Choosing apropriate Routing method

Hello dear civil/environmental engineering friends.

I need to find the best method to model this situation:
A waste water treatment plant discharges constantly into a small otherwise dry creek. There is flow in the creek that ends after ~1-2 km, depending on season (more and less evaporation). Meaning: the entire flow is absorbed eventually into the ground, in a varying distance from the source.
I need a model that I can use to estimate where will the flow end (distance from the source).

I have time series of discharge into the creek (constant) and at a point in the middle of the flow length.
I have atmospheric parameters for evaporation evaluation.
For the simple case I will assume constant creek properties.

What routing method could I use for this task?

Apreciate your help!

Nitzan