### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model Derivation
- 3. A Three-Dimensional Two-Phase Debris-Flow Model
- 4. A Reduced Two-Dimensional Two-Phase Debris Flow Model
- 5. The Model Equations
- 6. Discussion on the Important Features of the New Model Equations
- 7. Simulations of Two-Phase Debris Flows in an Inclined Channel
- 8. Summary
- Appendix A:: Functionality of Generalized Drag
- Appendix B:: Sign Convention and Solid Stresses
- Appendix C:: Reduction to Previous Models
- Appendix D:: Model Structure and Eigenvalues
- Acknowledgments
- References

[1] This paper presents a new, generalized two-phase debris flow model that includes many essential physical phenomena. The model employs the Mohr-Coulomb plasticity for the solid stress, and the fluid stress is modeled as a solid-volume-fraction-gradient-enhanced non-Newtonian viscous stress. The generalized interfacial momentum transfer includes viscous drag, buoyancy, and virtual mass. A new, generalized drag force is proposed that covers both solid-like and fluid-like contributions, and can be applied to drag ranging from linear to quadratic. Strong coupling between the solid- and the fluid-momentum transfer leads to simultaneous deformation, mixing, and separation of the phases. Inclusion of the non-Newtonian viscous stresses is important in several aspects. The evolution, advection, and diffusion of the solid-volume fraction plays an important role. The model, which includes three innovative, fundamentally new, and dominant physical aspects (enhanced viscous stress, virtual mass, generalized drag) constitutes the most generalized two-phase flow model to date, and can reproduce results from most previous simple models that consider single- and two-phase avalanches and debris flows as special cases. Numerical results indicate that the model can adequately describe the complex dynamics of subaerial two-phase debris flows, particle-laden and dispersive flows, sediment transport, and submarine debris flows and associated phenomena.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model Derivation
- 3. A Three-Dimensional Two-Phase Debris-Flow Model
- 4. A Reduced Two-Dimensional Two-Phase Debris Flow Model
- 5. The Model Equations
- 6. Discussion on the Important Features of the New Model Equations
- 7. Simulations of Two-Phase Debris Flows in an Inclined Channel
- 8. Summary
- Appendix A:: Functionality of Generalized Drag
- Appendix B:: Sign Convention and Solid Stresses
- Appendix C:: Reduction to Previous Models
- Appendix D:: Model Structure and Eigenvalues
- Acknowledgments
- References

[2] Debris flows are extremely destructive and dangerous natural hazards. There is a significant need for reliable methods for predicting the dynamics, runout distances, and inundation areas of such events. Debris flows are multiphase, gravity-driven flows consisting of randomly dispersed interacting phases [*O'Brien et al.*, 1993; *Hutter et al.*, 1996; *Iverson*, 1997; *Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005; *Takahashi*, 2007; *Hutter and Schneider*, 2010a, 2010b]. They consist of a broad distribution of grain sizes mixed with fluid. The rheology and flow behavior can vary and depend on the sediment composition and percentage of solid and fluid phases. Significant research in the past few decades has focused on single-phase, dry granular avalanches [*Savage and Hutter*, 1989; *Hungr*, 1995; *Hutter et al.*, 1996; *Gray et al.*, 1999; *Pudasaini and Hutter*, 2003; *Zahibo et al.*, 2010], single-phase debris flows [*Bagnold*, 1954; *Chen*, 1988; *O'Brien et al.*, 1993; *Takahashi*, 2007; *Pudasaini*, 2011], flows composed of solid-fluid mixtures [*Iverson*, 1997; *Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005], two-layer flows [*Fernandez-Nieto et al.*, 2008], and two-fluid debris flows [*Pitman and Le*, 2005]. However, a comprehensive theory accounting for all the interactions between the solid particles and the fluid is still out of reach.

[3] Two-phase granular-fluid mixture flows are characterized primarily by the relative motion and interaction between the solid and fluid phases. Although*Iverson and Denlinger* [2001] and *Pudasaini et al.* [2005]utilized equations that allow basal pore fluid pressure to evolve and include viscous effects, their mixture models are only quasi two-phase or virtually single-phase because they neglect differences between the fluid and solid velocities. Thus, drag force cannot be generated. A key and ad-hoc assumption in*Iverson* [1997], *Iverson and Denlinger* [2001], *Pudasaini et al.* [2005] and *Fernandez-Nieto et al.* [2008] is that the total stress (**T**) can be divided into solid and fluid constituents by introducing a factor Λ_{f} such that the partial solid and fluid stresses are given by (1 − Λ_{f})**T** and Λ_{f}**T**, respectively. In these models, Λ_{f} (ratio between the basal pore fluid pressure and the total basal normal stress, i.e., the pore pressure ratio [see *Hungr*, 1995]) is treated phenomenologically as an internal variable. Also, in these models, volume fraction of the solid is not a dynamical field variable [*Hutter and Schneider*, 2010a, 2010b].

[4] As observed in natural debris flows, the solid and fluid phase velocities may deviate substantially from each other, essentially affecting flow mechanics. Depending on the flow configuration and the material involved, several additional physical mechanics are introduced as mentioned below. Drag is one of the very basic and important mechanisms of two-phase flow as it incorporates coupling between the phases. In terms of modeling the relative motion between the solid and the fluid phases and the associated drag,*Pitman and Le* [2005]proposed a two-fluid debris flow model in which both the solid and fluid phases are considered as ‘fluids’. The*Pitman and Le* [2005] model was subsequently modified by *Pelanti et al.* [2008], but both models neglect viscous stresses, another important physical aspect of two-phase flows. In*Pitman and Le* [2005] model, the drag force depends on the terminal velocity of a freely falling solid particle through a less dense fluid, and there is no direct effect of fluid viscosity on drag. For the fluid phase, the *Pitman and Le* [2005] model and its variants [*Pelanti et al.*, 2008; *Pailha and Pouliquen*, 2009] retain only a fluid-pressure gradient and neglect the viscous effects of the fluid phase. However, the fluid phase in natural debris flows can deviate substantially from an ideal fluid (pure water, for example, but with negligible viscosity) depending on the constituents forming the fluid phase, which can include silt, clay, and fine particles. In many natural debris flows, viscosity can range from 0.001 to 10 Pas or higher [*Takahashi*, 1991, 2007; *Iverson*, 1997]. A small change in the fluid viscosity may lead to substantial change in the dynamics of the debris motion.

[5] Debris-flow dynamics depend on many different factors, including flow properties, topography, and initial and boundary conditions. Although fluid pressure [*Iverson*, 1997; *Iverson and Denlinger*, 2001; *Pitman and Le*, 2005; *Pudasaini et al.*, 2005], viscous effects [*Iverson*, 1997; *Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005] and simple drag between the two phases [*Pitman and Le*, 2005] have been included in various models, three important physical aspects often observed in the natural debris flows are not yet included in any models. (*i*) One phase (e.g., solid) may accelerate relative to another phase (e.g., fluid), thus inducing virtual mass. Relative acceleration between the phases is always present [*Ishii*, 1975; *Ishii and Zuber*, 1979; *Drew*, 1983; *Drew and Lahey*, 1987; *Kytoma*, 1991; *Ishii and Hibiki*, 2006; *Kolev*, 2007]. Hence, dynamic modeling and numerical simulation should include virtual mass effects. (*ii*) The amount and gradient of the solid particles considerably influences flow, which can enhance or diminish viscous effects. If the solid-concentration gradient is positive in the flow direction, then the viscous shear-stress will be enhanced by the increased number of solid particles in the downstream direction. Thus, fluid shear stress is enhanced (or suppressed) by the gradient of the volumetric concentration of the solid particles [*Ishii*, 1975; *Drew*, 1983; *Ishii and Hibiki*, 2006], and this effect should be included in dynamic models. (*iii*) Depending on the amount of grains and flow situation, I propose that drag should combine the solid- and fluid-like contributions in a linear (laminar-type, at low velocity) and quadratic (turbulent-type; e.g., Voellmy drag; at high velocity) manner. Here, a*Richardson and Zaki* [1954]relationship between sedimentation velocity and the terminal velocity of an isolated particle falling in a fluid, and the Kozeny-Carman packing of spheres are combined to develop a new generalized drag coefficient that can be applied to a wide range of problems from the simple linear drag to quadratic drag. There are two distinct contributions in the proposed drag force; one fluid-like, and the other solid-like, having different degrees of importance (sections2.2.1, 6.2, and Appendix A). A generalized drag coefficient, modeled by a linear combination of these two limiting contributions, is presented in this paper. Existing models are limited either to solid-like or to fluid-like drag contribution to flow resistance.

[6] The mathematical structure of equations can also be an important aspect of granular- and debris-flow modeling [*Pudasaini et al.*, 2005; *Pelanti et al.*, 2008]. Dynamical-model equations should be constructed in a standard, and preferably conservative, form. Such a form facilitates numerical integration of model equations even when shocks are formed as has been observed in natural and laboratory flows of debris and granular materials on inclined slopes [*Pudasaini et al.*, 2005, 2007; *Pudasaini and Kröner*, 2008; *Pudasaini*, 2011]. However, the final form of model equations depends largely on how one formulates a model and on how mathematical operators are applied. Here, I start with rigorously structured basic conservation equations, and maintain their structure to the final model expressions. This makes the new model unique, and the most generalized, two-phase mixture mass flow model that exists. Both three-dimensional and depth-averaged, two-dimensional two-phase model equations are presented.

[7] Starting from *Ishii* [1975], *Ishii and Zuber* [1979] and *Drew* [1983], I use phase-averaged mass and momentum balance equations for the solid and fluid components; adopt Mohr-Coulomb plasticity for the solid phase; use a non-Newtonian rheology for the fluid phase; utilize a solid-volume-fraction-gradient-enhanced viscous stress; include virtual mass force due to relative accelerations between the solid and fluid constituents; and introduce a generalized drag coefficient based on*Richardson and Zaki* [1954]and Kozeny-Carman [see, e.g.,*Kozeny*, 1927; *Carman*, 1937, 1956; *Kytoma*, 1991; *Ouriemi et al.*, 2009; *Pailha and Pouliquen*, 2009]. I derive a set of well-structured, hyperbolic-parabolic model equations in conservative form [*Pudasaini and Hutter*, 2003, 2007]. The model equations reveal strong coupling between solid and fluid momentum transfer, both through interfacial momentum transfer and the solid-concentration-gradient-enhanced viscous fluid stresses. Furthermore, the virtual-mass forces couple the momentum equations of the two components, which would be only weakly coupled (by the volume fraction of solid) in the absence of drag forces. The model presented unifies the three pioneering theories in geophysical mass flows, the dry granular avalanche model of*Savage and Hutter* [1989], the debris-flow model of*Iverson* [1997] and *Iverson and Denlinger* [2001], and the two-fluid debris-flow model of*Pitman and Le* [2005], and result in a new, generalized two-phase debris-flow model. The generalized model reduces to three special cases which are compared with the three (classical) avalanche and debris-flow models noted above. The similarities and differences between the reduced model and the relatively simple classical models are discussed in detail.

[8] To develop insight into the basic features of the complex governing equations, the new model is applied to simple, one-dimensional debris flows down an inclined channel. The influence of the generalized drag, buoyancy, virtual mass, Newtonian viscous stress and the enhanced non-Newtonian viscous stress on the overall dynamics of a two-phase debris flow is analyzed in detail. Furthermore, the influence of the initial distribution of the solid volume fraction on the evolution of the solid and fluid constituents, and on the fluid (or the solid) volume fraction is investigated. The simulation results demonstrate fundamentally new features of the proposed model as compared to the classical mixture [*Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005] and two-fluid [*Pitman and Le*, 2005] models. The results highlight the basic physics associated with the contributions of the viscous stresses (both Newtonian and non-Newtonian), virtual mass, generalized drag, and buoyancy, and thus imply the applicabilities of the new model to a wide range of two-phase geophysical mass flows.

### 7. Simulations of Two-Phase Debris Flows in an Inclined Channel

- Top of page
- Abstract
- 1. Introduction
- 2. Model Derivation
- 3. A Three-Dimensional Two-Phase Debris-Flow Model
- 4. A Reduced Two-Dimensional Two-Phase Debris Flow Model
- 5. The Model Equations
- 6. Discussion on the Important Features of the New Model Equations
- 7. Simulations of Two-Phase Debris Flows in an Inclined Channel
- 8. Summary
- Appendix A:: Functionality of Generalized Drag
- Appendix B:: Sign Convention and Solid Stresses
- Appendix C:: Reduction to Previous Models
- Appendix D:: Model Structure and Eigenvalues
- Acknowledgments
- References

[47] The conservative structure of the model equations (41)–(42) facilitates numerical integration even when shocks are formed in the field variables [*Pudasaini et al.*, 2005; *Pudasaini and Kröner*, 2008]. Model equations are applied for channel flows and are solved in conservative variables , where *h*_{s} = *α*_{s}*h* and *h*_{f} = *α*_{f}*h* are the solid and the fluid contributions to the flow heights, and *m*_{s} = *α*_{s}*hu*_{s}, *m*_{f} = *α*_{f}*hu*_{f}are the solid and fluid momentum fluxes, respectively. High-resolution, shock-capturing Total Variation Diminishing Non-Oscillatory Central (TVD-NOC) scheme is implemented to solve the model equations numerically [*Nessyahu and Tadmor*, 1990; *Tai et al.*, 2002; *Pudasaini et al.*, 2005; *Pudasaini and Hutter*, 2007; *Pudasaini and Domnik*, 2009] (Appendix D).

[48] *Simulation set-up and focus.*Model equations are integrated for a simple flow configuration in which a debris flow is released from a triangular dam and moves down an inclined one-dimensional channel (slope angle*ζ* = 45°, Figure 1). The initial triangular mass is divided into two parts: an upper triangle (UT), and a lower triangle (LT), which have either the same or different solid volume fraction. The idea of using different initial solid volume fractions in the front and the back of the debris body is motivated by field observations that the phases can be spatially non-uniformly distributed (see, e.g.,*Sano* [2011]). Initially, some parts of the mixture material may be fully saturated, whereas the other parts may be partially saturated. In addition, the material within each triangular zone is uniformly mixed. Internal and basal friction angles of the solid-phase are*ϕ* = 35° and *δ* = 15°, respectively. Other parameter values are: *ρ*_{f} = 1, 100 kgm^{−3}, *ρ*_{s} = 2, 500 kgm^{−3}, *N*_{R} = 150, 000, , Re_{p} = 1, , respectively. The values chosen for Re_{p}, are assumed to be typical for laminar debris flows, whereas other parameter values are similar to those measured in the field or used in literature, including *Takahashi* [1991, 2007], *Iverson and Denlinger* [2001], *Pudasaini et al.* [2005], and *George and Iverson* [2011]. Below, I investigate the spatial and temporal evolution of the solid (solid lines) and fluid (dashed lines) phases, and the fluid volume fraction as the two-phase debris flow moves down the slope. The influence of the initial solid-volume fraction on flow evolution is analyzed. The emphasis of the simulations is to analyze the overall dynamics of the two-phase debris-flow in detail with respect to the influence of the generalized drag, buoyancy, virtual mass, Newtonian viscous stress, and enhanced non-Newtonian viscous stress.

#### 7.1. Evolution of Solid and Fluid Phases, and Influence of Initial Volume Fraction

[49] Simulation results reveal strong influence of the initial distribution of solid volume fraction in the evolution of solid and fluid phases, and the debris dynamics as a whole. Initially, the upper and lower triangles are homogeneously and uniformly filled (50% solid, 50% fluid; Figure 2). After debris collapse the fluid rapidly spreads to both the leading and trailing edges of the debris. As a result, both the leading and trailing edges of the debris flow are dominated by fluid, whereas the central part is dominated by solids. With time, both phases are continuously elongated, and the flow shape changes. This occurs because the entire mass was initially uniformly mixed, and as soon as the mass collapses, the fluid can slide easily and faster in the downslope direction than can the solid grains. This results from the higher frontal resistance for the solid grains as compared to the fluid, which can move relatively easily. Consequently, the main part of the debris body loses some fluid so that it becomes dominated by solids. However, the tail is dominated by fluid. Since the central part of the flowing mass is dominated by solids, it increases resistance to fluid motion in two ways. First, due to the positive slope of the trailing edge of the initial mass, some fluid moves easily to the rear of the flow. Second, due to the induced higher solid volume fraction in the central part of the debris, the drag is increased. Hence, some fluid movement through the mass of debris is hindered.

[50] Another important aspect of the two-phase debris-flow simulation is the time evolution of the fluid volume fraction (Figure 2, bottom). Initially the solid and fluid volume fractions are 0.5. Following debris collapse, there evolves a strong, non-linear dynamics of the fluid volume fraction (*α*_{f}), and at all the times the front and tail are dominated by the fluid. As the debris mass collapses and moves downslope, *α*_{f} increases in the leading and trailing edges whereas it attains minimum value somewhere in the central part of the debris mass. This behavior is also reflected by the evolution of the solid and fluid phases in Figure 2 (top).

[51] In a second simulation, the initial mass is divided into uniform mixtures of 48% solids (upper triangle) and 75% solids (lower triangle) (Figure 3). In this simulation, both the front and central body of the flow are dominated by solids. This behavior results because the fluid can not easily escape from or pass through the more densely packed solids in the front. As with the simulation of a fully uniform initial distribution of solids (Figure 2), the tail remains dominated by fluid. This is a commonly observed phenomena in granular-rich debris flows, in which the front is solids-rich, and the main body is followed by a fluid-rich tail [*Iverson*, 1997; *Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005]. Both the solid and fluid phases are continuously elongated in time. However, the relative difference between the solid and fluid fractions that contribute to flow depth decrease in time, indicating more mixing as a flow proceeds downslope. Furthermore, Figure 3(bottom) explains the intrinsic dynamics of the debris mixture in terms of the fluid volume fraction. It is important to note that, right after the mass collapse, the jump in the initial fluid volume fraction is immediately transformed into a strong non-linearity. The fluid volume fraction decreases from the front to the middle portion of the flow, becomes minimum somewhere in the middle-right, and then increases non-linearly in the tail side of the flow.

[52] In a third simulation, the initial mass is divided into a more fluid-rich mass leading a more solids-rich mass. In this simulation, the mass is partitioned into uniform mixtures of 68% solids (upper triangle) and 32% solids (lower triangle) (Figure 4). In this simulation, the dynamics between the solid and fluid evolution is the opposite of that shown in the prior simulation (Figure 3). From the beginning, the flow front and much of the central body is dominated by fluid behavior. Since the initial amount of fluid in the lower triangle is much greater than the volume of solids, such behavior is explained because the solid grains are dispersed and the mixture is diluted, and thus fluid easily flows downslope. In contrast, the rear of the mass is initially dominated by solids. Whereas the fluid in the front of the mass moves easily downslope, the fluid in the rear of the mass passes slowly through the solid matrix. The debris mass continuously elongates and its shape changes in time, characterizing the gradual mixing between the phases in the central part of the flowing debris and phase separation in the front and tail. As before, Figure 4(bottom) explains the complex non-linear dynamics of the debris mixture in terms of the fluid volume fraction,*α*_{f}. However, the dynamical behavior of *α*_{f} here is quite different than in Figures 2 and 3. In the present simulation, *α*_{f} is maximum in the front of the flow, it attains the minimum value somewhere in the back side of the central body, and then increases in the tail.

[53] Fluid related longer travel distance discussed above is also observed in other debris flow simulations [*Pitman and Le*, 2005; *Pudasaini et al.*, 2005]. This reflects the higher strength of the debris material with higher amount of solid and other induced dynamical effects, such as the drag and friction. If the fluid volume fraction of the initial mass is much higher (particularly in the lower part, as in a fully saturated lower part of a mountain flank as compared to a partially saturated upper part of the same mountain flank) than the solids volume, then debris evolution shows that almost half of the frontal part is dominated by the fluid while the back side is dominated by the solid. In all simulations, the solid front and tail are tapered, whereas the fluid front and tail are parabolic, which is typical of granular and viscous deformation [*Pudasaini et al.*, 2005; *Pudasaini and Hutter*, 2007]. Therefore, there is a strong influence of the initial volume fractions leading to different deformation and different flow-margin geometries.

#### 7.2. Generalized Drag

[54] Drag is one of the most basic and important aspects of two-phase debris-flow, because it influences the relative motion between the solid and fluid phases. Increasing the value of the drag coefficient*C*_{DG} produces less relative motion (and separation) between the phases, slows down the motion, and hinders the front, because the flow front with higher drag intensity is behind the front with less drag intensity (Figure 5). Therefore, proper modeling of drag is required in order to adequately simulate two-phase debris flows.

#### 7.3. Buoyancy

[55] Buoyancy is an important aspect of two-phase debris flow, because it enhances flow mobility by reducing the frictional resistance in the mixture. Buoyancy is present as long as there is fluid in the mixture. It reduces the solid normal stress, solid lateral normal stresses, and the basal shear stress (thus, frictional resistance) by a factor (1 −*γ*). The effect is substantial when the density ratio (*γ*) is large (e.g., in the natural debris flow). If the flow is neutrally buoyant, i.e., *γ* = 1, [e.g., *Bagnold*, 1954] the debris mass is fluidized and moves longer travel distances (Figure 6). Compared to a buoyant flow (Figure 6, top) the neutrally buoyant flow (Figure 6, bottom) shows completely different behavior. For the latter case, the solid and fluid phases move together, the debris bulk mass is fluidized, the front moves substantially farther, the tail lags behind, and the overall flow height is also reduced.

#### 7.4. Virtual Mass

[56] Whereas drag force includes the phase-interaction in a uniform flow field, if the solid particles also accelerate relative to the fluid, part of the ambient fluid is accelerated, which induces a virtual mass force (thus, the solid particle induced kinetic energy of the fluid phase). Due to the virtual mass force, for present flow configuration, solid particles bring along more fluid mass with them and the fluid is pumped to the front. By focusing on the flow front, one observes that fluid flow is followed by the main debris surge (Figure 7). The solid mass loses some inertia, so it is pushed back by the fluid. The front led by a fluid flood (‘muddy water’) is an observable phenomena in some natural debris flows [*McArdell et al.*, 2007]. Previous debris flow models did not include the virtual mass effect.

#### 7.5. Newtonian Viscous Stress

[57] Fluid viscosity, which can vary depending on flow composition, can substantially affect flow dynamics. To investigate this, initial upper and lower triangles of the static mass are uniformly filled with the debris consisting of 48% solids (upper triangle), and 75% solids (lower triangle), respectively. An inviscid fluid flow is characterized by *η*_{f} ≈ 0, or equivalently *N*_{R} *∞*in our case for the fluid-phase. A typical viscous fluid in debris-flow can be represented by*N*_{R} = 150, 000 [see *Pudasaini et al.*, 2005]. For the present flow configuration, a typical choice of parameters g = 9.81 ms^{−2}, *L* = 350 m, *H* = 2 m, *ρ*_{f} = 1, 100 kgm^{−3}, and *α*_{f} = 0.5 suggests that the fluid-phase viscosity (*η*_{f}) is about 2 Pas. Comparing a flow with inviscid fluid to one with viscous fluid shows that viscous stress controls the propagation of the flow front and determines how the debris mass elongates and deforms (Figure 8). The amount of fluid in the tail of a flow is substantially higher without a viscous stress compared to a flow experiencing viscous stress. Even with a small amount of fluid in the mixture, the viscous stress effect is important as it substantially reduces the deformation. Therefore, the effect of viscous resistance should be taken into account in debris flow simulation. Previous models do not systematically include the effect of viscous stress (or fluid viscosity) in two-phase debris flow dynamics.

#### 7.6. Enhanced Non-Newtonian Viscous Stress

[58] The enhanced non-Newtonian viscous contribution to shear stress can play a significant role in appropriately controlling the two-phase debris flow dynamics. For the present configuration (in the central and the frontal part of the debris flow), (*u*_{f} − *u*_{s}) > 0 and ∂*α*_{s}/∂*x* > 0 (the way the solid volume fraction gradient evolves) (see Figure 2), and thus the viscous stress is down-played. BetweenFigures 9 (top) and 9 (bottom), there are large differences, mainly in fluid deformation, which is enhanced substantially. Figure 9(bottom) shows that, right after the debris collapse, a large amount of fluid is pumped to the front from the middle part of the debris, and that the central part is largely dried-out. Such typical behavior may be observed in dilute debris flows where the front is largely dominated by the fluid and the tail also exhibits dominant fluid mass. Such an important physical mechanism is not yet included in classical debris flow model. Furthermore,Figures 8 and 9reveal that the total viscous effect can substantially alter the deformation process in two-phase debris flows.

[59] Simulation results obtained for one-dimensional inclined channel flows demonstrate the differences between previously proposed Coulomb mixture and two-fluid models [*Iverson and Denlinger*, 2001; *Pudasaini et al.*, 2005; *Pitman and Le*, 2005; *Pelanti et al.*, 2008], and a new general, two-phase debris-flow model(41)–(42). The differences; as discussed in section 6, Appendices A–D, and displayed in above figures and associated texts; are substantial and are highlighted with respect to important physical aspects included in the new model, namely, the generalized drag, buoyancy, virtual mass, classical viscous stress and enhanced non-Newtonian viscous stress. Also investigated were the effects of the initial distribution of the solid volume fraction on the evolution of solid and fluid constituents, and the evolution of the fluid volume fraction.

### 8. Summary

- Top of page
- Abstract
- 1. Introduction
- 2. Model Derivation
- 3. A Three-Dimensional Two-Phase Debris-Flow Model
- 4. A Reduced Two-Dimensional Two-Phase Debris Flow Model
- 5. The Model Equations
- 6. Discussion on the Important Features of the New Model Equations
- 7. Simulations of Two-Phase Debris Flows in an Inclined Channel
- 8. Summary
- Appendix A:: Functionality of Generalized Drag
- Appendix B:: Sign Convention and Solid Stresses
- Appendix C:: Reduction to Previous Models
- Appendix D:: Model Structure and Eigenvalues
- Acknowledgments
- References

[60] In this paper, a new, general two-phase debris-flow model was developed, which includes many essential physical phenomena observable in debris flows. Mohr-Coulomb plasticity is used to close the solid stress. The fluid stress is modeled as a non-Newtonian viscous stress that is enhanced (or downplayed) by the solid-volume-fraction gradient. The model includes virtual mass induced by relative accelerations between the solid and fluid phases. A generalized interfacial momentum transfer includes viscous drag, buoyancy and virtual mass forces. The*Richardson and Zaki* [1954]terminal velocity of a solid particle and a Kozeny-Carman expression for fluid flow through densely packed grains are combined to develop a new generalized drag force that covers both solid-like and fluid-like drag contributions, and allows linear and quadratic drag contributions to flow resistance. This drag force is expressed explicitly in terms of the volume fractions and densities of the solid and fluid, the terminal velocity of solid particles, particle diameter, fluid viscosity, and the particle Reynolds number. There are strong couplings between solid and fluid momentum transfer both through the interfacial momentum transfer and the solid-concentration-gradient-enhanced viscous fluid stresses. The model includes both advection and diffusion of the solid-volume fraction. The proposed model unifies existing avalanche and debris flow theories including the single-phase avalanche model of*Savage and Hutte*r [1989], the debris-mixture model of*Iverson and Denlinger* [2001] and *Pudasaini et al.* [2005], and the two-fluid debris-flow model of*Pitman and Le* [2005].

[61] Simulation results are presented for two-phase debris flows down an inclined channel. They demonstrate the importance of properly modeling the parameters and physical aspects of new, two-phase debris-flow model. The magnitude of the generalized drag force determines whether the flow phases remain mixed or separated, and whether the flow contracts or expands. Buoyancy enhances flow mobility. The virtual mass force alters flow dynamics by increasing the kinetic energy of the fluid. Newtonian viscous stress substantially reduces flow deformation, whereas non-Newtonian viscous stress may move a large amount of fluid from the middle part of a debris flow toward the flow front. The initial volume fraction distribution of solids strongly influences overall flow dynamics. Gradual mixing during the debris flow is observed. Strong non-linear dynamics of the fluid volume fraction demonstrates typical dynamics of the two-phase debris flow as there is a strong coupling between the solid and fluid phases. These findings are consistent with observable phenomena in natural debris flows. The simulation results indicate the potential applicability of the full model equations to adequately describe the complex dynamics of debris flows, avalanches, particle-laden, and dispersive flows. Finally, proper modeling of two-phase debris-flow dynamics should include the five dominant physical mechanisms presented and discussed in this paper, namely, drag, buoyancy, virtual mass, Newtonian viscous stress and enhanced non-Newtonian viscous stress. These mechanisms can substantially control and change the debris flow dynamics.