- Email: [email protected]
A mechanism of pore pressure accumulation in rapidly sliding submerged porous blocks
Computers and Geotechnics 31 (2004) 209–226 www.elsevier.com/locate/compgeo
A mechanism of pore pressure accumulation in rapidly sliding submerged porous blocks A. Musso *, F. Federico, G. Troiano Department of Civil Engineering, University of Rome ‘‘Tor Vergata’’, Via del Politecnico, 1, Rome 00133, Italy
Abstract The generation and dissipation of pore pressure excess due to the rapid, mutual sliding of two water-saturated blocks which are composed of glued arrays of cylindrical rods is herein theoretically examined. The mathematical model consists of three ordinary differential equations describing the motion of a solid mass coupled with the continuity of a fluid mass. These equations are numerically solved for the following conditions: (a) motion induced by the constant horizontal velocity of the sliding mass; (b) constant drag force acting in a horizontal direction; (c) gliding on the irregular, inclined surface that envelops the rods. Theoretical predictions for case (a) favorably compare with experimental results reported in the literature from an investigation carried out on a physical model. Primary results show that the pore pressure excess depends upon physical and geometrical parameters, and furthermore, that displacement of the rods and pore pressure excess are strongly coupled. Maximum pore water pressure occurs when ‘‘fluidization’’, i.e. the complete balancing of submerged block weight by pore pressure excess, is achieved. The latter does not coincide with the rising of the upper block (hydroplaning). A cyclic motion is reached under condition (a), when dynamic pore water pressure increases from a negative to a maximum positive value and then falls again to a negative value. On the other hand, non-cyclical motion and monotonically increasing pore water pressure are obtained under sliding conditions (b) and (c). The response of the model under condition (c), which is of particular interest for run-out analyses of flow slides, is compared to the sliding response of ordinary submerged blocks on a slope, showing significant differences and pointing out the roles of particle diameter, layer extent and medium permeability. The occurrence of different pore pressure states for fluidization and hydroplaning is highlighted. This, hopefully, can open the way to a reliable treatment of the mechanics of sub-marine flow slide propagation. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Pore pressure excess generation; Solid–fluid interaction; Sliding of a block; Hydroplaning; Fluidization
1. Introduction The high mobility of ‘‘wet’’ flows, such as debris flows, sub-aerial and sub-marine landslides in loose sands and silts, is still a matter of considerable concern. In most cases, particularly when materials with a metastable structure are implied, static liquefaction is clearly present as a trigger mechanism (flow slide). Soil Mechanics’ aspects of the problem are usually limited to trigger analysis of the phenomenon, thus leaving to Fluid Mechanics the task of exploring the characteristics of the run-out. In some circumstances, this is accom*
Corresponding author. Tel.: +39-6-72-59-70-62; fax: +39-6-72-5970-05. E-mail address: [email protected] (A. Musso). 0266-352X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2004.02.001
plished by assuming, as flowing material, an equivalent, monophasic medium, whose governing rheologic parameters are to be assigned carefully, often by trial-anderror, in order to assure adequate stability to the solution of the problem in terms of covered displacements versus time [6]. In others, use is made of pseudo-biphasic approaches, where solid particles and pore filling fluid are both considered in the momentum equations. However, since no convincing relationship is introduced to describe the interaction between the fluid and solid particles so as to link pore fluid response to the deformation of the solid skeleton, the equations governing the evolution of stresses in the fluid are decoupled from those governing the stress evolution in the solids [15]. The first step towards a rational analysis of rapid flows of multi-phase materials could be represented by
210
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
Nomenclature List of symbols a efflux area A cross-sectional area b number of rods in a column Cf shape coefficient D rod diameter f frequency of contact-and-detachment cycle FR hydrodynamic resistance acting on a single column G0 buoyant weight of a column g gravity acceleration h mobile array thickness k efflux coefficient L rod length m mass of vertical rod column n rod block porosity N0 effective normal force at contact p pore pressure excess P resultant pore pressure excess acting on a single column q fluid discharge
an extension to a saturated medium in rapid motion as in the approach proposed by Savage and Hutter [13] to model the run-out of a mass of frictional particles down a slope. Presently, however, this extension is hindered by the results of the back-analysis of the generally long (even kms) and fast (up to tenths ms1 ) run-out phenomena, which reveal that unreasonably low frictional resistance values (mobilized) are involved in the motion, even though the friction angle still retains relatively high values which could be related to the steady state of the granular materials. Our understanding of the processes leading to the reduction of strength to such low values is still incomplete, although most researchers share the opinion that this high mobility should be sustained by the pressure excess of fluid inside the pores, generated during the flow itself [10]. Several hypotheses have been proposed regarding the mechanism which takes place during the rapid and long run-out of some landslide bodies. Some of these, based upon proper experimental investigation, assume that, after collapse, the rearrangement of soil particles following the destruction of the solid skeleton is accompanied by a contraction tendency in the soil layers. In the past, the most significant experimental research devoted to measuring pore water pressure excess in saturated, flowing, granular soils refer to laboratory investigations involving fluidization of small-scale model slopes, submitted to various inundation programs, coming either from a horizontal or vertical infiltration.
Q r R Ru t Tmax Ts ux uy u_ x u_ y €ux €uy mmax Vm a l h qs qw
horizontal drag force on single column number of column in a rod block rod radius ratio between pore pressure at complete fluidization and pore pressure at detachment time maximum tangential force at contact detachment instant mobile block displacement in direction x mobile block displacement in direction y mobile block velocity in direction x mobile block velocity in direction y mobile block acceleration in direction x mobile block acceleration in direction y maximum velocity reached by the rod array volume of the void co-ordination angle between the sliding rods friction coefficient between the rods inclination angle of the fixed rods rod material density pore fluid density
Eckersley [2] used a large, glass-sided tank in which a 1-m high coal stockpile was constructed and then brought to failure by the horizontal seepage of water towards the base. The coal, either dry or with low water content, was placed with no compaction to get a very loose stockpile. Pore pressure transducers and total stress cells (both shear and normal) were installed at the base of the flume, within the lower 200 mm of the stockpile, in order to record stresses, while the overall deformation-to-failure of the slope was monitored by video cameras. Spence and Guymer [14] investigated an induced flow slide in extremely loose sand poured into a perspex channel 4 m long, 0.1 m wide and 0.2 m high. The material was initially ‘fluidized’ by vertical seepage (from the bottom-up) induced by opening a water inlet valve at the base of the channel just behind a watertight barrier, whose rapid removal provoked the collapse of the earth structure. Measurement of the surface elevation of both the static and flowing materials was performed by video cameras and time-lapse photography, while pore water pressures were measured by transducers set into the channel base. Wang and Sassa [16] monitored the collapse of a loose sandy silt embankment built within a flume with transparent sides of perspex plate, 1.90 m long, 0.24 m wide, and 0.15 m high, acted upon by uniform, artificial rainfall and diffused by conveniently located spray nozzles. Measurement systems involved laser displacement sensors to record the displacement components of
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
the soil mass, video cameras to monitor the entire test process and pressure transducers to measure pore fluid and total stress at the base of the flume. The analysis of results obtained by Iverson and LaHusen [4] deserves special attention. They used a hydromechanical physical model of a porous medium, formed by a close-packed array of cylindrical fiberglass rods (length L ¼ 290 mm; diameter D ¼ 19 mm, density qs ¼ 2300 kg m3 ), each spaced 0.12 mm one-from-the other, submerged and water saturated (Fig. 1). Neighboring rods were connected at their ends with glue, except along one pre-determined slip surface. Thus, the model was composed of two rigid blocks of rods, one gliding over the other, with the upper block having the possibility to fluctuate freely along the vertical direction while forced to slide horizontally over the lower block. A steady horizontal speed variable in the range 0.1– 0.4 m s1 was imposed. As the sliding proceeded, the periodic irregularity of the slip surface determined a cyclic vertical displacement of the upper block of rods, which, in turn, induced pore-volume changes and waterpressure fluctuations. Miniature transducers, located near the centre of three selected pores (Fig. 1), allowed pore water pressure to be measured. High-speed photographs recorded the contemporary evolution of graincontacts. As a preliminary comment regarding all of the abovereferenced research-work, it is worth noting that the significance to be given to stresses measured in a sort of ‘‘Eulerian’’ framework, where the pressure device is fixed in space while solid and fluid materials are rapidly
211
gliding over it, still cannot be defined. It appears that the phenomenon should best be described, even from an experimental point of view, by adopting a ‘‘Lagrangian’’ framework. Apart from this concern, however, there are other valid issues which seem to arise from these investigations and which may be synthesized as follows: (a) significant pore pressure excesses do not exist before the beginning of movement; (b) pore pressures gradually increase before ‘‘failure’’; (c) during the flowslide the pore pressures grow and become larger than the minimum stationary value compatible with the hydrostatic water table; (d) pore pressure excesses last until well after motion stops. One important hypothesis of Iverson and LaHusen [4] is that the continuous rearrangement of soil particles leads pore water pressure to fluctuate periodically, thus reducing the strength connected with effective stress and allowing the sliding mass to travel a long distance. This, in turn, enhances new destruction of the ‘‘solid skeleton’’ resulting in new phases of pore pressure excess generation and so forth. Based upon these premises, the present theoretical investigation is aimed at throwing some light on how the pore pressures get so high and stay high within such a discrete saturated material which is being sheared at very high deformation rates. In this paper, starting from the short review of the most significant results available in lab experiments on pore pressure excess generation in flowing granular material, previously reported, a theoretical model along with solutions and results (analytically or numerically obtained) for three different sliding conditions is presented. Finally, after a comparison with Iverson and LaHusen’s [4] experimental data, a critical review of theoretical results accompanied by closing comments on the applicability to fluidization and run-out processes of submerged flow slides, follows.
2. Theoretical model 2.1. Problem setting
Fig. 1. Scheme of the physical model employed by Iverson and LaHusen [4]. Points 1, 2 and 3 indicate the position of water pressure transducers. Arrows show the average direction of relative motion.
An assembly of rigid, cylindrical rods, composing two distinct blocks, is considered. The rods making up each block are glued one-to-another and cannot undergo relative displacements. The lower block is kept fixed, while the upper block is forced to slide over it, along the wavy surface that connects its upper rods. The upper, mobile block may fluctuate vertically during the motion, the evolution of its displacements depending upon the diameter of the rods, their regular arrangement, the pressure rising in the fluid within the thin layer between both blocks and the conditions
212
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
(either ‘‘contact’’ or ‘‘contact-free’’) occurring during the sliding motion over the lower block. The goal of this analysis is mainly the determination of the fluid pressure excess in the referenced thin layer, whose thickness is subjected to modification (increase or decrease) due to both the vertical and horizontal displacements of the upper block, which cause it to collide and bounce over the lower one. Using the further assumption that both solid skeleton and fluid are incompressible and owing to the flow-resistance associated with the seepage flow through the upper block, any attempt to either reduce or increase the volume of the fluid-filled layer between the two blocks causes a relative, instantaneous pressure change in the fluid. The generated pore water pressure then dissipates due to the seepage flow towards the moving medium’s free surface. Thus, the entire process (void volume variation/generation of pore pressure excess) is confined in the thin layer of continously variable thickness interposed between the rigid blocks. This is a simplifying hypothesis, but it depicts quite closely the physical situation represented by the equipment devised by Iverson and LaHusen, whose investigation was motivated by several experimental observations concerning the rapid propagation of subaerial debris flows that repeatedly have shown how the debris body often behaves as a quasi-rigid mass sliding along a very thin zone: the pore pressure excess generation and dissipation within this zone govern the mechanics of run-out. This assumption, indeed, limits the possibilities of applying the model results to other types of natural phenomena. Thus, some caution has to be used in the application of the model to cases of submarine flow slides, since, both the distribution of the relatively uniform fine grain-size and the usually low consolidation degree of the involved sediments, if not underconsolidated at all, inhibit their movement as a whole mass. In this scenario, the dynamics of the upper block can be described taking into account pore pressure excesses generated by the block movement itself, giving due regard to specific conditions imposed upon the block (geometry or velocity or forces). For example, its horizontal speed may assume a constant, imposed value, or it may vary as a consequence of an applied constant force, high enough to produce the sliding. Furthermore, the irregular, bumpy surfaces whereby contact takes place between the blocks may be inclined enough to induce the sliding of the upper block over the lower one, without any imposition of force or velocity. These situations are herein analysed. Due to the regular array of the rods assembly, only a vertical column of the mobile block (characterized by constant width D being equal to the diameter pffiffiffi of the rods, length L and whole height H ¼ ½ðb 1Þð 3=2ÞD þ DÞ is considered (Fig. 2) with b the number of rods in the column.
Fig. 2. Geometrical characteristics of the hydromechanical model employed in the analysis.
For a sufficiently high value pffiffiffi for b, the whole height may be approximated by ð 3=2ÞbD, the error being less than 1% if b P 10. This simple approximation allows for the column mass m to be expressed as follows: pffiffiffi 3 m¼ q bD2 Lð1 nÞ ð1Þ 2 s with n the porosity of the rod array. For a dense arrangement ofpthe pffiffiffi ffiffiffi set of cylindrical rods, n ¼ 1pffiffiffi p= ½2 3 þ ð4 2 3Þ=b which becomes n ¼ 1 p=2 3 in the limit b ! 1. Putting b ¼ 10, the complete formula yields n ¼ 0:1069, while n ¼ 0:0931 is obtained by the simplified one. The difference between these values being very small, the approximate value n ¼ 0:1 may be assumed [4]. The behaviour of the hydromechanical model has been analysed under the following additional hypotheses: (1) the added-mass effect, correlated to rod acceleration within a fluid [1] is neglected; (2) the length L of the rods is sufficiently greater than the diameter D so as to allow referring to plane strain conditions; (3) the fluid pressure assumes uniform values within the zone between the two sliding blocks; and (4) the collision among the rods is inelastic, that is, no elastic energy is stored within the system. Therefore, after impact, the velocity component of the moving block, orthogonal to the contact plane, is set equal to zero. 2.2. Motion of the solid mass 2.2.1. Acting forces The forces acting on a single vertical column of glued rod are shown in Fig. 3. The driving force Q and the
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
213
Fig. 3. Forces acting on the vertical rod column: (a) two blocks in contact; (b) absence of contact between blocks.
hydrodynamic resistance FR act horizontally. Force Q may coincide with an external force applied to the sliding block or may derive from a kinematical condition (e.g., a constant rate of horizontal displacement) imposed to the sliding block. The buoyant weight G0 of the unit cell and the resultant P of the pore pressure excess p at the base of the unit cell acts vertically through the center of the mass. The appropriate components of the normal effective force N 0 acts orthogonally to the average direction of the sliding surface. The tangential contact force T acts at the bottom of the unit cell. 2.2.2. Law of motion Newton’s law of motion for the upper block, written along the x, y directions, yields, respectively: Q FR þ N 0 sin a Tmax cos a ¼ m€ ux ; q N 0 cos a þ Tmax sin a þ m w 1 g þ P ¼ m€ uy ; qs
ð2Þ ð3Þ
a being the angle between N 0 and the y-axis; qs the density of rods material; g the gravity acceleration and € ux and € uy the accelerations of the rods mass along x and y. During the motion, the tangential force T attains its maximum value Tmax ¼ lN 0 ;
Fluid resistance to the motion of a single column can be written as 1 A ð6Þ FR ¼ Cf qw u_ 2x ; 2 r where Cf is a ‘‘shape’’ coefficient; A the area of the projection of the body on a plane perpendicular to the direction of motion; r the number of columns composing the array of rods; qw the fluid density and u_ x the rate of displacement u_ x . 2.2.3. Sliding The trajectory of the moving block, sliding over the fixed one, is determined by their inelastic collisions, in accordance with the irregularity of the sliding surface; thus, it must follow the external profile of the lower block. This compatibility condition is expressed in terms of the displacement components ux and uy by the following formula: pffiffiffi !2 2 ux 1 uy 3 þ þ ¼ 1: ð7Þ 2 D 2 D This illustrates that the center of the lowest sliding rod belongs to a circumference having D aspradius and ffiffiffi center in the point of coordinates ð1=2D; 3=2DÞ. The displacement ux is subjected to the further condition
ð4Þ
l being the friction coefficient at contact between the rods. The force P is obtainable by integrating the pore pressure excess p acting at the bottom of the unit cell (Fig. 3) P ¼2
Z
p=2
RLp sin / d/ ¼ 2pRL 0
¼ 2pRL ¼ DLp:
Z
p=2
sin / d/ 0
ð5Þ
Fig. 4. Configuration of the basal rods in function of the angle a between the sliding rods (a ¼ 0°, maximum porosity; a ¼ 30°, minimum porosity).
214
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
ux 6 D, since the contact is limited to the range 30° < a < 30° (Fig. 4).
permeability coefficient KD (m s1 ) of the ‘‘porous’’ block could be recovered [8] as a function of parameter k
2.2.4. Contact-free motion If the combined effect of pore pressure excess and inertial force of the rod mass causes the upper block to rise in relation to the underlying fixed one, the motion equations then become simpler, the contact forces N 0 and Tmax being equal to zero in Eqs. (2) and (3). Thus
KD ¼
ux Q FR ¼ m€ and qw m 1 g þ P ¼ m€ uy : qs
ð20 Þ
ð30 Þ
2.2.5. Continuity of the fluid flow For rigid blocks and incompressible fluid, the continuity of fluid mass in the thin layer implies the volume continuity of the layer itself. Since the thin layer volume varies proportionally to the displacement along the direction orthogonal to the average sliding surface, geometrical compatibility imposes ð8Þ
Vm being the volume of the thin layer (void), beyond all other symbols already introduced. The difference between the water pressure excess (p) in the void at the base of the sliding block and the imposed nil value at the top of the same, generates a hydraulic gradient in direction y. This, in turn, induces a seepage flow towards the free surface with overall time discharge qðtÞ. Thus qðtÞ ¼ kapðtÞ;
ð10Þ
h being the length of the seepage path, equal to the thickness of the moving block. The fluid volume variation dVm is further related to the discharge qðtÞ dVm ¼ qðtÞ dt:
ð11Þ
After some algebra, one obtains
At the end of the contact-free motion, the upper block collides with the lower one and intergranular forces rise, their orientation depending on that of the impact plane. This orientation is changing during motion; the collision points on the sliding surface depend on the trajectory of the upper block and thus on the value of pore pressure excess prevailing at the moment of rising; therefore, its computation must be renewed for each impact.
dVm duy ¼ DL ; dt dt
ha k; DL
ð9Þ
k being an efflux coefficient and a an outflow area. As a consequence of the fluid discharge, both a volume reduction of the void at the base of the sliding block and a pore water pressure excess reduction take place simultaneously. The seepage flow along direction x is disregarded, owing to predominance of the gradient along direction y, orthogonal to the average sliding surface. Assuming that a laminar seepage flow is taking place and starting from a linearized relationship expressing the discharge from an orifice applied to the regular geometry of the porous medium at hand, the conventional
pðtÞ ¼
DL u_ y : ka
ð12Þ
Eq. (12) links the pore pressure excess in the fluid at the base of the mobile block with its vertical velocity, confirming that the interaction between the solid and fluid phases within the thin layer at the base of the sliding block, where all pore volume variations are concentrated, determines the pore pressure excess evolution. The pðtÞ, which depends on geometrical and physical parameters, constitutes the lower boundary condition for seepage within the moving block, when the boundary condition at its upper edge is fixed at a null value. 2.2.6. Model ingredients The motion of the saturated block in contact with the stationary one is generally described by six equations: two differential equations of motion (x and y-direction), one contact condition, one fluid flow continuity equation, one constitutive relation connecting the normal force N 0 to the shear resistance Tmax and one equation expressing the resistance force offered by the fluid against the mobile block. The unknown variables are also six: two displacements (ux , uy ), one normal effective contact force (N 0 ), one tangential contact force (Tmax ), one pore pressure excess (p) and one flow resistance force (FR ). To solve the problem, four initial conditions on ux , uy and on first time derivatives u_ x and u_ y must be assigned. On the other hand, the simpler case of motion in the absence of contact between the two blocks is described by only four equations, since the contact condition and the friction law are no more applicable. The unknowns are: ux , uy , p and FR , and again, four initial conditions must be assigned.
3. Sliding conditions and related governing equations The previous equations of motion are herein rewritten and solved for three conditions: Model M1 – motion at constant horizontal velocity, Model M2 – motion under an applied constant horizontal force and Model M3 – motion down a rough inclined surface.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
3.1. Model M1 – constant rate of horizontal displacement (u_ x ¼ const) 3.1.1. Sliding Since ux ¼ u_ x t, through the expression (7) it is possible to express vertical displacement (13), velocity (14) and acceleration (15) as follows: 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 !2 pffiffiffi u u 37 u_ x t 1 6 uy ¼ D4t1 ð13Þ 5; D 2 2
u_Dx t u_ x u_ y ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; u_ x t 1 1 D 2 1 2
ð14Þ
2 2 _ u_ 2x 12 uDx D ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi € uy ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 2 3 u_ x t 1 u_ x t 1 1 D 2 1 u_ x t D
D
ð15Þ
It is possible to put into evidence N in Eq. (3) and Q in Eq. (2) m€ uy m qqws 1 g DLp N0 ¼ ; ð16Þ b2
where b1 ¼
u_ x t 1 D 2
!
ð17Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u u_ x t 1 t l 1 ; D 2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ! u u _ _ t 1 t 1 u u x x b2 ¼ t1 : þl D 2 D 2
uy ðTs Þ ¼ uy1 ; u_ y ðTs Þ ¼ u_ y1
ð22Þ
at the instant of detachment Ts . After double integration, uy and u_ y are obtained as follows: 0 q 1 w m 1 g qs kam uy ¼ uy1 2 2 @ u_ y1 A D2 L2 DL ka 0 q 1 w m qqws 1 g m 1 g qs kam @ þ u_ y1 A ðt Ts Þ þ 2 2 D2 L2 D2 L2 DL ka ka D2 L2 exp ðt Ts Þ ð23Þ kam 0 q 1 m qqws 1 g m qws 1 g u_ y ¼ u_ y1 A @ D2 L2 D2 L2 ka
0
Q ¼ F R b1 N 0 ;
Eq. (21) can be solved by introducing initial conditions:
ka
2
ð18Þ
ð19Þ
Since the horizontal velocity is constant, even the hydrodynamic resistance (6) is constant. Finally, the pore pressure excess is obtained after substitution of (14) into Eq. (12) ! 1 u_ x t u_ x 2 D DL vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ p¼ !2 : ka u u _ t 1 u x t1 D 2 3.1.2. Contact-free motion In the absence of contact, Eq. (30 ), without the contribution of ‘‘intergranular’’ forces N 0 and Tmax , can be rewritten only in terms of uy and its time derivatives q D2 L2 uy : ð21Þ m w1 g u_ y ¼ m€ qs ka
215
exp
DL ðt Ts Þ : kam 2 2
Introducing u_ y in Eq. (12) yields 2 q 0 q 1 w w DL 4 m qs 1 g @ m qs 1 g u_ y1 A p¼ D2 L2 D2 L2 ka ka ka 3 2 2 DL exp ðt Ts Þ 5: kam
ð24Þ
ð25Þ
The instant of detachment, Ts , and the values describing the initial condition uy1 and u_ y1 , can be obtained by the previous equation (16) by imposing N 0 ¼ 0. For a submerged block, Ts can be interpreted as the instant in which the hydroplaning phase of motion begins [9]. It is characterized by a substantial increase of pore pressure excess, causing a dramatic reduction in frictional resistance. It is observed that pore pressure excess p is obtained as the algebraic addition of two terms. The first is a steady term, whilst the second depends on time. Starting from the instant of detachment Ts , the pore pressure excess grows, approaching, in the limit, a maximum value represented by the first term. This asymptotic value leads to a global force that balances the buoyant weight. It could be assumed to characterize the complete fluidization. 3.2. Model M2 – constant applied horizontal force (Q ¼ const) 3.2.1. Sliding Newton’s equations for the rod motion are: Q FR þ b1 N 0 ¼ m€ux ;
ð26Þ
216
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
qw b2 N þ m 1 g þ DLp ¼ m€ uy ; qs 0
b1 and b2 being represented by: pffiffiffi ! ux 1 uy 3 l b1 ¼ þ 2 D 2 D and b2 ¼
pffiffiffi ! uy ux 1 3 ; þl þ 2 D D 2
ð27Þ
2 ð28Þ
ð29Þ
respectively. Solving for N 0 in (26) and substituting the result in (27) gives qw D2 L2 b ux Q þ FR Þ ¼ m€uy : 1 mg u_ y þ 2 ðm€ qs ka b1 ð30Þ uy are written as From contact condition (7), u_ y and € follows: 3 2
ux 1 u_ x 2 D 7 6 D ð31Þ u_ y ¼ D4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi 5; 1 uDx 12 8 2 >
u_ x > € < ux ux 12 D D D €uy ¼ D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi > > : 1 uDx 12 1 uDx 12 h
ux
1 2
u_ x i2
9 > > =
D D : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h
u 1 2 i 3 > > x ; 1 D 2
After substituting the previous expressions into (30), an ordinary differential equation with constant coefficients in terms of the horizontal displacement ux is obtained
ux
1 2
3
7 6b D m€ux 4 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ux 1 2ffi 5 b1 1 D 2 2 3
ux 1 3 2 2 q b DL 6 7 u_ x D ¼ 1 w mg þ 2 Q 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 1 2ffi 5 D qs b1 ka 1 Dx 2 !2 !2 b2 u_ x u_ x 2 mD CR D b1 D D 8 9 > >
> > 2 < = ux 1 1 2 D ffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 h >
2 i3 > > > : 1 uDx 12 ; 1 ux 1 D
2
The constant terms figuring in (6) have been collected in CR . The initial conditions are: ux ð0Þ ¼ ux0 ; u_ x ð0Þ ¼ u_ x0 ;
ð33Þ
ux0 and u_ x0 being, respectively, the position and the horizontal velocity at the beginning of the sliding.
ð32Þ
3.2.2. Contact-free motion The motion of the rods in the absence of contact is governed by Eqs. (20 ) and (30 ), under the following initial conditions:
Fig. 5. Schematic representation of the upper system of rods sliding on a system of rigid rods disposed according to an average slope angle h.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
ux ðTs Þ ¼ ux1 ; u_ x ðTs Þ ¼ u_ x1 ; uy ðTs Þ ¼ uy1 ; u_ y ðTs Þ ¼ u_ y1 :
h being the slope angle of the fixed rod block. ð34Þ
The detachment instant, Ts , along with the values ux1 , u_ x1 , uy1 and u_ y1 can be recovered by imposing N 0 ¼ 0 in the motion equations. Once Eq. (30) is solved, Eq. (21) expresses the vertical motion of the mobile block, which is used to obtain the pore pressure excess Eq. (12). 3.3. Model M3 – constant slope angle (h ¼ const) 3.3.1. Sliding Newton’s second law equations are (Fig. 5): qw mg 1 ux ; sin h FR þ b1 N 0 ¼ m€ qs qw mg 1 uy ; cos h þ DLp þ b2 N 0 ¼ m€ qs
217
3.3.2. Contact-free motion The equations of motion become: q mg 1 w sin h FR ¼ m€ux ; qs qw mg 1 cos h þ DLp ¼ m€uy : qs
ð37Þ ð38Þ
The equations obtained for movement down a slope coincide with equations obtained for horizontal motion under constant force, after the following substitutions: q mg 1 w sin h for Q qs
ð35Þ
and qw mg 1 cos h qs
ð36Þ
Thus, the solution procedure is similar to the one implemented for the Model M2.
qw for mg 1 : qs
Fig. 6. Theoretical results (Model M1) for two values of horizontal velocity u_ x (u_ x1 ¼ 0:1 m s1 and u_ x2 ¼ 0:5 m s1 ). The meaning of symbols a, b, c, d, e and f is described in Table 1.
218
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
The non-dimensional, vertical displacement uy =D, the effective normal force N 0 and the pore pressure excess are represented in Fig. 6 vs. the non-dimensional, horizontal displacement ux =D. The instants and phases corresponding to the most significant events in the history of this motion are marked with the letters a, b, c, d, e and f. Their meanings in all models are briefly summarized in Table 1. The history of events can be described in more detail as follows: (a) System at rest: N 0 equals the static equilibrium value; the pore pressure excess is zero. (b) Instantaneous increase of force Q: the block tries to move upwards, in relation to the irregularity of the sliding surface; the pore volume between the blocks tends to increase more rapidly than fluid can fill it, so a negative pressure rises that interlocks the upper
4. Results The set of sliding Eqs. (2)–(4), (6), (7) and (12) along with the contact free motion Eqs. (20 ), (30 ), (4) and (12) have been solved taking into account the different conditions corresponding to the three models M1, M2, M3. The solutions have been sought following numerical procedures, using the fourth-order Runge–Kutta’s method to solve the system of governing first order differential equations [12]. 4.1. Model M1 – constant rate of horizontal displacement (u_ x ¼ const) Two simulations were carried out by imposing u_ x ¼ 0:1 m s1 and u_ x ¼ 0:5 m s1 with the values of the parameters used in computation reported in Table 3. Table 1 Phases of motion for models M1, M2 and M3 Instant or phases
Model M1
Model M2 0
0
Model M3 0
a b
System at rest N ¼ G =2 cos 30° Exertion of net force Q
b!c c c!d d
Upper rods slide over lower rods u_ x ¼ const Detachment of upper rods from lower rods Hydroplaning of upper rods p < pfluid Complete fluidization pfluid ¼ G0 ðDLÞ1
d!e e f
Fluidized motion of upper rods p ¼ pfluid Contact of upper rods with lower rods Beginning of a new cycle
0
System at rest N ¼ G System at rest N 0 ¼ G0 Exertion of force Q ¼ h ¼ const Plane inclination h ¼ const N 0 ¼ G0 N 0 ¼ G0 cos h p¼0 p¼0 Upper rods slide over lower rods Detachment of upper rods from lower rods Hydroplaning of upper rods p < pfluid Complete fluidization Complete fluidization pfluid ¼ G0 ðDLÞ1 pfluid ¼ G0 cos hðDLÞ1 Fluidized motion of upper block p ¼ pfluid Contact of upper rods with lower rods Beginning of a new phase of motion without cyclicity
Table 2 Values of geometrical, physical and mechanical parameters considered to simulate Iverson and LaHusen’s experiment [4] Model parameters
Values
Number of rods, b Diameter, D Length, L Coefficient of friction, l Efflux coefficient, k Horizontal rate, u_ x Efflux area, a Water density, qw Density of rods material, qs Mass of the cell, m
7 0.019 0.29 0.45 2.63–9.77 104 0.118 3.48 105 1000 2300 1.304
m m m2 s kg1 m s1 m2 kg m3 kg m3 kg
Table 3 Values of the main computation parameters Figure reference
D (m)
6, 8, 9 10-I, 11-I, 12-I, 13-I 10-II, 11-II, 12-II, 13-II 10-III, 11-III, 12-III, 13-III
0.02 0.02 0.02 0.02
0.05
h (m)
k (m2 s kg1 )
ka (m4 s kg1 )
0.12 0.12 0.12 0.12
1.00E ) 03 1.00E ) 03 1.00E ) 03 1.00E ) 04
4.00E ) 08 4.00E ) 08 4.00E ) 08 4.00E ) 09
1.00
1.00E ) 03
l 2.50E ) 07 4.00E ) 08
0.45 0.45 0.45 0.45
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
block to the base even more closely, and thus the normal force N 0 is increased. This occurs instantaneously, so N 0 jumps. (b ! c) The upper block slides over the lower one with constant horizontal velocity; the effective forces N 0 and Q progressively decrease to zero while the pore pressure monotonically increases. (c) Detachment of the upper block from the lower one. (c ! d) Hydroplaning of the upper block moving downwards: the pore pressure continues to increase; N 0 and Q assume zero values. (d) Complete fluidization: the submerged weight G0 of the column of rods is entirely balanced by the vertical force P resulting from the pore pressure excess that has attained its maximum value; vertical acceleration vanishes. (d ! e) Pore pressure excess p maintains its maximum value (plateau); the upper block moves downwards at constant vertical rate.
219
(e) The upper block comes again in contact with the lower one. (f) A new, identical cycle of motion and fluid–solid interaction begins. To check the validity of the proposed hydromechanical model, Iverson and LaHusen’s [4] experiment has been simulated, with the imposed constant value u_ x ¼ 0:118 m s1 . Measured pore pressure fluctuations generated by the sliding movement exhibit a repetitive time-series pattern, characterized by high-pressure plateaux (1.4 kPa at maximum) and separated by deep, negative pressure troughs (negative pressure, 12 kPa at minimum) (Fig. 7). Photographs indicate that the plateaux occur when the upper sliding block loses contact with the underlying fixed block and glides on a cushion of water. In the absence of ‘‘intergranular’’ forces, the buoyant weight of the moving rod array is entirely supported by the pore pressure excess acting along the sliding surface.
Fig. 7. (a) Comparison between theoretical values (grey lines) and measured values (black lines) at three transducer positions, after Iverson and LaHusen [4]. Broken lines refer to hydrostatic pressure. Case a: ka ¼ 3:4 108 m4 s kg1 ; case b: ka ¼ 9:15 109 m4 s kg1 .
220
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
A sudden pore pressure decrease occurs when the upper block of rods comes in contact with the underlying block and then starts again to slide, inducing a sudden increase of pore volume. Experimental results show appreciable fluctuations of the pore pressure excess when the upper block collides with the lower one. Pressure fluctuations propagate from their source (sliding zone) towards the free surface at the top of the model, undergoing both an amplitude attenuation and a phase shift. The values of the parameters used in the model, as reproduced from Iverson and LaHusen’s paper [4], are
listed in Table 2. It should be noted that, since parameters k and a do not figure separately in the model, the product ka has been assigned. In order to compute the values of pore pressure excess in points not located along the slip surface, due to the hypothesis of incompressible fluid and to the rigid assembly of the rod blocks, a linear distribution of pore pressure excess along y resulted. It has been found that for both values of ka, delimiting the range of experimental values (9.15 109 m4 s kg1 < ka < 3:4 108 m4 s kg1 ), an acceptable modelling of the experimental regime of pore pressure
Fig. 8. Theoretical results (model M2) for two values of applied horizontal constant force Q (Q1 ¼ 10 N, Q2 ¼ 15 N). The meaning of symbols a, b, c, d, e and f is described in Table 1.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
excess is obtained. In particular, by assuming the lower value (ka ¼ 9:15 109 m4 s kg1 ), the minimum pore pressure excess value (pmin ¼ 12:5 kPa) is fitted (Fig. 7(b)), while the maximum value (ka ¼ 3:4 108 m4 s kg1 ) leads to the maximum pore pressure excess (pmin ¼ 6 kPa) (Fig. 7(a)). In addition, theoretical and measured peak values of pore pressure excess and periods of constant values of fluid pressure significantly agree, although a small temporal shift of the measured plateaus at different depth is observed. These differences may be attributed to an imperfect-inelasticity condition occurring during the collisions.
221
4.2. Model M2 – constant applied horizontal force (Q ¼ const) The mobile block starts sliding over the fixed one if Q > Qmin ; Qmin is depending on the mass of the system, the initial arrangement and the friction coefficient; it is obtained from the limit equilibrium of the mobile block: Qmin ¼
sin a þ l cos a G0 cos a þ l sin a
ð39Þ
with
Fig. 9. Theoretical results (model M3) for two values of plane inclination hðh1 ¼ 40°, h2 ¼ 50°). The meaning of symbols a, b, c, d, e and f is described in Table 1.
222
G0 ¼
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
q 1 w mg qs
ð40Þ
representing the buoyant weight of the rod column. For a ¼ 0° (maximum initial porosity) and l ¼ 0:45, (39) results in Qmin ¼ 0:45G0 ; but if a ¼ 30° (see Fig. 4), that is, the geometrical condition corresponding to minimum porosity, Qmin ¼ 1:39G0 results. Fig. 8 depicts the motion history of model M2. For other parameter values see Table 3. It is observed that, in comparison with model M1 results, the trajectory of the rods becomes straighter, the contact–no contact cycles are more frequent, the no-contact phase lasts longer and the contact phase reduces with distance travelled.
At first, the horizontal velocity rises fast then, it tends towards a constant limiting value when the hydrodynamic force becomes more significant. The peak value of the normal contact force reduces and its frequency increases with elapsed time. The pore pressure excess behaves in a similar way as in case M1: it grows from a negative minimum value to its maximum value during the no-contact phase and tends to stabilize. It is worth noting the monotonic increase of pore pressure excess both before and after the detachment took place. It resembles the continuous increase of the same variable both before and during the flow slide, as recorded in the monitored, small-scale experiments quoted in points (b) and (c) (see Section 1). The model,
Fig. 10. Theoretical values of mmax (model M3) against slope angle h. For parameter values see Table 3.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
however, does not fit the conclusions reported in point (d), probably as it cannot properly account for the ‘‘consolidation’’ of the porous medium [3]. 4.3. Model M3 – constant slope angle (h ¼ const) The behaviour of the system, in this case, is analogous to case M2. The minimum slope angle for the motion to take place is h > hmin , where: tan hmin ¼
sin a þ l cos a : cos a þ l sin a
ð41Þ
The phases of motion are represented in Fig. 9, being parameter values reported in Table 3.
223
In Fig. 9, a straighter trajectory is observed, the velocity reaches higher values, N 0 peaks progressively decrease and the fluidization value of pore pressure excess slightly decreases as h increases. These results are quite significant if they are represented against to the following response variables: mmax – maximum velocity achieved by the rod array; Ru – ratio between pore pressure at complete fluidization and pore pressure at detachment; Ts – instant of detachment of the mobile array of rods; 1=f – cycle period of sliding and contact-free motion. Figs. 10–13, show the variables mmax , Ru , Ts and 1=f as functions of an ‘‘average’’ slope angle h using two rod
Fig. 11. Theoretical values of Ru (model M3) against slope angle h. For parameter values see Table 3.
224
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
diameters, two column heights and two efflux coefficient values. The parameters considered for the computations are reported in Table 3. The variable mmax increases with the slope angle h. It assumes higher values for D and h increasing; if k increases for a small slope angle, mmax decreases; conversely, for a high slope angle, mmax increases. Moreover, mmax is upper-limited by the velocity of the submerged upper block moving without any frictional resistance at the base and it is lower-limited by the velocity of the same block sliding with friction along the slope [7]. Its limiting velocities depend on the ratio between its mass and the cross-sectional area; this ratio in turn depends on D. For low values of h, the maximum velocity of the rod block approaches the ‘‘frictional’’ velocity; as h in-
creases, mmax tends towards ‘‘contact-free’’ velocity. This result offers a more interesting interpretation when read along with the picture of Ru . The ratio Ru increases as h increases. Values of Ru near unity indicate an easier fluidization of the rod system: this occurs with higher diameter, lower layer extent and lower efflux coefficient. This could mean that, for greater angles h, the system is more prone to sliding than to fluidization, at least in sub-aqueous landslides. As expected, the instant of detachment Ts decreases with increasing slope angle. It does not show any significant dependence on h and k, while a dependence on D is more marked: Ts increases if D increases. The period of the contact–detachment cycle, which is not constant, decreases as h increases. If D decreases, the
Fig. 12. Theoretical values of Ts (model M3) against slope angle h. For parameter values see Table 3.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
225
Fig. 13. Theoretical values of 1=f (model M3) against slope angle h. For parameter values see Table 3.
period decreases; if h increases, 1=f decreases. 1=f also decreases if, with low angle values, k decreases and if, with high angle values, k increases.
5. Closing remarks A simple model is presented which is believed to reveal some important aspects related to pore pressure evolution within saturated granular materials, while being sheared at a high rate. The distinctive feature modelled here is the dynamic interaction between grains and pore filling fluid beyond that of the grains themselves. The pore pressure excesses derive from a rapid modification of pores structure,
connected with the continuous rearrangement of grains involved in the flow motion. ‘‘Fluidization’’, i.e. the process occurring during movement due to a continuous rearrangement of structure, does not coincide with ‘‘liquefaction’’, which is a process marking a ‘‘phase-change’’. Significant increments of pore pressure occurring during flow could thus be justified and the roles of various geometrical parameters (i.e., grain diameter, height of flowing layer) as well as mechanical ones (permeability, friction coefficient) could be properly accounted-for. As a by-product, the build-up of pore pressure excess in coarse debris flows could also be interpreted as due to the weight of small particles carried along in suspension in the pore fluid saturating the body, as repeatedly invoked, although in purely speculative terms [11].
226
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
The extension to a ‘‘continuum’’ formulation [5], which is yet to be done, apparently presents major difficulties, due to the need of taking into account the peculiar characteristics of the phenomenon; particularly, the high rate of displacements and the associated, substantial, pore-structure modification. However, the presented model which succeeds in generating new amounts of pore pressure, together with diffusion equations which allow for the dissipation of pore pressure excess, and momentum equations for the fluid–solid mass, could bring about the improvement of models aimed at predicting sub-marine flow slide runout characteristics. References [1] Batchelor GK. An introduction to fluid mechanics. Cambridge: Cambridge University Press; 1967. [2] Eckersley JD. Instrumented laboratory flowslides. Geotechnique 1990;40(3):489–502. [3] Hutchinson JN. A sliding-consolidation model for flow slides. Can Geotech J 1986;23(2):115–26. [4] Iverson RM, LaHusen RG. Dynamic pore pressure fluctuations in rapidly shearing granular materials. Science 1989;246:796–9.
[5] Iverson RM. Differential equations governing slip-induced pore pressure fluctuations in water saturated granular medium. Math Geol 1993;25(8):1027–48. [6] Johnson AM. Physical processes in geology. San Francisco: W.H. Freeman; 1970. [7] Lambe TW, Whitman RV. Soil mechanics. New York: Wiley; 1978. [8] Leverett MC. Capillary behavior in porous solids. Trans, AIME 1941;142:152–69. [9] Mohrig D, Whipple KX, Hondzo M, Ellis C, Parker G. Hydroplaning of subaqueous debris flows. Geol Soc Am Bull 1998;110(3):387–94. [10] Musso A. Analysis of rapid debris flows, in Interventi di Stabilizzazione di Pendii, CISM, Udine, 1997 (In Italian). [11] Pierson TC. Dominant particle support mechanisms in debris flows at Mt Thomas, New Zealand, and implications for flow mobility. Sedimentology 1981;28:49–60. [12] Rosemberg D. Methods for numerical solution of partial differential equations. Elsevier; 1969. [13] Savage SB, Hutter K. The motion of a finite mass of granular material down a rough incline. J Fluid Mech 1989;199: 177–215. [14] Spence KJ, Guymer I. Small scale laboratory flowslides. Geotechnique 1997;47(5):915–32. [15] Takahashi T. Debris flow. Rotterdam: Balkema; 1991. [16] Wang G, Sassa K. Factors affecting rainfall-induced flowslides in laboratory flume tests. Geotechnique 2001;51(7): 587–99.
A mechanism of pore pressure accumulation in rapidly sliding submerged porous blocks A. Musso *, F. Federico, G. Troiano Department of Civil Engineering, University of Rome ‘‘Tor Vergata’’, Via del Politecnico, 1, Rome 00133, Italy
Abstract The generation and dissipation of pore pressure excess due to the rapid, mutual sliding of two water-saturated blocks which are composed of glued arrays of cylindrical rods is herein theoretically examined. The mathematical model consists of three ordinary differential equations describing the motion of a solid mass coupled with the continuity of a fluid mass. These equations are numerically solved for the following conditions: (a) motion induced by the constant horizontal velocity of the sliding mass; (b) constant drag force acting in a horizontal direction; (c) gliding on the irregular, inclined surface that envelops the rods. Theoretical predictions for case (a) favorably compare with experimental results reported in the literature from an investigation carried out on a physical model. Primary results show that the pore pressure excess depends upon physical and geometrical parameters, and furthermore, that displacement of the rods and pore pressure excess are strongly coupled. Maximum pore water pressure occurs when ‘‘fluidization’’, i.e. the complete balancing of submerged block weight by pore pressure excess, is achieved. The latter does not coincide with the rising of the upper block (hydroplaning). A cyclic motion is reached under condition (a), when dynamic pore water pressure increases from a negative to a maximum positive value and then falls again to a negative value. On the other hand, non-cyclical motion and monotonically increasing pore water pressure are obtained under sliding conditions (b) and (c). The response of the model under condition (c), which is of particular interest for run-out analyses of flow slides, is compared to the sliding response of ordinary submerged blocks on a slope, showing significant differences and pointing out the roles of particle diameter, layer extent and medium permeability. The occurrence of different pore pressure states for fluidization and hydroplaning is highlighted. This, hopefully, can open the way to a reliable treatment of the mechanics of sub-marine flow slide propagation. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Pore pressure excess generation; Solid–fluid interaction; Sliding of a block; Hydroplaning; Fluidization
1. Introduction The high mobility of ‘‘wet’’ flows, such as debris flows, sub-aerial and sub-marine landslides in loose sands and silts, is still a matter of considerable concern. In most cases, particularly when materials with a metastable structure are implied, static liquefaction is clearly present as a trigger mechanism (flow slide). Soil Mechanics’ aspects of the problem are usually limited to trigger analysis of the phenomenon, thus leaving to Fluid Mechanics the task of exploring the characteristics of the run-out. In some circumstances, this is accom*
Corresponding author. Tel.: +39-6-72-59-70-62; fax: +39-6-72-5970-05. E-mail address: [email protected] (A. Musso). 0266-352X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2004.02.001
plished by assuming, as flowing material, an equivalent, monophasic medium, whose governing rheologic parameters are to be assigned carefully, often by trial-anderror, in order to assure adequate stability to the solution of the problem in terms of covered displacements versus time [6]. In others, use is made of pseudo-biphasic approaches, where solid particles and pore filling fluid are both considered in the momentum equations. However, since no convincing relationship is introduced to describe the interaction between the fluid and solid particles so as to link pore fluid response to the deformation of the solid skeleton, the equations governing the evolution of stresses in the fluid are decoupled from those governing the stress evolution in the solids [15]. The first step towards a rational analysis of rapid flows of multi-phase materials could be represented by
210
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
Nomenclature List of symbols a efflux area A cross-sectional area b number of rods in a column Cf shape coefficient D rod diameter f frequency of contact-and-detachment cycle FR hydrodynamic resistance acting on a single column G0 buoyant weight of a column g gravity acceleration h mobile array thickness k efflux coefficient L rod length m mass of vertical rod column n rod block porosity N0 effective normal force at contact p pore pressure excess P resultant pore pressure excess acting on a single column q fluid discharge
an extension to a saturated medium in rapid motion as in the approach proposed by Savage and Hutter [13] to model the run-out of a mass of frictional particles down a slope. Presently, however, this extension is hindered by the results of the back-analysis of the generally long (even kms) and fast (up to tenths ms1 ) run-out phenomena, which reveal that unreasonably low frictional resistance values (mobilized) are involved in the motion, even though the friction angle still retains relatively high values which could be related to the steady state of the granular materials. Our understanding of the processes leading to the reduction of strength to such low values is still incomplete, although most researchers share the opinion that this high mobility should be sustained by the pressure excess of fluid inside the pores, generated during the flow itself [10]. Several hypotheses have been proposed regarding the mechanism which takes place during the rapid and long run-out of some landslide bodies. Some of these, based upon proper experimental investigation, assume that, after collapse, the rearrangement of soil particles following the destruction of the solid skeleton is accompanied by a contraction tendency in the soil layers. In the past, the most significant experimental research devoted to measuring pore water pressure excess in saturated, flowing, granular soils refer to laboratory investigations involving fluidization of small-scale model slopes, submitted to various inundation programs, coming either from a horizontal or vertical infiltration.
Q r R Ru t Tmax Ts ux uy u_ x u_ y €ux €uy mmax Vm a l h qs qw
horizontal drag force on single column number of column in a rod block rod radius ratio between pore pressure at complete fluidization and pore pressure at detachment time maximum tangential force at contact detachment instant mobile block displacement in direction x mobile block displacement in direction y mobile block velocity in direction x mobile block velocity in direction y mobile block acceleration in direction x mobile block acceleration in direction y maximum velocity reached by the rod array volume of the void co-ordination angle between the sliding rods friction coefficient between the rods inclination angle of the fixed rods rod material density pore fluid density
Eckersley [2] used a large, glass-sided tank in which a 1-m high coal stockpile was constructed and then brought to failure by the horizontal seepage of water towards the base. The coal, either dry or with low water content, was placed with no compaction to get a very loose stockpile. Pore pressure transducers and total stress cells (both shear and normal) were installed at the base of the flume, within the lower 200 mm of the stockpile, in order to record stresses, while the overall deformation-to-failure of the slope was monitored by video cameras. Spence and Guymer [14] investigated an induced flow slide in extremely loose sand poured into a perspex channel 4 m long, 0.1 m wide and 0.2 m high. The material was initially ‘fluidized’ by vertical seepage (from the bottom-up) induced by opening a water inlet valve at the base of the channel just behind a watertight barrier, whose rapid removal provoked the collapse of the earth structure. Measurement of the surface elevation of both the static and flowing materials was performed by video cameras and time-lapse photography, while pore water pressures were measured by transducers set into the channel base. Wang and Sassa [16] monitored the collapse of a loose sandy silt embankment built within a flume with transparent sides of perspex plate, 1.90 m long, 0.24 m wide, and 0.15 m high, acted upon by uniform, artificial rainfall and diffused by conveniently located spray nozzles. Measurement systems involved laser displacement sensors to record the displacement components of
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
the soil mass, video cameras to monitor the entire test process and pressure transducers to measure pore fluid and total stress at the base of the flume. The analysis of results obtained by Iverson and LaHusen [4] deserves special attention. They used a hydromechanical physical model of a porous medium, formed by a close-packed array of cylindrical fiberglass rods (length L ¼ 290 mm; diameter D ¼ 19 mm, density qs ¼ 2300 kg m3 ), each spaced 0.12 mm one-from-the other, submerged and water saturated (Fig. 1). Neighboring rods were connected at their ends with glue, except along one pre-determined slip surface. Thus, the model was composed of two rigid blocks of rods, one gliding over the other, with the upper block having the possibility to fluctuate freely along the vertical direction while forced to slide horizontally over the lower block. A steady horizontal speed variable in the range 0.1– 0.4 m s1 was imposed. As the sliding proceeded, the periodic irregularity of the slip surface determined a cyclic vertical displacement of the upper block of rods, which, in turn, induced pore-volume changes and waterpressure fluctuations. Miniature transducers, located near the centre of three selected pores (Fig. 1), allowed pore water pressure to be measured. High-speed photographs recorded the contemporary evolution of graincontacts. As a preliminary comment regarding all of the abovereferenced research-work, it is worth noting that the significance to be given to stresses measured in a sort of ‘‘Eulerian’’ framework, where the pressure device is fixed in space while solid and fluid materials are rapidly
211
gliding over it, still cannot be defined. It appears that the phenomenon should best be described, even from an experimental point of view, by adopting a ‘‘Lagrangian’’ framework. Apart from this concern, however, there are other valid issues which seem to arise from these investigations and which may be synthesized as follows: (a) significant pore pressure excesses do not exist before the beginning of movement; (b) pore pressures gradually increase before ‘‘failure’’; (c) during the flowslide the pore pressures grow and become larger than the minimum stationary value compatible with the hydrostatic water table; (d) pore pressure excesses last until well after motion stops. One important hypothesis of Iverson and LaHusen [4] is that the continuous rearrangement of soil particles leads pore water pressure to fluctuate periodically, thus reducing the strength connected with effective stress and allowing the sliding mass to travel a long distance. This, in turn, enhances new destruction of the ‘‘solid skeleton’’ resulting in new phases of pore pressure excess generation and so forth. Based upon these premises, the present theoretical investigation is aimed at throwing some light on how the pore pressures get so high and stay high within such a discrete saturated material which is being sheared at very high deformation rates. In this paper, starting from the short review of the most significant results available in lab experiments on pore pressure excess generation in flowing granular material, previously reported, a theoretical model along with solutions and results (analytically or numerically obtained) for three different sliding conditions is presented. Finally, after a comparison with Iverson and LaHusen’s [4] experimental data, a critical review of theoretical results accompanied by closing comments on the applicability to fluidization and run-out processes of submerged flow slides, follows.
2. Theoretical model 2.1. Problem setting
Fig. 1. Scheme of the physical model employed by Iverson and LaHusen [4]. Points 1, 2 and 3 indicate the position of water pressure transducers. Arrows show the average direction of relative motion.
An assembly of rigid, cylindrical rods, composing two distinct blocks, is considered. The rods making up each block are glued one-to-another and cannot undergo relative displacements. The lower block is kept fixed, while the upper block is forced to slide over it, along the wavy surface that connects its upper rods. The upper, mobile block may fluctuate vertically during the motion, the evolution of its displacements depending upon the diameter of the rods, their regular arrangement, the pressure rising in the fluid within the thin layer between both blocks and the conditions
212
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
(either ‘‘contact’’ or ‘‘contact-free’’) occurring during the sliding motion over the lower block. The goal of this analysis is mainly the determination of the fluid pressure excess in the referenced thin layer, whose thickness is subjected to modification (increase or decrease) due to both the vertical and horizontal displacements of the upper block, which cause it to collide and bounce over the lower one. Using the further assumption that both solid skeleton and fluid are incompressible and owing to the flow-resistance associated with the seepage flow through the upper block, any attempt to either reduce or increase the volume of the fluid-filled layer between the two blocks causes a relative, instantaneous pressure change in the fluid. The generated pore water pressure then dissipates due to the seepage flow towards the moving medium’s free surface. Thus, the entire process (void volume variation/generation of pore pressure excess) is confined in the thin layer of continously variable thickness interposed between the rigid blocks. This is a simplifying hypothesis, but it depicts quite closely the physical situation represented by the equipment devised by Iverson and LaHusen, whose investigation was motivated by several experimental observations concerning the rapid propagation of subaerial debris flows that repeatedly have shown how the debris body often behaves as a quasi-rigid mass sliding along a very thin zone: the pore pressure excess generation and dissipation within this zone govern the mechanics of run-out. This assumption, indeed, limits the possibilities of applying the model results to other types of natural phenomena. Thus, some caution has to be used in the application of the model to cases of submarine flow slides, since, both the distribution of the relatively uniform fine grain-size and the usually low consolidation degree of the involved sediments, if not underconsolidated at all, inhibit their movement as a whole mass. In this scenario, the dynamics of the upper block can be described taking into account pore pressure excesses generated by the block movement itself, giving due regard to specific conditions imposed upon the block (geometry or velocity or forces). For example, its horizontal speed may assume a constant, imposed value, or it may vary as a consequence of an applied constant force, high enough to produce the sliding. Furthermore, the irregular, bumpy surfaces whereby contact takes place between the blocks may be inclined enough to induce the sliding of the upper block over the lower one, without any imposition of force or velocity. These situations are herein analysed. Due to the regular array of the rods assembly, only a vertical column of the mobile block (characterized by constant width D being equal to the diameter pffiffiffi of the rods, length L and whole height H ¼ ½ðb 1Þð 3=2ÞD þ DÞ is considered (Fig. 2) with b the number of rods in the column.
Fig. 2. Geometrical characteristics of the hydromechanical model employed in the analysis.
For a sufficiently high value pffiffiffi for b, the whole height may be approximated by ð 3=2ÞbD, the error being less than 1% if b P 10. This simple approximation allows for the column mass m to be expressed as follows: pffiffiffi 3 m¼ q bD2 Lð1 nÞ ð1Þ 2 s with n the porosity of the rod array. For a dense arrangement ofpthe pffiffiffi ffiffiffi set of cylindrical rods, n ¼ 1pffiffiffi p= ½2 3 þ ð4 2 3Þ=b which becomes n ¼ 1 p=2 3 in the limit b ! 1. Putting b ¼ 10, the complete formula yields n ¼ 0:1069, while n ¼ 0:0931 is obtained by the simplified one. The difference between these values being very small, the approximate value n ¼ 0:1 may be assumed [4]. The behaviour of the hydromechanical model has been analysed under the following additional hypotheses: (1) the added-mass effect, correlated to rod acceleration within a fluid [1] is neglected; (2) the length L of the rods is sufficiently greater than the diameter D so as to allow referring to plane strain conditions; (3) the fluid pressure assumes uniform values within the zone between the two sliding blocks; and (4) the collision among the rods is inelastic, that is, no elastic energy is stored within the system. Therefore, after impact, the velocity component of the moving block, orthogonal to the contact plane, is set equal to zero. 2.2. Motion of the solid mass 2.2.1. Acting forces The forces acting on a single vertical column of glued rod are shown in Fig. 3. The driving force Q and the
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
213
Fig. 3. Forces acting on the vertical rod column: (a) two blocks in contact; (b) absence of contact between blocks.
hydrodynamic resistance FR act horizontally. Force Q may coincide with an external force applied to the sliding block or may derive from a kinematical condition (e.g., a constant rate of horizontal displacement) imposed to the sliding block. The buoyant weight G0 of the unit cell and the resultant P of the pore pressure excess p at the base of the unit cell acts vertically through the center of the mass. The appropriate components of the normal effective force N 0 acts orthogonally to the average direction of the sliding surface. The tangential contact force T acts at the bottom of the unit cell. 2.2.2. Law of motion Newton’s law of motion for the upper block, written along the x, y directions, yields, respectively: Q FR þ N 0 sin a Tmax cos a ¼ m€ ux ; q N 0 cos a þ Tmax sin a þ m w 1 g þ P ¼ m€ uy ; qs
ð2Þ ð3Þ
a being the angle between N 0 and the y-axis; qs the density of rods material; g the gravity acceleration and € ux and € uy the accelerations of the rods mass along x and y. During the motion, the tangential force T attains its maximum value Tmax ¼ lN 0 ;
Fluid resistance to the motion of a single column can be written as 1 A ð6Þ FR ¼ Cf qw u_ 2x ; 2 r where Cf is a ‘‘shape’’ coefficient; A the area of the projection of the body on a plane perpendicular to the direction of motion; r the number of columns composing the array of rods; qw the fluid density and u_ x the rate of displacement u_ x . 2.2.3. Sliding The trajectory of the moving block, sliding over the fixed one, is determined by their inelastic collisions, in accordance with the irregularity of the sliding surface; thus, it must follow the external profile of the lower block. This compatibility condition is expressed in terms of the displacement components ux and uy by the following formula: pffiffiffi !2 2 ux 1 uy 3 þ þ ¼ 1: ð7Þ 2 D 2 D This illustrates that the center of the lowest sliding rod belongs to a circumference having D aspradius and ffiffiffi center in the point of coordinates ð1=2D; 3=2DÞ. The displacement ux is subjected to the further condition
ð4Þ
l being the friction coefficient at contact between the rods. The force P is obtainable by integrating the pore pressure excess p acting at the bottom of the unit cell (Fig. 3) P ¼2
Z
p=2
RLp sin / d/ ¼ 2pRL 0
¼ 2pRL ¼ DLp:
Z
p=2
sin / d/ 0
ð5Þ
Fig. 4. Configuration of the basal rods in function of the angle a between the sliding rods (a ¼ 0°, maximum porosity; a ¼ 30°, minimum porosity).
214
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
ux 6 D, since the contact is limited to the range 30° < a < 30° (Fig. 4).
permeability coefficient KD (m s1 ) of the ‘‘porous’’ block could be recovered [8] as a function of parameter k
2.2.4. Contact-free motion If the combined effect of pore pressure excess and inertial force of the rod mass causes the upper block to rise in relation to the underlying fixed one, the motion equations then become simpler, the contact forces N 0 and Tmax being equal to zero in Eqs. (2) and (3). Thus
KD ¼
ux Q FR ¼ m€ and qw m 1 g þ P ¼ m€ uy : qs
ð20 Þ
ð30 Þ
2.2.5. Continuity of the fluid flow For rigid blocks and incompressible fluid, the continuity of fluid mass in the thin layer implies the volume continuity of the layer itself. Since the thin layer volume varies proportionally to the displacement along the direction orthogonal to the average sliding surface, geometrical compatibility imposes ð8Þ
Vm being the volume of the thin layer (void), beyond all other symbols already introduced. The difference between the water pressure excess (p) in the void at the base of the sliding block and the imposed nil value at the top of the same, generates a hydraulic gradient in direction y. This, in turn, induces a seepage flow towards the free surface with overall time discharge qðtÞ. Thus qðtÞ ¼ kapðtÞ;
ð10Þ
h being the length of the seepage path, equal to the thickness of the moving block. The fluid volume variation dVm is further related to the discharge qðtÞ dVm ¼ qðtÞ dt:
ð11Þ
After some algebra, one obtains
At the end of the contact-free motion, the upper block collides with the lower one and intergranular forces rise, their orientation depending on that of the impact plane. This orientation is changing during motion; the collision points on the sliding surface depend on the trajectory of the upper block and thus on the value of pore pressure excess prevailing at the moment of rising; therefore, its computation must be renewed for each impact.
dVm duy ¼ DL ; dt dt
ha k; DL
ð9Þ
k being an efflux coefficient and a an outflow area. As a consequence of the fluid discharge, both a volume reduction of the void at the base of the sliding block and a pore water pressure excess reduction take place simultaneously. The seepage flow along direction x is disregarded, owing to predominance of the gradient along direction y, orthogonal to the average sliding surface. Assuming that a laminar seepage flow is taking place and starting from a linearized relationship expressing the discharge from an orifice applied to the regular geometry of the porous medium at hand, the conventional
pðtÞ ¼
DL u_ y : ka
ð12Þ
Eq. (12) links the pore pressure excess in the fluid at the base of the mobile block with its vertical velocity, confirming that the interaction between the solid and fluid phases within the thin layer at the base of the sliding block, where all pore volume variations are concentrated, determines the pore pressure excess evolution. The pðtÞ, which depends on geometrical and physical parameters, constitutes the lower boundary condition for seepage within the moving block, when the boundary condition at its upper edge is fixed at a null value. 2.2.6. Model ingredients The motion of the saturated block in contact with the stationary one is generally described by six equations: two differential equations of motion (x and y-direction), one contact condition, one fluid flow continuity equation, one constitutive relation connecting the normal force N 0 to the shear resistance Tmax and one equation expressing the resistance force offered by the fluid against the mobile block. The unknown variables are also six: two displacements (ux , uy ), one normal effective contact force (N 0 ), one tangential contact force (Tmax ), one pore pressure excess (p) and one flow resistance force (FR ). To solve the problem, four initial conditions on ux , uy and on first time derivatives u_ x and u_ y must be assigned. On the other hand, the simpler case of motion in the absence of contact between the two blocks is described by only four equations, since the contact condition and the friction law are no more applicable. The unknowns are: ux , uy , p and FR , and again, four initial conditions must be assigned.
3. Sliding conditions and related governing equations The previous equations of motion are herein rewritten and solved for three conditions: Model M1 – motion at constant horizontal velocity, Model M2 – motion under an applied constant horizontal force and Model M3 – motion down a rough inclined surface.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
3.1. Model M1 – constant rate of horizontal displacement (u_ x ¼ const) 3.1.1. Sliding Since ux ¼ u_ x t, through the expression (7) it is possible to express vertical displacement (13), velocity (14) and acceleration (15) as follows: 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 !2 pffiffiffi u u 37 u_ x t 1 6 uy ¼ D4t1 ð13Þ 5; D 2 2
u_Dx t u_ x u_ y ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; u_ x t 1 1 D 2 1 2
ð14Þ
2 2 _ u_ 2x 12 uDx D ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi € uy ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 2 3 u_ x t 1 u_ x t 1 1 D 2 1 u_ x t D
D
ð15Þ
It is possible to put into evidence N in Eq. (3) and Q in Eq. (2) m€ uy m qqws 1 g DLp N0 ¼ ; ð16Þ b2
where b1 ¼
u_ x t 1 D 2
!
ð17Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u u_ x t 1 t l 1 ; D 2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ! u u _ _ t 1 t 1 u u x x b2 ¼ t1 : þl D 2 D 2
uy ðTs Þ ¼ uy1 ; u_ y ðTs Þ ¼ u_ y1
ð22Þ
at the instant of detachment Ts . After double integration, uy and u_ y are obtained as follows: 0 q 1 w m 1 g qs kam uy ¼ uy1 2 2 @ u_ y1 A D2 L2 DL ka 0 q 1 w m qqws 1 g m 1 g qs kam @ þ u_ y1 A ðt Ts Þ þ 2 2 D2 L2 D2 L2 DL ka ka D2 L2 exp ðt Ts Þ ð23Þ kam 0 q 1 m qqws 1 g m qws 1 g u_ y ¼ u_ y1 A @ D2 L2 D2 L2 ka
0
Q ¼ F R b1 N 0 ;
Eq. (21) can be solved by introducing initial conditions:
ka
2
ð18Þ
ð19Þ
Since the horizontal velocity is constant, even the hydrodynamic resistance (6) is constant. Finally, the pore pressure excess is obtained after substitution of (14) into Eq. (12) ! 1 u_ x t u_ x 2 D DL vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ p¼ !2 : ka u u _ t 1 u x t1 D 2 3.1.2. Contact-free motion In the absence of contact, Eq. (30 ), without the contribution of ‘‘intergranular’’ forces N 0 and Tmax , can be rewritten only in terms of uy and its time derivatives q D2 L2 uy : ð21Þ m w1 g u_ y ¼ m€ qs ka
215
exp
DL ðt Ts Þ : kam 2 2
Introducing u_ y in Eq. (12) yields 2 q 0 q 1 w w DL 4 m qs 1 g @ m qs 1 g u_ y1 A p¼ D2 L2 D2 L2 ka ka ka 3 2 2 DL exp ðt Ts Þ 5: kam
ð24Þ
ð25Þ
The instant of detachment, Ts , and the values describing the initial condition uy1 and u_ y1 , can be obtained by the previous equation (16) by imposing N 0 ¼ 0. For a submerged block, Ts can be interpreted as the instant in which the hydroplaning phase of motion begins [9]. It is characterized by a substantial increase of pore pressure excess, causing a dramatic reduction in frictional resistance. It is observed that pore pressure excess p is obtained as the algebraic addition of two terms. The first is a steady term, whilst the second depends on time. Starting from the instant of detachment Ts , the pore pressure excess grows, approaching, in the limit, a maximum value represented by the first term. This asymptotic value leads to a global force that balances the buoyant weight. It could be assumed to characterize the complete fluidization. 3.2. Model M2 – constant applied horizontal force (Q ¼ const) 3.2.1. Sliding Newton’s equations for the rod motion are: Q FR þ b1 N 0 ¼ m€ux ;
ð26Þ
216
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
qw b2 N þ m 1 g þ DLp ¼ m€ uy ; qs 0
b1 and b2 being represented by: pffiffiffi ! ux 1 uy 3 l b1 ¼ þ 2 D 2 D and b2 ¼
pffiffiffi ! uy ux 1 3 ; þl þ 2 D D 2
ð27Þ
2 ð28Þ
ð29Þ
respectively. Solving for N 0 in (26) and substituting the result in (27) gives qw D2 L2 b ux Q þ FR Þ ¼ m€uy : 1 mg u_ y þ 2 ðm€ qs ka b1 ð30Þ uy are written as From contact condition (7), u_ y and € follows: 3 2
ux 1 u_ x 2 D 7 6 D ð31Þ u_ y ¼ D4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi 5; 1 uDx 12 8 2 >
u_ x > € < ux ux 12 D D D €uy ¼ D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi > > : 1 uDx 12 1 uDx 12 h
ux
1 2
u_ x i2
9 > > =
D D : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h
u 1 2 i 3 > > x ; 1 D 2
After substituting the previous expressions into (30), an ordinary differential equation with constant coefficients in terms of the horizontal displacement ux is obtained
ux
1 2
3
7 6b D m€ux 4 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ux 1 2ffi 5 b1 1 D 2 2 3
ux 1 3 2 2 q b DL 6 7 u_ x D ¼ 1 w mg þ 2 Q 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 1 2ffi 5 D qs b1 ka 1 Dx 2 !2 !2 b2 u_ x u_ x 2 mD CR D b1 D D 8 9 > >
> > 2 < = ux 1 1 2 D ffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 h >
2 i3 > > > : 1 uDx 12 ; 1 ux 1 D
2
The constant terms figuring in (6) have been collected in CR . The initial conditions are: ux ð0Þ ¼ ux0 ; u_ x ð0Þ ¼ u_ x0 ;
ð33Þ
ux0 and u_ x0 being, respectively, the position and the horizontal velocity at the beginning of the sliding.
ð32Þ
3.2.2. Contact-free motion The motion of the rods in the absence of contact is governed by Eqs. (20 ) and (30 ), under the following initial conditions:
Fig. 5. Schematic representation of the upper system of rods sliding on a system of rigid rods disposed according to an average slope angle h.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
ux ðTs Þ ¼ ux1 ; u_ x ðTs Þ ¼ u_ x1 ; uy ðTs Þ ¼ uy1 ; u_ y ðTs Þ ¼ u_ y1 :
h being the slope angle of the fixed rod block. ð34Þ
The detachment instant, Ts , along with the values ux1 , u_ x1 , uy1 and u_ y1 can be recovered by imposing N 0 ¼ 0 in the motion equations. Once Eq. (30) is solved, Eq. (21) expresses the vertical motion of the mobile block, which is used to obtain the pore pressure excess Eq. (12). 3.3. Model M3 – constant slope angle (h ¼ const) 3.3.1. Sliding Newton’s second law equations are (Fig. 5): qw mg 1 ux ; sin h FR þ b1 N 0 ¼ m€ qs qw mg 1 uy ; cos h þ DLp þ b2 N 0 ¼ m€ qs
217
3.3.2. Contact-free motion The equations of motion become: q mg 1 w sin h FR ¼ m€ux ; qs qw mg 1 cos h þ DLp ¼ m€uy : qs
ð37Þ ð38Þ
The equations obtained for movement down a slope coincide with equations obtained for horizontal motion under constant force, after the following substitutions: q mg 1 w sin h for Q qs
ð35Þ
and qw mg 1 cos h qs
ð36Þ
Thus, the solution procedure is similar to the one implemented for the Model M2.
qw for mg 1 : qs
Fig. 6. Theoretical results (Model M1) for two values of horizontal velocity u_ x (u_ x1 ¼ 0:1 m s1 and u_ x2 ¼ 0:5 m s1 ). The meaning of symbols a, b, c, d, e and f is described in Table 1.
218
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
The non-dimensional, vertical displacement uy =D, the effective normal force N 0 and the pore pressure excess are represented in Fig. 6 vs. the non-dimensional, horizontal displacement ux =D. The instants and phases corresponding to the most significant events in the history of this motion are marked with the letters a, b, c, d, e and f. Their meanings in all models are briefly summarized in Table 1. The history of events can be described in more detail as follows: (a) System at rest: N 0 equals the static equilibrium value; the pore pressure excess is zero. (b) Instantaneous increase of force Q: the block tries to move upwards, in relation to the irregularity of the sliding surface; the pore volume between the blocks tends to increase more rapidly than fluid can fill it, so a negative pressure rises that interlocks the upper
4. Results The set of sliding Eqs. (2)–(4), (6), (7) and (12) along with the contact free motion Eqs. (20 ), (30 ), (4) and (12) have been solved taking into account the different conditions corresponding to the three models M1, M2, M3. The solutions have been sought following numerical procedures, using the fourth-order Runge–Kutta’s method to solve the system of governing first order differential equations [12]. 4.1. Model M1 – constant rate of horizontal displacement (u_ x ¼ const) Two simulations were carried out by imposing u_ x ¼ 0:1 m s1 and u_ x ¼ 0:5 m s1 with the values of the parameters used in computation reported in Table 3. Table 1 Phases of motion for models M1, M2 and M3 Instant or phases
Model M1
Model M2 0
0
Model M3 0
a b
System at rest N ¼ G =2 cos 30° Exertion of net force Q
b!c c c!d d
Upper rods slide over lower rods u_ x ¼ const Detachment of upper rods from lower rods Hydroplaning of upper rods p < pfluid Complete fluidization pfluid ¼ G0 ðDLÞ1
d!e e f
Fluidized motion of upper rods p ¼ pfluid Contact of upper rods with lower rods Beginning of a new cycle
0
System at rest N ¼ G System at rest N 0 ¼ G0 Exertion of force Q ¼ h ¼ const Plane inclination h ¼ const N 0 ¼ G0 N 0 ¼ G0 cos h p¼0 p¼0 Upper rods slide over lower rods Detachment of upper rods from lower rods Hydroplaning of upper rods p < pfluid Complete fluidization Complete fluidization pfluid ¼ G0 ðDLÞ1 pfluid ¼ G0 cos hðDLÞ1 Fluidized motion of upper block p ¼ pfluid Contact of upper rods with lower rods Beginning of a new phase of motion without cyclicity
Table 2 Values of geometrical, physical and mechanical parameters considered to simulate Iverson and LaHusen’s experiment [4] Model parameters
Values
Number of rods, b Diameter, D Length, L Coefficient of friction, l Efflux coefficient, k Horizontal rate, u_ x Efflux area, a Water density, qw Density of rods material, qs Mass of the cell, m
7 0.019 0.29 0.45 2.63–9.77 104 0.118 3.48 105 1000 2300 1.304
m m m2 s kg1 m s1 m2 kg m3 kg m3 kg
Table 3 Values of the main computation parameters Figure reference
D (m)
6, 8, 9 10-I, 11-I, 12-I, 13-I 10-II, 11-II, 12-II, 13-II 10-III, 11-III, 12-III, 13-III
0.02 0.02 0.02 0.02
0.05
h (m)
k (m2 s kg1 )
ka (m4 s kg1 )
0.12 0.12 0.12 0.12
1.00E ) 03 1.00E ) 03 1.00E ) 03 1.00E ) 04
4.00E ) 08 4.00E ) 08 4.00E ) 08 4.00E ) 09
1.00
1.00E ) 03
l 2.50E ) 07 4.00E ) 08
0.45 0.45 0.45 0.45
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
block to the base even more closely, and thus the normal force N 0 is increased. This occurs instantaneously, so N 0 jumps. (b ! c) The upper block slides over the lower one with constant horizontal velocity; the effective forces N 0 and Q progressively decrease to zero while the pore pressure monotonically increases. (c) Detachment of the upper block from the lower one. (c ! d) Hydroplaning of the upper block moving downwards: the pore pressure continues to increase; N 0 and Q assume zero values. (d) Complete fluidization: the submerged weight G0 of the column of rods is entirely balanced by the vertical force P resulting from the pore pressure excess that has attained its maximum value; vertical acceleration vanishes. (d ! e) Pore pressure excess p maintains its maximum value (plateau); the upper block moves downwards at constant vertical rate.
219
(e) The upper block comes again in contact with the lower one. (f) A new, identical cycle of motion and fluid–solid interaction begins. To check the validity of the proposed hydromechanical model, Iverson and LaHusen’s [4] experiment has been simulated, with the imposed constant value u_ x ¼ 0:118 m s1 . Measured pore pressure fluctuations generated by the sliding movement exhibit a repetitive time-series pattern, characterized by high-pressure plateaux (1.4 kPa at maximum) and separated by deep, negative pressure troughs (negative pressure, 12 kPa at minimum) (Fig. 7). Photographs indicate that the plateaux occur when the upper sliding block loses contact with the underlying fixed block and glides on a cushion of water. In the absence of ‘‘intergranular’’ forces, the buoyant weight of the moving rod array is entirely supported by the pore pressure excess acting along the sliding surface.
Fig. 7. (a) Comparison between theoretical values (grey lines) and measured values (black lines) at three transducer positions, after Iverson and LaHusen [4]. Broken lines refer to hydrostatic pressure. Case a: ka ¼ 3:4 108 m4 s kg1 ; case b: ka ¼ 9:15 109 m4 s kg1 .
220
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
A sudden pore pressure decrease occurs when the upper block of rods comes in contact with the underlying block and then starts again to slide, inducing a sudden increase of pore volume. Experimental results show appreciable fluctuations of the pore pressure excess when the upper block collides with the lower one. Pressure fluctuations propagate from their source (sliding zone) towards the free surface at the top of the model, undergoing both an amplitude attenuation and a phase shift. The values of the parameters used in the model, as reproduced from Iverson and LaHusen’s paper [4], are
listed in Table 2. It should be noted that, since parameters k and a do not figure separately in the model, the product ka has been assigned. In order to compute the values of pore pressure excess in points not located along the slip surface, due to the hypothesis of incompressible fluid and to the rigid assembly of the rod blocks, a linear distribution of pore pressure excess along y resulted. It has been found that for both values of ka, delimiting the range of experimental values (9.15 109 m4 s kg1 < ka < 3:4 108 m4 s kg1 ), an acceptable modelling of the experimental regime of pore pressure
Fig. 8. Theoretical results (model M2) for two values of applied horizontal constant force Q (Q1 ¼ 10 N, Q2 ¼ 15 N). The meaning of symbols a, b, c, d, e and f is described in Table 1.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
excess is obtained. In particular, by assuming the lower value (ka ¼ 9:15 109 m4 s kg1 ), the minimum pore pressure excess value (pmin ¼ 12:5 kPa) is fitted (Fig. 7(b)), while the maximum value (ka ¼ 3:4 108 m4 s kg1 ) leads to the maximum pore pressure excess (pmin ¼ 6 kPa) (Fig. 7(a)). In addition, theoretical and measured peak values of pore pressure excess and periods of constant values of fluid pressure significantly agree, although a small temporal shift of the measured plateaus at different depth is observed. These differences may be attributed to an imperfect-inelasticity condition occurring during the collisions.
221
4.2. Model M2 – constant applied horizontal force (Q ¼ const) The mobile block starts sliding over the fixed one if Q > Qmin ; Qmin is depending on the mass of the system, the initial arrangement and the friction coefficient; it is obtained from the limit equilibrium of the mobile block: Qmin ¼
sin a þ l cos a G0 cos a þ l sin a
ð39Þ
with
Fig. 9. Theoretical results (model M3) for two values of plane inclination hðh1 ¼ 40°, h2 ¼ 50°). The meaning of symbols a, b, c, d, e and f is described in Table 1.
222
G0 ¼
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
q 1 w mg qs
ð40Þ
representing the buoyant weight of the rod column. For a ¼ 0° (maximum initial porosity) and l ¼ 0:45, (39) results in Qmin ¼ 0:45G0 ; but if a ¼ 30° (see Fig. 4), that is, the geometrical condition corresponding to minimum porosity, Qmin ¼ 1:39G0 results. Fig. 8 depicts the motion history of model M2. For other parameter values see Table 3. It is observed that, in comparison with model M1 results, the trajectory of the rods becomes straighter, the contact–no contact cycles are more frequent, the no-contact phase lasts longer and the contact phase reduces with distance travelled.
At first, the horizontal velocity rises fast then, it tends towards a constant limiting value when the hydrodynamic force becomes more significant. The peak value of the normal contact force reduces and its frequency increases with elapsed time. The pore pressure excess behaves in a similar way as in case M1: it grows from a negative minimum value to its maximum value during the no-contact phase and tends to stabilize. It is worth noting the monotonic increase of pore pressure excess both before and after the detachment took place. It resembles the continuous increase of the same variable both before and during the flow slide, as recorded in the monitored, small-scale experiments quoted in points (b) and (c) (see Section 1). The model,
Fig. 10. Theoretical values of mmax (model M3) against slope angle h. For parameter values see Table 3.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
however, does not fit the conclusions reported in point (d), probably as it cannot properly account for the ‘‘consolidation’’ of the porous medium [3]. 4.3. Model M3 – constant slope angle (h ¼ const) The behaviour of the system, in this case, is analogous to case M2. The minimum slope angle for the motion to take place is h > hmin , where: tan hmin ¼
sin a þ l cos a : cos a þ l sin a
ð41Þ
The phases of motion are represented in Fig. 9, being parameter values reported in Table 3.
223
In Fig. 9, a straighter trajectory is observed, the velocity reaches higher values, N 0 peaks progressively decrease and the fluidization value of pore pressure excess slightly decreases as h increases. These results are quite significant if they are represented against to the following response variables: mmax – maximum velocity achieved by the rod array; Ru – ratio between pore pressure at complete fluidization and pore pressure at detachment; Ts – instant of detachment of the mobile array of rods; 1=f – cycle period of sliding and contact-free motion. Figs. 10–13, show the variables mmax , Ru , Ts and 1=f as functions of an ‘‘average’’ slope angle h using two rod
Fig. 11. Theoretical values of Ru (model M3) against slope angle h. For parameter values see Table 3.
224
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
diameters, two column heights and two efflux coefficient values. The parameters considered for the computations are reported in Table 3. The variable mmax increases with the slope angle h. It assumes higher values for D and h increasing; if k increases for a small slope angle, mmax decreases; conversely, for a high slope angle, mmax increases. Moreover, mmax is upper-limited by the velocity of the submerged upper block moving without any frictional resistance at the base and it is lower-limited by the velocity of the same block sliding with friction along the slope [7]. Its limiting velocities depend on the ratio between its mass and the cross-sectional area; this ratio in turn depends on D. For low values of h, the maximum velocity of the rod block approaches the ‘‘frictional’’ velocity; as h in-
creases, mmax tends towards ‘‘contact-free’’ velocity. This result offers a more interesting interpretation when read along with the picture of Ru . The ratio Ru increases as h increases. Values of Ru near unity indicate an easier fluidization of the rod system: this occurs with higher diameter, lower layer extent and lower efflux coefficient. This could mean that, for greater angles h, the system is more prone to sliding than to fluidization, at least in sub-aqueous landslides. As expected, the instant of detachment Ts decreases with increasing slope angle. It does not show any significant dependence on h and k, while a dependence on D is more marked: Ts increases if D increases. The period of the contact–detachment cycle, which is not constant, decreases as h increases. If D decreases, the
Fig. 12. Theoretical values of Ts (model M3) against slope angle h. For parameter values see Table 3.
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
225
Fig. 13. Theoretical values of 1=f (model M3) against slope angle h. For parameter values see Table 3.
period decreases; if h increases, 1=f decreases. 1=f also decreases if, with low angle values, k decreases and if, with high angle values, k increases.
5. Closing remarks A simple model is presented which is believed to reveal some important aspects related to pore pressure evolution within saturated granular materials, while being sheared at a high rate. The distinctive feature modelled here is the dynamic interaction between grains and pore filling fluid beyond that of the grains themselves. The pore pressure excesses derive from a rapid modification of pores structure,
connected with the continuous rearrangement of grains involved in the flow motion. ‘‘Fluidization’’, i.e. the process occurring during movement due to a continuous rearrangement of structure, does not coincide with ‘‘liquefaction’’, which is a process marking a ‘‘phase-change’’. Significant increments of pore pressure occurring during flow could thus be justified and the roles of various geometrical parameters (i.e., grain diameter, height of flowing layer) as well as mechanical ones (permeability, friction coefficient) could be properly accounted-for. As a by-product, the build-up of pore pressure excess in coarse debris flows could also be interpreted as due to the weight of small particles carried along in suspension in the pore fluid saturating the body, as repeatedly invoked, although in purely speculative terms [11].
226
A. Musso et al. / Computers and Geotechnics 31 (2004) 209–226
The extension to a ‘‘continuum’’ formulation [5], which is yet to be done, apparently presents major difficulties, due to the need of taking into account the peculiar characteristics of the phenomenon; particularly, the high rate of displacements and the associated, substantial, pore-structure modification. However, the presented model which succeeds in generating new amounts of pore pressure, together with diffusion equations which allow for the dissipation of pore pressure excess, and momentum equations for the fluid–solid mass, could bring about the improvement of models aimed at predicting sub-marine flow slide runout characteristics. References [1] Batchelor GK. An introduction to fluid mechanics. Cambridge: Cambridge University Press; 1967. [2] Eckersley JD. Instrumented laboratory flowslides. Geotechnique 1990;40(3):489–502. [3] Hutchinson JN. A sliding-consolidation model for flow slides. Can Geotech J 1986;23(2):115–26. [4] Iverson RM, LaHusen RG. Dynamic pore pressure fluctuations in rapidly shearing granular materials. Science 1989;246:796–9.
[5] Iverson RM. Differential equations governing slip-induced pore pressure fluctuations in water saturated granular medium. Math Geol 1993;25(8):1027–48. [6] Johnson AM. Physical processes in geology. San Francisco: W.H. Freeman; 1970. [7] Lambe TW, Whitman RV. Soil mechanics. New York: Wiley; 1978. [8] Leverett MC. Capillary behavior in porous solids. Trans, AIME 1941;142:152–69. [9] Mohrig D, Whipple KX, Hondzo M, Ellis C, Parker G. Hydroplaning of subaqueous debris flows. Geol Soc Am Bull 1998;110(3):387–94. [10] Musso A. Analysis of rapid debris flows, in Interventi di Stabilizzazione di Pendii, CISM, Udine, 1997 (In Italian). [11] Pierson TC. Dominant particle support mechanisms in debris flows at Mt Thomas, New Zealand, and implications for flow mobility. Sedimentology 1981;28:49–60. [12] Rosemberg D. Methods for numerical solution of partial differential equations. Elsevier; 1969. [13] Savage SB, Hutter K. The motion of a finite mass of granular material down a rough incline. J Fluid Mech 1989;199: 177–215. [14] Spence KJ, Guymer I. Small scale laboratory flowslides. Geotechnique 1997;47(5):915–32. [15] Takahashi T. Debris flow. Rotterdam: Balkema; 1991. [16] Wang G, Sassa K. Factors affecting rainfall-induced flowslides in laboratory flume tests. Geotechnique 2001;51(7): 587–99.