Erosion and Sedimentation in the Pacific Rim ( P r oceed ings of the C o r v a l l i s Symposium, Augus t , 1987). IAHS P u b l . n o . 165.

The physics of debris flows — a conceptual assessment

R. M. IVERSON & R. P. DENLINGER U.S. Geological Survey, Cascades Volcano Observatory 5400 MacArthur Blvd., Vancouver, Washington 98661, USA

ABSTRACT Debris flows exhibit conspicuous dynamic inter­actions among their solid and fluid constituents. Key features of the interactions are neglected in traditional theories that treat debris flows as viscoplastic continua or as uniformly dispersed grain flows, but improved under­standing of grain-grain and fluid-grain interactions has emerged from recent experimental and theoretical research. Grain-flow research has extended the concepts of statis­tical thermodynamics to consider inelastic grain colli­sions and to predict energy-dissipation, velocity, and grain-concentration distributions in flowing, granular materials. Research on fluid-grain interactions has focussed on fluctuating solid and fluid stresses in the vicinity of colliding grains and on energy dissipation in deforming solid-fluid mixtures. Insights born from these new approaches have practical ramifications for interpre­tive and predictive studies of debris flows.

INTRODUCTION

Debris flows are important geologic phenomena. Their widespread occurrence, both as present-day events and as deposits preserved in the geologic record, motivates some compelling scientific questions about the physics of flowing debris. What gives highly concentrated debris the remarkable mobility that allows it to flow on slopes as gentle as a few degrees? What physical processes can produce inverse grading of debris-flow deposits, and under what conditions is inverse grading most likely to occur? How is energy partitioned and dissipated in debris flows? What role is played by the fluid or matrix phase? How do conservative (elastic) and dissipative (flow-type) momentum-transfer processes influence macroscopic debris-flow dynamics? Even more compelling, perhaps, than the need to answer these basic scientific questions is the need to improve under­standing of debris flows so that well-founded engineering predic­tions and hazard mitigation procedures can be developed. Indeed, in Japan alone debris flows are responsible for the loss of many tens of lives each year (Takahashi, 1981).

Debris flows typically consist of discrete solid particles of rock, soil, and organic matter enveloped in a fluid-like matrix that may include liquid water, fine particles carried in suspension, dissolved solids, and bubbles of exsolved or entrained gas. Solid particles can collide, rub, rotate, vibrate, and possibly fracture as they translate downslope. The matrix phase, meanwhile, can flow, compress, vibrate, and cavitate. Adding to the complexity of these processes is the constant exchange of momentum between the solids and matrix, and the distinction between solid and matrix phases may itself not be obvious. Clearly, the complex, multi-phase character

155

156 R.M.Iverson & R.P.Denlinger

of debris flows makes their dynamics difficult to conceptualize and to express as a quantitative physical theory.

In this paper we describe the conceptual framework behind tradi­tional theories for the dynamics of debris flows, and we explain why these theories are only partly consistent with the physical behavior of flowing debris. We then discuss some new concepts that offer considerable promise for improving understanding of debris flows. To illustrate ramifications that new concepts have for field prob­lems, we comment briefly on the appropriateness of hydraulic formu­lae for debris flows and on the origin of inversely graded deposits. However, our focus here is on key physical issues, and we do not attempt a comprehensive account of field phenomena. For more thorough overviews of field phenomena as well as applications-oriented descriptions of traditional theories, the reader is referred to recent reviews by Takahashi (1981), Innes (1983), Costa (1984), and Johnson (1984).

CONCEPTS OF TRADITIONAL DEBRIS FLOW THEORIES

During the past quarter century two principal approaches emerged in the attempt to quantify the physics of debris flows. Prior to the mid-1960's the study of debris flows was wholly empirical, so the development of quantitatively explicit theories constituted a major advance in scientific efforts to understand this phenomenon.

¥i§Ç2El§§>tic iÇoulomb-viscous}^ theory

The first theory for debris flows was developed almost simultan­eously in the USA by Johnson (1965) and in Japan by Yano & Daido (1965). This theory postulates that debris-flow material behaves as a homogeneous, viscoplastic (Bingham) continuum. Viscoplastic materials have a finite yield strength and flow as a linearly viscous (Newtonian) fluid if the yield strength is exceeded. Their mechanical behavior has been thoroughly reviewed in a physical context by Bird and others (1982) and in a geological context by Johnson (1970). The viscoplastic conceptualization of debris flows is founded largely on the idea that a continuous, muddy, matrix phase gives the debris both strength and viscosity, and these matrix properties control the mechanical behavior of the flow. The yield strength may be generalized by adding a frictional strength compo­nent that depends on the normal stress acting on planes of shearing, in which case the viscoplastic model may be referred to as the Coulomb-viscous model (Johnson, 1965, 1970, 1984).

Table 1 depicts a complete set of equations for the application of the Coulomb-viscous model to steady, uniform, one-dimensional, gravitational shear flow of debris on an inclined plane. The three equations in this set reflect the influence of gravitational normal stress (equation la), gravitational shear stress (lb), and the dynamic shear stress that resists deformation (lc). These three equations contain three unknowns (two stress components and a down-slope velocity component), and thus they comprise a closed set that can be solved readily if appropriate boundary conditions and

Physics of debris flows 157

parameter values are specified. Viscoplasiic models for debris flows are easy to apply, and they

can be called upon to explain some key debris-flow features. For example, the viscous component of the viscoplastic model can explain the ability of debris to flow steadily over beds with diverse slopes. Concurrently, yield strength can explain why most deformation in channelized debris flows is localized in relatively thin bands along the flow margins, while a "rigid plug" of relatively undeformed material rides along in the channel center (Johnson, 1970; Pierson, 1986). Yield strength may also be used to explain the bluntly tapered shape of some debris flow snouts (Johnson, 1970), and it provides one mechanism for supporting large clasts of dense rock in

TABLE 1 Equations of traditional theories for steady, uniform debris flow on an infinite, inclined plane

COULOMB-VISCOUS (VISCOPLASTIC) EQUATIONS a = pmgy cos 6 (1a) 7 = pmgy sin d (1b) r = c +atari 0 + jufdvx/dy) (1c)

UNIFORMLY DISPERSED GRAIN-FLOW EQUATIONS

a = (pm -p f )gy cos 6 (2a) 7 = pmgy sin 8 (2b) 7 = a tan0 (2c) o= a p s [ ( C J C ) 1 / 3 - i r 2 d 2 ( d v x / d y ) 2 c o s 0 (2d)

UNKNOWNS DIMENSIONS a Effective normal stress [M/LT2 ] 7 Shear stress [M/LT2 ] vx Velocity in x-direction (downslope) [L/T]

PARAMETERS c Cohesive strength of matrix [M/LT2 ] C Volumetric concentration of solid grains

C* Maximum possible value of C d Grain diameter (assumed constant) [ L ] g Magnitude of gravitational acceleration [L /T 2 ] y Depth normal to flow surface [L ]

a Proportionality constant 6 Slope angle <p Static or dynamic friction angle n Dynamic viscosity of matrix [M/LT]

p f Mass density of fluid-like matrix [M/L3 ] ps Mass density of solid grains [M/L3 ] pm Mass density of mixture [M/L3]

= C p +(1 - C ) p f

158 R.M.Iverson & R.P.Denlinger

less-dense debris-flow matrix. The combination of yield strength and post-yield viscous flow allows the viscopiastic theory to simu­late a variety of vertical velocity profiles in debris flows, but the theory constrains all such profiles to have the shape of a basal, parabolic curve that connects to an upper, unsheared zone that represents the "rigid" plug (Fig. 1).

The fundamental shortcoming of the viscopiastic debris-flow theory is that it makes no provision for dynamic particle inter­actions with one another or with the fluid-like matrix. Such inter­actions are obvious to field observers, who commonly report thun­derous noises and great jostling of rocks during debris flows (e^g., Johnson, 1984), and the interactions probably produce sorting and inverse grading of deposits. Thus, we conclude that the viscopias­tic theory is at best incomplete. A more complete theory would explain observations that the viscopiastic theory explains, and it would also account for the interactions of discrete debris-flow constituents.

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Viscopiastic and uniformly dispersed grain-flow theories

Viscopiastic theory

Uniformly dispersed grain-flow theory

Grain-flow theories

FIG.l Vertical grain-concentration (A) and velocity (B) profiles according to different theories for steady, uniform, gravity flow. Grain-flow theories can produce a variety of profiles that encompass those of the other theories, so profile measurements are insufficient to constrain the theoretical options. In B we have arbi­trarily assigned the thickness of "rigid" regions for the viscopiastic and grain-flow theories.

Physics of debris flows 159

Uniformly dispersed grain^flow theory

The most prominent early efforts to construct a debris-flow theory that accounts for particle interactions were those of Takahashi (1978, 1980, 1981). The central feature of his theory is the con­cept of grain-flow dispersive stress, which was originally intro­duced through the seminal work of Bagnold (1954). Dispersive stress arises in shear flows of granular solids because adjacent grains collide and exchange momentum as they move past one another during the macroscopic shearing motion. Both the magnitude and frequency of linear momentum exchanges vary in direct proportion to the ambient shear rate in the material, so a stress that is proportional to the square of the shear rate and that acts normally to the planes of shear is generated as a result of the particle collisions. The magnitude of this stress has been predicted by both elementary (Bagnold, 1954) and sophisticated (Lun et al., 1984) analyses of the particle collisions, and the theoretical predictions have been corroborated by experimental measurements (Bagnold, 1954; Savage & Sayed, 1984). Takahashi's theory for debris flows therefore encom­passes a demonstrably important physical effect that is not repre­sented in the viscoplastic theory. Furthermore, his theory can be used to calculate realistic surface profiles of debris flows, and it is believed to provide an explanation for the segregation of coarse particles that leads to bouldery debris-flow snouts and to inverse grading in debris-flow deposits (Takahashi, 1980). The theory can appropriately be labeled the uniformly dispersed grain-flow (UDGF) theory, because it employs the assumption that colliding grains are uniformly dispersed throughout the thickness of the flow.

Despite its apparent advantages, the UDGF theory has some signi­ficant shortcomings. Perhaps most physically important is the fact that interstitial matrix effects enter the theory only by contribu­ting to the static weight and pore pressure in the flowing debris mixture. Neglect of dynamic matrix effects is based on the belief that the matrix, like an ideal fluid, is negligibly viscous and strong (Takahashi, 1978). In the case of muddy debris flows, this belief appears to be unfounded (cf. Johnson, 1984; Pierson, 1986).

Regardless of the role of the matrix phase, significant concep­tual problems arise from the UDGF assumption that solid grains are uniformly dispersed throughout a debris flow's thickness. The prob­lems can be traced by referring to Takahashi's (1978, p. 1156) basic equations for steady, uniform, one-dimensional, gravitational shear flow on an inclined plane. His theory requires that four equations (2a-d in Table 1) be satisfied by three unknowns. This mathematical overdeterminacy exists because the uniform-particle-concentration assumption demands that gravitationally induced normal stresses increase linearly with the flow depth (equation 2a), while the Bagnold dispersive-stress equation (2d) requires that these same normal stresses must be proportional to the square of the local shear rate or velocity gradient. This dual specification of the normal stress is unrealistically restrictive. Vertical velocity profiles calculated by integrating the dispersive-stress equation (2d), for example, are constrained to have a specific, concave-upward shape (Fig. 1), so the profiles cannot simulate the appar­ently "rigid" plugs that exist in many debris flows. Moreover, for

160 R.M.Iverson & R.P.Denlinger

solutions to exist that satisfy all four of the equations (2a-d) of the uniformly dispersed grain-flow theory, either some further restrictions must apply to the equations or a fourth unknown must be allowed to enter the theory. A restriction implicit in the equa­tions can be identified by combining equations 2a, b, and c of Table 1 to yield the relation, tan 0/tan 6 = pm/ifim-pt). This relation mandates that steady, uniform flows can exist only when the debris travels down a slope with a specific inclination. In contrast, if the grain concentration were treated as a variable quantity, the gravitational stresses expressed in equations 2a and b could vary nonlinearly with depth, there would be four equations in four unknowns, and velocity-profile and slope-inclination constraints would be unnecessary.

Another restriction of the UDGF theory is that it requires all particles to be of uniform size; thus it cannot rigorously explain the collisional dynamics that lead to inverse grading of debris-flow deposits. Because the momentum exchanged during each particle collision depends on the particles' mass (which is proportional to their diameter cubed, d3) and because the frequency of collisions (for a particular shear rate) is inversely proportional to d, the theory predicts that the resulting dispersive stress should be proportional to d3/d, or to d2 (cf. equation 2d). This d2-depen-dence of the dispersive stress has been offerred by Bagnold (1954) and subsequently by others as a fuzzy explanation for inverse grading. The fu^ziness is dispelled if we acknowledge that the uniform-concentration assumption demands that the flow velocity profile be concave-upward. In this event shearing of the mixture is more rapid beneath grains than above them, so grains experience a preponderance of collisions on their bottom side. This preponder­ance is accentuated if the grains are larger than average, so colli­sions could presumably drive large grains away from the rapidly shearing flow near the bed and toward the flow surface. The logic required to explain inverse grading in this way is necessarily circular and therefore unsatisfactory; it relies on uniform-concen­tration and uniform-size assumptions to explain nonuniform concentrations of nonuniformly sized grains.

The shortcomings of traditional debris-flow theories point to a need for more fundamental investigations.

NEW CONCEPTS AND THEIR APPLICATION TO DEBRIS FLOWS

Concepts introduced in recent studies of gravity flows of dry, granular solids and phase interactions in solid-fluid mixtures are highly relevant to understanding debris-flow behavior. Here we briefly assess these new concepts to see how they might augment and possibly supplant traditional debris-flow theories.

BEY granular fl°w§

Recent experimental and theoretical studies of dry, gravity-driven grain flows reveal much about the dynamic effects of interactions that occur when grains collide with one another and with flow boun-

Physics of debris flows 161

daries. In general these studies emphasize that grain collisions are inelastic, so each collision dissipates energy. Angular as well as linear momentum can be exchanged during grain collisions, especially if prolonged frictional contacts between grains occur. Individual grains have both a mean translational velocity component and a fluctuating, random velocity component as they jostle and collide while travelling within a flow. Furthermore, the concentra­tion of grains may vary from point to point within a flow.

Many physical experiments have been performed on rapid grain flows (Savage, 1984), but the most detailed and illuminating appear to be those of Drake and Shreve (1986). In these experiments high­speed movies were used to trace the translational motion of uniform, cellulose-acetate spheres flowing in an inclined, glass-walled channel that was one grain wide. The Drake-Shreve experiments show that steady, uniform flows probably can occur with diverse channel slopes and discharges and that the concentration of particles and frequency of particle collisions diminish with increasing distance from the channel bed. Direct interactions with the bed are the most prominent source of particle fluctuation energy. Near the bed, particles tend to slide over one another in irregular layers, and they occasionally form densely packed clusters that are three to five grains across. Farther from the bed particles tend to follow more chaotic paths. Detailed observations like these are very important for building a conceptual basis for theoretical work.

Most recent theories for grain flows employ the concepts of statistical thermodynamics and draw heavily on analogies with the kinetic theory of molecular fluids (Lun et al., 1984; Haff, 1986). The behavior of single-phase fluids composed of elastic molecules is governed by the thermodynamic properties of temperature and pres­sure, which control the random but very numerous molecular interac­tions that give these fluids their deterministic macroscopic proper­ties (e^g., Bird et al., 1960). In contrast, the basic building blocks of granular composites are macroscopic, inelastic grains that are typically 10ls to 102° times less numerous than liquid or gas molecules in any given volume (Haff, 1986). Owing to their rela­tively small numbers, random grain interactions in a typical grain-flow volume may produce a partly random macroscopic flow behavior. This fundamental concept implies that some aspects of grain-flow and debris-flow dynamics might defy deterministic prediction even if a perfect theoretical model were available. Statistically based methods therefore appear warranted.

The appropriate statistical energy balance for grain flows dis­tinguishes between the time-averaged mean flow velocity and the instantaneous, random fluctuation velocities of grains. The ratio between the fluctuation velocity and mean velocity depends on the average momentum loss per collision and is an important property of granular flows (Lun et al., 1984; Drake and Shreve, 1986). The fluctuation velocity determines the macroscopic pressure produced as a result of grain collisions, while the mean mass flux of particles determines the time-averaged, macroscopic velocity profile. Such velocity profiles can have a wide range of shapes, depending on the spatial distribution of grains and on how much kinetic energy is dissipated in grain collisions (Fig. 1).

162 R.M.Iverson & R.P.Denlinger

An energy-dissipation "length" (Haff, 1986) reflects the extent to which fluctuation energy derived from grain collisions with the flow boundary can diffuse into the interior of the flow. The dissi­pation length decreases as the grain collisions become less elastic, so grain flows consisting of soft clay aggregates would be expected to have narrow shear zones and steep gradients in particle fluctua­tion energy relative to flows consisting of hard quartz grains. This important theoretical inference shows that apparently "rigid" plugs may develop in grain flows solely as a consequence of inelas­tic particle collisions that are independent of any yield strength (Hui & Haff, 1986); it has been corroborated with numerical model results obtained by Campbell and Brennen (1985). Energy absorbed or reflected at the margins of the flow is also fundamentally impor­tant, particularly when the flow margins are the location of the greatest amount of shear deformation (Haff, 1986).

The grain-to-grain interaction effects described above appear capable of explaining some important properties of debris flows. However, more knowledge about energy dissipation during particle interactions with flow boundaries and with each other is needed, as is better understanding of the role of particle rotations and rubbing in the flow. Moreover, the presence of an interstitial, fluid-like matrix phase will mediate particle interactions and influence energy dissipation in virtually all debris flows.

Solid-fluid Interactions

The role of a fluid or matrix phase in the dynamics of debris-flow mixtures is potentially very diverse, and it might include the effects of viscosity, inertia, buoyancy, strength, compressibility, and interfacial tension. The key to understanding solid-fluid interactions lies in understanding the physical processes that operate on a single-grain scale and in rationally extrapolating from the single-grain scale to the scale of the mixture.

As solid grains in a debris flow approach each other prior to a collision, the fluid matrix that lies between the grains will be pressurized and squeezed from the gap while it simultaneously cushions the collision. The ease with which fluid is expelled will significantly influence the grain impact energy that might be dissipated during the collision, while the fluid flow itself will also dissipate energy. If grain collisions are elastic, the fluid flow will be the only source of dissipation. However, if the grain collisions are inelastic and involve prolonged frictional contact, fluid cushioning of grain collisions could cause a net reduction of energy dissipation and thus enhance the mobility of the flow.

Effects of fluid cushioning on single collisions of ideal, spherical grains have been studied numerically by Davis et al. (1986), who calculated fluid pressure fluctuations and grain defor­mations that occur as two elastic grains approach one another and then rebound. They found that during the grain approach, fluid is expelled from the gap between the grains until the pressure there becomes so high that the grains are repelled by elastic forces. The grains then rebound until they attain an equilibrium spacing that is controlled by the fluid viscosity and by negative fluid pressures

Physics of debris flows 163

that develop between the rebounding grains. Similar fluid pressure fluctuations are revealed by high-speed

pore-pressure measurements made in a rapidly sheared mixture of uniformly sized, coaxial fiberglass rods and water (Iverson and Lahusen, 1986). In the pores adjacent to shearing layers of rods, pressures measured at frequencies up to 10,000 Hz show pronounced positive and negative fluctuations that correspond with individual rod collisions and rebounds. The magnitude of the pressure fluctua­tions is of the same order as the gravitationally induced static pressure, so the fluctuations strongly influence rod-contact stresses. In addition, elastic energy propagates and fluid pressure diffuses away from the locus of rod collisions, so fluid pressure fluctuations caused by collisions can affect stresses and energy dissipation at a distance of several rod diameters away from colli­sion loci. For the simple geometry and localized shear conditions of these experiments, propagation and dissipation of energy far from the collision loci is predicted well by the dynamic, poroelastic mixture theory of Biot (1956).

An appropriate mixture theory for debris flows would allow for large deformations and arbitrary concentrations and contacts of particles — conditions not allowed by the poroelastic theory of Biot (1956). Mixture theories with this aim have been proposed, for example, by Shen and Ackerman (1982) and Shibata and Mei (1986), but each of these theories considers only simple, viscous drag effects produced by particles interacting with a moving fluid. More complex effects related to inertial solid-fluid interactions, pore-scale pressure fluctuations, and nonuniformly sized particles remain to be incorporated in a rigorous theoretical framework.

PRACTICAL RAMIFICATIONS OF NEW CONCEPTS

Despite their incomplete formulation, some of the new concepts described above have significant ramifications for interpretive field studies and engineering applications.

Inverse grain-size grading is generally regarded as diagnostic of sediment deposited by debris flows or other sediment gravity flows (e^g., Scott, 1985), but the physical cause for inverse grading is not clear. As detailed above, inverse grading can be produced by dispersive stress only if the flow velocity profile is concave upward. At least two other mechanisms for production of inverse grading are implied by statistical theories for granular flows. One mechanism, apparently first suggested by Middleton (1970), is termed kinetic sieving and requires simply that grains have velocity fluctuations in the presence of a gravity field. In this event small grains are more likely than are large grains to fall downward through voids that transiently open in the agitated mixture, so large grains eventually concentrate near the surface (cf. Bridg­water, 1980). The second mechanism requires that the grain mixture is sheared, so rotations are imparted to particles during frictional interactions. In a shear flow large particles will then tend to achieve greater angular momenta than will small particles, so the large particles will tend to climb over small particles when they have prolonged contacts. This process eventually will lead to an

164 R.M.Iverson & R.P.Denlinger

increased concentration of large grains toward the flow surface. Field workers might consider these diverse hypotheses in their interpretation of deposits, bearing in mind that all kinetic particle interactions cease during the deposition process.

Another important ramification of the new concepts described above concerns application of hydraulic formulae to debris flows. Hydraulic formulae for molecular fluids such as water are based on energy-conservation principles in which energy dissipation along the flow path is lumped into a boundary-roughness term. This approach is valid because the microscopic mechanism of energy dissipation can be represented well by macroscopic viscosity or eddy-viscosity effects, and simple parameters such as the Reynolds number and Froude number can be used to characterize the partitioning of energy in almost any flow.

In contrast, the mechanisms of energy dissipation in debris flows are multifarious and poorly understood. The dimensionless parame­ters that characterize the energy content of debris flows are not yet defined, and even a list of such parameters for dry grain flows is long (Savage, 1984). For debris flows, lumping energy-dissipa­tion effects into a boundary-roughness term may be radically incor­rect, as boundary energy absorption might be nearly as important as roughness. In light of these considerations, we are skeptical of the use of hydraulic formulae such as Manning's equation and Ber­noulli's equation to predict and interpret debris-flow behavior. More fundamental research on energy dissipation and transport in debris flows needs to be conducted before understanding is expanded beyond its current rudimentary level.

ACKNOWLEDGMENTS We thank T.G. Drake and J.J. Major for reviewing a preliminary manuscript and several colleagues at the Cascades Volcano Observatory for stimulating discussions.

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