We investigate the effects of the porosity (and thus permeability) of porous matrix and the fracture aperture on transport in fractured porous media. We do so with detailed lattice Boltzmann simulations on constructed porous media with a single fracture. Both plume spatial moments (the average plume velocity and the dispersion coefficients) and mass transfer coefficients between the facture and the porous matrix are analyzed. It is found that unlike transport in channels or in pure porous media, solute transport in fractured porous media is non-Gaussian with long tails and with time-dependent mean plume velocities and non-Fickian dispersion coefficients. Higher spatial moments may be needed to fully characterize such plumes and the conventional advection-dispersion equation with upscaled coefficients may be inadequate for describing this anomalous dispersive behavior. The long tailing stems from mass transfer between the fracture and the porous matrix and from the contrast in flow velocities of the two media. It is shown that the mass transfer coefficient is proportional to the matrix diffusivity and inversely proportional to the square of the grain size of the porous matrix. Even for low porosity and low permeability porous matrix whose contribution to flow in the fractured porous media can be neglected, mass transfer between the fractures and the matrix is usually non-negligible.