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

Abstract

Rock avalanches, debris flows, and related phenomena consist of grain-fluid mixtures that move across three-dimensional terrain. In all these phenomena the same basic forces govern motion, but differing mixture compositions, initial conditions, and boundary conditions yield varied dynamics and deposits. To predict motion of diverse grain-fluid masses from initiation to deposition, we develop a depth-averaged, threedimensional mathematical model that accounts explicitly for solid-and fluid-phase forces and interactions. Model input consists of initial conditions, path topography, basal and internal friction angles of solid grains, viscosity of pore fluid, mixture density, and a mixture diffusivity that controls pore pressure dissipation. Because these properties are constrained by independent measurements, the model requires little or no calibration and yields readily testable predictions. 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 avalanches. Analysis of intermediate phenomena such as debris flows and pyroclastic flows requires use of the full mixture equations, which can simulate interaction of high-friction surge fronts with more-fluid debris that follows. Special numerical methods (described in the companion paper) are necessary to solve the full equations, but exact analytical solutions of simplified equations provide critical insight. An analytical solution for translational motion of a Coulomb mixture accelerating from rest and descending a uniform slope demonstrates that steady flow can occur only asymptotically. A solution for the asymptotic limit of steady flow in a rectangular channel explains why shear may be concentrated in narrow marginal bands that border a plug of translating debris. Solutions for static equilibrium of source areas describe conditions of incipient slope instability, and other static solutions 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 analysis. Assessment of dimensionless scaling parameters also reveals that miniature laboratory experiments poorly simulate the dynamics of full-scale flows in which fluid effects are significant. Therefore large geophysical flows can exhibit dynamics not evident at laboratory scales. tigators have modeled these events mathematically by specifying rheological rules that govern flow behavior. In general, however, specified rheologies are neither well-constrained nor sufficient to explain flow dynamics, because steady, uniform, rheometric flows of grain-fluid mixtures do not occur in nature.Here we investigate a simpler hypothesis, which holds that most gravity-driven, grain-fluid flows obey no particular stressstrain rate relation. Instead, intergranular stresses satisfy the familiar Coulomb rule, and variation...

Particle-size segregation and diffusive remixing in shallow granular avalanches

“…The theory is designed to be compatible with the assumptions of shallowness, incompressibility and bulk lithostatic pressure, that are made by almost all current avalanche models (e.g. Grigorian et al 1967;Kulikovskii & Eglit 1973;Eglit 1983;Savage & Hutter 1989;Iverson 1997;Gray et al 1999Gray et al , 2003Iverson & Denlinger 2001;Gray 2001). The segregation-remixing theory can therefore be coupled to these models to compute segregation and remixing in dynamically evolving avalanches.…”

“…Grigorian, Eglit & Iakimov 1967;Kulikovskii & Eglit 1973;Eglit 1983;Savage & Hutter 1989;Iverson 1997;Gray, Wieland & Hutter 1999;Iverson & Denlinger 2001;Gray 2001;Gray et al 2003) that the bulk velocity u is incompressible and the overburden pressure p is lithostatic.…”

A three-phase mixture theory for particle size segregation in shallow granular free-surface flows

“…Grigorian et al 1967, Kulikovskii & Eglit 1973, Eglit 1983, Savage & Hutter 1989, Gray et al 1999, Gray et al 2003 to compute the development of the particle size distribution and allow feedback onto the flow. In particular, the inclusion of a passive fluid now allows the model to be incorporated into water saturated debris-flow models such as those of Iverson (1997) and Iverson & Denlinger (2001).…”

“…This extends the model to water-saturated debris-flows (e.g. Iverson 1997, 2003, Iverson & Denlinger 2001 and shows that for dense fluids significant buoyancy forces can be exerted on the particles, which can slow, prevent or reverse the segregation.…”

“…The models all compute the avalanche thickness h and the bulk velocity u and fall into three basic categories. Simple shallow water type avalanche theories of Grigorian et al (1967), Kulikovskii & Eglit (1973), Eglit (1983) and Gray et al (2003), Mohr-Coulomb models of Savage & Hutter (1989, 1991 and Gray et al (1999) and debris-flow models of Iverson (1997), Iverson & Denlinger (2001) that incorporate fluid in the pore space. These theories all assume that the granular material is incompressible, with a hydrostatic pressure distribution through the avalanche depth, which is consistent with the model presented here.…”