Permeability Modeling in Porous Media: Setting Archie and Carman-Kozeny Right

Abstract

Abstract Archie's empirical law constitutes the basis of quantitative Petrophysics; however, the physical significance of this law is poorly understood. The issue involves substantial uncertainty in oil in place. Similarly, Carman-Kozeny's (C-K) relation is source of several permeability models. C-K is derived from Poiseuille's equation, applicable in laminar-viscous-flow in straight-uniform non-communicating tubes. Neither C-K nor Poiseuille's formulae consider inertial accelerations, non-Darcy flow, caused by changes in either cross section or flow direction occurring in porous media. Implications include sizeable limitations in permeability modeling. Suitable hydrodynamic and electrical models can be created using fluid mechanics analytical methods. The models delineate the velocity and electrical potentials, streamlines and controls, represented by C-K and Archie's equations. This approach theoretically verifies both relations and reveals that: Assisted by the boundary layer theory, fluid circulation and permeability are optimally modeled invoking superposition of viscous and inertial regimes in nozzles, throats, diffusers, pipe networks, and arrays of solid particles. These hydraulic components, once assembled, emulate interconnected pore throats and bodies very well. The flow around a corner solution of Laplace's equation characterizes the changes in fluid and electric flow direction, mainly occurring in the pore bodies. The solution precisely defines rock frame, conductive phase, and hydraulic tortuosities, enabling direct ties to pore geometry. This facilitates permeability calculations utilizing the "perfect permeability transform" procedure. The aim of this work is to show how Archie's law ceases being merely empirical and C-K becomes thorough, thus gaining a full physical conception of their power law behavior. As a result, revised Archie's and C-K relations are proposed for water saturation and permeability computations. Main applications comprise the generation of rock catalogs and synthetic production logs to assist in history matching.