Some remarks on avalanches modelling: an introduction to shallow flows models

Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models Enrique D. Fernández-Nieto and Paul Vigneaux These notes are dedicated to D. Antonio Valle Sánchez (1930–2012). D. Antonio was the first Spanish PhD student of Jacques-Louis Lions. He can be considered as one of the founders of modern Applied Mathematics in Spain. Abstract The main goal of these notes is to present several depth-averaged models with application in granular avalanches. We begin by recalling the classical Saint-Venant or Shallow Water equations and present some extensions like the Saint-Venant–Exner model for bedload sediment transport. The first part is devoted to the derivation of several avalanche models of Savage–Hutter type, using a depth-averaging procedure of the 3D momentum and mass equations. First, the Savage–Hutter model for aerial avalanches is presented. Two other models for partially fluidized avalanches are then described: one in which the velocities of both the fluid and the solid phases are assumed to be equal, and another one in which both velocities are unknowns of the system. Finally, a Savage–Hutter model for submarine avalanches is derived. The second part is devoted to non-newtonian models, namely viscoplastic fluids. Indeed, a one-phase viscoplastic model can also be used to simulate fluidized avalanches. A brief introduction to Rheology and plasticity is presented in order to explain the Herschel–Bulkley constitutive law. We finally present the derivation of a shallow Herschel–Bulkley model. E.D. Fernández-Nieto () Dpto. Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain e-mail: [email protected] P. Vigneaux Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, Lyon, France e-mail: [email protected] C. Parés et al. (eds.), Advances in Numerical Simulation in Physics and Engineering, SEMA SIMAI Springer Series 3, DOI 10.1007/978-3-319-02839-2__2, 51 52 E.D. Fernández-Nieto and P. Vigneaux 1 Introduction: Shallow Water Equations The classical shallow water equations were first derived in 1871 by Saint-Venant (see [46]). This system of equations describe the motion of a shallow layer of fluid in a channel, a lake, coastal areas, etc. Several extensions of these classical equations have been proposed in the literature. For example, in [25] Gerbeau and Perthame propose a viscous Shallow Water model. They perform the asymptotic analysis of the Navier–Stokes equations where friction effects at the bottom have been taken into account. While in a first order approximation the viscous terms do not appear in the equations, a second order is needed to get them. In [35], a viscous one layer 2D Shallow-Water system is derived, by including a surface-tension term associated to the capillary effects at the free surface and a quadratic friction term at the bottom. These terms have been useful to prove the existence of global weak solutions in [13]. In the simplecase of a rectangularchannelwith constantwidth and a fixedbottom topography (see Fig.1), the Shallow Water equations are 8 < : @[email protected] u/D0; @t.hu/C@xhu2C1 2gh2Dgh@xzbghSf;(1) where xdenotes the horizontal variable through the axis of the channel and tis the time variable. u.x; t/ and h.x; t/ represent the velocity and the height of the water column, respectively. gis the gravity and zb.x/ the bottom topography (see Fig.1). The term Sfmodels the friction forces. In the particular case of the Manning law we have SfDg2juju R4=3 h ;(2) where is the Manning’s coefficient and Rhis the hydraulic radius, which can be approximated by h. The Saint-Venant–Exner equations take into account the bed-load sediment transport. In this case, we have the Shallow Water or Saint-Venant system coupled with a continuity equation to model the evolution of the sediment layer, 8 ˆ ˆ < ˆ ˆ : @[email protected] u/D0; @t.hu/C@xhu2C1 2gh2Dgh@xzbghSf; @tzbC@xqbD0; (3) where D1=.1  0/and 0is the porosity of the sediment layer. qbD qb.h; q/ represents the solid transport discharge. The definition of the solid transport Some Remarks on Shallow Avalanches Modelling 53 h(x,t) x z z (x) b Fig. 1 Shallow water equations with a fixed bottom discharge is set usually by empirical laws. In the Appendix some classical formulae are presented. Far to be exhaustive, several extensions of the Shallow Water equations can be mentioned: models that take into account varying bottom topography [11,18,24]; models to study erosion phenomena with local coordinate variable in space and time [12] or to study flows in rotating drums [27]; models that take into account dispersive effects [33]; models for two-layer stratified flows with viscosity and capillarity [39], turbidity currents models [38], multilayer shallow water models to incorporate tridimensional effects [7,23]::: In the pioneering work of Savage and Hutter [47], a shallow-water type model has been proposed to study aerial avalanches. In the following section we describe the derivation of the Savage–Hutter model. The classical Shallow Water system is the particular case of the Savage–Hutter model obtained by neglecting the Coulomb friction term. Therefore, its derivation from Navier–Stokes equations is a particular case of the general study presented in next section. In the following sections, the derivation of several shallow water type models to study three different types of avalanches are presented. Sections 2–6correspond to Savage–Hutter type models for aerial avalanches, partially fluidized aerial avalanches and submarine avalanches (see [21]). In Sect.7, we present a brief introduction to Rheology and plasticity in order to explain the constitutive equation of the Herschel–Bulkley model. A depth-averaged Herschel–Bulkley model is presented in Sect. 8. This model is a one-phase approach to study solid–fluid mixtures avalanches and represents an alternative to the two-phase Coulomb approach. 2 Savage–Hutter Model for Aerial Avalanches Numerical modelling of sub-aerial debris or snow avalanches has been extensively investigated during this last decade with application to both laboratory experiments dealing with granular flows and geological events (see for example [2,5,6,12,27, 31,34,50]). Most of the models devoted to gravitational granular flows describe the behavior of dry granular material following the pioneering work of Savage and Hutter (see [47]) in which a shallow water type model (i.e. thin layer approximation 54 E.D. Fernández-Nieto and P. Vigneaux θ h(X,t) (x,z) X x x z Z Fig. 2 Local coordinates for a continuum medium) is derived to describe granular flows over a slopping plane based on Mohr–Coulomb considerations: a Coulomb friction is assumed to reflect the avalanche/bottom interaction and the normal stress tensor is defined by a constitutive law relating the longitudinal and the normal stresses through a proportionality factor K. New Savage–Hutter models over a general bottom have been proposed. For example in [11], Bouchut et al., propose a Savage–Hutter type model for aerial avalanche which takes into account the curvature of the bottom. A two-layer Shallow Water type model with compressible effects has been introduced in [37] by Morales de Luna. He considers an upper compressible layer and a lower incompressible layer. In this section, we presentthe derivationof the Savage–Huttermodel over a plane with constant slope. First, we consider the Euler equations in Cartesian coordinates XD.x; z/, VDu w;rVD0; (4) @t.V/CVrXVDrPCrX.gX/; (5) where gD.0; g/,gbeing the gravity acceleration, Vis the velocity field and , the density of the granular layer. Moreover, we denote by Pthe negative Cauchy stress tensor, also named pressure tensor, PDpxx pxz pzxpzz ; with pxzDpzx. Let us rewrite first the Euler equations in local coordinates .X; Z/ on an inclined plain whose slope is tan./ (see Fig.2). Zis the distance between the points .x; z/ and .Nx;b.Nx//,where b.Nx/ Dtan./ Nx: Some Remarks on Shallow Avalanches Modelling 55 That is, Zis the distance to the bed, measured along the normal direction and X measures the arc length along the inclined plain. Nxis the x-Cartesian coordinate of the point .X; 0/ (see Fig. 2). Let h.x; t/ be the height of the granular layer along the normal direction to the bed. The domain is f.X; Z/IX2Œ0; L; 0 < Z < h.X; t/g:(6) The relation between the Cartesian coordinates XD.x; z/and the coordinates .Nx;Z/ is XDNxZsin ;b.Nx/ CZcos/;(7) where .Nx;b.Nx// is a point of the bed. The following definitions will also be used: •Uand Ware the tangential and normal velocities, respectively, U WDcossin sin cosV: •AndPis the rotated Cauchy stress tensor: PDcossin  sin cosPcossin  sin cosDPXX PXZ PZX PZZ : Note that, as pxz Dpxz,thenPXZ DPZX . Equations (4)and(5) are re-written in the new variables as follows: 8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ : @X.U / [email protected]/D0; @t.U/C@X.U 2/C@Z.WU/@X.gX/D@X.PXX/@Z.PXZ/; @t.W/C@X.U W / C@Z.W 2/@Z.gX/D@X.PZX /@Z.PZZ /: (8) In what follows, the derivation of the model proposed by Savage and Hutter in [47] to study aerial avalanches is described following the items: •Œ@ Boundary and kinematic conditions. •ŒQ A Dimensional analysis. •ŒlHydrostatic pressure and constitutive law. •ŒM  Momentum conservation law. 56 E.D. Fernández-Nieto and P. Vigneaux •ŒRIntegration process. •Œ,!Final system of equations. 2.1 Œ@ Boundary and Kinematic Conditions We denote by nhthe unit normal vector to the free granular surface ZDhwith positive vertical component, and by n0D.0; 1/ the unit normal vector to the bottom (ZD0). The following kinematic condition is considered @thCUjZDh@XhWjZDhD0; (9) which means that the particles at the free surface are transported with velocity .U jZDh;WjZDh/. The following boundary conditions are imposed: •OnZDh: nhPnhD0(10) Pnhnh.nhPnh/Dfrich.U / 0iD1; 2; (11) where frich.U / is the friction term between the granular layer and the air. For the sake of simplicity we will suppose that frich.U / D0. •OnZD0: .U; W / n0D0)WD0; (12) Pn0n0.n0Pn0/D0 @n0Pn0U jUjˇˇˇˇZD0 tanı0 01 A:(13) This last condition corresponds to a Coulomb friction law, defined in terms of the angle of repose ı0(see [47]). 2.2 ŒQ A Dimensional Analysis Next, a dimensional analysis of the set of Eqs.(8), the kinematic and boundary conditions is performed. The non-dimensional variables ( Q:) read: Some Remarks on Shallow Avalanches Modelling 57 .X;Z;t/D.L Q X;H Q Z;.L=g/1=2 Q t/; .U; W / D.Lg/1=2.Q U;" Q W/; hDHQ h; .PXX;PZZ /DgH.Q PXX;Q PZZ /; PXZ DgHQ PXZ; (14) where: •Dtan ı0,ı0being the angle of repose in the Coulomb term. •ByLand H, we denote, respectively, the tangential and normal characteristic lengths. •"DH=L, which is supposed to be small: the Savage–Hutter model has been shown to reproduce experimental granular collapse over horizontal plane for aspect ratio 0:5,see[34]. Using the above change of variables, the system of Eqs.(8) is re-written as follows: @X.U / [email protected]/D0; (15) @t.U /CU@XUCW @ZUC@X.bCZcos CPXX /" D@Z.PXZ/; (16) "f@t.W / C[email protected] / CW @Z.W / [email protected]/gC C@Z.b CZcos/ D@Z.PZZ /; (17) where tildes have been dropped for simplicity. The kinematic condition (9) is re-written as: @thCUjZDh@XhWjZDhD0: (18) Finally, the boundary conditions (10)–(13) are now given by: •OnZDh,wehavenhD."@Xh; 1/='Swith 'SDp1C"2.@Xh/2,then from (10)and(11) we obtain "@XhPXX CPZX D0; (19) "@XhPXZ CPZZ D0: (20) 58 E.D. Fernández-Nieto and P. Vigneaux •OnZD0,wehaven0D.0; 1/, then from (12)and(13) we obtain WjZD0D0; (21) PXZ DPZZ U jUjˇˇˇˇZD0 tan ı0:(22) 2.3 ŒlHydrostatic Pressure and Constitutive Law From (17) we obtain @Z.PZZ/DcosCO."/: (23) If we integrate (23) from Z>0to h, we have, up to order ", PZZ D.h Z/cos: (24) The following constitutive law is considered (see [47]) PXX DKPZZ ; where Kmeasures the anisotropy or normal stress effects: while KD1corresponds to isotropic conditions, K¤1makes ‘overburden pressures’ different from the normal stresses parallel to the basal surface. In the case of the Shallow Water equations, KD1is assumed. The coefficient Kis defined according to the motion of the granular layer (see [45]): KDKact if @XU>0; Kpas if @XU<0; with Kact=pas D2sec21.1 cos2sec2ı0/1=21; being the internal friction angle, defined in terms of the type of grains and size. The definition of Kcan be done in different ways. For example, while in [28] Heinrich et al. consider KD1, other definitions of Kcan be found in [29]. The effects related to the definition of Kin numerical modelling of experimental and natural flows is studied in [43,45]. Using the previous relations, we have, up to order , PXX DKPZZ DK.h Z/cos: (25) Some Remarks on Shallow Avalanches Modelling 59 2.4 ŒM Momentum Conservation Law: With Hydrostatic Pressure and Anisotropy of the Normal Stress By replacing (24)and(25)in(16) and using the incompressibility equation (15), we obtain, up to second order, @t.U /C@XU2C@Z.UW/C@XbCZcos CK.hZ/cos "D@Z.PXZ /: (26) 2.5 ŒRIntegration Process In this section, the mass equation (15) and the momentum equation (26) are depthaveraged in the normal direction. Let us introduce the following notation: we denote by N Uthe average of the velocity along the normal direction: N UD1 hZh 0 U.X;Z/dZ: We also introduce the notation: U2D1 hZh 0 U2.X; Z/dZ: If Eq.(15) is integrated from ZD0to ZDh, we obtain [email protected] N U/UjZDh@XhCWjZDhWjZD0: Now, using (18)and(21), the averaged mass equation is obtained @[email protected] N U/D0: Let us now integrate Eq.(26) from ZD0to ZDh. As in the previous case, we use the kinematic condition (18) to obtain @t.h N U/[email protected]/CZh 0 @XbCZcosCK.h Z/cos dZ" D .PXZ.h/ PXZ.0//: (27) 66 E.D. Fernández-Nieto and P. Vigneaux granular material near the front. The right column shows qfand qs. We can observe that the motion of the solid phase stops before. Figure 6shows the evolution of the avalanche for 'D0:4. The initial condition and the values of rand ıare the same. Let us remark that at the front the dotted line practically coincides with the horizontal axis, meaning that there is only fluid near of the front. If we compare these two simulations, we can also observe that the maximum heights and the total lengths of spreading of the avalanches are completely different. For 'D0:4,the fluid goes out the domain. The presence of an interstitial fluid in the avalanche, neglected in the Pitman– Le model, may have a strong influence in its evolution. The flow of fluidized avalanches can be much more complex than the ones simulated with the Pitman– Le model. For example, non-hydrostatic pressure effects, related to the pore fluid pressure, may appear. Iverson and Denlinger extended the Savage–Hutter model in [29] to study avalanches of fluidized granular masses where the pores between the grains are assumed to be filled with a fluid, under the assumption that the velocities of both phases coincide, and by including the bed pore fluid pressure as an unknown of the system. Let us study now the derivation of a simplified version of the model proposed by Iverson and Denlinger in [29] to study partially fluidized aerial avalanches (Fig.7). We consider a granular layer of density sand porosity 0. We assume that the pores in the granular layer are filled with a fluid of density w. Then, the density of the fluidized layer is defined as D.1  0/sC 0w:(38) As in the previous section, the model will be described in local coordinates over a plain with constant slope (see Fig.2). Again Uis the velocity parallel to the bottom; W, the velocity perpendicular to the bottom; and Pthe rotated pressure tensor. Let us consider again the system of equations given by Euler equations in local coordinates: 8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ : @X.U / [email protected]/D0; @t.U/C@X.U 2/C@Z.WU/@X.gX/D@X.PXX/@Z.PZX/; @t.W/C@X.U W / C@Z.W 2/@Z.gX/D@X.PXZ/@Z.PZZ /: (39) In a binary mixture model the pressure tensor of the mixture is given by PDPsCPfw.UfUb/˝.UfUb/s.UsUb/˝.UsUb/; where Uwis the velocity of the fluid phase, Usis the velocity of the solid phase and UbD.wUwCwUs/=.sCw/is the barycentric velocity. Some Remarks on Shallow Avalanches Modelling 67 0 5 10 15 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0 5 10 15 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0 5 10 15 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0 5 10 15 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0 5 10 15 0 0.2 0.4 0.6 0.8 1 0 5 10 15 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 qh 0 5 10 15 0 0.2 0.4 0.6 0.8 1 qh 0 5 10 15 0 0.2 0.4 0.6 0.8 1 qh 0 5 10 15 0 0.2 0.4 0.6 0.8 1 qh 0 5 10 15 0 0.2 0.4 0.6 0.8 1 q t = 0.1 s. t = 0.5 s. t = 1 s. t = 1.5 s. t = 2 s.h Fig. 6 Two-phase avalanche. 'D0:4.Left:continuous line: granular free surface. Dotted line: hs;Right: discharge. Continuous line:qs.Dashed line:qf 68 E.D. Fernández-Nieto and P. Vigneaux θx z h(X,t) Fig. 7 Partially fluidized avalanches In order to model the evolution of the granular layer using the Euler equations, we suppose, following [29], that the velocity of the fluid in the pores and the grains are the same, UsDUfDU.ThenPcan be written as PDPsCPf; where Psand Pfare the pressure tensor of the solid phase (grains) and the fluid phase. The derivation of the model follows the same items as in the previous section. 3.1 Œ@ Boundary and Kinematic Conditions Let us denote again by nhthe unit normal vector to the free granular surface ZD hwith positive vertical component and by n0D.0; 1/ the unit normal vector to the bottom (ZD0). The kinematic condition is defined by (9). For the boundary conditions, the only difference is the definition of the Coulomb friction term. The following boundary conditions are imposed: •OnZDh:PnhD0. •OnZD0: the non-penetration condition WD0and the following Coulomb friction law are imposed: Pn0n0.n0Pn0/D0 @n0.PPf/n0U jUjˇˇˇˇZD0 tan ı0 01 A: In this last condition, the difference between the stress tensor and the fluid stress tensor, PPf, is used to take into account the buoyancy effects, since PsD PPf. Some Remarks on Shallow Avalanches Modelling 69 3.2 ŒQ A Dimensional Analysis The same non-dimensional variables as in the previous section, defined in (14), are considered. Then, • The system of equations is defined by (15)–(17). • The kinematic condition by (18). • The boundary condition on ZDhby (11)and(20). • The boundary condition on ZD0is different from (22). In this case we have PXZ D.PZZ Pf ZZ /U jUjˇˇˇˇZD0 tan ı0;(40) and the non-penetration condition WD0. 3.3 ŒlHydrostatic Pressure and Constitutive Law The main difference between the Savage–Hutter model presented in the previous section and the model for partially fluidized avalanches appears in this item. From (17) we obtain @Z.PZZ/DcosCO."/: (41) If we integrate (41) from Z>0to ZDh, we have, up to order ", PZZ D.h Z/cos: (42) But, as PDPsCPf,wehave Ps ZZ CPf ZZ DPZZ Dcos.hZ/: (43) In order to consider the anisotropy of the solid phase the following constitutive conditions are again considered (see for example [29,45]): Ps XX DKPs ZZ ;Pf XX DPf ZZ ; where Kmeasures the anisotropy or normal stress effects in the solid phase (see previous section). The difference appears in this step because, in order to impose these two constitutive conditions, an expression for both Ps ZZ and Pf ZZ has to be known. But only the expression of the total pressure PZZ is known. 70 E.D. Fernández-Nieto and P. Vigneaux In order to model the flow of a grain-fluid mixture, Iverson and Denlinger (see [29]) assume a linear profile of the normal stress Pf ZZ , which is consistent with Eq.(43). Moreover they suppose that Pf ZZ takes its maximum value at ZD0and is proportional to the pressure in absence of the granular phase. They suppose Pf ZZ .Z/ Dcos.hZ/; (44) being a parameter of the model. In this case, by (43)wehave Ps ZZ .Z/ D.1 /cos.h Z/: (45) Remark 1. In [29], the authors propose not to set as a fixed parameter in time. Instead, they propose to rewrite the model in terms of the pore fluid pressure pbed Dhcos . Then, they assume that the evolution of pbed can be described by a convection-diffusion equation. For the sake of simplicity in these notes, is considered as a fixed parameter. Let us remark that we can set D 0, the porosity of the layer. 3.4 ŒM Momentum Conservation Law: With Hydrostatic Pressure and Anisotropy of the Normal Stress of the Solid Phase By replacing (44)and(45)in(16) and using the incompressibility equation (15), we obtain up to second order @t.U / C@[email protected] W / C@XbCZcosC.h Z/cos. CK.1 //" D @Z.PXZ/: (46) 3.5 ŒRIntegration Process As in the previous section, let us define: N UD1 hZh 0 U.X;Z/dZ and U2D1 hZh 0 U2.X; Z/dZ: As there is no difference in the integration process for the mass equation, we focus here on the momentum equation. Some Remarks on Shallow Avalanches Modelling 71 Let us integrate Eq.(46) from ZD0to ZDhand use the kinematic conditions (18) to obtain @t.h N U/[email protected]/CZh 0 @XbCZcos C.h Z/cos . CK.1 //dZ" D .PXZ.h/ PXZ.0//: (47) Moreover, Zh 0 @XbCZcosC.h Z/cos. CK.1 //dZ Dh@Xb C@Xh2 2cos.CK.1 //: Then, we obtain the following averaged momentum conservation law, @t.h N U/C@XhU2C"h2 2cos.CK.1//D"h@Xb .PXZ.h/PXZ.0//: (48) Now, we can use the boundary conditions and the constitutive laws to derive PXZ.h/ and PXZ.0/: •Using(19) and the constitutive laws Ps XX DKPs ZZ,Pf XX DPf ZZ we have PXZ.h/ D"@XhPs ZZ .K 1/: If we suppose again that is of order for some 2.0; 1/,(Dtanı0D O."/)then PXZ.h/ DO."1C/: (49) •UsingEq.(40), we obtain PXZ.0/ D.PZZ .0/ Pf ZZ .0// U jUjˇˇˇˇZD0 tanı0: Now, using (42)and(43)wehave .PZZ .0/ Pf ZZ.0// Dhcos .1/ CO."/: 72 E.D. Fernández-Nieto and P. Vigneaux Then,  PXZ.0/ Dhcos.1/ U jUjˇˇˇˇZD0 tanı0CO."1C/: (50) Finally, substituting (49)and(50)in(48), the averaged momentum equation is obtained @t.h N U/C@XhU2C"h2 2cos.CK.1 // D"h@Xbh.1 /cosU jUjˇˇˇˇZD0 tan ı0CO."1C/: 3.6 Œ,!Final System of Equations Coming back to the original variables, using (14), neglecting the terms of order "1C and supposing a constant profile of the velocities, the following system is obtained 8 < : @[email protected] N U/D0; @t.h N U/C@XhN U2Cgcosh2 2. CK.1 //Dgh@XbCT;(51) where Tis the Coulomb friction term. In this model this term must be understood as follows: If jTjc)TDgh.1 /cosN U jN Ujtanı0; If jTj< c)N UD0; where cDgh.1 / cos tan ı0. 4 Comparison with Pitman–Le Model Let us remark that if anisotropy is taken into account in the deduction of the solid phase momentum equation in the Pitman–Le model, Eqs.(37c)and(37d) will read as follows: Some Remarks on Shallow Avalanches Modelling 73 @t.'hUs/C@X.'hU 2 sCKgcos'h2 2/Dr 2gh2cos@ X' gh'@Xb Cˇh.UfUs/CTI (52a) @t..1 '/hUf/[email protected] '/hU2 fCgcos.1'/h2 2/Dgh2 2cos@X.1 '/ gh.1 '/@Xb 1 rˇh.UfUs/: (52b) Let us now consider that UsDUfDU, that is, the assumption considered in the deduction of the Iverson–Denlinger model and let us define D'sC.1 '/f: Then, from (37a)and(37b) we obtain @t.h/ C@X.hU / D0: And from (52a)and(52b) we obtain that @t.hU / C@X.hU 2Cgcos h2 2. Cs'.K 1/// Dgh@XbCT:(53) Note that in the two-phase model the pressure of the fluid phase evaluated at ZD 0is pPL bed Dgfcos.1'/h; while in the model proposed by Iverson and Denlinger the pressure at the bottom of the fluid phase is assumed to be: pbed Dgcos h: If we set Df .1 '/; 74 E.D. Fernández-Nieto and P. Vigneaux we have pPL bed D gcos h: On the other hand, we have Cs'.K 1/ DsC  rCK.s  r/: If we define finally by  DsC  r; we have the equality Cs'.K 1/ D CK. /: Then, we can rewrite system (53)as @t.hU/C@X.hU 2Cgcosh2 2. CK. /// Dgh@XbCT: That is, we have the same structure as the momentum equation of (51)for the Iverson–Denlinger model. This implies a relation between the hypothesis considered in [29] and the two-phase model when the velocities of the two phases coincide: ppl bed DpbedrC.1 '/.sf/rghcos: Let us remark finally that, while in the Pitman–Le model the pressure of the phases are considered to be hydrostatic, the inclusion of the pressure at the bed in terms of the parameter can be understood as a way to introduce a deviation from the hydrostatic pressure law in the Iverson–Denlinger model. 5 Submarine Avalanches In this section, we present a simplified version of the two-layer Savage–Hutter type model proposed in [21], with application to submarine avalanches and tsunami waves generated by them. Submarine avalanches or landslides have been poorly studied compared to their subaerialcounterparts.This is however a key issue in geophysics.Indeed, submarine granular flows driven by gravity participate in the evolution of the sea bottom and in particular of the continental margins. They also represent a threat to submarine Some Remarks on Shallow Avalanches Modelling 75 2 (x,z) X x x z Z θ h (X,t) h (X,t) 1 Fig. 8 Submarine avalanches infrastructures, especially for the oil or port industries as well as to many sea shore inhabitants due to the potential tsunamis that can be triggered by such landslides. In the model derived in this section, index 1refers to the upper layer, composed of an homogeneous inviscid fluid of constant density 1. Index 2refers to the lower layer, composed of a granular material of density sand porosity 0(see Fig. 8). The pores of the granular layer are assumed to be filled with the fluid of the upper layer. Accordingly, the density of layer 2is given by: 2D.1  0/sC 01:(54) We consider the incompressible Euler equations, with unknowns ViDui wi;iD1; 2; being uiand wi, the horizontal and vertical velocity components of each layer, respectively. Then, the incompressible Euler equations can be written as divViD0; i D1; 2; (55) i@tViCiVirViDdivPiCir.gX/; i D1; 2; (56) where Pi,iD1; 2, represent the pressure tensor of each layer PiDpi;x x pi;x z pi;zxpi;zz ;i D1; 2; with pi;x zDpi;zx,i,iD1; 2, the densities of each layer, XD.x; z/,theCartesian coordinates and gD.0; g/, the gravity. In order to model the evolution of the granular layer using the Euler equations, on the one hand we suppose following [29] (see Sect. 3) that the velocity of the fluid in the pores of the second layer coincides with that of the grains. On the other hand, P2is assumed to be decomposed as 82 E.D. Fernández-Nieto and P. Vigneaux [email protected] U1/C[email protected] 1/C1ZS h2 @X.bCScos /dZ"Dfric.U1;U 2/CO."2/: Next, the integral term in the last equality can be computed as follows: ZS h2 @X.b CScos/dZ Dh1@XbC@Xh2 1 2cos[email protected]h2/: Finally, we obtain the equation [email protected] U1/C1@Xh1U2 1C"h2 1 2cosD"1h1@Xb[email protected]h2/ Cfric.U1;U 2/CO."2/: Let us now integrate Eq.(86) from ZD0to ZDh2. As in the previous case, we use the kinematic conditions (69) to obtain [email protected] U2/C[email protected] 2/C2Zh2 0 @XbCZcosC1 2 Œh1cos 1.1CK.11// C.h2Z/cos.21CK.221//dZ"D2.P2XZ.h2/P2XZ.0//: (90) Let us introduce the densities ratio rD1 2 ; where 1is the density of the fluid and 2is defined by (54). We obtain Zh2 0 @XbCZcosC1 2 Œh1cos1.1CK.1 1// C.h2Z/cos.21CK.221//dZ Dh2@Xb Crh2.K.1 1/C1/@X.h1cos/ C@Xh2 2 2cos.r2CK.1 r2//: Some Remarks on Shallow Avalanches Modelling 83 Replacing this last expression in (90) and dividing by 2, the following equation is obtained: @t.h2N U2/C@Xh2U2 2C"h2 2 2cos.r2CK.1 r2//D"h2@Xb "rh2.1CK.1 1//@X.h1cos/2 2 .P2XZ.h2/P2XZ.0//: (91) Just like in the previous case, the boundary conditions and the constitutive laws are used to derive 2P2XZ.h2/and 2P2XZ.0/: •Using(73)andPs 2XX DKPs 2ZZ ,Pf 2XX DPf 2XX,wehave 2P2XZ.h2/Dfric.U1;U 2/C2"@Xh2Ps 2ZZ .K 1/: We suppose again that 2Dtanı0DO."/with 2.0; 1/. Under this assumption, we have 2P2XZ.h2/Dfric.U1;U 2/CO."1C/: (92) •UsingEq.(76), we obtain 2P2XZ.0/ D.P2ZZ .0/ P1ZZ .0// U2 jU2jˇˇˇˇZD0 tan ı0: Now, using (79)and(80)wehave .P2ZZ .0/ P1ZZ .0// Dh2cos.21/CO."/: Therefore, assuming tan ı0DO."/,wehave 2P2XZ.0/ D.21/h2cosU2 jU2jˇˇˇˇZD0 tanı0CO."1C/: (93) Finally, substituting (92)and(93)in(91), we derive the averaged momentum equation for the second layer @t.h2N U2/C@Xh2U2 2C"h2 2 2cos .r2CK.1 r2// D"h2@Xb"rh2.1CK.1 1//@X.h1cos / 1 2 fric.U1;U 2/..1 r/h2cos Ch2N U2 2dX/ U2 jU2jˇˇˇˇZD0 tan ı0CO."1C/: 84 E.D. Fernández-Nieto and P. Vigneaux 5.6 Œ,!Final System of Equations Reverting to the original variables [see (65)], neglecting the terms of order "1Cand supposing a constant profile of the velocities we obtain the following system: 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : @[email protected] U1/D0I @t.h1N U1/[email protected] U2 1Cgh2 1 2cos/ D Dgh1@Xbgcosh1@Xh2C1 1 fric.U1;U 2/I @[email protected] U2/D0I @t.h2N U2/C@Xh2N U2 2Cgh2 2 2cos.r2CK.1 r2//D Dgh2@Xbrgcosh2.1CK.1 1//@Xh11 2 fric.U1;U 2/CTI (94) where by T, we denote the Coulomb friction term. Again, this term must be understood as follows: If jTjc)TD.g.1 r/h2cos / N U2 jN U2jtanı0; If jTj< c)N U2D0; where cDg..1 r/h2cos tanı0. Recall that rD1 2 ; where 1is the density of the fluid and 2is defined in (54). We can define the friction term between the layers fric.U1;U 2/under the following structure fric.U1;U 2/DKin .N U1N U2/; with Kin D1KinjN U1N U2j; being Kin a positive constant. In Fig. 9, an example of application of the model is presented. As initial condition a rectangular granular layer is imposed at the middle of the domain. The water layer is initially at rest and its free surface is flat. As we can see in the different times shown in the figure, the avalanche produced by the granular layer interacts with the fluid and some waves appear. The stationary solution reached consists of water at rest with a flat free surface over a granular layer in equilibrium. This simulation has Some Remarks on Shallow Avalanches Modelling 85 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 t = 0.1 s.t = 0 s. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 t = 0.3 s.t = 0.2 s. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Stationary solutiont = 0.4 s. Fig. 9 Submarine avalanche test. Continuous line: granular free surface; Dashed-dotted line: water free surface been obtained by numerically solving System (94) with the finite volume method introduced in [21]. See also [32], where an application of the model to the case of tsunamis in the Alboran Sea is studied. 6 Entropy Inequality and Stationary Solutions of Savage–Hutter Type Models In this section we state without proof a result concerning the entropy inequality and the stationary solutions of the Savage–Hutter type models presented in previous sections for aerial, fluid–solid mixture and submarine avalanches. The following result can be proved for the submarine avalanche model: 86 E.D. Fernández-Nieto and P. Vigneaux Theorem 1. System (94) has the following properties: .i/ It admits an entropy dissipation inequality, @tr1h1N U2 1Ch2N U2 2 2Cgb.r1h1Ch2/Cgcos r1h2 1C2h2 2 2 Cgcos r 1h1h2 C@Xr1h1N U1 N U2 1 2Cgb Ccos .h1Ch2/!Ch2N U2 N U2 2 2Cgb Cgcos .rh1C2h2/! rKinjN U1N U2j.U2U1/.U21U1/g..1 r/h2cos  Ch2dX. N U2 2gh2cos  2//jN U2jtan ı0 Cgh2 2 2U2.1 2/sin @X: where 1D1CK.11/; 2Dr 2CK.1r 2/: .ii/It has the family of steady state solutions: N U1D0; N U2D0; (95) bC.h1Ch2/cos Dcst;(96) j.2r1/@x.b Ch2cos /C.1 2/.@Xbh2 2sin @x/j.1 r/tan ı0; (97) corresponding to water at rest over a stationary granular layer.  Note that the models presented in Sects. 2and 3can be seen as particular cases of this one. They can be obtained as follows: • Savage–Hutter model: set h1D0,N U1D0,2D0,rD1,1D1. • Iverson–Denlinger model: set h1D0,N U1D0,2D,rD1,1D1. Therefore, the properties presented in Theorem 1are also valid for the Savage– Hutter and Iverson–Denlinger models. Some Remarks on Shallow Avalanches Modelling 87 Note that the stationary solutions defined by (97) correspond to situations in which the free surface of the granularlayer is in equilibrium with the internalfriction angle. The angle ı0can be measured in laboratory experiments. 7 Rheology of Complex Avalanches Several differential models have been proposed in the literature to describe sediment mixtures: a review is presented in [2]. A possible approximation is given by the model presented in Sect.3, based on a two-phase approach and a friction law proportional to the normal stress and the tangent of the internal friction angle. Another possibility is the use of visco-plastic models. They represent an approximation of the rheological behaviors of complex flows, such as debris flows, lava flows and snow avalanches. In this section a brief introduction to non-Newtonian fluids is first given in order to motivate the definition of the stress tensor corresponding to the Herschel–Bulkley model.This model can be used to study debrisflows, fluid–solidmixture avalanches. As in the previous section we consider local coordinates on a plane slope with angle (see Sect.2for details on the notation). Let us denote the velocity vector as UDU W: Let us remember that the general system of Eqs.(4)and(5) can be re-written in the new variables as follows: 8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ : @X.U / [email protected]/D0; @t.U/C@X.U 2/C@Z.WU/@X.gX/D@X.PXX/@Z.PXZ/; @t.W/C@X.U W / C@Z.W 2/@Z.gX/D@X.PZX/@Z.PZZ/; (98) where the density is assumed to be constant. Let us remark that, although in these notes we are working with the negative stress tensor P, it is also usual to write the system of equations in terms of , the positive Cauchy stress tensor, where PD: The stress tensor is defined as the sum of the pressure component and the viscous one (cf. [9]), DpI C0; 88 E.D. Fernández-Nieto and P. Vigneaux where Iis the identity matrix. 0is called the deviatoric part of . Note that we can also write, PDpI 0: Let us use the notation: 0D0 XX 0 XZ 0 ZX 0 ZZ : A fluid is said to be Newtonian if 0is proportionalto the rate of deformation tensor D.U/,where D.U/DrUCrUtD2@XU@ XWC@ZU @XWC@ZU2@ ZW: Then for a Newtonian fluid, such as water, we have 0DD.U/; where is the viscosity coefficient, depending on the material. In theses notes, we suppose that is a constant value. Let us consider the case of a uniform flow such that D.U/D0@ ZU @ZU0 0D0 0 00; where 0DXZ DZX is the shear stress. In this case the relation which characterizes the fluid as Newtonian can be easily represented as a straight line in the plane .@ZU/.0/with slope (see Fig.10). The behavior of the flows of materials like honey, corn flour or paint cannot be modelled with such a linear relation. Moreover, the concept of Newtonian fluid is an idealization: there are always nonlinear relations between the shear stress and the shear rate. The study of the deviation (from the linear law) of 0as a function of D.U/belongs to the field called Rheology.ThetermRheology is due to Bingham in 1929. It comes from the Greek “"!”—“to flow”. It is related to the study of deformation and flow of complex fluids. Years around 1900s saw a significant increase of activity on these subjects, including authors like Maxwell (1868), Boltzmann (1877), Bingham, Blair, Reiner, Herschel–Bulkley, Weissenberg (all between 1900 and 1930, see [48]). Then Rheology became a research field of intense activity. Fluids in which is a function of D.U/,D.D.U//,i.e. 0D.D.U// D.U/: Some Remarks on Shallow Avalanches Modelling 89 ab Fig. 10 (a) Representation of a Newtonian fluid in the plane .0/.@ZU/ with viscosity ;(b) generalized Newtonian fluids (see Fig. 10) are called Generalized Newtonian fluids;cf[41]). For example: • Corn flour is a material whose behavior is fluid when it is gently mixed but it becomes very viscous if it is strongly mixed. That is, “viscosity ” increases with “shear”. Such materials are shear-thickening. In the case of a uniform shear-thickening flow, we have 0D.@ZU/@ ZU, where .@ZU/is an increasing function of @ZU. They can be represented in the .0/.@ZU/plane as follows: • There are many other materials, like paint, whose viscosity decreases with shear. These materials are shear-thinning. In the case of a uniform shear-thinning flow, we have 0D.@ZU/@ ZU, where .@ZU/is a decreasing function of @ZU. They can be represented in the .0/.@ZU/plane as follows: 90 E.D. Fernández-Nieto and P. Vigneaux Shear-thinning and shear-thickening fluids can be modelled using power-law fluids. The viscosity of power-law fluid is defined by .D.U// DNjD.U/jn1,for some positive constant Nand n0. For these, we have 0DNjD.U/jn1D.U/; Let us remark that: •Ifn<1the material is shear-thinning. •IfnD1the fluid is Newtonian and Nis the constant viscosity. •Ifn>1the fluid is shear-thickening. Nevertheless, the flow of some materials cannot be modeled by a power-law model. This is the case of clay, snow or lava that only flow when the shear stress is bigger than a critical value. These materials are example of what we can call “threshold” fluids. Below a stress cthe material present a rigid behavior but above cthe material begins to flow. They are visco-plastic materials. Bingham defined Plasticity as follows (see [10]): We may now define plasticity as a property of solids in virtue of which they hold their shape permanently under the action of small shearing stresses but they are readily deformed, worked or molded, under somewhat larger stresses. Bingham law follows this property and it depends on a threshold shear stress c. 0is defined as follows: (j0j< cif jD.U/jD0 0DD.U/CcD.U/ jD.U/jif jD.U/j¤0: (99) Note that this definition implies that, in the case that jD.U/jD0, we only know that 0is bounded by c.Thatis,0is a multivalued function in this case. It is easier to understand this law by considering the inverse function. Let us suppose that we perform two experiments of a uniform flow for a plastic material and we measure the shear rate @ZUin terms of the shear stress 0. We have marked with crosses in the following figure two points corresponding to the measures: Some Remarks on Shallow Avalanches Modelling 91 With black dashed line we plot the law corresponding to the case of Newtonian fluids, that is, straight lines passing by the point of measure and by .0; 0/.Let us remark that if the fluid is Newtonian then it is enough to look for only one experiment in order to measure its viscosity. Remember that the viscosity in a Newtonian fluid is the inverse of the slope of such a straight line. Performing a second experiment is a way to see if the fluid is Newtonian. If we have a graph as the one of previous figure in which the second measure does not lead to a point in the same straight line, then the fluid is not Newtonian. Conversely, if we consider a straight line passing through these two points, we obtain the value of the shear threshold c, as the point at which this line cuts the horizontal axe. Actually, measurements show that the real behavior of plastic materials does not follows exactly this straight line. They follow a curve that can be seen as a regularization of the corner around the point .c;0/ (dashed-blue line in previous figure). The model proposed by Bingham, defined by (99), corresponds to the graph defined by the union of the two blue straight lines in the previous figure. The general case combines power-law and plasticity. This is the Herschel– Bulkley constitutive equation. For the case of uniform flow it can be represented in the .0/.@ZU/plane as follows: Newtonian Herschel-Bulkley Bingham Herschel–Bulkley model is characterized by the following stress tensor: PDpI 0;(100) 98 E.D. Fernández-Nieto and P. Vigneaux where jD".U/jn1D4.@XU/ 2C2."@XWC1 "@ZU/ 2C4.@ZW/ 2.n1/=2 : Let us also remark that for the inclined plane case considered in these notes we have @XbDsin: 8.4 ŒRM .X/ 0 Variational Inequality for Test Functions Independent of Z In this section, we obtain the mass and momentum equationsof a Shallow Herschel– Bulkley model. To obtain it, we neglect the second order terms (O."2/)andwe consider test functions which are independent of Z. Let us remark that to consider test functions independent of Zis analogous to depth average the mass and momentum equations. In fact, we can see that if cD0, then we have a variational equality for the momentum conservation and the procedure described below is another way of deriving the Shallow Water equations. And, if the Coulomb friction law is considered at the bottom, the Savage–Hutter model deduced in Sect.2is recovered. First, note that if q2L.1Cn/0.˝.t// is independent of Zthen Z˝.t/ qdiv.X;Z/UDZL 0 q.X/Zh 0 div.X;Z/UdZdX D0: By using the kinematic conditions, we obtain ZL 0 q.X/@[email protected]/dX D0; 8q2L.1Cn/0.Œ0; L/: This gives a different way to obtain the mass conservation equation: @[email protected]/D0: Let us now consider test functions D. ; '/ where is independent of Z. Analogously to previous sections, we assume that the velocity parallel to the bottom Uis independent of Z. Then, if we neglect second order terms (O."2/)in(112), we obtain Some Remarks on Shallow Avalanches Modelling 99 ZL 0 [email protected] / CU@XU /. U/dXdZ CZ˝.t/ ".@Xb. U/Ccos.' W//dXdZ Z˝.t/ "p .@X. U/C@Z.' W //dXdZ CZ˝.t/ N2 2n2.@XU/ 2C.@ZW/ 2n12"@X.U /@X. U/ C2"@Z.W /@Z.' W/ dXdZ CcZ˝.t/ "p.@X /2C.@Z'/2p.@XU/ 2C.@ZW/ 2dXdZ CZL 0 ˛U. U/d 0: (113) Moreover, by using the incompressibility condition, by choosing also test functions with zero divergence whose vertical component vanishes at the bottom— to be consistent with boundary condition (104)—we have WDZ@XU; and 'DZ@X : (114) By using (114)weget: Z˝.t/ .@Xb. U/Ccos.'W//dXdZ DZL 0h@[email protected] 2cos/. U/dX: Finally, using this last equality and (114), we obtain from (113): ZL 0 [email protected] / CU@XU /. U/dXdZ CZL 0 h@[email protected] 2cos/. U/dX CZL 0 23n1Nj@XUjn1"@X.U /@X. U/dX C2cZL 0 "h.j@X jj@XUj/dX CZL 0 ˛U. U/d 0: (115) 100 E.D. Fernández-Nieto and P. Vigneaux 8.5 Œ,!Final System of Equations Coming back to the original variables and using (108), we obtain the following system: • Mass conservation: ZL 0@[email protected] /q.X/dX D0; 8q2L.1Cn/0.Œ0; L/: (116) • Momentum variational inequality: 8 2W1;1Cn.Œ0; L/, ZL 0 [email protected] / ChU@XUCgh@[email protected] 2cos/. U/dXdZ CZL 0 23n1Nj@XUjn[email protected] /@X. U/dX C2cZL 0 h.j@X jj@XUj/dX CZL 0 ˛U. U/d 0: (117) Note that the first line corresponds to convection and pressure terms in Shallow Water systems and the second one to viscous effects. The third line contains the terms associated to the rigidity properties of the material. Last line of previous equation correspond to the bottom friction term. Let us remark that (116)and(117) corresponds to the weak formulation of the following partial differential system: 8 ˆ ˆ < ˆ ˆ : @[email protected] / D0; h@tUCU@ xUCg.b Chcos/C˛U @X.h0/D0 (118) where 8 ˆ ˆ ˆ < ˆ ˆ ˆ : 0D23n1Nj@XUjn1@XUC2c @XU j@XUjif j@xUj¤0 j0j2cif j@xUjD0: (119) As mentioned before, it is easy to see that if cD0,nD1and the linear friction law is replaced by the Coulomb friction law (13) then the Savage–Hutter model deduced in Sect. 2—with standard viscous terms—is obtained. Some Remarks on Shallow Avalanches Modelling 101 –2 –1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 Fig. 11 Initial condition. Dashed-dotted line: free surface of the viscoplastic material. Continuous line: bottom We refer to [1] for the discretization of this shallow-Herschel–Bulkley model. Let us present an example. As initial condition a rectangular layer is considered on a closed domain with a plain with a slope of 30ı(see Fig.11). Figure 12 shows the evolution of the avalanche of the visco-plastic material corresponding to nD1,cD4,ND102,˛D102. The left column shows the evolution of the free surface at times t2f0.4, 1, 2.4, 4 gs. The right column shows the velocity profile. This simulation shows a typical behavior of visco-plastic fluids: at the beginning it moves as a rigid body and then it starts to flow as a viscous fluid. Indeed, notice that for tD0:4 and tD1s the velocity profile is nearly constant on all the domain filled by the avalanche, but this is no more the case for tD2:4 s. For tD4s. the material is at rest. These different behaviors are due to the definition of the stress tensor (119). Appendix: Bed-Load Sediment Transport Formulae In this appendix we present several possible definitions of the solid transport discharge, qb, that allow one to close the Saint-Venant Exner system (see Sect.1). The study of the definition of the solid transport discharge can be seen as a deterministic problem or a probabilistic one. For example, deterministic methods have been proposed by Meyer-Peter & Müller [36] and probabilistic methods by Einstein [20]. In general, the models take into account the fact that motion of the granular sediment begins when the shear stress () is bigger than a certain critical shear stress (c). Moreover, shear stress can be written in terms of the hydrodynamic unknowns hand uby DRhjSfj:(120) 102 E.D. Fernández-Nieto and P. Vigneaux –2 –1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 012345678910 –5 –4 –3 –2 –1 0 1 2 –2–10123456789 0 1 2 3 4 5 6 012345678910 –5 –4 –3 –2 –1 0 1 2 –2 –1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 012345678910 –5 –4 –3 –2 –1 0 1 2 –2 –1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 012345678910 –5 –4 –3 –2 –1 0 1 2 t=0.4 s. t=1 s. t=2.4 s. t=4 s. Fig. 12 Complex avalanche: Herschel–Bulkley model. Left: free surface; Right: velocity Here Sfis defined by (2)andis the specific weight of fluid Dgw,wherewis the water density. Shear stress appears usually in non-dimensional form in the formula of qb.If and crepresent the non-dimensional shear stress and the critical shear stress, respectively, then Some Remarks on Shallow Avalanches Modelling 103 D .s/d ; cDc .s/d:(121) Here dis the sediment grain size and sis the specific sediment weight sDgs, where sis the sediment density. Using (120)and(121), can be written as a function of the specific gravity or the relative density of fluids rDs=w. Dg2u2 .r 1/dR1=3 h : To determine cmany experiments have been performed in different works. Concretely, Shields proposed the well-known Shields-diagram (cf. [40], p. 107). Some usual formulae for rivers are the following: • Grass (see [26]) proposed the following formula for the solid transport discharge, qbDAgujujmg1;16mg64; where the constant Ag(s2=m) must take into account the grain diameter and the kinematic viscosity. It is usually obtained by experimental data. The usual value of exponent mgis set to mgD3. • Meyer-Peter & Müller (see [36]) developed one of the most popular formulae for the solid transport discharge, qbDq.r 1/gd3sgn .u/8 .c/3=2 ; where cusually is set to 0:047. • Van Rijn (see [49]) developed the following formula for the solid transport discharge, qbDq.r 1/gd30:005 C1:7 Dd h0:2 1=2 sgn .u/1=2 1=2 c2:4 ; where CDis the drag coefficient. • Nielsen (see [40]) developed the following formula qbDq.r 1/gd3sgn .u/12p.c/: In this case the usual value of cis set equal to cD0:05. 104 E.D. Fernández-Nieto and P. 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