A general state-based peridynamic formulation is presented for convective single-phase flow of a liquid of small and constant compressibility in heterogeneous porous media. In addition to local fluid transport, possible anomalous diffusion due to non-local fluid transport is considered and simulated. The governing integral equations of the peridynamic formulation are computationally easier to solve in domains with discontinuities than the traditional conservation models containing spatial derivatives. A bond-based peridynamic formulation is also developed and demonstrated to be a special case of the state-based formulation. The non-local model does not assume continuity in the field variables, satisfies mass conservation over an arbitrary bounded body and approaches the corresponding local model as the non-local region goes to zero. The exact solution of the local model closely matches the non-local model for a classical two- dimensional flow problem with fluid sources and sinks and for both Neumann and Dirichlet boundary    conditions.    The    model    is    shown    to    capture    arbitrary    flow discontinuities/heterogeneities without any fundamental changes to the model and with small incremental computational costs.