This paper describes the application of a single relaxation time (SRT) lattice Boltzmann scheme to the transport in porous media with large spatial variations of diffusion coefficients. Effective diffusion coefficients can vary substantially within porous media because of their dependence on porosity and tortuosity which can span over several orders of magnitude, depending on pore size and connectivity. Moreover, when mass is transported with pore-water in porous media, the hydrodynamic dispersion, which depends on Darcy's velocity, contributes additionally to the usually anisotropic variation of the dissipative term. In contrast to the traditional treatment of spatially variable diffusion coefficient by the variation of a SRT, here the variability is accommodated through the use of diffusion velocity formulation which allows for larger variabilities of diffusion coefficient. The volume averaged properties of mass transport in macroscopic porous media are resolved through the additional source term which is similar to the existing force adjusting methods. The applicability of both the proposed schemes is demonstrated on two examples. The first demonstrates that the method is accurate for the large variation of diffusion coefficients and porosities. The second example introduces mass diffusion in a real, geometrically complex system with spatially contrasting properties.