  The Bialas-Bzdak model of elastic proton-proton scattering assumes a purely
imaginary forward scattering amplitude, which consequently vanishes at the
diffractive minima. We extended the model to arbitrarily large real parts in a
way that constraints from unitarity are satisfied. The resulting model is able
to describe elastic $pp$ scattering not only at the lower ISR energies but also
at $\sqrt{s}=$7~TeV in a statistically acceptable manner, both in the
diffractive cone and in the region of the first diffractive minimum. The total
cross-section as well as the differential cross-section of elastic
proton-proton scattering is predicted for the future LHC energies of
$\sqrt{s}=$13, 14, 15~TeV and also to 28~TeV. A non-trivial, significantly
non-exponential feature of the differential 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
profiles is discussed and related to saturation at small impact parameters.
