Prediction of impact force of debris flows based on distribution and size of particles

ORIGINAL ARTICLEPrediction of impact force of debris flows based on distribution and size of particlesSiming He 1,2,3•Wei Liu 1,2•Xinpo Li 1,2Received:30November 2014/Accepted:20October 2015/Published online:11February 2016ÓSpringer-Verlag Berlin Heidelberg 2016Abstract A debris flow is a solid–liquid two-phase flow;the composition and gradation of the particles within have a significant influence on its impact force.This paper proposes a new method for studying the impact force according to the composition of a debris flow based on analysis of existing calculation methods.The impact force is divided into three parts:(1)the dynamic pressure pro-vided by the debris flow slurry,which is composed of fine particles and water;(2)the impact force of coarse particles;and (3)the impact force of boulders.This paper analyzes the established formulations used to calculate the impact force by using hydrodynamic theory and contact mechanics to propose a debris flow impact model according to the debris flow type.The results show that the impact force is closely related to the solid volume fraction,composition of particle materials,motion velocity,and depth of a debris flow.Among all the components of the impact force,the boulder impact force is the largest followed by the impact force of coarse particles;the dynamic pressure is minimal.Keywords Debris flow ÁSolid–liquid two-phase flow ÁGradation of particles ÁImpact forceIntroductionAs natural phenomena,debris flows play an important role in sediment transfer and erosion in mountainous zones throughout the world.Debris flows are water-saturated masses of soil and fragmented rock that are intermediate in character between rock avalanches and flash floods according to their mechanical behavior.Debris flows have a solid volume fraction that exceeds 0.4,peak speeds that surpass 10m/s,and volumes that approach 109m 3(Iverson and George 2014).Therefore,debris flows commonly have high mobility and extremely destructive natures,posing a high risk to human life,safety,and property in mountain-ous regions (Iverson 1997;Iverson and Denlinger 2001;Iverson et al.2011;Iverson and George 2014;Luna et al.2012;Ouyang et al.2014;Canelli et al.2012;Tang et al.2012;Eidsvig et al.2014).Debris flows cause damage mainly in three ways:deposition,entrainment,and direct impact (Hu et al.2011).However,impact is the main factor that causes structural destruction (Mizuyama 1979;Hungr et al.1984;Armanini 1997;Zhang 1993;Shieh et al.2008;Hu¨bl et al.2009;Moriguchi et al.2009).There is an increasingly greater need for predicting the impact force of debris flows for vulnerability and risk assessments as well as for designing mitigation measures to withstand such impact forces (Bugnion et al.2012;Scheidl et al.2013).However,the impact mechanism is poorly understood,owing partly to difficulties in measuring impact force (Hu et al.2011).At present,two different approaches exist for calculating the impact force of debris flows:hydrostaticmodels (Lichtenhahn 1973;Armanini 1997;Hu¨bl et al.2009)and hydrodynamic formulas (Watanabe and Ike 1981;Zhang 1993;Egli 2000;Wendeler et al.2006;Bugnion et al.2012).A hydrostatic model is assumed to be proportional to the hydrostatic pressure exerted on a fixed&Wei Liusponlol@1Key Laboratory of Mountain Hazards and Surface Process,Chinese Academy of Science,Chengdu 610041,Sichuan,China2Institute of Mountain Hazards and Environment,Chinese Academy of Sciences,Chengdu 610041,Sichuan,China 3CAS Center for Excellence in Tibetan Plateau Earth Sciences,Chengdu,ChinaEnviron Earth Sci (2016)75:298DOI 10.1007/s12665-015-5180-2barrier by a debris flow.In general,the hydrostatic formula of a debris flow can be expressed as p max ¼k q gh ;ð1Þwhere p max denotes the maximum debris flow impact pressure,k is an empirical parameter,q is the density of the debris flow,h is the total flow height,and g is the gravi-tational acceleration.However,impact force is related to factors other than flow depth and velocity.The hydrostatic model theoreti-cally does not consider the fact that dynamic impact is related to hydrodynamic action rather than hydrostatic pressure (Moriguchi et al.2009).A hydrodynamic formula is assumed to be proportional to the square of the flow velocity from the change in fluid momentum (Mizuyama 1979).In general,the impact hydrodynamic formula of a debris flow can be expressed as p max ¼a q v 2;ð2Þwhere p max denotes the maximum debris flow impactpressure,a is the empirical coefficient,and v is the average velocity of a debris flow.Mizuyama (2008)reported that a limitation of hydro-dynamic formulas is that the impact process of debris flows is transient and unsteady and thus cannot be described by a certain constant velocity distribution.In summary,the existing impact force formulas of debris flows,both hydrostatic and hydrodynamic,consider the flow as a single-phase homogeneous bo-ratory testing and in situ monitoring data are incorporated into hydraulics formulas,thereby synthetically reflecting the overall impact force of a debris flow.During laboratory testing,however,the influence of the scale effect and the layout and size of the impact sensors for in situ monitoring prevents impact force analysis from accurately reflecting the actual debris flow conditions.In particular,the impact of boulders is two to three orders of magnitude greater than the dynamic pressure of the slurry and is the main factor contributing to barrier damage (Suwa and Okuda 1983).A debris flow is a typical solid–liquid two-phase fluid flow that consists of a broad distribution of grain sizes from 65l m to several meters.Fine particles and water form a relatively uniform slurry,and coarse particles form slurry packages.Their mobility differs;therefore,the impacts of debris flows should be calculated separately based on their compositions and the differences in their mobility characteristics.The objective of this study is to investigate the impact forces of debris flows based on grain size distributions and debris components.We divide the impact force of a debris flow into three parts:(1)the dynamic pressure of slurry,(2)the impact force of coarse particles,and (3)the impact force of boulders.The corresponding calculation formula is then pre-sented.In this study,the sizes of solid particles are classi-fied into three categories:fine particles with grain sizes less than 10mm,boulders with grain sizes greater than 500mm,and coarse particles with grain sizes between the two (Fig.1).Dynamic pressure formula of debris flow slurryThe slurry characteristics of a debris flow deviate from that of an ideal fluid and depend on the constituents forming the slurry phase such as silt,clay,and fine particles.Assuming a debris flow slurry with average velocity v f ,flow depth h ,and width b on an impact barrier (Fig.2),the momentum theorem is p f bh ÀÁD t ¼q f bhv f D t ÀÁv f ;ð3Þwhere p f denotes the dynamic pressure of the slurry of adebris flow,q f is the density of the slurry,v f is the average velocity of the slurry,and D t is the impact time.Then,the dynamic pressure exerted by the slurry of a debris flow on a barrier can be expressed as p f ¼q f v 2f :ð4ÞImpact pressure exerted by coarse particlesIn this paper,coarse particles refer to the transition size between fine particles and boulders.Owing to the broad range in the size of coarse particles,we use theaverageFig.1Classification standard for particle sizes of debrisflowsFig.2Computation mode of dynamic pressure of debris flow slurry298Page 2of 8Environ Earth Sci (2016)75:298particle size,d 50,as a representative for calculating theimpact force exerted by coarse particles.Because d 50is relatively small,the impact is correspondingly limited;therefore,the influence of non-linear characteristics on the impact contact of the particles can be ignored.The impact force of a single coarse particle may be computed according to the Hertz contact theory (Johnson 1985;Vu-Quoc et al.2001;Braccesi and Landi 2007).Consider two spheres,sphere (1)and sphere (2),in contact with each other as shown in Fig.3,and subjected to a normal contact force P .Let R 1and R 2be the radii of sphere (1)and sphere (2),respectively.The material properties of sphere (1)are denoted by E 1for the Young’s modulus and l 1for Pois-son’s ratio;similar properties are given for sphere (2).The equivalent Young’s modulus E can be expressed as (Johnson 1985):1E ¼1Àm 21E 1þ1Àm 22E 2and the relative radius R of contact curvature as (Johnson 1985)1R ¼1R 1þ1R 2:The normal contact force–displacement relation can be expressed as P ¼43ER 12d 32;ð5Þwhere P denotes the contact force,and d is the normal approach or interference.Assume that the concrete barrier is in half-space (R 2!1)relative to a single coarse particle and that the concrete barrier and coarse particle have the same Young’s modulus and Poisson ratio (E 1¼E 2,l 1¼l 2).Then,the impact force of a single coarse particle may be expressed asP g ¼43Ed 50212d 32;ð6Þwhere P g denotes the impact force of a single particle,d 50is the average size of the coarse particles,E is the equiv-alent elastic modulus,and d is the contact deformation.Assuming v s and q s represent the velocity and density of coarse particles,respectively,the following equation can be derived according to the conservation of energy:1243p d 502 3q s "#v 2s ¼Z d max 043E d 50212d 32d d :ð7ÞThen,the impact force formula of a single coarse par-ticle can be expressed asP g ¼4354p 0:6E 0:4d 5022q 0:6s v 1:2s ¼3:03E 0:4d 502 2q 0:6s v 1:2s :ð8ÞAccording to the volume fraction of a solid,the total number of coarse particles per unit area is expressed as (Iverson et al.2010)N ¼up d 50ÀÁ2;ð9Þwhere N is the total number of coarse particles per unit area,and u is the solid volume fraction of the debris flow.The average impact pressure formula of coarse particles can be written asP g ¼0:964E 0:4q 0:6s v 1:2s u :ð10ÞImpact force of bouldersHuang et al.(2007)applied the theory of elastic collision to build a boulder impact model based on the Hertz law.In their studies,the two balls in the collision represent a concrete barrier and a boulder.Assuming the collision is elastic and the radius and mass of the concrete barrier are significantly larger than those of the boulder,the impact force of the boulder can be expressed asF b ¼43p k 1þk 2ðÞ0:45q s p 24g0:6U 1:2R 2;ð11Þwhere k 1¼1Àm 11,k 2¼1Àm 22,F b represents for the impact force of the boulder,U is the moving velocity of the boulder,R is the radius of the boulder,q s is the density of the boulder,and E 1,E 2,l 1and l 2denote Young’s elastic modulus and the Poisson ratio of the concrete barrier and boulder,respectively.Fig.3Hertz contact of two nonconforming elastic bodiesEnviron Earth Sci (2016)75:298Page 3of 8298Huang et al.(2007)used experimental results to modify the impact models and obtained a more reasonable and universal model:F b¼30:8U0:5R2:ð12ÞHowever,the impact interaction between a boulder and a barrier involves complicated elastic–plastic deformation and energy conversion.Due to the Hertz elastic contact theory overestimates the impact of the boulder,Kuwabara and Kono(1987),He et al.(2007),and Thornton(1997) have suggested alternative models that consider viscous–elastic and elastic–plastic behavior.In addition,applying the impact force formula to actual situations based on a laboratory model impact test gives inaccurate results.Considering the Hertz elastic contact theory,a nonlinear contact model has been developed on the basis of detailed studies of the characteristics of the materials and current engineering practices.This model stresses the influences of elastic–plastic properties of materials on the mechanical contact properties.The Meyer nonlinear contact theory assumes that upon normal loading,the contact deformation between a spher-ical particle of a given dimension and a planar surface and the contact pressure have the following relationship(Haz-izan and Cantwell2002;Andrews et al.2002):F¼c d nb;ð13Þwhere F is the normal pressure acting on the contact mass, d b is the corresponding contact deformation,and c and b are experimental regression coefficients that can be obtained through an indent test or thefinite element method (FEM).The following equation can be derived by considering the conservation of energy at the impact of the boulder and the barrier(He et al.2007):1 2M b U2¼Z d maxc d nbd d;ð14Þwhere M b stand is the mass of the boulder,U is the moving velocity of the boulder,and d max represents the maximum contact deformation.Further,Eq.(14)can be rewritten as1 243p R3q sU2¼Z d maxc d n b d d:ð15ÞThus,the maximum contact deformation formula can beobtained asd max¼nþ13cp R3U2q s1nþ1:ð16ÞThe impact force formula of a boulder is presented asF b¼cnþ13cp R3U2q sn:ð17ÞVelocity of slurry,coarse particles,and bouldersIn the above impact force formula of debrisflow,it isnecessary to determine the moving velocity of each phase,including the velocities of the debrisflow,slurry,andcoarse particles in addition to the moving velocity of theboulders.The mixture mass density of a debrisflow can be definedas(Iverson1997;Iverson and Denlinger2001)q¼1ÀuðÞq fþ/q s;ð18Þwhere q is the mass density of the debrisflow,u is thesolid volume fraction,and q f and q s are the densities of thefluid and solid phases,respectively.The mixture moving velocity of a debrisflow can bedefined as(Iverson1997;Iverson and Denlinger2001)v¼1ÀuðÞq f v fþuq s v sq;ð19Þwhere v is the velocity of the debrisflow,and v f and v s arethe velocities of thefluid and solid phases,respectively.Iverson(1997)determined that solid velocity can besubstituted forfluid velocity:v¼v f¼v s:ð20ÞHowever,the velocity of a boulder is associated with thetype and state of motion of the debrisflow according to thefollowing conditions:1.When a boulderfloats in a debrisflow slurry during aviscous debrisflow,it is reasonable to assume that theboulder has the same velocity as that of the debrisflow:U¼v¼v f¼v s;ð21Þ2.When a boulder moves mainly by rolling during adilute debrisflow,the velocity of the boulder is less thanthat of the debrisflow and is related to the depth and thediameter of the boulder.Impact force load pattern of a debrisflowThe temporal and spatial distribution of solid phase parti-cles is closely related to the type of debrisflow.In general,solid-phase particle distribution is more uniform in viscousdebrisflows,and the boulders within theflow move by298Page4of8Environ Earth Sci(2016)75:298floating.However,the solid particle concentration is dis-tributed at the bottom of a dilute debris flow,and the boulders within move by rolling along the gully bed.Based on the distribution characteristics of solid parti-cles in different types of debris flows,the present study adopts methods reported previously (Kwan 2012)and proposes a model for estimating the impact force of a viscous debris flow on a containment barrier with an empty reservoir under continuous impact (Fig.4).In addition,a model for estimating the impact force of a dilute debris flow on a containment barrier is provided (Fig.5).Sample calculation analysisBased on the calculation method proposed in this study for estimating the impact force of debris flows,in addition to the model for loading effects,calculations were performed on the impact of a viscous debris flow against a contain-ment barrier.The following assumptions were made:(1)The height of the containment barrier is 6m,(2)the depth of the debris flow is 2m,and (3)the average flow velocityof the debris flow is 5m/s.Related parameters required for calculating debris flow impact are shown in Table 1.The values of c and n were obtained through static indentation testing or finite element analysis (Wang et al.2013).The values of the other parameters were determined through field measurements or laboratory experiments.The relationship between the impact force and the velocity of boulders within a debris flow upon impact with the containment barrier was analyzed.The results of the calculations,shown in Fig.6,indicate a linear relationship between the impact force and velocity.Higher velocity relates to a greater impact force when the boulders hit the containment barrier.The impact force based on the calculation formula for a typical debris flow was derived by using the aforemen-tioned calculation parameters.The calculation results are shown in Table 2.Our proposed calculation method combines the influ-ence of the slurry,coarse particles,and boulders within a debris flow.The calculation results show a sum of 169.44kPa for the impact forces of the debris flow slurry and coarse particles.Even without the introduction ofaFig.4Impact mode of viscous debrisflowFig.5Impact mode of dilute debris flowEnviron Earth Sci (2016)75:298Page 5of 8298correction factor,the calculation results are equivalent to that derived using other currently available formulas for computing the impact force of a debrisflow.However,the impact force of the boulders within a debrisflow deter-mined by our method is significantly smaller than that determined by Huang et al.(2007).The calculation results for the three types of impact force of a viscous debrisflow in addition to a model illustrating the loading effect on an empty reservoir are shown in Fig.7.The dynamic pressure of the debrisflow slurry and the impact intensity of the coarse particles,60and278.8kN, respectively,were then converted to an impact force on a per unit width of the barrier.The total impact force exerted on the containment barrier by the debrisflow was3106.6kN.In terms of the proportions of the total impact force,that produced by the boulders,coarse particles,and slurry was88,9,and3%, respectively.The impact force of the boulders was nearly30 times that of the slurry and nearly ten times that of the coarse particles.The impact force of the coarse particles was in turn three times that of the slurry.Hence,when designing con-tainment barriers and structures,appropriate consideration must be given to the boulders within the debrisflows because they constitute the main load of the impact force.Naturally,the three types of forces in a debrisflow do not all act on the containment barrier at the same time. Among the three,the impact forces of the slurry and coarse particles are continuous loads,whereas that of the boulders is a stochastic load.However,when designing protective structures such as containment barriers,the combined effect of all three types of loads must be considered.Only then can the protective structures be guaranteed to provide safety from debrisflows.ConclusionThe existing impact force formulas of debrisflows always need to determine some empirical parameters which rele-vant to the distribution and sizes of particles in a debris flow.However,the determination of empirical parameter may have some errors due to the complex components of debrisflow.In order to avoid the arising of error,a novelTable1Calculation parameters of the impact force of a debrisflow u q f(kg/m3)q s(kg/m3)v(m/s)d50(m)c n R(m)E1(MPa)E2(GPa)H(m)0.31200240050.110 1.4 1.010256 Fig.6Relationship of impact force and velocity of boulders in adebrisflowTable2Comparison of results by other calculation methods for a typical debrisflowAuthor Calculation formula Modified parameter Results NotesLichtenhahn(1973)p max¼k q gh 2.8–4.485.6–134.5kPa Hydrostatic formulaScotton and Deganutti(1997)p max¼k q gh 2.5–7.576–229kPa Hydrostatic formula Watanabe and Ike(1981)p max¼a q v2 2.0–4.078–156kPa Hydrodynamic modelZhand(1993)p max¼a q v2 3.0–5.0117–195kPa Hydrodynamic modelHu¨bl and Holzinger(2003)pmax¼0:5q v0:8ðghÞ0:6168.5kPa Hydrodynamic modelArmanini and Scotton(1992)p max¼912q gy2d275.18kPa Hydrodynamic modelCanelli et al.(2012)F¼K q v2A 1.5–5.5117–429kN Hydrodynamic modelHuang et al.(2007)F b¼43p k1þk2ðÞ0:45q s p24g0:6U1:2R25526kN Impact force of bouldersPresent model p f¼qfv2fp f=30kPa Dynamic pressureP g¼0:964E0:4q0:6s v1:2s u P g¼139:44kPa Impact force of coarse particlesF b¼c nþ13cp R3U2q sÀÁn Fb=2737.76kN Impact force of boulders298Page6of8Environ Earth Sci(2016)75:298impact force formulas which contain three parts:dynamic pressure of slurry,impact pressure of course particles and impact force caused by boulders is suggested on the basis of the distribution and sizes of particles in a debris flow.The proposed formulas were used for theoretical calcula-tion,and the results were compared with those derived by using other currently available formulas.The results have shown that the impact forces caused by the slurry and coarse particles are quite close to the results derived by using existing formulas for calculating the impact flow of a debris flow and among the components of the debris flow impact load,boulders produced the greatest impact force,followed by coarse particles and,to a much smaller extent,slurry.When designing mitigation structures for debris flows,the impact force caused by boulders should be the primary consideration.A pattern for the impact forces and loading effects of viscous and dilute debris flows was proposed.This mode is realistic because it considers the distribution characteristics of the solid particles within the different types of debris flow.It is important to further validate the presented model by comparing the predictions and observations by using small model experiments and measurements of actual-scale debris flow events.Acknowledgments This research has received financial support from the NSFC (Grant No.41272346),the STS project of Chinese Academy of Sciences (project No.KFJ-EW-STS-094)and The Research Rlan of Shaanxi Provincial Transport Department.ReferencesAndrews EW,Giannakopoulos AE,Plisson E,Suresh S (2002)Analysis of the impact of a sharp indenter.Int J solid struct 39(2):281–295Armanini A (1997)On the dynamic impact of debris flows.In:Armanini A,Masanori M (eds)Recent developments on debrisflows,lecture notes in earth sciences.Springer,Berlin,pp 208–226Armanini A,Scotton P (1992)Experimental analysis on the dynamicimpact of a debris flow on structures.In:Internationales symposion interpraevent 1992,vol 6,Bern,pp 107–111Braccesi C,Landi L (2007)A general elastic–plastic approach toimpact analisys 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