## 1. Introduction

[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.