One workflow would be this:
1. Run Merge on the two shapefiles. Merge—Help | ArcGIS Desktop
2. Run Dissolve on the result of the Merge. Dissolve—Help | ArcGIS for Desktop
3. Then field calculate the lengths (Calculate Geometry) to update them (as you are using shapefiles, which don't auto-update like feature classes in a geodatabase).
Assumptions:
1. The shapefiles are coincident in that the river segments that do overlap are in the same location. If they are not, for example say they were collected at different accuracy's and are off by a foot from each other, this process will not work. However, If they are off, there are processes to try to fix that. If so, post that and we can suggest a few.
2. The shapefiles only are of polluted features - ie they do not contain segments of unpolluted features. If they do contain unpolluted segments, a pre-processing step will be needed to thin down the shapefiles to just the polluted segments before running this process.
Chris Donohue, GISP