We use fine-grid streamline simulation to demonstrate that anomalous (non-Gaussian) transport arises from purely advective movement through heterogeneous systems.The simulation model represents a channeled sandstone North Sea oil field and contains over one million grid blocks. We find the average location and spreading of the plume and the breakthrough curves. These features are consistent with anomalous transport described by an exponent ¿ that characterizes the long-time tail of the transit time distribution. ¿ depends on the degree of heterogeneity and whether the tracer was injected or originally uniformly distributed across the domain. Incorporating dispersion to account for sub-grid-block heterogeneity does not affect the results.