For analyzing shallow-water flows over the uneven bottom, the HLLL scheme and the divergence form for bed slope source term (DFB) technique, respectively were applied to the flux gradient and the bottom gradient source terms in a finite-volume model for the shallow water equations. And also the model incorporated the volume/free-surface relationship (VFR) to consider the partially submerged cells (PSC). It was identified that a simpler version of the weighted surface-depth gradient method in the MUSCL was equivalent to the original one in the accuracy for 1D steady flows. It was verified that the flux gradient term and the bottom gradient source term were well-balanced exactly by the VFR for the 1D PSC. The VFR for the triangular PSC settled the problem which the governing equations were not well-balanced by the DFB technique for the 2D PSC. There were good agreements in simulations and experiments for 2D dam-break flows over a triangular sill and a round bump. In addition, the partial dam-break flow was successfully simulated for flooding of roughnesses in an irregular bottom as well as a sloping one. Therefore, this model is expected to be applied to the real river with uneven topography.