  Providing a sound appraisal of the nature of the relationship between flow
$(Q)$ and pressure drop $(\Delta P)$ for porous media is a long-standing
fundamental research challenge. A wide variety of environmental, societal and
industrial issues, ranging, e.g., from water-soil system remediation to
subsurface energy optimization, is affected by this critical issue. While such
dependence is well represented by the Kozeny-Carman formulation for homogeneous
media, the fundamental nature of such a relationship ($Q$ vs $\Delta P$) within
heterogeneous porous systems characterized by a broad range of pore sizes is
still not fully understood. We design a set of controlled and complex porous
structures and quantify their intrinsic permeability through detailed high
quality microfluidics experiments. We synthesize the results upon deriving an
original analytical formulation relating the overall intrinsic permeability of
the porous structure and their key features. Our formulation explicitly embeds
the spatial variability of pore sizes into the medium permeability through a
conceptualization of the system as a collection of smaller scale porous media
arranged in series. The resulting analytical formulation yields permeability
values matching their experimentally-based counterparts without the need of
additional tunable parameters. Our study then documents and supports the strong
role played by the micro-structure on the overall medium permeability.
