The phenomenon of heat flux bifurcation inside a porous medium is analyzed by studying the convective heat transfer process within a channel partially filled with a porous medium under Local Thermal Non-Equilibrium (LTNE) conditions. Either the thermal dispersion effect or the inertial effect is considered in the physical model. Exact solutions are derived for both the fluid and solid temperature distributions for three interface thermal models at the porous-fluid interface. The required conditions for validity of each interface thermal model are obtained, and the equivalence correlations between different interface thermal models are developed. The range of validity of the Local Thermal Equilibrium (LTE) condition is established, and the phenomenon of heat flux bifurcation inside a porous medium is established and demonstrated for the first time in the literature. Furthermore, the Nusselt number is obtained and investigated for pertinent parameters. The ranges of physical parameters in which the thermal dispersion effect and the inertia effect are important are established. (C) 2011 Elsevier Ltd. All rights reserved.