A two-phase shallow debris flow model with energy balance

ESAIM: M2AN 49 (2015) 101–140 ESAIM: Ma hema ical Modelling and Nume ical Analysis DOI: 10.1051/m2an/2014026 www.esaim-m2an.o g A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE F. Bouchu 1,E.D.Fe n ´ andez-Nie o2, A. Mangeney3,4 and G. Na bona-Reina5 Abs ac . This pape p oposes a hin laye dep h-a e aged wo-phase model p o ided by a dissipa i e ene gy balance o desc ibe a alanches o solid- luid mix u es. This model is de i ed om a 3D wo-phase model based on he equa ions p oposed by Jackson [The Dynamics o Fluidized Pa icles. Camb idges Monog aphs on Mechanics (2000)] which akes in o accoun he o ce o buoyancy and he o ces o in e ac ion be ween he solid and luid phases. Jackson’s model is based on mass and momen um conse a ion wi hin he wo phases, i.e. wo ec o and wo scala equa ions. This sys em has i e unknowns: he solid olume ac ion, he solid and luid p essu es and he solid and luid eloci ies, i.e. h ee scala s and wo ec o s. As a esul , an addi ional equa ion is necessa y o close he sys em. Su p isingly, his issue is inadequa ely accoun ed o in he models ha ha e been de eloped on he basis o Jackson’s wo k. In pa icula , Pi man and Le [Philos.T ans.R.Soc.A363 (2005) 799–819] eplaced his closu e simply by imposing an ex a bounda y condi ion. I he p essu e is assumed o be hyd os a ic, his condi ion can be conside ed as a closu e condi ion. Howe e , he co esponding model canno accoun o a dissipa i e ene gy balance. We p opose he e a closu e equa ion o comple e Jackson’s model, imposing incomp essibili y o he solid phase. We p o e ha he esul ing whole 3D model is compa ible wi h a dissipa i e ene gy balance. F om his model, we deduce a 2D dep h-a e aged model and we also p o e ha he ene gy balance associa ed wi h his model is dissipa i e. Finally, we p opose a nume ical scheme o app oxima e he dep h-a e aged model. We p esen se e al nume ical es s o he 1D case ha a e compa ed o he esul s o he model p oposed by Pi man and Le. Ma hema ics Subjec Classi ica ion. 65C20, 81T80, 91B74, 97M10. Recei ed Ap il 9, 2013. Re ised May 5, 2014. Published online Janua y 14, 2015. Keywo ds and ph ases. G anula flows, wo-phase flows, hin laye app oxima ion, ene gy balance, non-conse a i e sys ems, p ojec ion me hod, fini e olume schemes. 1Uni e si ´e Pa is-Es , Labo a oi e d’Analyse e de Ma h´ema iques Appliqu´ees (UMR 8050), CNRS, UPEMLV, UPEC, 77454 Ma ne-la-Vall´ee, F ance. ancois.bouchu @uni -ml . 2Depa amen o de Ma em´a ica Aplicada I, Uni e sidad de Se illa. E.T.S. A qui ec u a. A da, Reina Me cedes, s/n. 41012 Se illa, Spain. [email p o ec ed] 3Uni e si ´e Pa is Dide o , So bone Pa is Ci ´e, Ins i u de Physique du Globe de Pa is, Seismology g oup, 1 ue Jussieu, 75005 Pa is, F ance. [email p o ec ed] 4ANGE g oup INRIA, Jacques Louis Lions, CETMEF 5Depa amen o de Ma em´a ica Aplicada I, Uni e sidad de Se illa. E.T.S. A qui ec u a. A da, Reina Me cedes, s/n. 41012 Se illa, Spain. [email p o ec ed] A icle published by EDP Sciences c EDP Sciences, SMAI 2015 102 F. BOUCHUT ET AL. (b) (a) Figu e 1. (a) Deposi s o se e al deb is flows in Iceland. (b) Close-up o a c oss-sec ion o he deposi o a deb is flow co e ing a oad in Canada. 1. In oduc ion Landslides, deb is a alanches o deb is flows play a key ole in e osion p ocesses on he su ace o he Ea h and o he ellu ic plane s. On Ea h, hey ep esen one o he majo na u al haza ds. G a i a ional ins abili ies a e also closely ela ed o olcanic, seismic and clima ic ac i i y and hus ep esen po en ial p ecu so s o p oxies o he change o hese ac i i ies wi h ime. Resea ch in ol ing he dynamic analysis o g a i a ional mass flows is ad ancing apidly. One o i s ul ima e goals is o p oduce ools o de ec ion o na u al ins abili ies and o p edic ion o eloci y and unou ex en o apid landslides. The heo e ical desc ip ion and physical unde s anding o hese p ocesses in a na u al en i onmen a e s ill open and ex emely challenging p oblems o ea h scien is s, gi ing ise o equally challenging mechanical, ma hema ical and nume ical issues. In ecen yea s, significan p og ess in he ma hema ical, physical and nume ical modelling o g a i a ional flows has made i possible o de elop and use nume ical models o in es iga e geomo phological p ocesses and assess isks ela ed o such na u al haza ds. Howe e , key ques ions s ill emain unanswe ed, o ins ance conce ning he eason o he high mobili y o na u al landslides (e.g. [21,22]). Se e e limi a ions p e en a ull unde s anding o physical p ocesses in ol ed in landslide dynamics and he de elopmen o ools o de ec ion o ins abili ies and p edic ion o hei eloci y and ex en . Indeed, nume ical models do no ake in o accoun complex na u al phenomena such as he s a ic/flowing ansi ion in g anula flows o he co-exis ence and in e ac ion o fluid (wa e , gas) (e.g. [7,8,14,18,23,26–28,42,43]). Wa e is almos always in ol ed in na u al landslides (e.g. [15,16,29]) (Fig. 1b). In e ac ion o ces be ween he solid and fluid (wa e ) phases may play an impo an ole in flow mobili y and deposi ex en . Diffe en app oaches can be used o simula e fluid-solid mix u es, ex ending om disc e e elemen models based o example on con ac dynamics o molecula dynamics (e.g. [32,47]), and aking in o accoun indi idual pa icles, o con inuum models ha deal wi h a fluid phase and a solid phase. The disc e e elemen app oach is ha d o use in geophysical applica ions due o he high compu a ional cos s equi ed o ake in o accoun he b oad-size dis ibu ion o pa icles in eal flows, which is c i ical in such simula ions. Exis ing models used o desc ibe he beha iou o fluid-solid mix u es a e mainly based on Jackson’s model [19]. This model akes in o accoun solid and fluid s esses, he in e ac ion o ce be ween he fluid and solid phases and he buoyancy o ce, h ough mass and momen um conse a ion wi hin he wo phases. This model hus in ol es ou equa ions ( wo scala and wo ec o equa ions). Howe e , he sys em has fi e unknowns: he solid olume ac ion, he solid and fluid p essu es and he solid and fluid eloci ies ( h ee scala s A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 103 and wo ec o s). As a esul , an addi ional equa ion is necessa y o close he sys em. Su p isingly, his issue is inadequa ely accoun ed o in he models ha ha e been de eloped on he basis o Jackson’s wo k. Sol ing he 3D wo-phase equa ions leads o high compu a ional cos s. Fo his eason, mos ly dep h-a e aged models ha e been p oposed o deal wi h na u al geophysical flows (e.g. [10,35,36,38]). I e son [15]was hefi s o add ess he need o include in e s i ial fluid effec s in he cons i u i e beha iou o he mass flow and de eloped a hin laye model o a solid-fluid mix u e mo ing on ealis ic e ain, unde he simpli ying assump ions o cons an po osi y and equali y o he fluid and solid eloci y. The flow is desc ibed by a single se o equa ions o he densi y and momen um o he mix u e, which is o mally ep esen ed by a single-phase model wi h a s ess e m accoun ing o con ibu ions om he wo cons i uen s. Due o he lack o an explici equa ion o he po e fluid p essu e in his model, a po e p essu e ad ec ion-diffusion equa ion was added based on expe imen al measu emen s. Va ious e sions and applica ions o his g ain-fluid mix u e model ha e since been p esen ed (e.g. Pudasaini e al. [39]; Geo ges and I e son, [10]). Taking ano he s ep o wa d, Pi man and Le p oposed in [38] a no el dep h-a e aged wo-fluid model o deb is flows, based on Jackson’s model, ha con ains mass and momen um equa ions o bo h he fluid and solid phases, hus p o iding equa ions o he eloci ies o he wo phases and o solid olume ac ion. In he model p oposed by Pi man and Le and he modified e sion p oposed by Pelan i e al. [36], he au ho s do no p o ide a closu e equa ion o he wo-phase model. On he o he hand, hey impose wo bounda y condi ions in ol ing anishing su ace ension condi ions a he ee su ace, i.e. he p essu e o bo h he solid and he fluid phases anish a he ee su ace. Two kinema ic bounda y condi ions a e also imposed a he ee su ace, because he wo phases a e assumed o fill a common domain, his gi es an o e de e mined p oblem a he ee su ace. Howe e , in he hin laye app oxima ion, because o he hyd os a ic p essu e assump ion, he ex a bounda y condi ion makes i possible o exp ess a dep h-a e aged model, e en hough no closu e ela ion o he whole sys em is p o ided. Howe e , bounda y condi ions ob iously do no eplace a closu e equa ion inside he domain. This a ificial compensa ion o he missing closu e equa ion by o e de e mined bounda y condi ions leads o a physically i ele an ene gy equa ion in he Pi man−Le model (see Sec . 4.1). A physically meaning ul ene gy equa ion is essen ial o ob ain ealis ic models. A key issue in wo-phase flow models is hus o p opose a sui able closu e ela ion ha is compa ible wi h he ene gy balance. Some new and e y use ul ways o close he sys em o equa ions ha e been p oposed by Roux and Radjai [44], Pailha and Pouliquen [35] and Geo ge and I e son [10]. The gene al idea is o ake in o accoun he dila ion/comp ession o he g anula phase and i s in e ac ion wi h he p essu e o he fluid filling he po es o he g anula ma e ial. Indeed, hese effec s ha e been shown o be c ucial a he ini ia ion o mass des abiliza ion and o ha e a s ong impac on he gene a ed flow dynamics [e.g. [17,40]]. Roux and Radjai [44] p oposed an equa ion o desc ibe he e olu ion o he olume ac ion and o he shea s ess in a g anula ma e ial in e ms o he shea -induced dila ancy (a p ope y o he g anula ma e ial ela ed o i s dila ion when he ma e ial is submi ed o a shea o ce). Pailha and Pouliquen used his equa ion o close hei model, based on he wo-phase app oach p oposed by Jackson [35]. They also imposed ha bo h he solid and fluid p essu es anish a he ee su ace. Mo eo e , hey in oduced a closu e equa ion, ela ed o dila ancy effec s (Eq. (3.18) in [35]). Bu , as he esul ing sys em is o e de e mined, a condi ion had o be elaxed. Indeed, hey elaxed mass conse a ion o one o he wo phases, ha hey jus ified by assuming ha he hickness o he flowing mix u e is nea ly cons an . Al e na i ely, Geo ge and I e son [10] de i ed a model using he mass and momen um equa ions o he mix u e. In hei model, he unknowns a e he o al heigh and eloci y o he mix u e, he solid olume ac ion and he po e fluid p essu e. As a closu e ela ion, hey used a sligh ly diffe en equa ion han ha p oposed in [44] o desc ibe dila ancy effec s, ha includes he ime de i a i e o he effec i e no mal s ess and he po e fluid p essu e. This ela ion is de i ed om he mass conse a ion o he solid phase by assuming ha he a e aged mix u e eloci y is equal o he a e aged solid eloci y (Eqs. (6) and (7) o hei pape ) and a Da cy law. Howe e , he final model does no impose explici ly he mass conse a ion o he solid phase. We p opose he e o sol e he mass and momen um equa ions o bo h phases, oge he wi h he ele an numbe o bounda y condi ions and a closu e equa ion ha p o ides a possibly physically ele an ene gy 104 F. BOUCHUT ET AL. equa ion. In a fi s s ep owa d his objec i e, we use he simples closu e equa ion (i.e. incomp essibili y o he solid phase). We impose a anishing s ess condi ion a he ee su ace o he mix u e (no o each phase) and kinema ic su ace bounda y condi ions ( he wo phases a e supposed o fill he same domain), o ming a well-posed 3D sys em. The analysis o he hyd os a ic app oxima ion sugges s ha a a iable ela ed o he p essu e field emains in he hin-laye asymp o ics. On choosing a s a ic cons ain as a closu e ela ion, his ex a a iable can be de e mined as he associa ed Lag ange mul iplie . The esul ing model has a buil -in ene gy balance. 2.The3D wo-phasemodel In his sec ion we p esen he h ee-dimensional model used o desc ibe he mix u e o solid and fluid ma e ials. No e ha we do no conside he e he ole o he ai (i.e. a hi d phase) ha can be c i ical in some cases due o capilla y o ces, especially a he labo a o y scale [16]. As a esul , hese equa ions a e only alid when he g anula media is sa u a ed wi h fluid so ha he e is no ai wi hin he po es o he g anula ma e ial. In Sec ion 2.1, he mass and momen um equa ions o Jackson’s model a e p esen ed and a closu e equa ion is p oposed. In Sec ion 2.2, he bounda y condi ions a e desc ibed. In Sec ions 2.3 and 2.4, we exp ess he d ag o ce and he assump ions conce ning he s ess enso . Finally, in Sec ion 2.5,weexp ess hecomple emodel in local coo dina es. 2.1. Mass and momen um equa ions We conside geophysical mass flows made o a mix u e o solid and fluid ma e ials. The wo fluid model p esen ed below is de i ed in he Jackson’s book [19]. I is based on he dynamics o an assembly o solid pa icles imme sed in a New onian fluid. The wo-fluid model is ob ained by a e aging in he whole egion he undamen al equa ions o bo h componen s, he fluid and pa icles. Namely, he Na ie -S okes equa ion o he mo ion o he fluid and he equa ions o linea and angula momen um o each pa icle o he solid pa . These equa ions a e coupled by he no-slip bounda y condi ion imposed on he su ace o each pa icle. The mass and momen um conse a ion equa ions o fluid and pa icle phases a e deduced by an a e aging p ocedu e. Bu , some e ms linked o he mic oscopic le el o he indi idual pa icles a e neglec ed. Consequen ely, a e he a e aging p ocedu e he e a e mo e unknowns han equa ions in he de i ed sys em. And a closu e o he sys em mus be se . The wo-phase model is defined by he ollowing mass and momen um equa ions o he solid and fluid phases: ∂ (ρsϕ)+∇·(ρsϕ )=0,(2.1a) ∂ (ρ (1 −ϕ)) + ∇·(ρ (1 −ϕ)u)=0,(2.1b) ρsϕ(∂ +( ·∇) )=−∇ · Ts+ 0+ρsϕg,(2.2a) ρ (1 −ϕ)(∂ u+(u·∇)u)=−∇ · T − 0+ρ (1 −ϕ)g,(2.2b) whe e he subsc ip “s” e e s o he solid phase and he subsc ip “ ” e e s o he fluid phase. The eloci ies a e o he solid phase and u o he fluid phase. Tdeno es he s ess enso and ρ he densi y. Accele a ion due o g a i y is deno ed by gand 0 ep esen s he a e age alue o he esul an o ce exe ed by he fluid on a solid pa icle. The solid olume ac ion is ϕ. Fo monodispe se beads, he maximal olume ac ion is ϕmax ≃0.6, while i can be highe han 0.9 o highly polydispe se ma e ials because he small pa icles can fill he po e space be ween la ge pa icles ([3,13,48]). The solid ac ion is p ac ically ne e equal o 1. The case o d y g anula flows can be ob ained by se ing all he a iables ela ed o he fluid phase (fluid s ess and 0) A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 105 o ze o and olume ac ion o one in equa ions (2.1a)and(2.2a). The minus sign on he s ess enso e ms ag ees wi h he sign con en ion used in soil mechanics, whe e s ess is defined as posi i e in comp ession. No e ha bo h he g ain densi y ρsand he fluid densi y ρ a e cons an , so ha each ma e ial is incomp ess- ible. Howe e , he densi y o he solid phase ϕρs(i.e. densi y o he o al amoun o g ains pe uni olume) and he densi y o he fluid phase (1 −ϕ)ρ (densi y o he o al amoun o fluid filling he po es o he g anula assembly pe uni olume) can change because ϕ a ies wi h space and ime. In his sense, he solid and fluid phase could be comp essible. No e ha he combina ion o (2.1a)and(2.1b) defines mass conse a ion o he mix u e: ∂ (ρm)+∇·(ρm m)=0,(2.3) whe e ρm=ρsϕ+ρ (1 −ϕ)and m=ρsϕ +ρ (1 −ϕ)u ρsϕ+ρ (1 −ϕ) a e espec i ely he densi y and eloci y o he mix u e. Mul iplying (2.1a)byρ and (2.1b)byρsgi es: ∇·(ϕ +(1−ϕ)u)=0.(2.4) This ela ion is diffe en om he one exp essing incomp essibili y o he mix u e because i does no imply ha ∇· mis equal o ze o. The a e aged alue o he in e ac ion o ce be ween fluid and pa icle is collec ed in 0. This o ce is decom- posed in o he sum o he buoyancy o ce Band all emaining con ibu ion . The main componen s o his o ce a e a e m depending on he pa icle concen a ion and he ela i e eloci y (u− )− he d ag o ce −, a e m depending on he concen a ion and he ela i e accele a ion − he i ual mass o ce −and he hi d con ibu ion due o he o ce no mal o he di ec ion (u− )− he li o ce −. Se e al exp essions o he buoyancy o ce a e discussed in [19]. In he simples case his o ce is w i en as B=−ϕ∇p wi h p he fluid p essu e ha esume he o ce exe ed by he fluid a es on an imme sed body ha is also a es . The gene aliza ion o mo e gene al mo ions leads us o he exp ession B=−ϕ∇T , howe e he app oxima ion o −ϕ∇p gi es equi alen equa ions assuming ha he ac ion on he pa icles due o he g adien o he de ia o ic pa o T is collec ed by he o he e ms o he o al o ce .Sowew i e 0= B+ =−ϕ∇p + . (2.5) In he case when he ine ia associa ed o he ela i e mo ion o fluid and pa icles can be neglec ed, e e ed o as he sho elaxa ion ime app oxima ion, he i ual mass o ce may be neglec ed compa ed o he d ag o ce in he equa ions o mo ion. Rega ding o he li o ce, he algeb aic exp ession o his con ibu ion no mal o he ela i e eloci y is unce ain because i akes qui e diffe en o ms o diffe en flow egimes. This o ce will no be conside ed in his pape . Thus, we assume ha can be exp essed simply by he d ag o ce. The d ag o ce ac s in he di ec ion o he ela i e eloci y (u− ) and also depends on he pa icle concen a ion. So in gene al i can by w i en as: β(ϕ, |u− |)(u− ). Fo small alues o |u− |, his o ce is p opo ional o he ela i e eloci y so we can w i e =˜ β(u− ) (2.6) ˜ βbeing he d ag coefficien (see Sec . 2.3). The no a ion ˜ βis used o dis inguish his coefficien om he d ag coefficien s deno ed by βin some o he publica ions (see o example [35]). The effec i e s ess enso s Tsand T a e ela ed o he in e ac ions be ween fluid and pa icles and be ween pa icles hemsel es. In [19] is poin ed ou he difficul y o w i ing hem in e ms o a e aged a iables in o de o close he sys em. Fu he mo e an ex ensi e discussion is included o gi e some explici and empi ical closu es o diffe en egimes in e ms o he S okes numbe . In his wo k, we ake a symme ic solid s ess enso Ts and deno e psi s fi s in a ian , i.e., he p essu e o he solid phase (see Sec . 2.4). 106 F. BOUCHUT ET AL. The iscosi y o he fluid ac s a he “mac oscopic” scale h ough iscous e ms o o de μU/L2,whe eU and La e cha ac e is ic alues o espec i ely he fluid eloci y and flow leng h. On he o he hand, he fluid iscosi y ac s a he “mic oscopic” scale du ing he ela i e mo ion be ween he fluid phase and he g anula po ous media commonly desc ibed by he Da cy law. This mic oscopic con ibu ion is o he o de o μΔU/κ, whe e κis he in insic hyd aulic pe meabili y o he g anula media and ΔU is he ypical ela i e eloci y o he fluid phase wi h espec o he solid phase. He e we assume ha he “mac oscopic” iscous o ces ela ed o he fluid a e negligible, so ha he fluid s ess enso educes o he p essu e e m, ∇·T =∇p .(2.7) By subs i u ing hese exp essions in o (2.2a)and(2.2b), we ob ain he sys em (2.1a), (2.1b), and ρsϕ(∂ +( ·∇) )=−∇ · Ts−ϕ∇p + +ρsϕg,(2.8a) ρ (1 −ϕ)(∂ u+(u·∇)u)=−(1 −ϕ)∇p − +ρ (1 −ϕ)g.(2.8b) This sys em o equa ions is he same as he sys em conside ed in [16,38]. Only he bounda y condi ions a e diffe en om hose used he e. As discussed abo e, his sys em o ou equa ions (2.1a), (2.1b), (2.8a), (2.8b) has fi e unknowns ϕ,Ts,p , uand . To close he sys em, we p opose o add a supplemen a y scala equa ion, based on he physical p ocesses in ol ed. S a ing om he simples closu e ela ion, we p opose o impose he incomp essibili y o he solid phase: ∇· =0.(2.9) In eal g anula ma e ials he dila ancy effec may induce changes o he olume o he solid phase, e en i he mass o he g anula ma e ial emains cons an . This means ha he di e gence o he eloci y o he solid phase may no be ze o (see [11]). The comp ession/dila ion o he g anula phase changes he in e s i ial fluid p essu e ha in u n couples wi h he solid momen um equa ions. This coupling appea s in he non-hyd os a ic p essu e e ms (see [30]), no included in he app oxima ions made in his wo k. The consis ency o he whole model can be e alua ed by he local ene gy balance equa ion. To ob ain i , we mul iply (2.8a), (2.8b)by and u espec i ely, combine wi h (2.1a)and(2.1b), and add he esul s. This yields ∂ ρsϕ| |2 2+ρ (1 −ϕ)|u|2 2+∇·ρsϕ| |2 2 +ρ (1 −ϕ)|u|2 2u =− ·(∇·Ts)−ϕ +(1−ϕ)u·∇p + ·( −u)+ρsϕ +ρ (1 −ϕ)u·g. (2.10) Deno ing X he space posi ion and once again using (2.1a)and(2.1b)alongwi h(2.4), we ob ain ∂ ρsϕ| |2 2+ρ (1 −ϕ)|u|2 2−(g·X)ρsϕ+ρ (1 −ϕ) +∇·ρsϕ| |2 2 +ρ (1 −ϕ)|u|2 2u−(g·X)ρsϕ +ρ (1 −ϕ)u +p ϕ +(1−ϕ)u+Ts  =(Ts−psId) : ∇ +ps∇· + ·( −u),(2.11) whe e psdeno es he solid pa icles p essu e. F om equa ion (2.6), he d ag con ibu ion ·( −u) is non-posi i e. Wi h he assump ion ha he solid phase is incomp essible, he second e m on he igh -hand side ps∇· is equal o ze o and i is na u al o assume ha he ic ion dissipa ion (Ts−psId) : ∇ is non-posi i e. As a esul , he sum o he h ee e ms in he igh -hand side o (2.11) is non-posi i e. A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 107 The model defined by (2.1a), (2.1b), (2.8a), (2.8b) wi h closu e (2.9) has a locally dissipa i e ene gy bal- ance (2.11). No e ha in he ini ial sys em conside ed by Pi man and Le, he e m ps∇· does no anish and we canno ensu e he non-posi i eness o he igh -hand side e m in (2.11). We will show in Sec ion 4.1 ha he e m esul ing om he closu e equa ion also makes i possible o ob ain a dissipa i e ene gy balance in he model. 2.2. Bounda y condi ions 2.2.1. A he ee su ace We conside he usual geome ic se ing, which is ha he mix u e lies in a spa ial domain limi ed by a fixed opog aphy a he bo om and by a ee su ace a he op. We assume ha he fluid and he solid fill he same domain ha is mo ing wi h he eloci y o bo h. This gi es he simul aneous kinema ic condi ions (1,u)·N=0,(1, )·N=0 a he eesu ace,(2.12) whe e N=(N ,N X) is he ime-space no mal. I can be ew i en u·NX= ·NX=−N a he ee su ace.(2.13) No e ha his is a s ong assump ion ha plays a key ole in he de i a ion o he equa ions and in he esul ing model p esen ed below. In [16,35,38], bo h he fluid and he solid p essu es a e se o ze o a he ee su ace. Howe e , as discussed in he in oduc ion, only one dynamic bounda y condi ion can be imposed a he eesu aceo hemix u e: (Ts+p Id)NX=0 a he eesu ace.(2.14) Rema k 2.1 (abou he o al s ess enso ).To ob ain he o al s ess o he mix u e we can combine equa- ions (2.8a)and(2.8b). F om he e he o al s ess o he mix u e becomes mo e complica ed han he sum o he wo s ess enso s o each phase. Namely, i can be w i en as T=Ts+T +T=Ts+p Id +T,wi hT a con ibu ion coming om he non-linea con ec i e e ms w i en h ough he ela i e eloci ies o he solid and he fluid wi h espec o he eloci y o he mix u e m: T=−ρsϕ( − m)( − m)−ρ (1 −ϕ)(u− m)(u− m). Ne e heless, o many geophysical flows one can assume ha his e m is negligible, by assuming in pa icula ha he ela i e eloci y o he fluid wi h espec o he solid is small compa ed o he solid eloci y (see p. 540 o [16] o de ails.). Thus, we can see condi ion (2.14) as a simplifica ion whe e he o al s ess o he mix u e a he ee su ace o ou sys em is defined as he sum o he fluid and solid phase s ess enso s. 2.2.2. A he bo om The condi ions a he bo om a e classically he non-pene a ion condi ions u·n=0, ·n= 0 a he bo om,(2.15) whe e nis he upwa d space uni no mal (i.e. he no mal o he opog aphy). This mus be comple ed by u he condi ions o he solid, in pa icula we conside a Coulomb ic ion law, ollowing [45] Tsn−(Tsn)·nn=− an δsign( )(Tsn)·na he bo om,(2.16) whe e δis he in e g anula Coulomb ic ion angle. 108 F. BOUCHUT ET AL. Rema k 2.2. The sys em (2.1a), (2.1b), (2.8a), (2.8b), (2.9) wi h he bounda y condi- ions (2.13), (2.14), (2.15), (2.16), is o mally well-posed. Mo eo e , we can check ha he p e ious bounda y condi ions ensu e ha all he bounda y con ibu ions anish in he ene gy balance o he model, excep he one coming om he Coulomb condi ion (2.16), which dissipa es a he bo om. The main diffe ence be ween his sys em and hose conside ed by Pi man and Le (see [38]) and Pailha and Pouliquen (see [35]) is he defini ion o he bounda y condi ions. Ins ead o conside ing ha he o al p essu e anishes a he ee su ace (Eq. (2.14)), hey conside ha bo h he p essu e o he solid phase and he p essu e o he fluid phase anish a he ee su ace. Pi man and Le do no conside any closu e equa ion, consequen ly we canno check he well-posedness o his sys em. Pailha and Pouliquen conside a closu e equa ion in e ms o he di e gence o he solid phase eloci y. Ne e heless, gi en ha he sys em is o e de e mined in his case and hey elax he mass conse a ion o one o he wo phases. 2.3. Assump ions conce ning he d ag o ce Diffe en empi ical ela ions a e p oposed in he li e a u e o he d ag o ce. As al eady men ioned, he d ag o ce exp ession is assumed o be =˜ β(u− ).(2.17) The d ag coefficien ˜ βcan be defined in diffe en ways: •Pi man and Le [38] used he d ag o ce p oposed by Richa dson and Zaki (see [41]): ˜ β=(ρs−ρ )ϕg T(1 −ϕ)m−1,(2.18) whe e Tis he e minal eloci y o an isola ed ep esen a i e solid pa icle alling in he fluid unde g a i y. This o ce has been calcula ed by Richa dson and Zaki, based on labo a o y expe imen s measu ing Tand S,whe e Sis he sedimen a ion eloci y o he dispe sion o pa icles in a fluid. Expe imen s gi e he empi ical law: S=(1−ϕ)n T. The alue o he empi ical exponen nlies in he ange [2.4,4.65]. Pi man and Le [38] (Appendix A) show ha m=n−2, so ha m∈[0.4,2.65]. Depending on he espec i e oles o iscous and ine ial o ces, S/ Tdepends o does no depend on he Reynolds numbe (see [41] o mo e de ails). Fo example, wi h he ypical alues o he expe imen s done by I e son [17], he ypical Reynolds numbe is Re= Tdρ /μ≈50. F om Table VI o Richa dson and Zacki, his gi es n≈3and henm≈1. •Pailha and Pouliquen [35] use he ollowing defini ion o he d ag coefficien : ˜ β=(1−ϕ)2μ αd2,(2.19) μbeing he dynamic iscosi y, d he mean g ain diame e and α=(1 −ϕ)3 150 ϕ2· This is de i ed om he Ca man–Kozeny ela ion o he pe meabili y o he po ous media o med by he pa icles (see [12,33]). Ano he way o es ima e ˜ βis o assume ha he ic ion be ween he wo phases is simila o he Da cy law. In deb is flows, pa o he e ical displacemen s and o he fluc ua ions o he ho izon al displacemen a e induced by he dila ion o he compac ion o he g anula media. These effec s impac on he fluid p essu e field ha in u n affec s he momen um conse a ion o he solid and fluid phases. The coupling be ween he fluid and solid phases comes om he d ag o ce (see [35]). This can be unde s ood by conside ing he de ia ion A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 109 om he hyd os a ic fluid p essu e. Le us deno e p =ph +pe ,whe eph co esponds o he hyd os a ic fluid p essu e, sa is ying ∂zph =−ρ gcos θ,andpe is he excess po e-fluid p essu e. I he igh hand side o (2.8b) is conside ed p edominan (small ine ia), he ho izon al a ia ion o ph is negligible and he g adien o he excess po e-fluid p essu e pe sa isfies ∇pe =−˜ β (1 −ϕ)(u− ).(2.20) This o mula has he same s uc u e as he linea Da cian d ag o mula desc ibing fluid flow wi hin po ous media. This law, conside ed in [10], ela es u− o he g adien o he excess po e-fluid p essu e, ∇pe =−μ κ(1 −ϕ)(u− ),(2.21) whe e μis he po e-fluid iscosi y and κis he in insic hyd aulic pe meabili y o he g anula deb is. Geo ge and I e son [10] poin ou ha e en i his linea d ag o mula may o e simpli y he effec s o complex phase- in e ac ion o ces in deb is flows, se e al esea ch pape s, such as [20,46], indica e ha i p obably p o ides a sui able fi s app oxima ion. Compa ing (2.20) wi h he Da cian law (2.21)leads o: ˜ β=(1−ϕ)2μ κ,(2.22) κbeing he pe meabili y o he g anula media. The alue o he effec i e pe meabili y de i ed om (2.18) and (2.19), when compa ed o (2.22) gi es espec i ely: •Fo Pi man and Le (2.18): κ=μ T(1 −ϕ)m+1 (ρs−ρ )gϕ ·(2.23) •Fo Pailha and Pouliquen (2.19): κ=d2(1 −ϕ)3 150ϕ2·(2.24) These wo diffe en alues o pe meabili y de i ed om (2.23)and(2.24) a e compa ed in Figu e 2.Fo his compa ison we se μ=0.001 Pa s, d=10 −3m, (ρs−ρ ) = 1500 kg m−3,g=9.81 m s−2,m=1, T= 0.143 m s−1. We obse e ha bo h models gi e ela i ely close app oxima ions o he pe meabili y o alues o ϕg ea e han 0.4. Geo ge and I e son [10] ha e simula ed he expe imen s pe o med in [17]. In hese expe imen s, he alue o κwas app oxima ely 10−12 m2, whe eas he nume ical simula ions whe e pe o med wi h a cons an alue o κ≈10−8m2. No e ha , wi h he defini ion o κgi en by (2.23)o (2.24), κ≈10−8m2when ϕ≈0.5. A alue o κ≈10−12 m2co esponds o ϕ≈0.9. 2.4. Assump ions conce ning he s ess enso To ob ain he final sys em, a cons i u i e ela ion should be s a ed o he fluid and g anula phases. •Fluid s ess enso T . As men ioned be o e, we assume ha he fluid s ess enso can be exp essed by he fluid p essu e: Txy =Txz =Tyz =0,T xx =Tyy =Tzz =p .(2.25) •Solid s ess enso Ts. We assume ha all i s componen s a e p opo ional o he no mal s ess pe pendicula o he opog aphy, i.e. he s ess componen Tzz s, Tjk s=αjkTzz s,j,k=x, y, z. (2.26) 116 F. BOUCHUT ET AL. In his sec ion, we will fi s es ablish a local ene gy equa ion o his model and hen desc ibe some o i s p ope ies. 4.1. Local ene gy In he ollowing lines we p o e ha he model (4.1) is compa ible wi h a dissipa i e ene gy balance. Fi s , om he mass equa ions (4.1a), (4.1b)weha e ∂ h+di (hϕ +h(1 −ϕ)u)=0,(4.2) ∂ (h(ρsϕ+ρ (1 −ϕ))) + di (h(ρsϕ +ρ (1 −ϕ)u)) = 0.(4.3) We can w i e sin θ(1,0) =cosθ∇˜ bwi h ˜ b=x an θ, so ha he sin θ e ms in (4.1c)and(4.1d)canbe g ouped wi h he ∇b e ms o gi e ∇(b+˜ b). Then we mul iply equa ion (4.1c)by(h )and(4.1d)by(hu) and sum up he esul s. Using he mass equa ions o simpli y he le -hand side, we ob ain ∂ ρsϕh| |2 2+ρ (1 −ϕ)h|u|2 2+di ρsϕh| |2 2 +ρ (1 −ϕ)h|u|2 2u =−(1 −ϕ)h(u− )·∇(p bed −ρ ghcosθ)   (a) −ghcosθρsϕ +ρ (1 −ϕ)u·∇(b+˜ b+h)   (b) −h2 2gcos θ(ρs−ρ ) ·∇ϕ   (c) −˜ βh|u− |2−| | an δϕ(ρs−ρ )ghcosθ. Ou objec i e is o compu e each e m on he igh -hand side o he p e ious equa ion and y o w i e i as a ime de i a i e o a di e gence o some hing explici . •Te m (a). Using (4.1e), (a)=−di (1 −ϕ)h(u− )(p bed −ρ ghcosθ).(4.4) •Te m (b). Taking in o accoun (4.3), (b)=−ghcosθ∇(b+˜ b)·(ρsϕ +ρ (1 −ϕ)u)−gcos θ∇h2 2·(ρsϕ +ρ (1 −ϕ)u) =−di gh(b+˜ b)cosθ(ρsϕ +ρ (1 −ϕ)u) +g(b+˜ b)cosθdi h(ρsϕ +ρ (1 −ϕ)u) −di 1 2gh2cos θ(ρsϕ +ρ (1 −ϕ)u)+1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u) =−di ghcosθ(b+˜ b+h 2)(ρsϕ +ρ (1 −ϕ)u) −∂ gh(b+˜ b)cosθ(ρsϕ+ρ (1 −ϕ))+1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u). A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 117 •Te m (c). (c)=−di 1 2gh2cos θ(ρs−ρ )ϕ +ϕ(ρs−ρ )gcos θdi h2 2 . Ga he ing all he e ms we ge ∂ ρsϕh| |2 2+ρ (1 −ϕ)h|u|2 2+gh(b+˜ b)cosθ(ρsϕ+ρ (1 −ϕ)) +di ρsϕh| |2 2 +ρ (1 −ϕ)h|u|2 2u+(1−ϕ)h(u− )(p bed −ρ ghcosθ) +ghcosθ(b+˜ b+h 2)(ρsϕ +ρ (1 −ϕ)u)+1 2gh2cos θ(ρs−ρ )ϕ  =T1,(4.5) whe e T1canbeexp essedas T1=1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u)+ϕ(ρs−ρ )gcos θdi h2 2  −˜ βh|u− |2−| | an δϕ(ρs−ρ )ghcos θ. (4.6) The fi s e m can be exp essed as 1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u)= 1 2ghcosθdi (h(ρsϕ +ρ (1 −ϕ)u)) −1 2ghcosθ(ρsϕ +ρ (1 −ϕ)u)·∇h. (4.7) Howe e , acco ding o (4.2)and(4.3)weha e ∂ 1 2gh2cos θ(ρsϕ+ρ (1 −ϕ))=−1 2ghcos θdi (h(ρsϕ +ρ (1 −ϕ)u)) −1 2ghcos θ(ρsϕ+ρ (1 −ϕ)) di (h(ϕ +(1−ϕ)u)) , Thus using his in (4.7)weob ain 1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u) =−∂ 1 2gh2cos θ(ρsϕ+ρ (1 −ϕ))−1 2ghcosθ(ρsϕ+ρ (1 −ϕ)) di (h ) −1 2ghcos θ(ρsϕ +ρ (1 −ϕ)u)·∇h. 118 F. BOUCHUT ET AL. Adding he second e m in (4.6) yields 1 2gh2cos θdi (ρsϕ +ρ (1 −ϕ)u)+ϕ(ρs−ρ )gcos θdi h2 2  =−∂ 1 2gh2cos θ(ρsϕ+ρ (1 −ϕ))−1 2ghcosθ(ρsϕ+ρ (1 −ϕ)) di (h ) −1 2ghcosθ(ρsϕ +ρ (1 −ϕ)u)·∇h+ϕ(ρs−ρ )gcos θdi h2 2  =−∂ 1 2gh2cos θ(ρsϕ+ρ (1 −ϕ))−1 2ρ ghcosθ(di (h )+(ϕ +(1−ϕ)u)·∇h). (4.8) Then we can compu e −1 2ρ ghcosθ(di (h )+(ϕ +(1−ϕ)u)·∇h) =−1 2ρ ghcosθdi (h )−di 1 2ρ gh2cos θ(ϕ +(1−ϕ)u) +1 2ρ ghcosθdi (h(ϕ +(1−ϕ)u)) =−di 1 2ρ gh2cos θ(ϕ +(1−ϕ)u)+1 2ρ ghcosθdi (h(1 −ϕ)(u− )) . Plugging his in o (4.8)and(4.6), we ob ain T1=−∂ 1 2gh2cos θ(ρsϕ+ρ (1 −ϕ))−di 1 2ρ gh2cos θ(ϕ +(1−ϕ)u) −˜ βh|u− |2−| | an δϕ(ρs−ρ )ghcos θ. Using his esul in (4.5) finally yields he ene gy iden i y ∂ ρsϕh| |2 2+ρ (1 −ϕ)h|u|2 2+ghcos θ(b+˜ b+h 2)(ρsϕ+ρ (1 −ϕ)) +di ρsϕh| |2 2 +ρ (1 −ϕ)h|u|2 2u+(1−ϕ)h(u− )p bed −ρ (1 −ϕ)gh2cos θ(u− ) +ghcosθ(b+˜ b+h 2)(ρsϕ +ρ (1 −ϕ)u)+1 2ϕgh2cos θ(ρs−ρ ) +1 2ρ gh2cos θ(ϕ +(1−ϕ)u)=Re, (4.9) wi h Re=−˜ βh|u− |2−| | an δϕ(ρs−ρ )ghcos θ. (4.10) The e o e he model (4.1) has a locally dissipa i e ene gy balance, since he esidual Reis non-posi i e. A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 119 Rema k 4.1. Iden i y (4.9) can be ob ained (up o O(3)) by in eg a ion o (2.11) wi h espec o z.This shows ha he le -hand side con ains he physically ele an ene gy and ene gy flux. Rema k 4.2. Le us ecall ha he Pi man−Le model [38] does no use any closu e equa ion (4.1e). Ins ead, he Pi man−Le model, in he o m p oposed by Pelan i e al. [36], can be seen as (4.1)whe ewese p bed =ρ ghcosθ (o equi alen ly ps|b+h= 0 acco ding o (3.10)). Consequen ly, he ene gy equa ion sa isfied by he Pi man−Le model is (4.9) wi h a igh -hand side Re ha is no always non-posi i e. The esidual e m o he Pi man−Le model is Re=−1 2ϕ(ρs−ρ )ghcos θdi (h(1 −ϕ)(u− )) −˜ βh|u− |2−| | an δϕ(ρs−ρ )ghcos θ and has no fixed sign (we will s udy his e m in Tes 1 p esen ed in Sec . 5.2.1). The in insic eason why he Pi man−Le model has a physically i ele an ene gy equa ion is ha i is de i ed om a 3D model ha does no ha e an ene gy dissipa ion p inciple (see Eq. (2.11)). 4.2. O he p ope ies Model (4.1) is a balance law ype sys em wi h non-local e ms ela ed o p bed .No efi s ha i ispossible o elimina e p bed om he sys em. Indeed, p bed appea s only in (4.1c)and(4.1d). We can hus e ain he sum o (4.1c)and(4.1d) and i we exp ess ∇p bed om (4.1c) o example and w i e ha he cu l o he esul anishes, we ob ain he missing ela ion. P oposi ion 4.3. Sys em (4.1)has he ollowing p ope ies. (i) The wo mass equa ions a e conse a i e. The momen um equa ions ake he quasi-conse a i e o m ρs(∂ (hϕ )+di (hϕ ⊗ )) = h(1 −ϕ)∇p bed −(1 −ϕ)ρ gcos θ∇h −ϕρsgcos θ∇(b+h)−1 2(ρs−ρ )ghcos θ∇ϕ −ϕρsgsin θ(1,0) +˜ β(u− ) −sign( ) an δϕ(ρs−ρ )g cos θ, (4.11a) ρ (∂ (h(1 −ϕ)u)+di (h(1 −ϕ)u⊗u)) = h−(1 −ϕ)∇p bed −(1 −ϕ)ρ gcos θ∇b −(1 −ϕ)ρ gsin θ(1,0) −˜ β(u− ).(4.11b) The o al momen um akes he conse a i e o m ρs(∂ (hϕ )+di (hϕ ⊗ )) + ρ (∂ (h(1 −ϕ)u)+di (h(1 −ϕ)u⊗u)) =−∇ (ρsϕ+ρ (1 −ϕ))gh2 2cos θ−gcos θ(ρsϕ+ρ (1 −ϕ)) h∇(b+˜ b) −sign( ) an δϕ(ρs−ρ )gh cos θ. (4.12) (ii) The hickness h emains non-nega i e, and 0≤ϕ≤1. 120 F. BOUCHUT ET AL. (iii) Special solu ions a e he s eady s a es a es , cha ac e ized by u= =0,b+˜ b+h=Cs , ϕ =Cs , (4.13) whe e we ecall ha ˜ b=x an θ. Indeed his is a solu ion o ou sys em wi h p bed =ρ ghcosθ. (i ) The classical single luid shallow wa e sys em is ob ained when u= ,ρ =ρs=ρand p bed =ρgh cosθ, and ei he ϕ=1in equa ion (4.11a)o ϕ=0in equa ion (4.11b). 5. Nume ical app oxima ion In his sec ion we desc ibe a nume ical me hod o app oxima e he p oposed wo-phase model (4.1)inone dimension. Then we pe o m diffe en es s, including a compa ison wi h he solu ion p o ided by he Pi man−Le model. We ocus on he one-dimensional si ua ion. As poin ed ou p e iously, he model can be ew i en in e ms o hesolidp essu ea he eesu aceps|b+h, he fluid p essu e a he ee su ace p |b+h=−ps|b+ho he solid p essu e a he bed psbed, ins ead o he fluid p essu e a he bed p bed, ia ela ions (3.9)and(3.10). In his sec ion we conside he o mula ion in e ms o he solid p essu e a he ee su ace ps|b+h,whichcanbe w i en ∂ (hϕ)+∂x(hϕ )=0,(5.1a) ∂ (h(1 −ϕ)) + ∂x(h(1 −ϕ)u)=0,(5.1b) ∂ (hϕ )+∂x(hϕ 2)=−h(1 −ϕ)∂xψ−ϕgh cos θ∂ x(b+h) −1 2(1 − )gh2cos θ∂ xϕ −ϕgh sin θ+ˆ βh(u− ), −sign( ) an δgcosθ(1 − )hϕ, (5.1c) ∂ (h(1 −ϕ)u)+∂x(h(1 −ϕ)u2)=h (1 −ϕ)∂xψ−(1 −ϕ)ghcos θ∂ x(b+h) −(1 −ϕ)ghsin θ−1 ˜ βh(u− ),(5.1d) ∂x(h(1 −ϕ)(u− )) = 0,(5.1e) whe e =ρ /ρs,ˆ β=˜ β ρs ,ψ=ps|b+h ρs =ρ ghcos θ−p bed ρs ·(5.2) I we conside ha he d ag coefficien ˜ βis defined by (3.12), hen ˆ β=(1 − )ϕg T(1 −ϕ)m−1·(5.3) 5.1. Nume ical me hod We apply a spli ing algo i hm, simila o he Teman-Cho in me hod o incomp essible Eule equa ions, in o de o impose he cons ain (5.1e). We obse e ha a he fi s s ep, when we neglec he ex a unknown ψ(which can be seen as a Lag ange mul iplie associa ed wi h he cons ain ) in (5.1a)−(5.1d), we ob ain he Pi man−Le model in he o m p oposed in [36], which is hype bolic whene e u− is no oo la ge. We conside he space domain [0,L] di ided in cells Ii=(xi−1/2,x i+1/2). Fo simplici y, we assume ha hese cells ha e a cons an size Δx. We define xi+1 2=iΔx and xi=(i−1/2)Δx, he cen e o he cell Ii.Le Δ be he ime s ep and define n+1 = n+Δ .Wis he ec o o he ollowing unknowns o he p oblem, W=[hϕ, h(1 −ϕ),hϕ ,h(1 −ϕ)u].(5.4) A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 121 The e o e Wn ideno es he app oxima ion p o ided by he nume ical scheme o he cell a e ages o he solu ion, Wn i∼ =1 Δx xi+1/2 xi−1/2 W( n,x)dx, (5.5) and by ψn i+1/2, an app oxima ion o ψ( n,x i+1/2). Assuming ha he alues o Wn ia e known, he sys em can be disc e ized in wo s eps. •Fi s s ep. We compu e he s a e W∗=[h∗ϕ∗,h ∗(1−ϕ∗),h ∗ϕ∗ ∗,h ∗(1−ϕ∗)u∗] by a semi-implici disc e iza- ion o he d ag W∗ i=Wn i−Δ ΔxLWn i−1,Wn i,Wn i+1,Δ(b+˜ b)i−1/2,Δ(b+˜ b)i+1/2 +0,0,Δ ˆ β∗ ih∗ i(u∗ i− ∗ i),−Δ ˆ β∗ i h∗ i (u∗ i− ∗ i),(5.6) whe e L(Wn i−1,Wn i,Wn i+1,Δ(b+˜ b)i−1/2,Δ(b+˜ b)i+1/2) defines he space disc e iza ion ope a o applied o model (5.1a)−(5.1d)wi hψ=0andˆ β= 0. In his wo k, we ha e conside ed he gene alized Roe me hod p oposed in [34]. Ano he possibili y would be o use he elaxa ion sol e p oposed by Pelan i e al. [37]. •Second s ep. In o de o en o ce he cons ain , we se hn+1 i=h∗ iand ϕn+1 i=ϕ∗ iand n+1 i,un+1 iand ψn+1 i+1/2 a e solu ions o he ollowing coupled sys em, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (hϕ )n+1 i=(hϕ )∗ i−Δ Δx(1 −ϕ∗ i)h∗ i(ψn+1 i+1/2−ψn+1 i−1/2), (h(1 −ϕ)u)n+1 i=(h(1 −ϕ)u)∗ i+Δ Δx(1 −ϕ∗ i)h∗ i (ψn+1 i+1/2−ψn+1 i−1/2), (h(1 −ϕ)(u− ))n+1 i+1 −(h(1 −ϕ)(u− ))n+1 i=0. (5.7) By ex ac ing n+1 iand un+1 i om he wo fi s equa ions o (5.7) and by subs i u ion in he hi d equa ion, we ob ain he ollowing sys em wi h unknowns {ψn+1 i+1/2}i, −a∗ i+1ψn+1 i+3/2+(a∗ i+a∗ i+1)ψn+1 i+1/2−a∗ iψn+1 i−1/2=h(1 −ϕ)(u− )∗ i+1 −h(1 −ϕ)(u− )∗ i,(5.8) wi h a∗ i=Δ Δxh∗ i(1 −ϕ∗ i)1 +1−ϕ∗ i ϕ∗ i·(5.9) Thus, in his second s ep, we mus sol e sys em (5.8) (wi h Di ichle bounda y condi ions, ψ=0) oob ain {ψn+1 i+1/2}iand use hese alues o upda e n+1 iand un+1 iby he wo fi s equa ions o (5.7). The ob ained scheme is ob iously well-balanced wi h espec o he s eady s a es a es (4.13) i he hype bolic sol e Lis well-balanced, and p ese es he na u al bounds h≥0and0≤ϕ≤1. 5.2. Nume ical es s We will now p esen some nume ical es s in o de o compa e he solu ion o he p oposed model (5.1a)−(5.1e) wi h he solu ion o he modified Pi man and Le p oblem p oposed in [36] (wi h he same d ag coefficien ˜ β). We simula e he collapse o a column made o a mix u e o g ains and wa e fi s o e a ho izon al plane and hen o e an inclined plane, a si ua ion widely in es iga ed o d y g anula flows (see o example [8,25,27]). Fi s , we simula e he flow o he mix u e o e a ho izon al bed o he wo diffe en d ag o ces gi en by (2.18) and (2.19). In he second es , we simula e he flow o he mix u e o e an inclined bed o cons an slope o a fixed choice o hese pa ame e s. 122 F. BOUCHUT ET AL. As gene al conside a ions, we fix he CFL numbe as 0.8, accele a ion due o g a i y g=9.81 m s−2and he ma e ial densi ies ρ = 1000 kg m−3and ρs= 2500 kg m−3, espec i ely. The e o e he a io o densi ies is =0.4. 5.2.1. Tes 1: Fla bo om In his expe imen , he space domain is Ω=[0,10]m and we conside 200 poin s. A ime = 5 s, he solid phase is s opped and some small eloci ies appea in he fluid phase. The ini ial condi ions a e defined as ollows h( =0s)=0.5m4m≤x≤6m 0.1mo he wise ;u( =0s)= ( =0s)=0ms −1;ϕ( =0s)=ϕ0. Fo he ini ial solid olume ac ion, we conside wo alues o ϕ0(0.3 and 0.6) o see he effec he ini ial s a e o mix u e fluidiza ion. We conside he in e g anula Coulomb angle o be δ=18 o. The objec i e o his es is o check he influence o he d ag o ce and ini ial solid olume ac ion on he flow and deposi . Remembe ha he d ag ic ion laws used he e (see Sec . 2.3)a egi enby: =˜ β(u− ), whe e he d ag coefficien ˜ βcan be se acco ding o: •Richa dson and Zaki [41]: ˜ β=(ρs−ρ )ϕg T(1 −ϕ)m−1,wi h m∈[0.4,2.65]. •Pailha and Pouliquen [35]: ˜ β=150μϕ2 d2(1 −ϕ). We also se T=0.143 m s−1,μ=10 −3Pa s,d=10 −3m and we a y he coefficien m, using he alues m=0.4,1and2.65. In he ollowing, we will e e o “RZ” and “PP” o he Richa dson and Zaki and o he Pailha and Poliquen d ag o ces espec i ely. In luenceo hed ag o ce. Figu es 4and 5show he hickness o he mass (i.e. a ime = 5 s) and he associa ed olume ac ion simula ed wi h diffe en d ag o ces bo h wi h he Pi man−Le (PL) model and wi h he new model p oposed he e. A ha ime, he solid phase has comple ely s opped. In Figu e 4we also compa e hese wo-phase flow models wi h he esul s ob ained wi h he Sa age-Hu e model, whe e he fluid phase is no conside ed (i.e. d y g anula flows). E en o ϕ0=0.6, Figu e 4show he s ong influence o he fluid phase on he a alanche hickness p ofile. The fi s obse a ion is ha he PL model and ou model ha e he same quali a i e beha iou . Howe e , o ϕ0=0.3, he PL model is mo e sensi i e o he diffe en d ag o ces in oduced in he model (Figs. 4aand5a). In pa icula , he final olume ac ion is highe o he PP d ag o ce han o he RZ d ag o ce a a cen e ed mass in e al, x∈(2m, 8m). They coincides jus in he cen e o mass (x=5m) o he case m=0.4, eaching up o ϕ=0.65. Fo he RZ d ag o ces, he a ia ion o he olume ac ion in space is smoo he han o he PP d ag o ce. Fo ou model, he d ag o ce only sligh ly affec s he esul s o ϕ0=0.3. Fo ϕ0=0.6, he sensi i i y o he wo models o he diffe en d ag o ces is quali a i ely simila e en hough he olume ac ion calcula ed wi h he PL model is s ill mo e sensi i e han ha calcula ed wi h ou model (Figs. 5b,d). Fo ϕ0=0.6, he olume ac ion a he cen e o he column eaches e y high alues wi h he PL model A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 123 Figu e 4. Tes 1: hickness p ofile o he mass h(m) as a unc ion o he dis ance x(m) a ime = 5 s, when he g anula phase has al eady s opped, o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e, simula ed wi h diffe en ic ion laws (“RZ” e e s o he Richa dson and Zaki and “PP” o he Pailha and Pouliquen d ag o ces). The ini ial olume ac ions a e: (le ) ϕ0=0.3; ( igh ) ϕ0=0.6. The hickness p ofile ob ained using he d y g anula flow model o Sa age and Hu e (ob ained by se ing all he e ms ela ed o he fluid phase equal o ze o) is also ep esen ed o compa ison. (abou 0.9), while ϕ<0.75 wi h ou model. The o e all lowe sensi i i y o ou model o he diffe en d ag o ces sugges s ha he diffe ence be ween he eloci ies o he wo phases is lowe wi h ou model han wi h he PL model, as shown in Figu e 8. Compa ison o he wo models. Le us now compa e he wo models o a gi en ic ion law, i.e., he Richa dson and Zaki law wi h m=1 (co esponding o he da a used in [10]). We also conside wo diffe en alues o he Coulomb ic ion angle: δ=18 ◦and δ=28 ◦. Fo bo h he PL model and ou model, he highe he ini ial fluidiza ion, he la ge he sp eading o he ma e ial and he smalle he aspec a io o he deposi (Fig. 6). Fu he mo e, he olume ac ion is highes a he cen e o mass and dec eases owa d he on , leading o ϕ<ϕ 0a he on . 124 F. BOUCHUT ET AL. Figu e 5. Tes 1: solid olume ac ion o he mass ϕas a unc ion o he dis ance x(m) a ime = 5 s, when he g anula phase has al eady s opped, o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e, simula ed wi h diffe en ic ion laws (“RZ” e e s o he Richa dson and Zaki and “PP” o he Pailha and Pouliquen d ag o ces). The ini ial olume ac ions a e: (le ) ϕ0=0.3; ( igh ) ϕ0=0.6. A ime = 5 s, when he solid phase is al eady a es , he hickness o he mass is e y simila in bo h models, e en hough he maximum hickness is sligh ly smalle wi h ou model (Fig. 6). Howe e , he dis ibu ion o he phases (i.e. olume ac ion) is diffe en . As obse ed p e iously, he olume ac ion is mo e uni o mly dis ibu ed in he simula ions wi h ou model. The peak o high olume ac ion a he cen e o mass is highe o he PL model and he dec ease in olume ac ion owa d he on o he mass is la ge han in ou model (Fig. 6c). The eloci y o he fluid phase a an in e media e ime =1.5 s, when he g anula phase is s ill flowing, is sligh ly highe wi h ou model o δ=18 ◦ o bo h ϕ0=0.3andϕ0=0.6 (see Figs. 7aand7b). Fo δ=28 ◦, he fluid eloci ies a e almos he same in he wo models owa ds he on and he fluid eloci y wi h ou model is lowe a ound he cen e han ha calcula ed wi h he PL model. The diffe ence be ween he wo models is g ea e o he solid eloci y o bo h ϕ0=0.3andϕ0=0.6, (see Figs. 7cand7d). Fo he p oposed model, he solid phase mo es as e han o he PL model. We also obse e ha o δ=18 ◦, he eloci ies o bo h phases a e g ea e han o δ=28 ◦. Mo eo e , o la ge alues o δ, he diffe ence be ween he eloci ies o he A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 125 Figu e 6. Tes 1: compa ison be ween he solu ions ob ained wi h he Pi man−Le and p o- posed models o he hickness o he mass h(m) and he olume ac ion ϕas unc ions o he dis ance x(m)a ime = 5 s when he solid phase has al eady s opped, o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e. The ini ial olume ac ions a e: (le ) ϕ0=0.3; ( igh ) ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. wophasesisg ea e (seeFig.8). This diffe ence is much la ge in he PL model han in ou model, leading o highe d ag o ces in he PL model. This explain why ou model is less sensi i e o he defini ion o he d ag o ce (see Fig. 4). E olu ion in ime. Le us now look a he changes o he diffe en quan i ies wi h ime (Figs. 9−13 in which imes =1,2,3and 5 s a e ep esen ed wi h diffe en colou s). Fo hese simula ions, we use he Richa dson and Zaki ic ion law wi h m= 1 and he Coulomb ic ion angle δ=18 ◦. No e ha e en hough he mass p ofiles change wi h ime in a simila way o he wo models (Fig. 9), he e is s ong diffe ence be ween he wo models o he changes o he olume ac ion wi h ime (Fig. 10), 132 F. BOUCHUT ET AL. Figu e 13. Tes 1: he a iable ψ(x, )inm 2s−2in he p oposed model a diffe en imes o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e. The ini ial olume ac ions a e: (le ) ϕ0=0.3; ( igh ) ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. Figu e 14. Tes 1: alues o he e ms in ol ed in he esidual ene gy (see Eq. (5.10)) as a unc ion o he dis ance x(m), o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e a he diffe en imes =0.5kwi hk=1s,...10 s. The ini ial olume ac ion is ϕ0=0.3 and he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 133 Figu e 15. Tes 1: alues o he o al esidual e m as a unc ion o he dis ance x(m) o he collapse o a ec angula g anula mix u e o e a ho izon al laye made o he same mix u e a he diffe en imes =0.5kwi hk=1s,...10 s. The ini ial olume ac ion is ϕ0=0.3and he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. Figu e 16. Tes 2: compa ison o he hickness p ofiles o he mass h(x, ) in me e s as a unc ion o he dis ance x(m) o he Pi man−Le and p oposed models, a ime =1,3,5,10 s (a = 10 s he solid phase has al eady s opped), o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. 134 F. BOUCHUT ET AL. Figu e 17. Tes 2: compa ison o he solid olume ac ion o he mix u e ϕ(x, ) as a unc ion o he dis ance x(m) o he Pi man−Leandp oposedmodels,a ime =1,3,5,10 s (a = 10 s he solid phase has al eady s opped), o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. added o Jackson’s model o ob ain a well-posed sys em. This may no be appa en wi h a dep h-a e aged model wi h hyd os a ic p essu e, such as he one p oposed by Pi man and Le [38]. Indeed, i we assume hyd os a ic p essu e o bo h phases, hey a e ela ed by hei bounda y condi ion. In his case, imposing ze o a mosphe ic p essu e o bo h phases can be seen as he co esponding closu e equa ion. Ne e heless, he model ha is de- duced does no ha e a dissipa i e ene gy balance. The main diffe ence be ween he model ha we p opose in his pape and he Pi man−Le model comes om he bounda y condi ion on he ee su ace. In he p oposed model, we only impose ha he sum o he p essu es o he wo phases is ze o and no each o hem. This in oduces new unknown in he simplified model. As a closu e equa ion o he 3D sys em, we conside incomp essibili y o he solid phase. This closu e ela ion is consis en wi h he hyd os a ic p essu e assump ion. The nume ical es s p esen ed he e show ha , o e all, he changes o he p ofiles o he flowing mass wi h ime a e simila o he Pi man−Le model and he model p oposed he e. The quali a i e beha iou o he solid olume ac ion and A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 135 Figu e 18. Tes 2: compa ison o he eloci y o he fluid phase u(x, )inms −1as a unc ion o he dis ance x(m) o he Pi man−Le and p oposed models, a ime =1,3,5,10 s (a =10s he solid phase has al eady s opped), o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. he solid and fluid eloci ies is he same o bo h models. Howe e , wi h he model p esen ed he e, he solid olume ac ion a ies less, he solid phase eloci y is gene ally highe and he diffe ence be ween he eloci ies o he wo phases is smalle , leading o smalle d ag o ces be ween he wo phases. This induces significan diffe ences in he p ofile o he sp eading mass and o he deposi wi h unou dis ances mo e han 10% la ge and eloci ies ha could be mo e han 20% heighe in he simple es o g anula collapse o e inclined plane pe o med he e. While i is qui e difficul o measu e expe imen ally o in he field he fluid and solid eloci ies, he use o seismic wa es gene a ed by deb is flows o a alanches may be a new way o disc imina e be ween hese wo models ([9,31]). An ad an age o ou model is ha he closu e equa ion (i.e. incomp essibili y o he solid phase) is explici ly imposed, making i possible o de i e physical in e p e a ion o ou esul s while in he PL model, he beha io is dic a ed by he imposed ze o-p essu e a he su ace o each phase, wi hou any desc ip ion o he mechanical 136 F. BOUCHUT ET AL. Figu e 19. Tes 2: compa ison o he eloci y o he solid phase (x, )inms −1as a unc ion o he dis ance x(m) o he Pi man−Leandp oposedmodels,a ime =1,3,5,7s(a =10s he solid phase has al eady s opped), o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. p ope ies o he solid phase. I is in e es ing o see he significan diffe ences be ween he wo models e en hough he su ace p essu e o each phase is e y small in ou model. This analysis is la gely d i en by he kinema ic bounda y condi ions ha imposes he wo phases o fill he same domain. Howe e in deb is flows, he fluid phase su ace can be highe o lowe han he solid phase su ace, due o he ela i e mo ion be ween hese wo phases. This is expec ed o be significan in pa icula when comp ession/dila ion o he solid phase occu . Because he models seem o be e y sensi i e o wha happens a he su ace, e en small a ia ions o he fluid and solid su aces could ha e a s ong impac on he esul s. Fu he analysis o he equa ions should be pe o med wi h elaxa ion o hese bounda y condi ions and including a mo e ealis ic closu e ela ion ela ed o he comp ession/dila ion o he g anula phase. A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 137 Figu e 20. Tes 2: compa ison o he eloci y diffe ence o he fluid u(x, ) and solid phase (x, )inms −1as unc ions o he dis ance x(m) o he Pi man−Le and p oposed models, a ime = 1 and 5 s, o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. Figu e 21. Tes 2: he mass hickness h(x, ) in me e s as a unc ion o he dis ance x(m) (a-b) and o he solid olume ac ion ϕ(x, ) as a unc ion o he dis ance x(m) (c-d) a diffe en imes o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦)made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. 138 F. BOUCHUT ET AL. Figu e 22. Tes 2: he eloci y o he fluid phase u(x, )inms −1as a unc ion o he dis ance x(m) (a-b) and o he solid phase (x, )inms −1as a unc ion o he dis ance x(m) (c-d) a diffe en imes o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. The ini ial olume ac ion is ϕ0=0.6. He e he ic ion law is he Richa dson and Zaki d ag o ce (RZ) wi h m=1. Figu e 23. Tes 2: le : he su ace p essu e o he solid phase ψ(x, )inm 2s−2as a unc ion o he dis ance x(m) a diffe en imes o he collapse o a ec angula g anula mix u e o e an inclined laye (θ=10 ◦) made o he same mix u e. Righ : compa ison be ween ψand g(b+h1+h2)a =10s. A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE 139 Acknowledgemen s. We hank Fa hang Radjai, Renaud Toussain and Oli ie Pouliquen o in e es ing discussions du - ing his wo k. We hank Nicolas Mangold and Old ich Hung o leading he ield ips in which we ook he pic u es in Figu e 1. The wo k o E.D. Fe n´andez-Nie o and G. 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