We show that for a large class of finite dimensional input-output positive systems that represent networks of transport and diffusion of solute in geological media, there exist equivalent multirate mass transfer and multiple interacting continua representations, which are quite popular in geosciences. Moreover, we provide explicit methods to construct these equivalent representations. The proofs show that controllability property is playing a crucial role. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute.