A precise numerical method is developed for the simulation of contaminant transport in dual-well flow. Our main motivation is the determination of model parameters by means of dual-well tests. The method presented here guarantees a very low level of numerical dispersion. We have applied a bipolar transformation to the transport equation so that the convective term is parallel to one of the axes. In the time stepping, we split the transport and diffusion parts: the transport part is semianalytically solved (thanks to the bipolar transformation) via a multiple Riemann problem, and the diffusion part is solved using a finite volume method. Examples are provided to show that the method is suitable for parameter identification.