Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.

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The physics of debris flows

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, 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 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.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JB900329

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

A watercolor model based on an ordered set of translucent glazes, which are created independently usinig a shallow water fluid simulation. A Kubelka-Munk compositing model is used for simulating the optical effect of the superimposed glazes. The computer generated watercolor model is used as part of an interactive watercolor paint system, or as a method for automatic image “watercolorization.”

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Computer-generated watercolor

Numerical solutions of the equations describing flow of variably fluidized Coulomb mixtures predict key features of dry granular avalanches and water-saturated debris flows measured in physical experiments These features include time-dependent speeds, depths, and widths of flows as well as the geometry of resulting deposits Three-dimensional (3-D) boundary surfaces strongly influence flow dynamics because transverse shearing and cross-stream momentum transport occur where topography obstructs or redirects motion Consequent energy dissipation can cause local deceleration and deposition, even on steep slopes Velocities of surge fronts and other discontinuities that develop as flows cross 3-D terrain are predicted accurately by using a Riemann solution algorithm The algorithm employs a gravity wave speed that accounts for different intensities of lateral stress transfer in regions of extending and compressing flow and in regions with different degrees of fluidization Field observations and experiments indicate that flows in which fluid plays a significant role typically have high-friction margins with weaker interiors partly fluidized by pore pressure Interaction of the strong perimeter and weak interior produces relatively steep-sided, flat-topped deposits To simulate these effects, we compute pore pressure distributions using an advection-diffusion model with enhanced diffusivity near flow margins Although challenges remain in evaluating pore pressure distributions in diverse geophysical flows, Riemann solutions of the depth-averaged 3-D Coulomb mixture equations provide a powerful tool for interpreting and predicting flow behavior They provide a means of modeling debris flows, rock avalanches, pyroclastic flows, and related phenomena without invoking and calibrating rheological parameters that have questionable physical significance

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JB900330

Flow of variably fluidized granular masses across three‐dimensional terrain: 2. Numerical predictions and experimental tests

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large-scale inundations. Copyright © 2003 John Wiley & Sons, Ltd.

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

Preface. 1. Shallow-water flows. 2. Equations. 3. Some properties. 4. Behaviour of solutions. 5. Boundary conditions. 6. Discretization in space. 7. Effect of space discretization on wave propagation. 8. Time integration methods. 9. Effects of time discretization on wave propagation. 10. Numerical treatment of boundary conditions. 11. Three-dimensional shallow-water flow. List of notations. References. Index.

Numerical methods for shallow-water flow

A great deal of literature has been devoted to gravity currents in estuaries. However, more or less detailed theoretical models of these phenomena are scarce. This is partly due to the fact that the equations have been difficult to solve if they describe the situation with some generality. This difficulty is surmounted by the use of digital computers. A more fundamental drawback is the lack of knowledge concerning the physical processes of turbulent flow in a stratified fluid. This precludes a detailed two- or three-dimensional description of the flow-pattern. Some schematical models exist which give an overall picture of the flow, still taking variations in space (along the estuary) and time (with the tide) into account. One of these is the two-layer model that is the subject of the present study. It is found that a great part of the information required for engineering applications can be obtained from it. A salt water and a fresh water layer are assumed to be present, either with or without mixing between them. Although flow in most estuaries is not strictly stratified, the two-layer schematization can be useful. This follows from an investigation of the approximations involved in the derivation of the equations. Empirically, the same fact is demonstrated by applying the two-layer model to the partly mixed Rotterdam Waterway. Knowledge of the turbulent flow processes, though in a less detailed form, is still required for a two-layer model, mainly to describe turbulent friction and mixing at the interface, as well as convection through it. If the interface is assumed to be impermeable, only the turbulent friction remains as an empirical parameter. Although the dynamical processes are reproduced less well in this case, the applicability is found to be superior, due to the small number of empirical parameters. Too little is yet known concerning the exchange of salt and water between the layers to permit a more detailed reproduction by means of the model with mixing. The latter therefore will be applied only if information on the salinity is required. The two-layer models result in mean velocities in each layer, and for the case with mixing also in mean densities. These parameters can be applied to define a family of velocity and density profiles. Combined with a crude model of the turbulent structure, this turns out to give reasonably realistic profiles. Therefore as an extension of the two-layer model an estimate of the velocity (and density) profiles can be given. The theory is verified by means of the 1956 measurements in the Rotterdam Waterway. A satisfactory correspondence is found, especially for the case without mixing. An estimate of the interfacial frictional coefficient as a function of the global conditions is obtained by hindcasting a number of flume tests. Although the present study is concerned mainly with estuaries, the two-layer model can be applied to several other cases of stratified flow, notably those concerned with thermal stratification. Such applications, however, require specific descriptions of empirical quantities, like mixing, friction, radiation.

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Cornelis Boudewijn Vreugdenhil

Papers

Numerical methods for shallow-water flow

Computation of gravity currents in estuaries