It is impossible to march through a river. Movement is otherwise unaffected. a unit at least partially in a river can never gain rank bonus.
Most rivers have at least one place at which they can be crossed, such as a bridge or a ford. Such crossing points should normally be treated as open terrain, but only if the unit attempting to use the crossing point is in a narrow enough formation to fit across, otherwise treat the entire unit as being in the river. The width of the bridge or ford should be determined upon placing the terrain feature.
Below is a selection of different rivers that you can choose to include in your games: