualization. Leveraging the TextFlow-like visual metaphor, SurfRiver untangles the convoluted individual stream surfaces along the flow direction and maps them along the horizontal direction of the abstract river view. It stacks multiple surfaces along the vertical direction of the river view. This visual mapping makes it easy for users to track along the flow direction and align stream surfaces for comparative study. Through brushing and linking, the river view is connected to the spatial surface view for collective reasoning. SurfRiver can be used to examine a single stream surface, investigate seeding sensitivity or variability of a family of surfaces from a group of related seeding curves, or explore a collection of representative surfaces. We describe our optimization solution to achieve the desirable mapping, present SurfRiver interface and interactions, and report results from different flow fields to demonstrate its efficacy. Feedback from a domain expert also indicates the promise of SurfRiver.