Spontaneous stratification of granular mixtures has been reported by Makse et al. [Nature (London) 386, 379 (1997)] when a mixture of grains differing in size and shape is poured in a quasi-two-dimensional heap. We study this phenomenon using two different approaches. First, we introduce a cellular automaton model that illustrates clearly the physical mechanism; the model displays stratification whenever the large grains are rougher than the small grains, in agreement with the experiments. Moreover, the dynamics are close to those of the experiments, where the layers are built through a "kink" at which the rolling grains are stopped. Second, we develop a continuum approach, based on a recently introduced set of coupled equations for surface flows of granular mixtures that allows us to make quantitative predictions for relevant quantities. This approach includes amplification (i.e., static grains entering the flow of rolling grains), a phenomenon neglected in the cellular automaton model. We study the continuum model in two limit regimes: the large flux or thick flow regime, where the percolation effect (i.e., segregation of the rolling grains in the flow) is important, and the small flux or thin flow regime, where all the rolling grains are in contact with the surface of the sandpile. (1) In the thick-flow regime, where most experiments are carried out, the flowing grains are segregated in the rolling phase; as they are flowing down, the large rolling grains are convected to the top of the rolling layer, and only the small rolling grains interact with the sandpile. We include this effect in the continuum model and find results very close to the experiments. (2) In the thin-flow regime, we find interesting results that are close to the thick-flow limit. However, due to the presence of cross-amplification processes, we find a small regime that shows stratification when the small grains are slightly rougher than the large grains, but stratification is much more pronounced if the large grains are rougher. We study in detail the dynamical process for stratification, where the layers are built through a "kink" mechanism, and find the dependence of the size of the layers on the parameters of the system. We find that the wavelength of the layers behaves linearly with the flux of grains. We also find a crossover behavior of the wavelength at the transition from the thin-flow to the thick-flow regime. We obtain analytical predictions for the shape of the kink giving rise to stratification as well as the profile of the rolling and static species when segregation of the species is observed.
