The Bialas-Bzdak model of elastic proton-proton scattering assumes a purely imaginary forward scattering amplitude, which consequently vanishes at the diractive minima. We extended the model to arbitrarily large real parts in a way that constraints from unitarity are satised. The resulting model is able to describe elastic pp scattering not only at the lower ISR energies but also at p s =7 TeV in a statistically acceptable manner, both in the diractive cone and in the region of the rst diractive minimum. The total cross-section as well as the dierential cross-section of elastic proton-proton scattering is predicted for the future LHC energies of p s =8, 13, 14, 15 TeV and also to 28 TeV. A non-trivial, signicantly nonexponential feature of the dierential cross-section of elastic proton-proton scattering is analyzed and the excitation function of the non-exponential behavior is predicted. The excitation function of the shadow proles is discussed and related to saturation at small impact parameters.