Shallow water 2D on irregular bed
 

I'm beginning to code a 2D Shallow water solver using Godunov finite volume method with the Roe's Riemann solver.
I observed that there's quite a few publications on how to balance fluxes and source terms in the case of a non-horizontal bed surface.
I'm facing the following problem:
When the simulation begins with still water, Roe's solver outputs a non-zero flux for the first component of the flux vector (uh) due to different flow depths in both sides of the cell face. In other words, the horizontal water surface with irregular bed surface is solved in the same way as a horizontal bed with different water surface elevation, which results in a non-physical flux.
The result is a change in the water depth which can't be balanced by the source term, since the first component of the latter is zero.
I guess there must be a straightforward solution to this problem, but my search hasn't been successful. Can anyone please help me?
Thanks
Ricardo