This is the stochastic root-finding problem, as in The stochastic root-finding problem: Overview, solutions, and open questions.
I don't have electrostatics training but your second picture looks OK to me. Your conductive inclusion is essentially shorting out the plates u1 and u2. I was taught that there is no electric ...
I'm not a river-ologist but you may want to account for the depth at which you've measured the velocity. http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/RiverViscosity.htm
this is a multidimensional array C++ library https://github.com/ContinuumIO/libdynd
np.apply_along_axis(f, 2, x).T