Summary: The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the upstream point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequence of these non-standard boundary conditions is that dispersion-reaction equations on both oxygen indicators are strongly coupled and the resulting system becomes ill-posed. The main purpose here is a finite element space-discretization of the variational problem. Mixed finite elements turn out to be well fitted and yield a non-symmetric saddle point system. The obtained semi-discrete problem is a differential algebraic equation that needs specific tools for its analysis. Combining analytical calculations and theoretical justifications, we try to elucidate the main properties of this ill-posed dynamical problem and understand its mathematical structure.