On the role of pore-fluid pressure evolution and hypoplasticity in debris flows

European Journal of Mechanics / B Fluids (

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On the role of pore-fluid pressure evolution and hypoplasticity in debris flows ∗

Julian Heß , Yongqi Wang Chair of Fluid Dynamics, Technical University Darmstadt, Otto-Berndt. Str. 2, 64287 Darmstadt, Germany

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Article history: Received 3 April 2018 Received in revised form 19 September 2018 Accepted 21 September 2018 Available online xxxx Keywords: Debris flow Hypoplasticity Pore-fluid pressure Granular material Two-phase flow

a b s t r a c t For debris flows, two characteristics play a crucial role in their behaviors: (i) the non-linear deformational behavior that stems from the internal contact stress between grains, and (ii) the pore-pressure feedback, which reduces the intergranular friction and, therefore, enhances the mobility of the whole mixture. In a previous work, the constitutive equations of a general grain–fluid multiphase mixture are deduced by exploiting the entropy principle based on the formulation of Müller and Liu. In the work at hand, this thermodynamically consistent and general model is applied to a two-phase debris flow. For this, it is scaled, depth-integrated, compared with other known models and employed for numerical investigations. The extra pore-fluid pressure and the hypoplastic stress are described as internal variables that are, respectively, governed by a pressure diffusion equation and a transport equation related to the hypoplastic material. Numerical investigation of a grain–fluid material, sliding down a slope into a runout, is carried out using a non-oscillatory, shock-capturing central-upwind scheme with the total variation diminishing property. Comparison of the obtained results with those from classical debris flow models shows that the proposed thermodynamic model provides a phenomenological insight into the influence of the porepressure feedback and intergranular friction in the flow dynamics. The extra pore-fluid pressure enhances the flow dynamics, while the effect of the intergranular stress is more significant during the settling of the flow and affects the shape of the granular bulk. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction Debris flows are moving mixtures of granular materials and fluids. They can travel long distances in mountainous regions due to their rapid mass movement driven by gravity. Furthermore, exhibiting immense masses, these kind of flows are dangerous for people and structures, causing severe casualties and destruction every year. Protection measures require an advanced understanding of the complex flow dynamics, arising from the interaction of multiple phases, which are hard to examine. Since debris flows occur both in nature and technical processes, they represent an important case of granular–fluid flows. Unlike mud flows, which tend to be rather fluid-like, or dry rock avalanches, being hindered in their dynamics by the relatively large frictional forces, in debris flows, the concurrence of a fluid phase, both driving and lubricating, and a ponderous solid phase conditions the flow dynamics. The present work tries to gain insight into these dynamics and the underlying physical mechanisms. Modeling faces the problem of depicting complex interactions by fairly simple equations and the need to disregard insignificant mechanisms. Nonetheless, simplistic models can fail to grasp the ∗ Corresponding author. E-mail address: [email protected] (J. Heß).

core of what they seek to describe. To propose a new model, careful investigation of the apparent physical mechanisms is necessary and needs to concur with the understanding of the depicted internal behavior as well as for the conceivable exclusion of particular mechanisms. On account of this, in our model, the incorporation of additional variables, an internal frictional stress and the extra porefluid pressure, goes back to well established findings in experiments and resulting attempts to model the behavior of granular materials and debris flows. To grasp the dynamics of debris flows, the granular material is described by hypoplastic deformational behavior, while the viscous fluid exhibits the evolution of an extra pore-fluid pressure. These fields are incorporated in the present model in a thermodynamically consistent manner. The development of this thermodynamically consistent model according to the entropy principle in its formulation by Müller and Liu [1], see also [2] and [3], has been carried out in a previous publication, see [4]. Firstly considering these two quantities in the modeling process, we proceed here with the derivation of an applicable model for debris flows. For this, the system’s equations are scaled and depth-integrated due to the flow shallowness, as it was suggested in [5] and afterwards established as the Savage–Hutter-type model class. However, unlike in most Savage–Hutter-type models, the stress tensor is given as a result of the modeling process and no Mohr– Coulomb closure condition is required. Instead, hypoplasticity is

https://doi.org/10.1016/j.euromechflu.2018.09.005 0997-7546/© 2018 Elsevier Masson SAS. All rights reserved.

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.

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introduced to allow for plastic material behavior. Hypoplastic models for the description of the deformational behavior of granular materials, taking the different paths of loading and unloading into account, have been developed in a series of publications, by Kolymbas [6,7,8] and are applied in recent works by Guo et al. [9] and Peng et al. [10]. Hypoplasticity itself is a rather simple model, developed in soil mechanics to facilitate previous, more sophisticated concepts, while retaining the core ability of accurately predicting the deformation of granular materials as both anelastic and non-linear. Hypoplasticity predicts a certain relation between the strain rate, i.e. the deformation due to shear, and an internal contact-stress, modeled as a symmetric second-order tensorial variable. In addition to viscosity, for the fluid phase, another well-known concept from soil mechanics is considered because of its explanatory power in terms of the increased mobility of debris flow. According to Terzaghi’s concept of effective stress in porous media, see [11,12], a dynamic extra pore-fluid pressure is considered to mediate emerging load on the solid structure to the fluid. This introduces a dynamic pressure, additional to the hydrostatic fluid pressure, that influences the solid structure, the intergranular friction and therefore also the flow dynamics. This line of tradition in debris flow modeling goes mainly back to the works of Iverson et al., see [13–16]. Furthermore, the concept of a pore pressure feedback is taken into account by many actual approaches, often in conjunction with dilatancy, see [17] and [18]. We are taking up both the concept of a dynamic pore-fluid pressure and hypoplasticity and merge them into a new depth-integrated model that is able to depict some of the core features and properties of debris flows. Beginning with a short summary of the general results of the thermodynamically consistent derivation in Section 2 and some further information on the underlying physical mechanisms, scaling and depth integration are applied in Section 3. Subsequently, a comparison with previous models for debris flows and discussion of the resulting set of equations follow in Section 4. In Section 5, the set of depth integrated equations is utilized for the numerical simulation of a simple debris flow with a Central upwind scheme and the influence of the distinct terms is examined, before Section 6 concludes the work. It is shown that especially during the onset and the settling of flows, the extra pore-fluid pressure and the internal friction capture significant aspects of debris flow behavior that have not been considered in this form yet. 2. Fundamental governing equations Here we give a short insight into the prerequisites of the underlying equations and show the specific structure of the results in [4] that we will use in Section 3. 2.1. General modeling In the previous work, see [4], we developed restrictions, posed by the second law of thermodynamics, on the constitutive relations of a system of governing equations within the framework of mixture theory, depicting a most general granular–fluid flow. With this, we tie up first and foremost to the work of Schneider and Hutter [19]. The point of origin of the modeling process is the definition of a set of governing equations for a range of fields that describe the physical properties of the mixture system, composed of α = 1, . . . , n constituents with fluid-like viscous and solid-like hypoplastic behavior. The fields and all functions refer to spatial

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coordinates xi , i = 1, 2, 3, and time t. This set of equations consists of: Balance of mass for constituent α

⎫ ⎪ ⎪ ⎪ ∂ν ρ vi ⎪ ∂ν ρ α α α α ⎪ ⎪ R = + − ν ρ c = 0; ⎪ ⎪ ∂t ∂ xi ⎪ ⎪ ⎪ Balance of momentum for constituent α ⎪ ⎪ α α α α α ⎪ α α α ⎪ ∂ν ρ v ∂ T v ∂ν ρ v i j ij ⎪ i α α α α Mi = + − − ρ ν gi − mi = 0; ⎪ ⎪ ⎪ ⎪ ∂t ∂ xj ∂ xj ⎪ ⎪ ⎪ Balance of energy for the mixture ⎪ ⎪ ⎪ ∂ρϵ ∂ρϵvi ∂ qi ∂vi ⎪ ⎪ ⎪ E = + + − Tij − ρ r = 0; ⎪ ⎪ ∂t ∂ xi ∂ xi ∂ xj ⎪ ⎪ Evolution equation for the volume fraction for constituent α ⎪ ⎪ ⎪ ⎪ α α α ⎪ ∂ν vi ∂ν ⎪ α α ⎬ N = + − n = 0; ∂t ∂ xi Evolution equation for the inner stress for constituent α ⎪ ( α ) ⎪ ⎪ n α α α β ⎪ ∑ ∂ Z ∂v Z ⎪ ρ − ρ ij k ij ⎪ α α α α α α ⎪ + + Zik Wkj − Wik Zkj − ν Zij = ⎪ ⎪ ⎪ ∂t ∂ xk ρ ⎪ β=1 ⎪ ⎪ α ⎪ ∂vk α ⎪ α ⎪ ⎪ × Zij − Φij = 0; ⎪ ⎪ ∂ xk ⎪ ⎪ ⎪ Evolution equation for a partial pressure for constituent α ⎪ ⎪ ⎪ α α ( 2 α ) ⎪ ∂ϖ ∂ϖ ∂ ϖ ⎪ α,ϖ α α α α ⎪ W = + vi − ∆i − κϖ − γϖ = 0; ⎪ ⎪ ∂t ∂ xi ∂ xi ∂ xi ⎪ ⎪ ⎪ Entropy Balance for the mixture ⎪ ⎪ ⎪ ρη ⎪ ∂φ ∂ρη ∂ρηvi ⎪ i ρη ρη ⎪ ⎭ P = + + − σ = π ≥ 0. ∂t ∂ xi ∂ xi α α

α α α

(1) The fields governed by these balance equations are the true density ρ α and the volume fraction ν α , describing the mass and structural distribution of the constituents α , the partial velocity viα , the inner stress Zijα , giving the dynamics and an inner stress state, the partial pressure ϖ α , and, for the mixture, the entropy η and the internal energy ϵ , respectively. For source and production terms, while c α and nα account for mass transfer within the phases, e.g. due to chemical reactions, the production term in the evolution equation for the inner stress, Φijα , specifies the required hypoplastic behavior, as γϖα describes the causal mechanisms for the evolution of the partial pressure. The partial stress tensor Tijα describes the mechanisms of load transmission within the material, as does the momentum interaction term mαi between the different phases. α Furthermore, there is a pressure flux coefficient κϖ and a further α,ϖ pressure flux quantity ∆i . There also is a range of external influences, with the gravity vector gi and the radiation r. Quantities without superscript α∑are mixture ( α like ) the mixture ∑n quantities, n α α α stress tensor Tij = vi − vi , the mixture α Tij − αρ ν ∑ ∑n α α n velocity vi = ρ −1 α ρ α ν α viα , the mixture density ρ = αρ ν and the heat flux vector of the mixture qi . The spin tensor Wijα = (∂xj viα − ∂xi vjα )/2 and the strain tensor Dαij = (∂xj viα + ∂xi vjα )/2 are made up of the gradients of the partial velocities, where we use the abbreviation ∂xi = ∂∂x . Finally, the entropy balance of the mixture i ρη consists of a term φi , accounting for the flux, a supply rate σ ρη and a production rate density π ρη . For entropy and internal energy, the mixture quantities are employed instead of the partial quantities. This is done because of the assumptions that (i) the distinct partial temperatures do not differ, since, in mixture theory, at every point in space all constituents are equally present, and (ii) the requirement of every partial entropy production being greater or equal to zero is regarded as too strong. Based on the evolution of the above fields, the material behavior has to be specified. For example, while both solid and fluid constituents underly the balance of momentum, they differ in their material behavior, that is first of all, the corresponding stress tensor. The entropy principle in its formulation by Müller and Liu helps to develop the distinct structure in conjunction with

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.

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the considered field variables, since constitutive relations for the material behavior need to be described by the dependences on a certain set of variables ΦC . Thus, the presumed material class is given as

( ∂θ ∂θ α ∂ρ α α ∂ν α α ˆ ˆ , ,ρ , ,ν , ,v , Π = Π (ΦC ) = Π θ , ∂ t ∂ xi ∂ xi ∂ xi i ) ∂ϖ α Dαij , Wijα , Bαij , Zijα , ϖ α , , ∂ xi

(2)

which denotes general elasto-visco-plastic material behavior. These variables define the dependences of the set of constitutive quantities Π

} { ρη α , ∆α,ϖ . Π ∈ η, φi , ϵ, qi , Tijα , mαi , c α , nα , Φijα , γϖα , κϖ i

(3)

It is assumed that all constituents have the same temperature, so incorporation of the thermal temperature of the mixture and its derivatives (in time and space) in the material class, here denoted as θ , ∂t θ and ∂xi θ , accounts for the ability of the material to transfer heat. While the dependence on the true density ρ α implies compressibility, the consideration of ν α gives a material that is dependent on the (micro)structure. Dαij and Wijα allow for the description of viscous effects and the general dependence on mechanical deformation. Furthermore, the incorporation of the intergranular stress tensor Zijα and the left Cauchy–Green deformation tensor Bαij in the material class gives a material that exhibits plastic and elastic properties, respectively. The partial pressure ϖ α reflects the influence of the extra pressure. With these presuppositions given, and the exploitation of the entropy principle, we obtain a set of restrictions posed upon the material behavior of the system. Roughly summarized, this kind of derivation consists of two steps: First, with the entropy principle in the formulation by Müller and Liu, a strict algorithm yields a set of derived equations. These provide insights in the appearance of terms, linking general laws of thermodynamics and the balance equations of the system to their constitutive relations. Thus, both the apparent arbitrariness of a priori modeling as well as the neglect of fundamental basis of thermodynamics can be avoided. By these means, in [4], thermodynamically consistent constitutive equations were developed most notably for the stress tensor and the momentum interaction term with regard to the set of Eqs. (1) and the material behavior (2). In consequence, the property of consistency here also means that the structure of the constitutive functions reflects the system of Eqs. (1), for which the constitutive equations themselves are developed. This provides insight into the influence of the new set of internal variables on the respective rheological properties. In a second step, these resulting equations have to be interpreted and closed by certain assumptions. In the context of debris flow modeling, this often tends to be the weak point of modeling efforts — the rather complex and theoretical manner of modeling in conjunction with the exploitation of the entropy principle often stops here. At the same time, modeling aiming at practical applicability mostly starts here, with a priori alignment of constitutive assumptions. Avoiding these two patterns, we tie in with closure suggestions made by de Boer and Ehlers [20] and Liu [21], linking the results of the Müller–Liu derivation to the pore-fluid pressure. After this step, a closed set of equations has been developed. As described above, in a second step, the results for the constitutive equations are further completed by a set of closing assumptions with the postulation of non-equilibrium parts. For the stress tensor, it then follows that Tijα = − ϖ α ν α δij +

e,α Tij

= − ϖ α ν α δij + ρδZ Zijα + aα1 Dαkk δij + aα2 Dαij + aα3 Dαik Dαkj ∂ν α ∂ν α + aα4 ∂ xi ∂ xj ( α )2 ( α )2 ∂ν ∂ν + aα5 Dαij + aα6 Dαil Dαlj + aα7 viα vjα , ∂ xk ∂ xk

)

–

3

and, for the momentum interaction term, that α

mi =

n−1 ∑ ∂ν β

β=1

+

(

∂ xi

n ∑

α

ϖh (1 − δαn ) − ϖ δ

n h αn

−

ϖνβ

ρα να ρ

) (5)

αβ

cD

(

β

α

vi − vi

)

.

β=1

The stress tensor exhibits the classical splitting with a pressure-like e,α spherical part ϖ α ν α and a deviatoric extra stress Tij . aα1 , aα1 , . . . , α a7 are the coefficient functions of the terms entering the stress αβ tensor and cD denotes the drag coefficient between the conαβ βα stituents α and β with cD = cD . The partial pressure ϖ α can be split up into a hydrostatic part ϖhα and an extra part. In conjunction with the hypoplastic stress, a coefficient δZ arises. It should be mentioned that the momentum interaction terms of all constituents sum up to zero, i.e. vanish for the mixture momentum equation. And while this interaction term also possesses classical αβ β structure, including a drag term cD (vi −viα ) and a buoyant term in α ϖhf , conjunction with pressure times volume fraction gradient, ∂ν ∂ xi it is worth noting here that in the latter, an additional term in conjunction with the configurational pressure ϖνα arises. This term is a direct result from the thermodynamically consistent derivation, usually not included in postulated buoyant terms of debris flow models. In Eqs. (4)–(5) and in the following, we restrict the original system of balance laws (1), by neglecting changes in the true densities (ρ α = const) and phase transitions (c α = nα = 0). We also assume that changes in the mixture temperature θ are insignificant, hence excluding the energy equation, and assume that the further influence of the left Cauchy–Green deformation tensors Bαij is negligible, thus omitting elastic behavior in the following. ∑n Also, the saturation condition is applied, i.e. α=1 ν α = 1. It may seem redundant, but for modeling, it is reasonable to start with a most general system of equations, comprising a wide range of quantities, even if these may later be omitted for practical application. The advantage of coming from a most general model to a simpler one by explicitly imposing restrictions is both the awareness of working with limitations, which always means abstraction from real processes, and the possibility to investigate the influence of further terms by re-taking them into account in future applications. Furthermore, our depth-integrated model equations are – with the help of further limitations – equivalent to well-known models in the context of debris flow modeling, as shown in Section 4. While this was the final result of our preceding work, for most approaches in the field of debris flow modeling, the postulation of set of equations is the very beginning of their studies. 2.2. Two-phase flows: a solid–fluid mixture We now concretize the general model with n constituents to a two-constituent solid–fluid mixture, i.e. α = s, f . For this and as a central closing assumption, by identifying the nth partial pressure ϖ n with the pore-fluid pressure ϖ f , consisting of a hydrostatic ϖhf f and a dynamic extra part ϖe , we specify the partial pressure terms in Eq. (4) to

ϖ s = ϖhs − Cef ϖef − ϖZs = ϖhf + ϖνs − Cef ϖef ( ) − ρ s − ρ f δZ Ziis ,

(6)

ϖ f = ϖhf + Cef ϖef , (4)

f

with a constant Ce , accounting for the influence of the extra poref s s fluid pressure, the ( ) static solid pressure ϖh = ϖh + ϖν and ϖZs = ρ s − ρ f δZ Ziis . A configurational pressure ϖνs and a term ϖZs , accounting for the spherical influence of intergranular friction,

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are present in the solid pressure-like term. Previous attempts to establish a link between the pore-fluid pressure and the resultant terms of the thermodynamic consistent exploitation of the entropy principle stem from the work of de Boer and Ehlers [20] and later Liu [21], in which the results of the exploitation of the Müller and Liu entropy principle are connected to the pore-fluid pressure of a granular–fluid mixture. Since we refrain from the assumption of pressure equilibrium, which states that ϖ f = ϖ s and is often applied to close systems in which the configurational pressure is omitted, a configurational pressure arises, which we define as

ϖνs =

(

) ρs − 1 ϖhf . ρf

(7)

This allows the derivation of the well-known equations of momentum apparent in debris flow modeling, see Section 4, while the thermodynamic coherence of these equations has been proved. It is important to note that Eq. (6) is a direct result of the thermodynamically consistent derivation in [4], but also in accordance with previous considerations on the pressure structure considered in debris flows, i.e. the fluid pressure in [20] and an excess pore pressure, see [16], and the spherical intergranular friction term, see [22,19]. The stress tensors are applied in a reduced form, in which aα3 = aα4 = aα5 = aα6 = aα7 = 0, i.e. the non-linear terms are omitted. From Eqs. (4) and (6), it follows that the reduced solid and fluid stress tensors are given as

(

f

)

s Tijs = − ϖh + ϖνs − Cef ϖef − ρ s − ρ f δZ Zkk ν s δij

(

)

+ ρδZ Zijs + as1 Dskk δij + as2 Dsij , ( ) f f f f f f Tij = − ϖh + Cef ϖef ν f δij + a1 Dkk δij + a2 Dij ,

as2 = 2µs

(

νs s s ν − νmax

)2

( )2 , af2 = 2µf 1 − ν s .

(9)

These relations have been proposed by Passman et al. [24] and Wang and Hutter [25], accounting for a growing viscous resistance s of the solid material due to dense packing, i.e. for ν s → νmax , s it follows that a2 → ∞. These formulations account for viscous behavior, dependent on the microstructure, with a maximum vols ume fraction νmax and the solid and fluid viscosities µs , µf . For a solid volume fraction reaching its maximum packing, a locking phenomenon hinders the motion of the granular particles. A viscous, non-plastic fluid is assumed, so for the fluid stress, the hypoplasticity term vanishes. The drag coefficient is taken as sf

cD = cDs ν s ν f ,

(10)

which assures the drag force vanishes when the fluid phase or the granular phase is absent. At this point, we follow Meng and Wang [26] and refrain from the implementation of a more complex drag term, as postulated for example by Pudasaini [27], to focus on investigating the newly introduced quantities. For the second order tensor Zijs , following Teufel [22] and Fang et al. [28], we employ the expression of the source term

( Φijs

= fs

asZ Dsij

with |Dsij | =

Zijs Zkls Dskl

+ (

s Zmm

)2 +

fD asZ

Dsij

|

( |

Zijs s Zmm

+

dev(Zijs ) s Zmm

))

, (11)

√

Dsik Dskj δij and the deviatoric part dev(Zijs ) = Zijs −

( e0 )pe0 δ Also notice that there is a stiffness factor fs = 1−ν ,a s √ ν s −ν s density factor fD = ν smax−ν s , see [29], and asZ = 8/27 sin (φint ), max C as in [28]. Here, e0 and pe0 are material constants and φint is the s internal friction angle. νC denotes the critical solid volume fraction. 1 s Z . 3 kk ij

ε

⏐ ⏐ ⏐ dεs ⏐ ⏐ ⏐, = C1 Z + C2 Z ⏐ dt dt dt ⏐

dZ s

sd s

s

(12)

with a stress Z s , the strain rate dt εs and two coefficients C1 , C2 . The strain rate is equivalent to the strain tensor in our model, dt εs ∼ Dsij and the material temporal derivative is replaced by an objective stress tensor, the Jaumann stress rate. The temporal development of the stress is determined by the product of stress and strain rate. A nonlinear term, including the absolute value of the strain rate, implicates that the relation between stress and strain rate is different for positive or negative strain rate, i.e. rate dependency is considered. This is also the basic structure of the applied source term in Eq. (11). Note that, to simplify further calculations, the coefficients fs and fD are set to unity, i.e. fs = 1, fD = 1. The incorporated evolution equation for a partial pressure, (1)6 , f is applied for the extra pore-fluid pressure ϖe . As in [16], the f pore-pressure diffusivity κϖ = km /(αD µf ) is composed of the a debris compressibility αD = ν s (σ D+σ ) , the fluid viscosity µf and 0

e

the hydraulic permeability km = k0 exp f ,ϖ ∆i

(

0.6−ν s 0.04

)

, with constants

k0 and aD , while = ∂xi κϖ . σ0 and σe denote the normal stress and the mean effective stress, respectively. With basis on the work of Iverson and George [16], the source term is composed of one term accounting for the variation of the extra stress and one related to dilatancy,

(

γϖf =

f

–

A simple form of this hypoplastic law is derived in [30] for 1D, exhibiting the basic structure

(8)

where as1 , a1 are constants, see [23]. The remaining coefficientfunctions in conjunction with the strain tensors are given as

)

f

d σ − ϖh dt

) −

f

γ˙ tan(ψ ). αD

(13)

The source term γϖf accounts for the changes in the extra porefluid pressure. These changes occur due to development of the f effective stress σe = σ − ϖh , referring to an applied additional load on the solid structure that is firstly mediated by the porefluid, and also due to changes in the pore volume induced by shear, i.e. dilatancy. σ refers to the mean total normal stress. The latter term is defined as the difference of the solid volume fraction s and its equilibrium value, tan(ψ ) = ν s − νeq , times the shear rate γ˙ , to depict the change in the pore space due to shearing. s The equilibrium volume fraction νeq is estimated with the help of the critical volume fraction νcs . For more details on the physical interpretation of these parameters, we refer to Iverson and George [16]. With the equations given above, the system is closed. To the best of the authors’ knowledge, for the first time, a model possessing evolution equations for both a dynamic extra pressure and an intergranular contact friction has been derived in a thermodynamically consistent manner. Before it is depth-integrated and applied to debris flows, some of the introduced key mechanisms are discussed. 2.3. Physical background in pore-fluid pressure and hypoplasticity In the context of Savage–Hutter-type models, the rheological behavior of debris flows is classically described with a system of depth integrated equations in conjunction with the Mohr– Coulomb criterion, applying the so-called earth-pressure coefficients. This simple relation between stress and shear and between normal and horizontal stresses, dependent on the internal friction angle and the bed friction angle, can roughly capture the behavior of dynamic debris flows, but disregards important aspects of the mechanical behavior, see [13]. The deformational paths of soil and variations in stiffness and strength due to the stress levels and density are disregarded, see [31]. Even more, the pressure is regarded as hydrostatic and well known phenomena like liquefaction are

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.

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ignored. Although rheology is thought to describe the observable macroscopic behavior in rater simple terms, we seek to overcome the apparent lack of descriptiveness by accounting for both the role of a dynamic pore pressure, elaborated first of all in the works of Iverson et al., see [13–16], and a hypoplastic relation between stress and strain, see [6–8]. Within the framework of soil mechanics, the investigation of interdependency between pore pressure in porous media and the state of stress has long tradition, starting with the idea of effective stress and the consolidation theory in the work of Terzaghi [11,12]. Basically, increasing in its pressure, the pore-fluid is thought to absorb additional load from the solid structure, before the fluid is squeezed out of the pores and imparts this extra pressure again to the solid structure. Analogously, the pore pressure in debris flows is both dependent on the pressuring load of the structures formed by the solid grains and, in return, influential on the solid behavior. High pore pressures can even cause the onset of debris flows by liquefying porous structures and inducing their collapse, see [32]. The segregation of larger grains and boulders at the front of such flows also induces greater permeability and decreases pore pressures, affecting the mobility of the front. This leads to the importance of an appropriate description of the pore pressure, according to Iverson and George [16], who apply an evolution equation for the pore-fluid extra pressure and substantiated their model by experimental findings. Furthermore, the rheological model in this work is set up with respect to the anelastic, non-linear deformational behavior of granular material. This especially means that granular materials exhibit different paths of deformation during loading and unloading. To capture this effect, a quantity that accounts for intergranular friction is introduced, depending non-linearly on the strain tensor and modeling the influences of the changes in the internal structure due to deformation on the intergranular force transmission. This concept, also well-known in soil mechanics, was thought to enhance previous elasto-plastic proposals. In the context of the entropy principle by Müller and Liu, it has been applied by Svendsen et al. [33] and Schneider and Hutter [19], Hutter and Schneider [34,35]. The intergranular contact stress not only governs deformational behavior but also gives a contact stress, balancing the pressure for static cases, since Zijs is a part of the equilibrium stress tensor. With the physical background for the incorporation of these two additional internal variables given above, it should be pointed out that the incorporation of new internal variables in the framework of continuum mechanics has been discussed in a variety of works, since the work of Goodman and Cowin [36]. Here, in order to describe the microstructure of distinct materials, the evolution of the volume fraction as a new internal variable has been coupled to physical observations. As another example, Kirchner [37], Kirchner and Teufel [38] introduced an internal variable accounting for abrasive behavior. For further discussion, see [39]. 3. Scalings and depth-integrated model equations We now proceed towards an application for two-dimensional two-phase debris flow, following the general approach for a Savage–Hutter-type model, in which the system of Eqs. (1) is scaled and depth-integrated due to the assumption of shallowness. For this purpose, a curved slope is taken into account and we introduce an orthogonal curvilinear coordinate system. The x coordinate represents the downslope direction, while the cross-slope direction is assumed uniform and the z coordinate is oriented in the direction normal to the slope. The slope angle ϑ changes with x. The curvature of the reference surface is given as κ = −∂x ϑ . After the scaling analysis, a set of depth integrated equations is derived by employing the shallow layer assumption. This vertical

)

–

5

integration is a preliminary work to efficient numerical simulation in the following. Unlike in the model of Savage and Hutter [5] and many successive works, this reduction is not strictly necessary for closure in the present model, since a constitutive model for the stress tensors of the solid and the fluid has been given in Eq. (8). 3.1. Scaling analysis To non-dimensionalize the quantities entering the system of equations, we presume a characteristic horizontal length L, a characteristic depth H and a typical radius of curvature R. Following Savage and Hutter [5], we then introduce a range of nondimensional quantities, labeled by a superscript asterisk, viz. x∗ =

x L

z∗ =

;

vα vxα∗ = √ x ;

cDs∗ =

Φijs∗ =

Tijα

ρg H

κ∗ =

√ ; g L Φijs

0 Zkk

t t∗ = √ ; L g

ϖ α∗ =

ϖα ; ρg H

Zijs∗ =

Zijs 0 Zkk

;

;

cDs

ρf

;

vα vzα∗ = √z ; ϵ Lg

Lg

Tijα∗ =

z H

κ R

;

κf f∗ κϖ = √ ϖ2 H g√

γϖf ∗ =

L

γϖf √

ρf H

; g3 L

√ . g L

As first non-dimensional parameters, the aspect ratio arises as ϵ = H/L and the characteristic curvature as λ = L/R. Also see [16] f∗ on the non-dimensional pore-fluid pressure ϖe and Teufel [22], Fang et al. [28] on the non-dimensionalization of Zijs , for which a 0 0∗ 0 referential intergranular contact stress Zkk , with Zkk = Zkk /(ρ g H), is introduced. The magnitude of the gravitational force is g. The system of equations is now specified for a solid and a fluid constituent, so α = s, f . For the sake of brevity, and since the procedure is well known, we refrain from giving details on the derivation of the non-dimensional system of equations and just state the results. Due to the bed curvature, with the spatial derivatives, additional terms arise. Here, we introduce the abbreviation

Ψ = 1/(1 − λϵκ z ) ,

(14)

for the term arising in conjunction with the curvilinear downslope derivatives, and also κx∗ = ∂x κ ∗ . For the volume fraction balance N α = 0, it follows that

∂ν α ∂ν α vxα∗ Ψ ∂ν α vzα∗ + + − λϵκx∗ z ∗ ν α vxα∗ Ψ 2 ∂t∗ ∂ x∗ ∂ z∗ − λϵκ ∗ ν α vzα∗ Ψ = 0, α = s, f .

(15)

The non-dimensional downslope and normal components of the momentum conservation equation Mαi = 0 for the granular phase and the fluid phase are

∂ν α vxα∗ ∂ν α vxα∗ vxα∗ Ψ ∂ν α vxα∗ vzα∗ + + ∗ ∗ ∂t ∂x ∂ z∗ −λϵκx∗ z ∗ ν α vxα∗ vxα∗ Ψ 2 − 2λϵκ ∗ ν α vxα∗ vzα∗ Ψ ϵ ∂ T α∗ Ψ 1 ∂ T α∗ T α∗ = α xx ∗ + α xz∗ − λϵ 2 κx∗ z ∗ xxα Ψ 2 ρ ∂x ρ ∂z ρ −2λϵκ ∗

α∗ Txz

ρα

Ψ + ν α gx∗ +

mα∗ x

ρα

,

(16)

α = s, f ,

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6

J. Heß, Y. Wang / European Journal of Mechanics / B Fluids (

∂ν α vzα∗ ∂ν α vxα∗ vzα∗ Ψ ∂ν α vzα∗ vzα∗ +ϵ +ϵ ∗ ∗ ∂t ∂x ∂ z∗ ( ( ) ( )2 ) 2 −λϵ 2 κx∗ z ∗ ν α vxα∗ vzα∗ Ψ 2 − λκ ∗ ν α ϵ 2 vzα∗ − vxα∗ Ψ (17)

Likewise, with W f = 0, it follows for the evolution of the extra pore-fluid pressure that

( ) f∗ 2 f∗ 2 ∂ϖef ∗ Ψ ∂ϖef ∗ 2 ∂κϖ Ψ f∗ 2 f ∗ ∂ ϖe Ψ − ϵ + v − ϵ κ x ϖ ∂t∗ ∂ x∗ ∂ x∗ ∂ x∗ ∂ x∗ ( ) f∗ f∗ 2 ∂ϖef ∗ ∂κ ∂ ϖ e f∗ + vzf ∗ − ϖ∗ − κϖ = γϖf ∗ , ∂ z∗ ∂z ∂ z∗∂ z∗

(18)

and finally for the hypoplastic stress in the xx direction, with the s evolution equation Zxx = 0,

ϵ

∂ ∂v +ϵ ∗ ∂t ∂ x∗

s∗ s∗ x Zxx Ψ

s∗ Zxx

+ϵ

∂v ∂ z∗

s∗ s∗ z Zxx

s∗ α∗ 2 vx Ψ − λϵκx∗ z ∗ Zxx s∗ α∗ −2λϵκ ∗ Zxx v)z Ψ

∂v ∂v s∗ − ϵ 2 ∗ − λϵκx∗ z ∗ vzα∗ Ψ 2 + λϵκ ∗ vxα∗ Ψ Zxx ∗ ∂ z ∂x ) ( ( ρs − ρf ∂vxs∗ Ψ ∂v s∗ − 1 + νs + z∗ − λϵκx∗ z ∗ vxα∗ Ψ 2 ∗ ρ ∂x ∂z )

(

−

s∗ x

s∗ z

− λϵκ ∗ vzα∗ Ψ

(19)

s∗ s∗ Zxx = ϵ Φxx .

For the sake of simplicity, hypoplasticity in the two-dimensional xz case is restricted to the xx-balance equation, assuming isotropic s s intergranular stress with √ Zxx =s Zzz and applying the steady-state s correlation Zxz = −(3/ 2)asZ Zxx . The correlation is a simplification, which can be derived for a simple shearing state, in which only one stress component must be known, see [28] and [22]. Thus, the 6 components of the full symmetric tensorial variable are reduced to one component. Therefore, the evolution of the hypoplastic stress is described by an equation similar to (12). For the momentum interaction term (5), it follows that msx∗

ρs

( ) ∂ν s Ψ = cDs∗ ν s ν f vxf ∗ − vxs∗ + ϵ ∂ x∗

(

ϖhs∗ −

ν s ρ s s∗ ϖν ρ

)

ν s ρ s s∗ ϖν ρ

)

f∗

=− msz∗ s

ρ

√ =

mx

ρs

,

( ) ∂ν s ϵ cDs∗ ν s ν f vzf ∗ − vzs∗ + ∗ ∂z

(

ϖhs∗ −

f∗

=−

mz

ρs

,

and for the gradients of the stress tensors (8) emerging in Eqs. (16)– (17), we have

( ϵ ∂ Txxs∗ ∂ = −ϵ ϖ f ∗ + ϖνs∗ − Eu ϖef ∗ ρ s ∂ x∗ ( ∂ x∗ ) h ) ρf ∂ Z s∗ s∗ − 2NZ 1 − s Zxx ν s + ϵ NZ ρ ∗ xx∗ ρ ∂x s∗ ( ) ϵ ∂ ϵ ∂ D xx +√ as1∗ ∗ Dsxx∗ + Dszz∗ + √ as2∗ , ∗ ∂x ∂ x Gas Ga s ( ∂ 1 ∂ Tzzs∗ =− ϖ f ∗ + ϖ s∗ − Eu ϖef ∗ ρs ∂ z∗ ( ∂ z∗ ) h ) ν ρf ∂ Z s∗ s∗ − 2NZ 1 − s Zxx ν s + NZ ρ ∗ xx∗ ρ ∂z s∗ ( s∗ ) 1 1 s∗ ∂ s∗ s∗ ∂ Dzz +√ a1 Dxx + Dzz + √ a2 , ∗ ∂z ∂ z∗ Gas Gas

–

( ) ∂ ϵ ϵ ∂ Txxf ∗ f∗ ∂ a1 = −ϵ ∗ ν f ϖhf ∗ + Eu ϖef ∗ + √ f ∗ ρ ∂x ∂x ∂ x∗ Gaf f∗ ( ) ϵ f ∗ ∂ Dxx a2 × Dfxx∗ + Dfzz∗ + √ , ∂ x∗ Gaf f∗ ) 1 1 ∂ Tzz ∂ f ( f∗ f∗ ∂ f∗ a1 +√ = − ν ϖ + Eu ϖ e h ρf ∂ z∗ ∂ z∗ ∂ z∗ Gaf f∗ ) ( f∗ 1 f ∗ ∂ Dzz a2 . × Dxx + Dfzz∗ + √ ∂ z∗ Gaf

ϵ

T α∗ ϵ ∂ T α∗ Ψ 1 ∂ T α∗ = α xz ∗ + α zz∗ − λϵ 2 κx∗ z ∗ xzα Ψ 2 ρ ∂x ) ρ ∂z ρ ( α∗ α∗ mα∗ Txx z ∗ Tzz α ∗ −λϵκ − α Ψ + ν g z + α , α = s, f . ρα ρ ρ

)

(21)

A range of the dimensionless numbers, Gaα =

(ρ α ) 2 g L 3 (µα )2

f

,

Eu =

Ce ρ f g H

ρg L

∼

∆ϖ , ρvi2

( 0 ) ρδZ Zxx + Zzz0 NZ = , ρg L is introduced in the non-dimensional stress tensor, describing the influence of different terms on the stress evolution. The ratio of gravitational forces to inner, viscous friction is characterized by the Galilei number Gaα . It appears with the viscosity terms. For the dimensionless intergranular friction, we introduce a number NZ in conjunction with the hypoplastic frictional stress terms. It is assembled with the proportion of hypoplastic forces to inertial forces. Furthermore, the Euler number Eu gives the influence of f a partial pressure Ce ρ f g H ∼ ∆ϖ in relation to inertial forces 2 ρ g L ∼ ρvi , where ∆ϖ = ϖ f − ϖhf is a pressure difference. It specifies the influence of the pore-fluid extra pressure and appears in conjunction with the non-dimensional extra pore-fluid pressure in the following. We also introduce Eub to denote the influence of the dynamic pore pressure on the solid bed friction, with f

Eub =

Ce,b ρ f g H

ρg L

,

f

where Ce,b is a constant, accounting for the influence of the basal excess pore pressure. 3.2. Depth integration For further simplification, the superscript asterisk is omitted below and the shallow flow assumption is applied, based on the small ratio of flow depth to downslope length being apparent in debris flows, i.e. ϵ ≪ 1. Similar to the procedure in the derivation of Savage and Hutter [5], depth averaging incorporates the integration of all equations over the z-coordinate, from the bed z = b(x) to the free surface z = s(x, t). b(x) represents a shallow basal topography that can be overlapped on the reference surface. It is assumed to vanish here, b(x) = 0. In this connection, a crucial information on the vertical dimension is maintained: the height h(x, t) = s − b, and with this, the information on the flow depth is projected on the x-coordinate. For this, the actual 1D flow can be still mapped as quasi-2D. Depth integration of a quantity φ gives the corresponding depth-integrated quantity φ , s

∫

φ dz = hφ, b

and following Pitman and Le [40] and Meng and Wang [26], it is assumed that (20)

φν α =

1 h

s

∫ b

φν α dz ∼

1 h

να

s

∫

φ dz = φ ν α , b

where the volume fraction is assumed to be less dependent on the depth direction. During the derivation, the Leibniz rule is applied to interchange differentiation and integration. Also following Pitman and Le [40], a simplifying assumption for the depth-integration of

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the drag term is applied. Furthermore, with ordering arguments prevalent in literature, see [41] and [26], we assume that λ = O(ϵ χ ) with 0 < χ < 1, so it follows that Ψ = 1 + O(ϵ χ ). Since the single steps of the procedure are well-known for the balance of mass, we only give the results. For the mass balance of the solid and fluid phases (15), it follows that

∂ ( s s) ∂ ( s) hν + hν vx = 0 + O(ϵ 1+χ ), ∂(t ∂ x ) ∂ ∂ ( f f) hν f + hν vx = 0 + O(ϵ 1+χ ), ∂t ∂x

(22)

–

7

and (18) as

]s [ f ∂ ( f) ∂ ( f f) f ∂ϖe hϖe + hϖe vx − κϖ ∂t ∂x ∂z b ( ( ( ( )) ) ) ∂ ∂ f f f = h σ − ϖh h σ − ϖh vx + ∂t )∂ x ( ∂ h ∂vxf h + − hγ˙ tan(ψ ) + O(ϵ 2 ). + ϖef ∂t ∂x

(26)

f For this, it is assumed that the diffusion coefficient κϖ is not changing with depth. To close this Eq. (26), we consider the following relations

giving the development of the height

∂ ( f f) ∂h ∂ ( s s) + hν vx + hν vx = 0 + O(ϵ 1+χ ). ∂t ∂x ∂x

)

]s ∂ϖef κf βf km β f (i) κϖ = 2 ϖ ϖef = 2 f ϖef , ∂z h hµ αD )b ) ( ( f f ∂ σ − ϖh ∂ σ − ϖh ) 1 ( f (ii) h+ hv x = h ρ − ρ f ∂t ∂x 2 ( ) ∂ hν s (vxs − vxf ) ∂vxf , × gz +h ∂x ∂x ) ( ∂ h ∂ hvxf ∂ hν s (vxs − vxf ) f (iii) ϖe + = ϖef , ∂t ∂x ∂x ( ) √ s ν C f √ (iv ) hγ˙ tan(ψ ) = β (vxs )2 κϖ 1 ν s − κϖ 2 , 1 + Nψ [

(23)

For the derivation of the momentum balances in x and z direction, see Appendix A, since we only give the results for the downslope momentum balances here. The depth-integrated solid x-momentum balance follows as

∂ ( s s) ∂ ( s s s) hν vx + hν vx vx ∂t ∂(x ϵ ∂ h2 h2 =hν s gx − s ν s ρ f gz + ν s (ρ s − ρ f )gz − Euν s ϖef h ρ ∂x 2 2 ) ( s ) s − 2NZ ρ − ρ f hZxx ( ( )) s ϵ ∂ρ hZxx ∂ − 1 ∂v s + s NZ + ϵ Gas 2 x h as1 + as2 ρ ∂x ∂x ∂x ( ) 2 2 s ϵ ∂ν h f h ρ s ν s h2 s s f f + s ρ gz + (ρ − ρ )gz − (ρ − ρ )gz ρ ∂x 2 2 ρ 2 s ) ( ) cD hν s ν f ( f 1 + vx − vxs − s sgn vxs ν s µsb s ρ ρ )] ( [ 2 s s2 f f s f + O(ϵ 1+χ ), × ϖh − ϖ + λκ h ρ vx − ρ vx (24) and the fluid x-momentum balance yields

∂ ( f f) ∂ ( f f f) hν vx + hν v x v x ∂t (∂ x 2 ) ϵ ∂ h f f f f f =hν gx − f ν ρ gz + Euν ϖe h ρ ∂x 2 ) ( f ( ) ∂ − 12 ∂vx f f + ϵ Gaf h a1 + a2 ∂x ∂x ( ) ϵ ∂ν s h2 f h2 s ρ s ν s h2 s f f − s ρ gz + (ρ − ρ )gz − (ρ − ρ )gz ρ ∂x 2 2 ρ 2 s ) cD hν s ν f ( f 1 f f f s − − v − v α ν v + O(ϵ 1+χ ). x x f ρ ρf b x (25) Depth-integration needs to be also performed with the evolution equation for the pore-fluid extra pressure (18) and the hypoplastic stress (19). For more details on the depth-integration of the pressure evolution equation, please see [16]. The assumptions with which we work are given below. The depth-integrated evolution equation for the pore-fluid extra pressure follows from Eqs. (13)

f

(27)

where we have

Nψ =

√

µf γ˙ ρ (γ˙ δP ) + σe 2

s

,

γ˙ =

2 (vxs )2 h

,

( ) σe = ρ − ρ f ggz h, (28)

two coefficients κϖ 1 , κ size δP and the function (ϖ 2 , the particle ) x −x β f (x) = 1 + 100 exp −10 fx , that simulates the decrease of L

the pore pressure at the flow front xf due to the accumulation of larger grains there, and, therefore, the increased permeability. xL is the actual length of the flow. This additional function accounts for segregational effect in a first, rudimentary way. It was proposed by Savage and Iverson [14]. (ii) and (iii) are derived with the depth f

integrated extra stress σe = σ − ϖh = 12 ρ − ρ f gz h and the mass balance Eq. (23). (iv) is developed in [16]. For the evolution equation of the xx-component of the hypoplastic stress, Eq. (19), it follows that

(

)

( ( )) s ρs − ρf s s [ s ]s ν ∂ hZxx ∂ hvxs Zxx s v ϵ +ϵ − hZxx 1+ x b −ϵ ∂t ∂x ρ ( ) s ∂ h ∂vx h s s h + O (ϵ 2 ), × Zxx + = ϵ Φxx ∂t ∂x

(29)

with a depth-integrated, non-dimensional source term

[ ] ( )2 ∂vxs s Z s h vs s s Zxx Zxx h ∂x ∂vxs xz x b = fs ϵ +ϵ( )2 + ( )2 ∂x s + Zs s + Zs Zxx Z zz xx zz ) [ ] ⏐ s⏐ s Z s h vs s s h ⏐ ⏐ Zxx Zxx zz z b s ⏐ ∂vx ⏐ +ϵ( + O(ϵ 2 ). ) 2 + ϵ f D aZ ⏐ ∂ x ⏐ s s s s Z + Z Zxx + Zzz xx zz (

ϵ

s h Φxx

asZ h

(30)

Here, we omitted ϵ 2 -terms and worked with the assumption of

∫s b

Zijs

∂vk dz ∂x

=

[ s ∂ h ]s s x vx ∂ x b ≈ h ∂v . ∂x

1 h

∫s b

Zijs dz

∫s b

∂vk dz. ∂x

Also, we assumed that

∂vxs h ∂x

−

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Next, we will omit the overbar-symbol, since all quantities are regarded as depth-integrated. 4. Discussion and comparison In this section, we seek to analytically compare the developed set of depth-integrated momentum balances with some known formulations in the field of debris flow modeling and discuss possible limiting cases. By considering the pioneering work of Savage and Hutter [5], which proposed a set of depth-integrated equations for dry granular flow with a Mohr–Coulomb criterion, Pitman and Le [40] generalized the approach for a two-phase flow that allows for the consideration of an additional fluid phase. The normal stress component of the solid is linked to the down- and crossslope stress via the introduction of an earth pressure coefficient to close the model. Also based on the work of Iverson [13], Pitman and Le [40] similarly emphasized the role of the pore-fluid, while retaining two full momentum equations instead of merging them to a mixture one. Pudasaini [27] developed an improved drag coefficient and an enhanced model for the fluid phase by including viscous terms. A further prosecution of the modeling efforts for debris flow has been presented by Meng and Wang [26], discussing the aforementioned works, extending them to simulate debris flows on a curvilinear surface and proposing corrections on the buoyant term in [27]. Next we want to show that our model can be reduced to these previous ones by appropriate simplifications. This demonstrates that the present model includes their features in a more general way. An underlying assumption is that, although this does not represent the actual development of the model, the results of our modeling efforts can be seen as an improvement of a basic model by taking more effects into account. This implies in turn that the model can be reduced to this basic model and the focal point of the following research has to be the proof of the meaningfulness of the additionally introduced terms. For this, the gravity shares gx and gz are inserted as functions of the inclination angle ϑ , as given in Appendix A. Hence, we will reveal this basic structure of the equations of Pitman and Le [40], before discussing the equations of the most recent model of Meng and Wang [26]. Also, we will discuss several limiting cases, since these cases disclose possible limitations of the model due to unphysical properties. 4.1. Pitman–Le model

)

–

while the fluid x-momentum balance yields

∂ ( f f f) ∂ ( f f) hν vx + hν v x v x ∂t ∂x ( ) sf ) c h( ∂ h2 f f =hν sin(ϑ ) − ϵν cos(ϑ ) − D f vxf − vxs . ∂x 2 ρ

The solid equation (31) yields, after a few transformations, the momentum balance of the Savage–Hutter model, see [5], by assuming that ρ f → 0, no phase interaction cDs = 0 and ν s = const.

4.2. Meng–Wang model The model of Meng and Wang [26], similar to Pudasaini [27], considers in particular the fluid rheology by adding a viscous term as well as a basal fluid friction term that is linearly dependent on the velocity. While in comparison to the Pitman–Le model, the solid momentum balance in the downslope direction is enhanced only in the basal friction term by accounting for the bed curvature, f λκ h((vxs )2 − ρρ s (vxf )2 ),

∂ ( s s) ∂ ( s s s) hν v x + hν vx vx ∂t ∂x ) ( ( ) ρf h2 ∂ =hν s sin(ϑ ) − ϵ kap 1 − s cos(ϑ )ν s ∂x ρ 2 sf ) ρf s ∂ h cD h ( f vx − vxs ν cos( ϑ )h + ρs ∂x ρs (( ) ( )) ρf ρf f 2 s 2 − 1 − s cos(ϑ ) + λκ (vx ) − s (vx ) ρ ρ ( ) × hν s sgn vxs tan(φbed ),

−ϵ

∂ ( s s s) ∂ ( s s) hν v x + hν vx vx ∂t ∂x ( ( ) ) ∂ ρf h2 =hν s sin(ϑ ) − ϵ kap 1 − s cos(ϑ )ν s ∂x ρ 2 ρf s ∂h − ϵ s ν cos(ϑ )h ρ ∂x ( ) sf ( ) ) c h( ρf − 1 − s ν s h cos(ϑ )sgn vxs tan(φbed ) + D s vxf − vxs , ρ ρ (31)

(33)

the new fluid momentum balance in downslope direction follows as

∂ ( f f f) ∂ ( f f) hν vx + hν v x v x ∂t ∂x =hν f sin(ϑ ) − ϵν f cos(ϑ )

The fundamental two-phase model of Pitman and Le [40] includes an ideal fluid phase to account for the role of the interstitial fluid in debris flows. The solid is modeled as an incompressible Coulomb granular material. The interaction of the distinct mixture phases are modeled with the help of averaging models, following Anderson and Jackson [42], which allows for buoyancy due to different densities and drag between the phases. We give an account of the momentum balances in the downslope direction. In [40], for the solid x-momentum balance it follows that

(32)

+

ϵ NR

(

∂ ∂x

αbf vxf ∂ 2 vxf 2h − ∂ x2 ϵ2

(

h2 2

) −

sf cD h (

ρ

f

vxf − vxs

)

(34)

) .

The last term in (34) contains a viscous part in conjunction with the second derivative of the fluid x-velocity and the fluid basal friction. Due to the scaling of these terms, the dimensionless variable NR is introduced with

√ ρf H g L . NR = ν f µf

(35)

4.3. Comparison If Eqs. (24) and (25) are rearranged and the solid viscous terms are omitted with as1 = as2 = 0, it follows for the granular phase that

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∂ ( s s s) ∂ ( s s) hν vx + hν v x v x ∂t ∂x (( ) ∂ ρf h2 =hν s sin(ϑ ) − ϵ 1 − s ν s cos(ϑ ) ∂x ρ 2 ν s ϖef h ρf − 2NZ 1 − s s ρ ρ

(

− Eu

)

1 NR ν s

−

ρ ρs

) f

1−

cos(ϑ ) − Eub

s hZxx

ϖef ρs

+ ϵ Gaf

+

f a2

( ) ) ∂ αf ν f ∂vxf h2 ∂ν s h − b f vxf − ϵ cos(ϑ ) ∂x ∂x ρ 2 ∂x

( ) ν f ρ s − ρ f h2 ∂ν s −ϵ cos(ϑ ) . ρ 2 ∂x (37) f

If the additional influence of the pore-fluid extra pressure, ϖe , and s the hypoplastic stress part, Zxx , in the momentum equations are neglected, the remaining differences between our model and Eqs. (36)–(37) of Meng and Wang [26] can be found in the buoyant term. a An additional term τby appears here, accounting for the configurational pressure. It should be stated that this term is a direct result of the thermodynamically consistent modeling in conjunction with the entropy principle of Müller and Liu, as derived in [4]. This term

ϵρ f

( ) ν f ρ s − ρ f h2 ∂ν s a cos(ϑ ) := τby , ρ 2 ∂x

represents an additional driving force due to buoyancy rate, coupled to changes in the micro-structure, i.e. differences in density times the gradient of the solid volume fraction. Apart from this, the system of equations developed into Eqs. (36)–(37) can be converted into Eqs. (31)–(34) as long as an appropriate assumptions for the fluid basal friction and the fluid viscous terms are considered. Also note that,(for the simple investigations here, we ( )) ( ) f

will now assume that ∂x h∂x vx

≈ ∂x2 hvxf

f

f

a1 + a2

)

= 1.

We now seek to investigate the compatibility of our model with well-known models and basic principles of fluid and soil mechanics.

∂ ( s) ∂ ( s s) ∂ hv x + hvx vx = h sin(ϑ ) − ϵ ∂t ∂x ∂x

∂ ( f f) ∂ ( f f f) hν vx + hν vx vx ∂t ∂x ( ) sf ) cD h ( f h2 ν f ϖef h ∂ f f ν cos(ϑ ) + Eu − vx − vxs =hν sin(ϑ ) − ϵ ∂x 2 ρf ρf f a1

ρf H g L

(

Dry granular flow. The governing equations for a dry granular flow can be obtained from the two-phase fluid-granular model by f choosing ν s = const, ν f = 0, ρ f , and ϖe = 0. With this, the dragand momentum interaction forces vanish. If we also leave out the centrifugal forces due to the curvature of the flow chute, it follows for Eq. (36) that

and for the fluid phase

(

− 21

= Gaf

4.4. Limiting cases (36)

( )) ( ) ρf hν s sgn vxs tan(φbed ), + λκ (vxs )2 − s (vxf )2 ρ

− 21

µf √

=

√

( ) ∂ν s ρ f ν f ρ s − ρ f h2 cos(ϑ ) s ρ ρ 2 ∂x

((

9

With that, the fluid viscosity can be determined as µf = ρ f H Lg, depending on the chosen scaling H and L and the values of the fluid density and the gravitational acceleration.

)

s ) c h( ϵ ∂ρ hZxx ρf ∂h NZ + D s vxf − vxs − ϵ s ν s cos(ϑ )h s ρ ∂x ρ ρ ∂x

+ϵ

–

introduced as in [26], see Eq. (35), but without the dependence on ν f , that

sf

+

)

and that the

coefficient of the ( ) fluid viscous term in Eq. (37) is simplified by − 12 f f a1 + a2 = 1. This assumption implies, if the respective Gaf terms in Eqs. (34) and (37) are compared and the formula for NR is

− ϵ NZ

(

h2 2

cos(ϑ ) −

s 2NZ hZxx

)

s ( ) ∂ hZxx − sgn vxs h cos(ϑ ) tan(φbed ). ∂x

(38) By further omitting the hypoplastic part of the internal friction, i.e. NZ → 0, we obtain the classical Savage–Hutter model. Furthermore, the ability of the model to depict the internal friction of static granular materials at a horizontal plane is evaluated. For this case, the gravity in the x-direction and velocities are set to zero, vxs = 0 and sin(ϑ ) = 0, cos(ϑ ) = 1, while NZ ̸= 0 and NZ = const, so ∂x NZ = 0. With this, the remaining momentum balance (38) is reduced to

∂ ∂x

(

h2

) = NZ

2

s ∂ hZxx , ∂x

(39)

showing that the gradient of height ∂x h is balanced with the aps parent internal friction in conjunction with Zxx . Giving a linear s correlation between h and Zxx in conjunction with the bed friction angle, this shows the importance of hypoplasticity for the settling of granular material on a pile under a corresponding angle of repose. Hypoplasticity and Mohr–Coulomb friction. Staying with the influs ence of hypoplasticity — if we identify the shear stress as τ = Zxz s and the normal√ stress as σ = Zxx , with the applied assumption s s of Zxz = −(3/ 2)asZ Zxx , we obtain a Coulomb-like friction law τ ∼ C σ sin(φ ). While this comes here as a presumption, it can be derived from hypoplastic equations for the state of simple shearing. If we reduce the depth-integrated evolution equation for the internal granular friction in the xz-plane to the order ϵ 0 s and also presume isotropic stress distribution, i.e. Zxx = Zzzs , the evolution of the intergranular stress yields

( s )2 [ s ]s Zxz h vx b ) [ s ]s [ s ]s s − h vx b = fs aZ vx b + fs ( ) s + Zs 2    Zxx zz =0 √ √ ) asZ ( s asZ ( s ) s s → Zxz = − Zxx + Zzz = −2 Zxx .  h h  (

s Zzz

s Zxx

τ

(40)

σ

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Therefore, the depth-integrated equations within a hypoplastic rheology can be interpreted as a higher-ϵ -order enhancement of a Coulomb-friction-like material, containing the basic relations between shear stress and normal load. This higher-order hypoplasticity becomes important when the granular flow decelerates and tends to accumulate. Equal densities. Equal densities ρs = ρf imply the extinction of the configurational pressure ϖνs , as well as the vanishing of the buoyancy, i.e. solid particles suspended in the fluid. If we transfer the model equations to the static case again, i.e. with vanishing velocities, bed friction and extra pressure, the momentum balance equation reduces to hydrostatic balance between pressure gradient and gravity. As it is pointed out by Meng and Wang [26], physically reasonable systems of equations should always reproduce this limiting case of hydrostatic and lithostatic pressure. The influence of the extra pressure. As described in Section 2.3, a dynamic extra pressure enhances the mobility of the debris flow. Therefore, in this work, it is introduced into the solid bed friction f term by ϖ s − ϖ f = ϖνs − Euϖe , decreasing the resistance due to friction. If the bed curvature is omitted, the bed friction term is

[

] s

Txz

b

( ) = −sgn(vxs )ν s µsb ϖνs − Euϖef ,

(41)

Fig. 1. Debris flow problem geometry: Initial pile on a slope.

part and a fluid part as

( ) ( ) ( ) ( ) ( ) ∂t Us + ∂x Fs = Ss + Cs1 ∂x Cs2 + Cs3 ∂x Cs4 + Cs5 ∂x Cs6 , ( ) ( ) ( ) ( ) ∂t Uf + ∂x Ff = Sf + Cf1 ∂x Cf2 + Cf3 ∂x Cf4 ,

where, if the extra pressure equalizes the pressure, [ configurational ] ϖef = ϖνs , the bed-friction vanishes, Txzs b → 0. Furthermore, it reduces the spherical solid stress and enhances the fluid pressure. 5. Numerical simulation

(42) with the two vectors of unknowns in terms of conservative variables

( Below, the partial differential equation system assembled by Eqs. (22), (26), (29), (36), (37) is numerically solved in order to get a first impression of the influence of the extra pore-fluid pressure and the hypoplastic intergranular stress. The numerical instabilities apparent in the simulations of these kind of systems have already been discussed by Savage and Hutter [5] and call for a sophisticated scheme. Subsequently, different schemes for systems with large spatial gradients and dominant convection terms have been compared in [43]. According to their results, a high-resolution scheme with total variation diminishing property is advisable. We deploy a central-upwind (CU) scheme, developed in the works of Kurganov and Tadmor [44], Kurganov et al. [45], and further deployed with special focus on depth-integrated Saint–Venant systems and Savage–Hutter type models in [46]. This robust scheme deploys Godunov-type finite volume methods, capable of solving systems of balance laws by specially treating source terms, nonconservative products and flux terms in conjunction with local wave speeds. The scheme is able to handle the discretization of nonconservative terms, which appear only proportional to the small parameter ϵ and are hence of minor importance, as is elaborated in [46]. As a core property, the scheme is both shock-capturing and non-oscillatory due to the satisfaction of the total variation diminishing property. The Nessyahu– Tadmor (NT) scheme, a forerunner of these schemes, has been proposed by Nessyahu and Tadmor [47]. As for discretization in time, a classical two-step Runge–Kutta scheme is applied. For the explicit time stepping, the time step size is restricted by the CFL condition, with Cmax = 0.05 in all computations. For details on the derivation of the scheme, its properties and limitations, see [46], where also the ability to capture surface waves is shown. For the sake of brevity, we do not give details on the discretization here. Information about the applied wave speed and the CFL number are given in Appendix B. To solve the system, the balance equations, Eqs. (22), (26), (29), (36) and (37), are rewritten in vectorial form, subsumed in a solid

Us =

hν s hν s vxs s hZxx

hν f f f ⎝ U = hν f v x ⎠ , f hϖe

⎞

⎛

) ,

(43)

and the solid downslope flux ⎞ hν s vxs ) ( ) ( f ⎜ ⎟ s 2 f f ν ϖe h h ρ ρ ⎜ s ⎟ Fs = ⎜hν s vxs vxs + ϵ 1 − s ν s cos(ϑ ) − ϵ Eu − 2ϵ NZ 1 − s hZxx ⎟, ρ 2 ρs ρ ⎝ ⎠ ⎛

s vs hν s Zxx x

(44) as well as the fluid flux

⎛

⎞

f x

hν v ⎜ ⎟ ⎜ h2 ν f ϖef h ⎟ f f f f ⎜ ⎟. F = ⎜hν vx vx + ϵν cos(ϑ ) + ϵ Eu 2 ρf ⎟ ⎝ ⎠ f f hϖe vx f

f

(45)

The source-terms Ss , Sf denote

⎛ ⎜ ⎜

s Ss = ⎜ ⎜hν sin(ϑ ) +

⎝

⎞ 0 sf cD h (

ρs

⎟ ⎟ v − v − sτ ⎟ ⎟ ⎠ f x

s x

)

s

s ϵ Φxx h

and

⎛

⎞ 0

⎜ ⎟ sf ⎜ ⎟ ) ( f) cD h ( f f s 2 f ⎟ ⎜ hν sin(ϑ ) − f vx − vx + ϵ∂x hvx − sτ ⎟ Sf = ⎜ ρ ⎜ ⎟, ( ) √ ⎟ ⎜ s νC ⎝ ⎠ √ β f (vxs )2 κϖ 1 ν s − κϖ 2 1 + Nψ

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J. Heß, Y. Wang / European Journal of Mechanics / B Fluids (

ssτ =

ρ ρs f

1−

(

)

+ λκ (v

s 2 x)

cos(ϑ ) − Eub

ρf − s (vxf )2 ρ

))

f e

ϖ ρs

11

Parameter

Dimension and value

Description

ρs ρf

2500 kg/m3 1000 kg/m3 9.81 m/s2 40◦ 30◦ 33◦ 20 18 783 7.5 · 10−9 1715 10 kg/m s2 0.001 m 0.64 0.75 1.1 3.3 0.54 kg m−1 s 0.5

Solid density Fluid density Gravitational acceleration Inclination angle Internal friction angle Bed friction angle Navier fluid friction coefficient Drag Coefficient Hydraulic permeability coefficient Compressibility factor Intergranular basic stress Average particle size Critical volume fraction Maximum volume fraction solid Dilatancy Coefficient 1 Dilatancy Coefficient 2 Dynamical viscosity Initial volume fraction solid

ν0f vx0

0.5

Initial volume fraction fluid

0

Initial velocity

0 Zxx

85.75 kg/m s2 0 and 2572.5 kg/m s2 1 1

Initial intergranular stress Initial extra pressure Height versus length ratio Hypoplastic coefficients

g

( ) hν sgn vxs tan(φbed ), s

f

αb ν f ρf

vxf . The hypoplastic source term ( ) s,f s,f s Φxx is given in Eq. (30). The nonconservative terms C1 ∂x C2 , ( ) ( ) s,f s,f C3 ∂x C4 and Cs5 ∂x Cs6 can be decomposed as ⎛ ⎞ 0 ⎛ ⎞ f s f 2 0 ρ f ρ −ρ ⎟ ⎜ h ⎜ ⎟ ϵ cos( ϑ ) ν s ⎟ , Cs = ⎝ ( ν ρs ρ )⎠ , Cs1 = ⎜ 2 ⎜( 2 ⎟ ) f s s s f f ⎝ ⎠ h v − v ρ nu − ρ nu x x s 1+ ν s Zxx ρ ⎛ ⎞ 0 ( ) 0 f 2 ρ ⎟ ⎜h Cs3 = −ν s , Cs4 = ⎝ ϵ cos(ϑ ) ⎠ , 2 ρs 0 and the fluid bed friction sfτ =

–

Table 1 Dimensional parameter values.

with the solid bed friction

((

)

ϑ0 φint φbed αbf cDs k0 aD

σ0 δP νCs s νmax κϖ 1 κϖ 2 µs ν0s

ϖe0 ϵ fs , fD

0

⎛

Cs5

0

⎞

(

⎜ϵ ⎟ = ⎝ s NZ ⎠ , ρ

Cs6

0

=

0 s ρ hZxx 0

)

f

,

and for the fluid

⎛

⎞

0

⎛ ⎞

⎜ ⎟ h2 ⎜ ⎟ f − ϵ cos(ϑ ) C1 = ⎜ ⎟, ⎝ h2 2 ⎠ ) ( f − cos(ϑ ) ρ − ρ

0 f s C2 = ⎝ ν ⎠ ,

vxf

2

⎛ f C3

0

⎞

s f ⎜ h2 ⎟ f ρ −ρ ⎟ ⎜ = ⎜ − 2 ϵ cos(ϑ )ν ⎟, ρ ⎝ ⎠ ( ) h ϖef + cos(ϑ ) ρ − ρ f

⎛ f C4

0

⎞

s (ν )⎠ . =⎝ f s s hν vx − vx

2 In contrast to the NT-scheme, the flux is propagated with local wave speeds, which leads to less numerical dissipation. Furthermore, no staggered grid is needed. For the treatment of the nonconservative products in the form of C1 ∂x (C2 ), described in [46], numerical derivatives are constructed with nonlinear limiters. We apply a weighted essentially non-oscillatory (WENO) limiting scheme of third order, see [48] for details on WENO schemes and Wang et al. [49] for a discussion of the application of TVD limiters to Savage–Hutter type flows. Below, we present the results of various numerical studies for the equation system (42). We apply numerical simulations for the case of a simple inclined plane without complex terrain, that runs out into the horizontal plane, see Fig. 1. The transition between ramp and horizontal plane is smooth. The granular–fluid mixture pile is released out of rest at time t( = 0, with a parabolic initial ) geometry, given as h(t = 0, x) = h0 1 − (x − x0 )2 /16 , with h0 = 1 m, x0 = 5 m. The computational domain is x ∈ [− 10 m, 80 m], discretized by a number of grids, nx = 600. The basal inclination angle is given as

{ ϑ (x) =

40◦ , 40◦ (1 − (x − 24) /10) , 0◦ ,

0 ≤ x ≤ 24, 24 < x < 34, x ≥ 34.

The chosen basic parameters are as listed with dimensions in Table 1. Following Iverson, we give a relation between the basal

extra pore pressure and the averaged pressure, proposing ϖe |b = 3 ϖef and additionally assume an exaggeration of the influence of 2 the pore pressure at the bed, Eub = 10 Eu. The latter accents the influence of the basal excess pore pressure, referring to a lubricating basal fluid layer. Since the simulation only aims at a first investigation of the influence of extra pore-fluid pressure and hypoplasticity, the chosen values and parameters are most simple. Most of the parameters are either physical values and/or have been suggested in the literature, most notably in [26], and are just adopted here. We refrain from a detailed discussion. As elaborated in [50], we choose ϵ = 1 and λ = 1. This means that the aspect ratio of the physical debris flow is preserved. 5.1. Development of height The height profile for the basic case, i.e. without the influence of both the extra pore-fluid pressure (Eu = 0) and the hypoplastic stress (NZ = 0), is displayed in Fig. 2, respectively, for the curvilinear coordinate system and the projection on the topography. We reproduce the results of Meng and Wang [26]: Released from rest and driven by gravity, the pile accelerates and elongates. In the runout, the front decelerates and piles up. The bulk contains a smoothly curved form, only at t = 21, there appears a slight breaking off of the front, resulting in an advanced, platform-like extension. As shown in [26] and also Fig. 7, this extension vanishes with higher drag numbers, i.e. increased phase-coupling. The dimensional value cDs = 18783 is applied in the code and equal to the non-dimensional cDs∗ = 6: cDs∗ =

cDs

ρf

√

= g L

18783

√

1000

9.8 1

= 6.

Note that the resulting height profile in [26] is shown for a 2D flow with an ellipsoidal shell, so the pile also propagates in the cross-slope direction. The difference of the height profile between a flow with uniform and variable cross slope is not shown here. To compare our results with theirs, further simulations were performed with their code for a 1D case. The similarity of the obtained results also shows that, for this particular case, the ada ditional buoyant term τby plays an insignificant role. It is the only

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Fig. 3. Height-profile development for times t = 0, 3, 9, 15 and t = 21, with NZ = 0 and varying influence of the extra pore-fluid pressure, i.e changing Eu. The results under the influence of the dynamic pore pressure are plotted with dashed lines, Eu = 2.5 · 10−2 , the solid lines are for the results without dynamic pore pressure, Eu = 0. The initial extra pore-fluid pressure is considered as ϖe0 = 0. The height exaggeration is 40 times.

Fig. 2. Height-profile development for times t = 0 (dashed line), t = 3, 9, 15 and t = 21, of the basic equation system, i.e. without the influence of hypoplastic stress, NZ = 0, and without extra pore-fluid pressure, Eu = 0. Part (a) displays the height in the on-slope coordinate system, (b) shows the projection on the topography. To enhance the pile visibility, the height has been enlarged by 40 in comparison to the length scale in (a), and by 20 times in (b).

term that distinguishes our momentum balance equations, for the case without further influence of extra pore-fluid pressure and hypoplastic stress, from the momentum balance equations in [26]. The apparent height profile is considered as a referential basis in order to compare results in the following and from which changes can be distinguished. 5.2. Influence of extra pore-fluid pressure We now impose the influence of the pore-fluid extra pressure via the variations of the Euler number Eu and the initial value ϖe0 . While the consideration of a dynamical extra pore-fluid pressure in the development of the velocities is adjusted through changes in Eu, i.e. in the relation of the particular dynamic pressure to the inertial forces, an increased initial pressure represents the destabilization of soil, for example after heavy rainfall. As a result of the extra pore-fluid pressure, the solid bed friction is reduced and the driving pressure gradients are affected. As it can be seen in Fig. 3, this increases the mobility of the flow, i.e. the pile moves further. It is also apparent that the impact of the extra pressure is most visible at later time steps. Taken all together, this implies firstly, that the extra pore-fluid pressure prolongates the movement of the debris flow, and secondly, that this effect takes place after the debris has accelerated, i.e. increases with the flow process. Both observations match expectations: The excess pore pressure rises during the settling because the granular material, after being extended and stretched during the movement, is finally compressed again. Before the water is squeezed out of the pores, the excess pore pressure builds up and pushes the grains apart. With that, the excess pore pressure hinders the granular settling due to internal friction, i.e. prolongates the movement by pushing the granular material apart at the moment the bulk tends to settle

and rest. Fig. 4 shows the evolution of the pore-fluid extra pressure for two time steps, t = 9 and t = 21, and the respective solid volume fraction ν s . The pore-fluid extra pressure forms a curve with the top in the middle of the flow and small peaks at the front and rear end at t = 9, while for the later times, e.g. t = 21, these develop into sharp peaks, representing the highest values of the extra pore-fluid pressure. Qualitatively, these peaks can also be found in the results of George and Iverson [15]. Nonetheless, in real debris flows, the water could be squeezed out of the pores at the front easier, since particle size segregation would lead to larger pore diffusivity and therefore decrease the pore fluid pressure there. For the solid volume fraction, the highest values are at the tail of the flow, as shown in Fig. 4. The dynamic pore-fluid pressure influences the solid-distribution only marginally. Next, to investigate on the influence of the initial pore-fluid pressure, ϖe0 is increased. With the mixture density ρ = 0.5· 2500 kg/m3 +0.5· 1000 kg/m3 = 1750 kg/m3 , the dimensional value of ϖe0 = 2572.5 kg/ms2 applied here denotes a non-dimensional initial extra pore pressure of

ϖ α∗ =

ϖα 2572.5 = = 0.15. ρg H 1750 · 9.1 · 1

This represents a triggering mechanism that can condition the onset of avalanches in nature, see [32,51]. Thus, the choice of the initial extra pore-fluid pressure depends on the investigated case and its physical implications. Fig. 5 shows that the resulting increased mobility due to the extra-pore fluid pressure can be observed earlier. In contrast to Fig. 3, in which the influence of the dynamic pore pressure only evolves with the flow dynamics, the initial level of the extra pore-fluid pressure significantly influences the acceleration from the start in Fig. 5 and also increases the impact at t = 21. 5.3. Influence of hypoplastic stress Neglecting the extra pore-fluid pressure (Eu = 0), we now investigate the influence of the hypoplastic stress by setting NZ to a non-zero value. This dimensionless number NZ gives the relation of the hypoplastic stress to the inertial forces, denoting the influence of intergranular friction. As it can be seen in Fig. 6, due to the introduction of an intergranular stress, the debris piles up with a steeper front as well as more peaked, rather copped than rounded, while the running distance is not significantly changed. At the front, a platform-like extension develops distinctively. Here, the fluid phase is dominant and runs out of the granular skeleton. For higher drag factors, this kind of extension changes into a smooth and rounded flow front, see Fig. 7 and [26], since the fluid and the

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Fig. 6. Influence of the hypoplastic stress on the flow dynamics. A comparison of the height-profile development for times t = 0, t = 3, t = 9, t = 15, and t = 21 with 0 changed NZ = 0.15 (dashed line) and an initial hypoplastic stress Zxx = 5 · 10−3 . No pore-fluid extra pressure, Eu = 0. The height exaggeration is 40 times.

Fig. 7. Comparison of the height for times t = 9 and t = 21 with changing influence of the drag. With hypoplasticity, NZ = 0.15, and no pore-fluid extra pressure, Eu = 0. The height exaggeration is 40 times. Fig. 4. Extra pore pressure development and volume fraction for times t = 9 and t = 21. Panel (a) displays the extra pore-fluid pressure, panel (b) shows a comparison of the solid volume fraction for Eu = 0 and Eu = 2.5 · 10−2 , together with the flow front for Eu = 2.5 · 10−2 . Results are derived for ϖe0 = 0, NZ = 0.

at the flow end, where the granular material piles up again, while the fluid tends to flow further, dominating the larger, platform-like front. The initial intergranular stress applied here, see Table 1, has the non-dimensional value Zij0∗ =

Fig. 5. Influence of changing initial extra pore-fluid pressure ϖe0 . Comparison of the height-profile development for selected times t = 9 and t = 21 with NZ = 0 and changing Eu. The height exaggeration is 40 times.

solid do not separate. Also, the influence of hypoplasticity is most visible during the settling at times t = 15 and t = 21, when intergranular contacts become more stable. s Fig. 8 displays the intergranular stress Zxx and a comparison of the solid volume fraction ν s with and without the intergranular stress, both for times t = 9 and t = 21. The intergranular stress runs approximately linear. The solid volume fraction exhibits a more distinct solid-dominated rear, ending with a sharp peak. The solid volume fraction exhibits an unrealistically high value here, as demonstrated also in other saturated mixture models, e.g. in [27] and [26]. This drawback points out to the necessity to either resort to countering mechanisms or apply a multi-layer approach, as in [52]. In general, differences in the distribution are most visible

Zij0

ρg H

=

85.75 1750 · 9.1 · 1

= 0.005.

Dam-break scenario. In order to further investigate the influence of the hypoplastic stress, a dam-break case is considered with x ∈ [0, 20], nx = 200 and NZ = 0.15. In a horizontal plane, ϑ (x) = 0, a rectangular pile with flat heading, supported on one side, is released out of rest and dissolves to the other side. This corresponds to the deposition of granular material after a dam break. In this connection, the structure-preserving influence of hypoplasticity is examined for a less dynamic case. This flow regime is most likely affected by the intergranular stress since hypoplasticity is considered less important for fast moving flows. Fig. 9(a) displays the height of the pile for a series of subsequent time steps and Fig. 9(b) a comparison between simulations with and without the influence of hypoplastic internal friction, i.e. for different values of NZ . It can be seen that in general, the qualitative behavior of such a process can be captured, see e.g. the experimental results of Wang et al. [53]. As the granular material settles down and due to a resisting intergranular contact stress, it does not dissolve as smoothly as without intergranular friction, but retains some sort of sharp bend. Pastor et al. [54] show that this sort of bend appears for dam-breaks with a wetted bed. It should be mentioned here that this case is outside of the actual scope of our depth-integrated model but can nonetheless be tackled.

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Fig. 9. Dam-break scenario. Height-profile development for NZ = 0.15 with subsequent times t = 0, t = 0.5, 1.0, 1.5, 2.5, 3.5 and t = 5.5 in Subfigure (a) and comparison of the height for times t = 0.5 and t = 2.5 with and without the 0 influence of hypoplasticity in (b). The initial hypoplastic stress is Zxx = 5 · 10−3 . The height exaggeration is 2 times. Fig. 8. Development hypoplastic stress and volume fraction for times t = 9 and s t = 21. Panel (a) shows the development of Zxx , panel (b) displays a comparison of the solid volume fraction together with the flow front for NZ = 0.15 at t = 21 . No influence of the dynamic pore-fluid pressure is considered, Eu = 0. The initial 0 hypoplastic stress is Zxx = 5 · 10−3 .

5.4. Combination of hypoplastic stress and extra pore-fluid pressure To investigate the capability of the new model, we now combine the influences of the intergranular stress and the extra pore-fluid pressure. This results in the superposition of the shape-effects of hypoplastic friction and the dynamic enhancement by the extra pore-fluid pressure. Fig. 10 shows the development of the height at different times, Fig. 11 a comparison of the volume fractions at two different times. The pile moves further due to the extra pressure and again, a copped peak is visible as well as an extension due to out-flowing fluid at the front. Furthermore, the distribution of the solid volume fraction, showing an almost dry rear and fluid domination at the front, points out further possibilities for improvement, since depicting the leakage of the fluid during the settlement with a two-layer approach may be advantageous. In a first attempt to investigate the influence of an extra porefluid pressure and an intergranular contact stress, implemented in terms of hypoplasticity, the expected results are obtained. The extra pore-fluid pressure enhances the mobility of the mixture, as it reduces the friction and affects the driving force. The implementation via an own evolution equation enables the model to consider the temporal development in conjunction with temporal alterations in the soil structure, i.e. dilatancy. Furthermore, via the setting of initial conditions for the extra pore-fluid pressure, geophysical triggering mechanisms can be taken into account as well.

Fig. 10. Case study with the influence of both the intergranular stress and the dynamic pore-fluid pressure. The height-profile development for times t = 0, t = 3, t = 9, t = 15 and t = 21 is shown with changing NZ and changing Eu. For the influence of hypoplastic stress and pore-fluid pressure, NZ = 0.15 and Eu = 2.5 · 10−2 is assumed (dashed lines). The height exaggeration is 40 times.

The intergranular contact friction shows its influence on the shape of the height profile. For deposition, the intergranular friction prevents the granular body from dissolving in a fluid-like way. Since a saturated mixture is considered, however, as a limiting case of the model, a resting pile of purely granular material with flown out fluid cannot be depicted. 6. Conclusions In this work, we proposed a debris flow model that accounts for the evolution of both the extra pore-fluid pressure and the intergranular friction and investigated on the influence of these features. Based on the results of the exploitation of the entropy principle in its formulation by Müller and Liu, we non-dimensionalized

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.

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15

Beside these modeling aspects, further investigation of 3D flow with a system of depth integrated equations and a full set of hypoplastic equations will be considered in future works as well as comparison with experimental findings. Also here, detailed insight in the stress–strain relations and the modeling of distinct coefficients will take place. In subsequent works, the model will be tested for 3D cases and compared to findings with other models and experimental data. Extended parametric be applied. Acknowledgments We want to acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) for this work within the project number WA 2610/3-1. Appendix A. Derivation: Depth-integrated momentum balances Fig. 11. Case study with the influence of both the intergranular stress and the dynamic pore-fluid pressure, showing the comparison of the solid volume fraction for times t = 9 and t = 21. The flow front for NZ = 0.15 and Eu = 2.5 · 10−2 is depicted at time t = 21.

and depth-integrated the balance equations. Comparison shows that the model is capable of reproducing previous modeling attempts but improves the basic framework of momentum balances with additional terms, accounting for the above mentioned physical mechanisms. The extra pore-fluid pressure increases the mobility and reduces the apparent bed friction, whereas the hypoplastic stress incorporates the anelastic deformational behavior of granular flows, counteracting fluid-like dissolving. This changes the shape of the granular pile during deceleration. Both quantities represent important physical mechanisms of granular media and debris flow. Numerical simulation corroborate the adequacy of these additional quantities and give a first impression of their influence on the flow dynamics. First investigations show the potential of the comprehensive consideration of these specific physical mechanisms. The combined influences of hypoplasticity and pore-fluid pressure determine both the shape of the bulk and its dynamics. Nonetheless, further work, both in modeling the interdependency and in numerical investigations, are considered fruitful to fully understand the impact of the new modeling approach and to refine the concurrence of the newly introduced mechanisms. Since, to the best of the authors’ knowledge, the combined integration of both hypoplasticity and extra pore-pressure evolution has not been carried out in previous works in a comparable way, the modeling of the mutual interferences is still in an early stage. An important link is granular dilatancy and, in general, the interplay between the deformational behavior of granular materials and the pore-pressure enhancement. Dilatancy describes the relation between shear and volumechanging, which depends on the intergranular forces. In return, these intergranular contact forces themselves depend on the pore pressure. With this, the modeling of the dilatancy part, tan(φ ), in the extra pore-fluid pressure source term could be connected to the hypoplastic stress tensorial variables Zijs , and the implementation of the hypoplastic intergranular friction in the solid-stress could be linked to pore-fluid pressure. Furthermore, consideration of the source term in the balance equation for the development of the solid volume fraction nα could account for the specific peculiarity of the microstructures of granular materials in their development, and account for e.g. dilatancy, as it is done by Iverson and George [16]. Also of importance is the investigation on the possible cases in which the additional buoyant term gains significant influence. However, these refinements are not the scope of this work and are left for further investigations.

The depth-integrated solid momentum balance in downslope direction (16) is

∂ ( s s) ∂ ( s s s) ϵ ∂ ( s) 1 [ s] hν v x + hν vx vx = s hTxx + s Txz b ∂t ∂x ρ ∂x ρ s ( ) s f cD hν ν + hν s g x + vxf − vxs s ρ ( ) ϵ ∂ν s νsρs + s hϖhs − hϖνs + O(ϵ 1+χ ). ρ ∂x ρ

(46)

For the fluid in the downslope x direction, it follows that

∂ ( f f f) ∂ ( f f) hν vx + hν v x v x ∂t ∂x ϵ ∂ ( f) 1 [ f ] = f hTxx + f Txz b ρ ∂x ρ s ) sνf ( s c h ν ϵ ∂ν D vxf − vxs − f +hν f gx − f ρ ∂x ( ρ ) × hϖhs −

νsρs hϖνs ρ

(47)

+ O(ϵ 1+χ ).

The depth-integrated vertical solid and fluid momentum balance yield

ϵ

∂ ( s s) ∂ ( s s s) ϵ ∂ ( s) 2 hν v z + ϵ hν vx vz + λκ hν s vxs = s hTxz ∂t ∂x ρ ∂x s ( ) √ cD h 1 [ ] + s Tzzs b + hν s gz + ϵ s vzf − vzs + O(ϵ 1+χ ), ρ ρ (48)

2 ∂ ( f f) ∂ ( f f f) ϵ ∂ ( f) f ϵ hν v z + ϵ hν vx vz + λκ hν f vx = f hTxz ∂t ∂x ρ ∂x s ( ) √ cD h 1 [ ] + f Tzzf b + hν f gz − ϵ f vzf − vzs + O(ϵ 1+χ ). ρ ρ

(49) For the zz fluid and solid stress at the bed we assume that Tzzf

[

] b

= ν f ϖ f |b ,

Tzzs

[

] b

= ν s ϖhf |b +ν s τRs ,

with a solid bed friction τRs . The reduction of Eqs. (48)–(49) to the order O(ϵ ) yields 2

λκ hν s vxs = f x

λκ hν f v

2

1 [ s] Tzz b + hν s gz + O(ϵ ),

ρs

(50)

1 [ ] = f Tzzf b + hν f gz + O(ϵ ).

ρ

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16

J. Heß, Y. Wang / European Journal of Mechanics / B Fluids (

If we now balance (50)1 and (50)2 2

λκ hvxs − λκ h

ρf f 2 ρ vx = ϖhf + τRs − ϖ f + hgz 1 − s ρs ρ

(

) f

+ O (ϵ ) , (51)

it follows for the solid bed friction that

( ) ρf f 2 2 s τ = −λκ h vx − s vx − ϖhf + ϖ f ρ ( ) ρf − hgz 1 − s + O(ϵ ), ρ s R

(52)

If Eq. (49) is further reduced to order O(ϵ χ ), the fluid pressure equalizes gravity, which yields the hydrostatic pressure balance if no extra pore-fluid pressure is regarded, so 0=

1 [

ρ

f

Tzzf

] b

cxs = abs(vxs ) +

√

cxf = abs(vxf ) +

√

Cmax = h 2

f

sin(ϑ ) and gz

= cos(ϑ ). For the solid bed friction Txzs , by [ ]b s using a Coulomb relation for the shear stress at the bed Txz = b ( s) s s sgn vx µb τR , it follows that [ ( ) ( s) s s ρf 1[ s] T = − sgn v ν µ h cos( ϑ ) 1 − x b ρ s xz b ρs ( )] ρf 2 2 − Eub ϖef + λκ h vxs − s vxf , (54) ρ

and likewise, for the fluid resistance at the bed, a Navier slip condition is applied with

ρf

and minimal eigenvalue can be evaluated. Since only the maximum wave speeds determine the time-stepping, we applied a simplified ansatz here, derived for a system without the additional fields. Such a system is similar to the equations of Meng and Wang [26], given in Section 4.2. The derivation of such wave speeds for a system has been shown in [52], there for a two-layer approach, which can easily be transferred to our system if the additional fields of extra pore-fluid pressure and hypoplastic stress are neglected. We follow their approach, giving not expressions for the actual eigenvalues of the fluxes in Eqs. (44)–(45) but bounds that can be determined via a perturbation argument. With these, we have the two wave speeds:

(53)

Now, we define a hydrostatic fluid-pressure part ϖ = ρ gz , which is consistent with Eq. (53), and a configurational pressure ϖνs = 2h (ρ s − ρ f )gz , see Eqs. (6)–(7), with gravity shares [ ]gx =

f Txz

]

=−

b

αbf ν f f v . ρs x

(55)

The bed-friction µsb is connected to the bed friction angle φbed by µsb = tan (φbed ) and a fluid bed-friction coefficient αbf arises. The last term in (54) arises due to the curvilinear coordinate system, pointing out the centrifugal forces. It was first incorporated by Savage and Hutter [55], and while Pitman and Le [40], Pudasaini [27] disregard this term, Meng and Wang [26] again incorporate it as it follows from derivation in curvilinear coordinates for the solid bed friction in the vertical momentum balances. Appendix B. Local propagation speeds and CFL condition The speeds of local propagation, or so-called wave speeds, describe the maximum characteristic speeds via the apparent fluxes. These wave speeds can be determined as the eigenvalues of the Jacobians of the flux-term. Here, we are not interested in deriving all eigenvalues exactly, i.e. a number of wave speeds equivalent to the number of variables (or fields), but only in the maximum wave speed. For this, estimated values are derived according to Kurganov and Miller [46] and, closer to the equations handled here, in [52]. The estimated maximum wave speed ax = max cxs , cxf ,

(

–

g cos ϑ hν s + 0.5hν f ,

(

)

g cos ϑ hν f + 0.5hν s .

(

)

The employed Courant–Friedrichs–Lewy (CFL) condition states

+ hν f gz + O(ϵ χ ) → ϖ f = hρ f gz . f h

1 [

)

)

is composed of the maximum wave speeds for both phases, respectively. In order to get these, as elaborated in [46], not the actual eigenvalues are determined, but bounds can be found with the coefficients of the characteristic equation, so that the maximum

ax ∆ t

∆x

+

ay ∆ t

∆y

= 0.05,

with the grid length in x and y directions, ∆x and ∆y, respectively, and the time-step size ∆t. During the computation, the CFL condition serves to condition the time-step size according to the apparent wave speeds. A small CFL number is chosen to increase the stability of this complex system with additional fields. References [1] I. Müller, I.-S. Liu, Thermodynamics of mixtures of fluids, in: C. Truesdell (Ed.), Rational Thermodynamics, Springer, 1984. [2] I. Müller, A thermodynamic theory of mixtures of fluids, Arch. Ration. Mech. Anal. 28 (1) (1968) 1–39. [3] I.-S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal. 46 (2) (1972) 131–148. [4] J. Heß, Y. Wang, K. Hutter, Thermodynamically consistent modeling of granular-fluid mixtures incorporating pore pressure evolution and hypoplastic behavior, Contin. Mech. Thermodyn. 29 (1) (2017) 311–343. [5] S.B. Savage, K. Hutter, The motion of a finite mass of granular material down a rough incline, J. Fluid Mech. 199 (1989) 177–215. [6] D. Kolymbas, A rate-dependent constitutive equation for soils, Mech. Res. Commun. 4 (6) (1977) 367–372. [7] D. Kolymbas, A generalized hypoelastic constitutive law, in: Proc. XI Int. Conf. Soil Mechanics and Foundation Engineering, Vol. 5, San Francisco, 1985. [8] D. Kolymbas, An outline of hypoplasticity, Arch. Appl. Mech. 61 (3) (1991) 143–151. [9] X. Guo, C. Peng, W. Wu, Y. Wang, A hypoplastic constitutive model for debris materials, Acta Geotech. 11 (6) (2016) 1217–1229. [10] C. Peng, X. Guo, W. Wu, Y. Wang, Unified modelling of granular media with smoothed particle hydrodynamics, Acta Geotech. 11 (6) (2016) 1231–1247. [11] K.v. Terzaghi, Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen, Sitz.ber. Akad. Wiss. Wien Math-Naturwiss. Kl. Abt. IIa 123 (1923) 125–138. [12] K.v. Terzaghi, The shearing resistance of saturated soils and the angle between the planes of shear, in: Proceedings of the 1st International Conference on Soil Mechanics and Foundation Engineering, Vol. 1, 1936, pp. 54–56. [13] R.M. Iverson, The physics of debris flow, Rev. Geophys. 35 (3) (1997) 245–296. [14] S. Savage, R.M. Iverson, Surge Dynamics Coupled to Pore-Pressure Evolution in Debris Flows, Vol. 1, Millpress, 2003, pp. 503–514. [15] D.L. George, R.M. Iverson, A two-phase debris-flow model that includes coupled evolution of volume fractions, granular dilatancy, and pore-fluid pressure, Ital. J. Eng. Geol. Environ. 10 (2011) 415–424. [16] R.M. Iverson, D.L. George, A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis, in: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 470, The Royal Society, 2014. [17] F. Bouchut, E.D. Fernández-Nieto, A. Mangeney, G. Narbona-Reina, A twophase two-layer model for fluidized granular flows with dilatancy effects, J. Fluid Mech. 801 (2016) 166–221. [18] X. Meng, Y. Wang, Modeling dynamic flows of grain–fluid mixtures by coupling the mixture theory with a dilatancy law, Acta Mech. 229 (6) (2018) 2521–2538. [19] L. Schneider, K. Hutter, Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context, in: Advances in Geophysical and Environmental Mechanics and Mathematics, Springer, 2009.

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.

J. Heß, Y. Wang / European Journal of Mechanics / B Fluids ( [20] R. de Boer, W. Ehlers, The development of the concept of effective stresses, Acta Mech. 83 (1) (1990) 77–92. [21] I.-S. Liu, A solid–fluid mixture theory of porous media, Int. J. Eng. Sci. 84 (2014) 133–146. [22] A. Teufel, Simple Flow Configurations in Hypoplastic Abrasive Materials (Master’s thesis), Technische Universität Darmstadt, 2001. [23] Y. Wang, K. Hutter, Comparison of two entropy principles and their applications in granular flows with / without fluid, Arch. Mech. 51 (5) (1999) 605– 632. [24] S.L. Passman, J.W. Nunziato, P.B. Bailey, K.W. Reed, Shearing motion of a fluid saturated granular material, J. Rheol. 30 (1) (1986) 167–192. [25] Y. Wang, K. Hutter, A constitutive theory of fluid-saturated granular materials and its application in gravitational flows, Rheol. Acta 38 (1999) 214–223. [26] X. Meng, Y. Wang, Modelling and numerical simulation of two-phase debris flows, Acta Geotech. 11 (5) (2016) 1027–1045. [27] S.P. Pudasaini, A general two-phase debris flow model, J. Geophys. Res. 117 (2012). [28] C. Fang, Y. Wang, K. Hutter, Shearing flows of a dry granular material— hypoplastic constitutive theory and numerical simulations, Int. J. Numer. Anal. Methods Geomech. 30 (14) (2006) 1409–1437. [29] E. Bauer, Calibration of a comprehensive hypoplastic model for granular materials, Solids Found. 36 (1) (1996) 13–26. [30] W. Fellin, Ein erster Einblick in die Hypoplastizität, ÖIAZ 144 (11–12) (1999) 434–436. [31] M. Arnold, I. Herle, Hypoplastic description of the frictional behaviour of contacts, Numer. Methods Geotech. Eng. (2006) 101–106. [32] R.M. Iverson, M.E. Reid, R.G. LaHusen, Debris-flow mobilization from landslides, Annu. Rev. Earth Planet. Sci. 25 (1) (1997) 85–138. [33] B. Svendsen, K. Hutter, L. Laloui, Constitutive models for granular materials including quasi-static frictional behaviour: Toward a thermodynamic theory of plasticity, Contin. Mech. Thermodyn. 11 (4) (1999) 263–275. [34] K. Hutter, L. Schneider, Important aspects in the formulation of solid–fluid debris-flow models. Part I. Thermodynamic implications, Contin. Mech. Thermodyn. 22 (5) (2010) 363–390. [35] K. Hutter, L. Schneider, Important aspects in the formulation of solid–fluid debris-flow models. Part II. Constitutive modelling, Contin. Mech. Thermodyn. 22 (5) (2010) 391–411. [36] M. Goodman, S. Cowin, A continuum theory for granular materials, Arch. Ration. Mech. Anal. 44 (4) (1972) 249–266. [37] N.P. Kirchner, Thermodynamically consistent modelling of abrasive granular materials. I. Non-equilibrium theory, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 458, The Royal Society, 2002, pp. 2153–2176. [38] N.P. Kirchner, A. Teufel, Thermodynamically consistent modelling of abrasive granular materials. II. Thermodynamic equilibrium and applications to steady shear flows, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 458, The Royal Society, 2002, pp. 3053–3077.

)

–

17

[39] B. Svendsen, K. Hutter, On the thermodynamics of a mixture of isotropic materials with constraints, Int. J. Eng. Sci. 33 (14) (1995) 2021–2054. [40] E.B. Pitman, L. Le, A two-fluid model for avalanche and debris flows, Phil. Trans. R. Soc. A 363 (1832) (2005) 1573–1601. [41] J. Gray, M. Wieland, K. Hutter, Gravity-driven free surface flow of granular avalanches over complex basal topography, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 455, 1999, pp. 1841–1874. [42] T.B. Anderson, R. Jackson, Fluid mechanical description of fluidized beds. Equations of motion, Ind. Eng. Chem. Fundam. 6 (4) (1967) 527–539. [43] Y. Wang, K. Hutter, Comparisons of numerical methods with respect to convectively dominated problems, Internat. J. Numer. Methods Fluids 37 (6) (2001) 721–745. [44] A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (1) (2000) 241–282. [45] A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740. [46] A. Kurganov, J. Miller, Central-upwind scheme for Savage–Hutter type model of submarine landslides and generated tsunami waves, Comput. Methods Appl. Math. 14 (2) (2014) 177–201. [47] H. Nessyahu, E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (2) (1990) 408–463. [48] X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1) (1994) 200–212. [49] Y. Wang, K. Hutter, S.P. Pudasaini, The savage-hutter theory: A system of partial differential equations for avalanche flows of snow, debris, and mud, ZAMM Z. Angew. Math. Mech. 84 (8) (2004) 507–527. [50] M.-C. Chiou, Y. Wang, K. Hutter, Influence of obstacles on rapid granular flows, Acta Mech. 175 (1) (2005) 105–122. [51] R.M. Iverson, Regulation of landslide motion by dilatancy and pore pressure feedback, J. Geophys. Res.: Earth Surf. 110 (F2) (2005). [52] X. Meng, Y. Wang, C. Wang, J.-T. Fischer, Modeling of unsaturated granular flows by a two-layer approach, Acta Geotech. 12 (3) (2017) 677–701. [53] C. Wang, Y. Wang, X. Meng, C. (n.d.) Peng, Dilatancy effects on the submerged granular column collapse. [54] M. Pastor, B. Haddad, G. Sorbino, S. Cuomo, V. Drempetic, A depth-integrated, coupled SPH model for flow-like landslides and related phenomena, Int. J. Numer. Anal. Methods Geomech. 33 (2) (2009) 143–172. [55] S.B. Savage, K. Hutter, The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis, Acta Mech. 86 (1991) 201–223.

Please cite this article in press as: J. Heß, Y. Wang, On the role of pore-fluid pressure evolution and hypoplasticity in debris flows, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.09.005.