before the compute terrain, you add a displacement shader. Into the child port of the displacement shader, you add an image map shader (to load and project your river).
You would use projection on x/z and remember to scale the image, maybe like 5000 / 5000 or so. I don't know you'll have to try this out.
Then, in the displacement shader, you would set the displacement multiplier to - say - minus 20. THis is how deep your river will be, where the river mask is pure white.
And that's it.
Regards,
Frank