  In this paper we study the generalized Buckley-Leverett equation with
nonlocal regularizing terms. One of these regularizing terms is diffusive,
while the other one is conservative. We prove that if the regularizing terms
have order higher than one (combined), there exists a global strong solution
for arbitrarily large initial data. In the case where the regularizing terms
have combined order one, we prove the global existence of solution under some
size restriction for the initial data. Moreover, in the case where the
conservative regularizing term vanishes, regardless of the order of the
diffusion and under certain hypothesis on the initial data, we also prove the
global existence of strong solution and we obtain some new entropy balances.
Finally, we provide numerics suggesting that, if the order of the diffusion is
$0< \alpha<1$, a finite time blow up of the solution is possible.
