Geophysical granular materials display a wide variety of behaviors and features. Typically, granular flows (1) are multiphase flows, (2) are very dissipative over many different scales, (3) display a wide range of grain concentrations, and (4), as a final result of these previous features, display complex nonlinear, nonuniform, and unsteady rheologies. Therefore the objectives of this manuscript are twofold: (1) setting up a hydrodynamic model which acknowledges the multiphase nature of granular flows and (2) defining a comprehensive rheological model which accounts for all the different forms of viscous dissipations within granular flows at any concentration. Hence three important regimes within granular flows must be acknowledged: kinetic (pure free flights of grain), kinetic-collisional, and frictional. The momentum and energy transfer will be different according to the granular regimes, i.e., strain rate dependent in the kinetic and kinetic-collisional cases and strain rate independent in the frictional case. A universal granular rheological model requires a comprehensive unified stress tensor able to adequately describe viscous stress within the flow for any of these regimes, and without imposing a priori what regime will dominate over the others. The kinetic-collisional viscous regime is defined from a modified Boltzmann's kinetic theory of dense gas. The frictional viscous regime is defined from the plastic potential and the critical state theories which account for compressibility of granular matter (e.g., dilatancy, consolidation, and critical state). In the companion paper <Dartevelle et al., 2004> we will introduce a multiphase computer code, (G)MFIX, which accounts for all the granular regimes and rheology and present typical simulations of diluted (e.g., plinian clouds) and concentrated geophysical granular flows (i.e., pyroclastic flows and surges).