Flow of variably fluidized granular masses across three-dimensional terrain 1. Coulomb mixture theory

Rock avalanches, debris flows, and related phenomena consist of grain-fluid mixtures that move across three-dimensional terrain. In all these phenomen a the same basic forces govern motion, but differing mixture compositions, initial conditions, and boundary conditions yield varied dynamics and depos its. To predict motion of diverse grain-fluid masses from initiation to dep osition, we develop a depth-averaged, three-dimensional mathematical model that accounts explicitly for solid- and fluid-phase forces and interactions . Model input consists of initial conditions, path topography, basal and in ternal friction angles of solid grains, viscosity of pore fluid, mixture de nsity, and a mixture diffusivity that controls pore pressure dissipation. B ecause these properties are constrained by independent measurements, the mo del requires little or no calibration and yields readily testable predictio ns. In the limit of vanishing Coulomb friction due to persistent high fluid pressure the model equations describe motion of viscous floods, and in the limit of vanishing fluid stress they describe one-phase granular avalanche s. Analysis of intermediate phenomena such as debris flows and pyroclastic flows requires use of the full mixture equations, which can simulate intera ction of high-friction surge fronts with more-fluid debris that follows. Sp ecial numerical methods (described in the companion paper) are necessary to solve the full equations, but exact analytical solutions of simplified equ ations provide critical insight. An analytical solution for translational m otion of a Coulomb mixture accelerating from rest and descending a uniform slope demonstrates that steady flow can occur only asymptotically. A soluti on for the asymptotic limit of steady flow in a rectangular channel explain s why shear may be concentrated in narrow marginal bands that border a plug of translating debris. Solutions for static equilibrium of source areas de scribe conditions of incipient slope instability, and other static solution s show that nonuniform distributions of pore fluid pressure produce bluntly tapered vertical profiles at the margins of deposits. Simplified equations and solutions may apply in additional situations identified by a scaling a nalysis. Assessment of dimensionless scaling parameters also reveals that m iniature laboratory experiments poorly simulate the dynamics of full-scale flows in which fluid effects are significant. Therefore large geophysical f lows can exhibit dynamics not evident at laboratory scales.