A sandpile model describing some of the features of plasma turbulent transport dynamics in the L-mode is extended, by adding appropriate new dynamics, to exhibit a transition to enhanced confinement modes. As a result, H-modes with and without edge localized modes (ELMs) can both be obtained by varying the appropriate parameters. Each exhibits features reminiscent of what is observed in confined plasmas. The interplay between an avalanche and a diffusive transport mechanism is shown to be essential, in this context, for the system to display periodic edge ELMing.