A review of Beliche Dam

Alonso, E. E., Olivella, S. & Pinyol, N. M. (2005). Ge ´otechnique 55, No. 4, 267–285 267 A review of Beliche Dam E. E. ALONSO*, S. OLIVELLA* and N. M. PINYOL* Beliche Dam, a zoned earthdam with rockfill shoulders and a central clay core, experienced large collapse settlements due to reservoir impounding and direct action of rainfall. Long-term field records of vertical and horizontal displacements are available as well as a set of largescale laboratory tests on rockfill specimens. It has been the subject of several numerical analyses that failed to capture the relevant effect of weather conditions on the behaviour of the dam. Recent developments in the constitutive modelling of rockfill allow a substantial improvement of modelling capabilities, and this is illustrated in the paper. Laboratory test results under dry and flooded conditions were interpreted, and material parameters were identified. The complete history of dam construction, impoundment and rainfall was then simulated by means of a coupled flow–deformation model. Deformations during construction and impoundment have been reproduced. Long-term deformations have consistently been related to rainfall records. In general, long-term deformations are controlled by the varying wetting history of the dam shoulders and by an intrinsic deformation component. The wetting action comes to an end when the relative humidity of the rockfill reaches 100% for the first time. The paper also discusses scale effects and the role of rockfill permeability in the development of deformations. KEYWORDS: case history; collapsed settlement; constitutive relations; dams; laboratory tests; numerical modelling and analysis Le barrage de Beliche, digue de terre zone ´eavece ´paulements en enrochement en vrac et noyau central en argile, a connu de grands affaissements en raison de l’endiguement du re ´servoir et de l’action directe des eaux de pluie. Il existe des donne ´es de terrain a `long terme sur les de ´placements verticaux et horizontaux ainsi qu’un lot d’essais grandeur nature en laboratoire re ´alise ´s sur des spe ´cimens d’enrochement en vrac. Ceci a e ´te ´le sujet de plusieurs analyses nume ´riques qui n’ont pas re ´ussi a ` expliquer pertinemment l’effet des conditions climatiques sur le comportement du barrage. Les progre `sre ´cents en mode ´lisation constitutive des enrochements permettent d’ame ´liorer de manie `re substantielle les capacite ´sdela mode ´lisation ; c’est ce que nous illustrons dans cet expose ´. Nous avons interpre ´te ´les re ´sultats des essais en laboratoire en conditions se `ches et noye ´es et nous avons identifie ´les parame `tres mate ´riels. Nous avons alors simule ´l’historique complet de la construction du barrage, de l’endiguement et de l’eau de pluie au moyen d’un mode `le couple ´flux-de ´formation. Nous avons reproduit les de ´formations pendant la construction et l’endiguement. Les de ´formations a `long terme ont e ´te ´lie ´es de manie `re constante aux pre ´cipitations. En ge ´ne ´ral, les de ´formations a `long terme sont contro ˆle ´es par l’historique de mouillage variant des e ´paulements du barrage et par une composante de de ´formation intrinse `que. L’action mouillante arrive a `sa fin lorsque l’humidite ´relative de l’enrochement atteint 100% pour la premie `re fois. Dans cet expose ´, nous e ´tudions e ´galement les effets d’e ´chelle et le ro ˆle de la perme ´abilite ´de l’enrochement sur le de ´veloppement de de ´formations. INTRODUCTION Beliche Dam has attracted wide interest from geotechnical researchers working in the field of earth and rockfill dam behaviour. The reason for the continued interest in Beliche probably lies in the availability of a set of large-scale laboratory tests (oedometer and triaxial) on the rockfill materials, as well as a comprehensive set of field monitoring data, which covers the construction and impoundment, and several years of dam operation. Published papers and doctoral dissertations that have addressed the analysis of the dam constitute a valuable source of information concerning both the laboratory and field behaviour of rockfill materials and the evolution of numerical techniques for dam analysis. Of particular relevance are the papers published by Naylor et al. (1986, 1997). These papers are the immediate reference for the work presented here. Before an evaluation of the achievements and limitations of the previous work is presented, the main features of the dam and its behaviour will be briefly presented. Comprehensive information is provided in the two references mentioned and in the additional publications mentioned later. Beliche Dam is a 54 m high earth and rockfill dam located in the Algarve, Portugal. It has a moderately curved shape in plan view. A central cross-section showing the position of stress cells, open pipe piezometers, extensometers and surface settlement markers is reproduced in Figs 1 and 2. The design is classic: a central core of low-plasticity clay is stabilised by two rockfill shells. Two zones are distinguished on each shell: an inner one, made of lightly compacted fractured schists, and a harder outer one, made of compacted greywacke. Schists and greywackes are relatively soft rocks, and large deformations (short and long- Manuscript received 19 July 2004; revised manuscript accepted 7 February 2005. Discussion on this paper closes on 1 November 2005, for further details see p. ii. * Department of Geotechnical Engineering and Geosciences, Universitat Polite`cnica de Catalunya, Barcelona, Spain. Stress cell Piezometer 050 m G8 G9G10 G6 G1 G2 G3 G4 G5 Fig. 1. Central cross-section (P7) of Beliche Dam: position of stress cells and piezometers (Naylor et al., 1997) term) may be expected. Naylor et al. (1986, 1997) state that the inner rockfill was made deformable on purpose, in order to avoid arch effects on the core, and therefore to ensure a significant compression state of the core, with the purpose of avoiding cracking phenomena in the core. Construction started in 1984. When the dam height reached 47 m (January/February 1986) the reservoir level reached an unexpected temporary elevation of 29 m over foundation level due to heavy rains. Significant collapse of the upstream shell was recorded. This is shown in Fig. 20, which shows the vertical displacements observed in extensometer I1, located within the upstream inner shell (line B corresponds to December 1985, and line C to February 1986). Maximum accumulated vertical displacements, in excess of 0.80 m, were recorded at mid-height. Note also (Fig. 20, extensometer I6) that the downstream shell, not affected by the impoundment, reached similar settlements. Once the dam was completed in March 1986, it was subjected to the rainfall and reservoir level evolution reproduced in Fig. 3(a). Settlements and horizontal displacements of the surface markers were recorded against time. Fig. 3(b) shows the recorded evolution of vertical displacements at marker M54 in the three centre sections of the dam (6, 7 and 8) (see Fig. 1 of Naylor et al., 1997) and the calculated values reported by Naylor et al. (1997). The figure shows a distinct correlation between measured settlement rates and extreme periods of rainfall intensity during the first five years of operation of the dam. Further data concerning the behaviour of the dam will be presented later in connection with the analysis reported in this paper. Review of previous work The paper by Naylor et al. (1986) is a prediction exercise of the construction of Beliche Dam. It was performed before actual field data became available. Two non-linear elastic models were used (a hyperbolic model and a K-G model (where K is bulk coefficient and G is shear modulus)). Model parameters were derived from back-analysis of triaxial and oedometer tests on compacted ‘dry’ samples. Specimens tested under triaxial conditions were flooded once they had reached the maximum deviatoric load. A sudden reduction in deviatoric load was measured. The vertical strain rate was maintained, and a reduced limiting strength was finally recorded, this time for saturated conditions. No attempt was made in Naylor’s paper to model this flooding stage of the tests. Five material parameters were required for the K-G model and eight for the hyperbolic model. The analysis was performed in terms of total stresses and material parameters corresponding to the dry stage of the tests performed. Small quantitative differences were found between the two models. A maximum settlement of 40 cm was predicted at the end of construction, for the centre of the dam. This was a clear underestimation of actual values, which were later attributed to the collapse effects associated with the unexpected partial impoundment of January–February 1986. Collapse effects in rockfill dam behaviour were first introduced into finite element analysis by Nobari & Duncan (1972). The idea is to measure collapse strains by performing dry and wet oedometer and triaxial tests on rockfill samples. The calculation starts by performing the finite element (FE) analysis using the set of material parameters corresponding to dry conditions; then collapse is numerically simulated in two stages. First, the stress change due to saturation (at constant strain) is determined on the basis of the experimental data. In the second stage nodal forces that restore equilibrium are applied. The computed strains reproduce the collapse strains. The procedure has been followed by some authors: Veiga Pinto (1983); Justo & Saura (1983), who performed a 3D linear elastic analysis of El Infiernillo Dam; and Soriano et al. (1990). Naylor et al. (1989) generalised the method for arbitrary constitutive models. It required the knowledge of two sets of constitutive parameters for dry and saturated conditions. The method is only approximate even if only the effects of full saturation (which is the case of the upstream shell) are required, because it does not guarantee that the actual stress path experienced in the field is followed by the calculation procedure. The point to underline here, recognised also by Naylor et al. (1997), is that the procedure is essentially a computational device, not necessarily associated with a physical reality. Naylor et al. (1997) describe a back-analysis of the behaviour of Beliche Dam. Impounding effects were now considered. Materials were described by means of an elastoplastic critical-state model, and the analysis was performed in terms of effective stresses. A known distribution of pore water pressures (derived from piezometer data) was required to perform the analysis. Collapse effects induced by full saturation were introduced by means of a numerical algorithm, which reproduces a change in material properties (from a dry state to a saturated state). It requires two sets of constitutive parameters: one set for the dry material and a second one for the wet material. For the critical-state model adopted, a total of 14 model parameters were needed. Some settlement predictions are reproduced in Fig. 3(b). The Surface marker Extensometer 0 50 m M45 M54 I3 J54 J40 J25 I1I2 I4I5 I6 Fig. 2. Central cross-section (P7) of Beliche Dam: position of extensometers and surface settlement markers (Naylor et al., 1997) Rainfall Reservoir level 85 86 87 88 89 90 91 92 93 94 85 86 87 88 89 90 91 92 93 94 Rainfall: mm/month 600 400 200 0 0 0·2 0·4 0·6 0·8 Section 6 Section 7 Section 8 Computed Estimated Settlement: m 60 40 20 0 Elevation: m Fig. 3. (a) Rainfall intensity and reservoir level after dam construction; (b) vertical settlements at marker M54 (continuous line) and predictions by Naylor et al. (1997) (broken line) (after Naylor et al., 1997) 268 ALONSO, OLIVELLA AND PINYOL predicted settlement values included in Fig. 3(b) correspond to full impoundment of the upstream shell and to a simulated impoundment (without uplift) of the downstream shell. The latter was introduced in an attempt to model the overall effect of rainfall events over an extended period (1988– 1994). The theses of Azevedo Filho (1990), Farias (1993), Pagano (1996) and Maranha (1997) include analyses of Beliche Dam using basically the procedure proposed by Naylor et al. (1989). Maranha das Neves (2002), in a comprehensive review of rockfill mechanics, describes the procedure followed by Mateus da Silva (1996). As wetting of an initially dry rockfill may imply that points initially at yield would remain outside the ‘shrinking’ yield surface, a viscoplastic algorithm is invoked to bring back the stress point to the ‘current’ yield surface. The current yield surface is located by assuming that material parameters evolve linearly from an initial dry state to a final saturated state. The construction and impoundment of Beliche Dam were also simulated by Mateus da Silva (1996) using the procedure outlined. The analysis of zoned earth dams has also been approached from the perspective of unsaturated soil mechanics. The formal similarities between wetting tests performed on unsaturated low-plasticity soils and rockfill provide a justification for these approaches, which are, however, not supported by physically convincing evidence. Placing emphasis on volumetric deformations Alonso et al. (1988), and Alonso & Batlle (1995) have presented several analyses of dam construction and impoundment. The mechanical analyses were performed in a fully coupled mode with the flow phenomena, and therefore transient flow boundary conditions could be properly introduced. Additional fully coupled unsaturated-like analysis of earth and rockfill dams, using suction as an additional independent stress variable and describing the unsaturated materials by means of elastoplastic models, has been described by Costa (2000) and Kogho (2003). The question of the applicability of unsaturated soil analysis to rockfill behaviour was raised by Naylor et al. (1997). In the discussion included in their paper they state: No attempt was made in the analysis to model the effects of partial saturation as, for example, has been proposed by Alonso et al. (1990). This effect is thought to be small in the rockfill... The opinion that partial saturation has a small effect in the rockfill is probably based on a capillary interpretation of suction effects in granular materials. In fact, approximate calculations of capillary induced stresses in rockfill and gravels (Alonso, 2003) indicate that their absolute values are extremely small, and therefore their reduction (due to wetting) may hardly affect gravel or rockfill behaviour. However, the effects of partial saturation in rockfill should be approached from a different perspective, as discussed below. Aims of the paper In recent papers (Oldecop & Alonso, 2001, 2002, 2003; Cha´vez & Alonso, 2003), the effect of water on rockfill behaviour has been linked to fracture propagation phenomena. This underlying mechanism, which controls particle breakage, and the results of large-scale oedometer and triaxial tests on gravel-like aggregates, in which the relative humidity prevailing at the large voids of the specimens was controlled, led to the formulation of new constitutive models. They were formulated within the framework of hardening plasticity, and they offer a comprehensive description of several features of rockfill behaviour, including collapse phenomena. These developments bring the opportunity of performing more accurate and consistent analysis of rockfill structures. From this perspective, Beliche Dam offers an interesting case record, with the added advantage of being a case extensively analysed in the recent past. New elements for an improved understanding will hopefully be added. As mentioned before, a fundamental aspect of the behaviour of Beliche Dam is the rockfill collapse induced by partial, as well as full, wetting. Partial wetting, induced by rainfall, was shown to have the same relevance as full flooding. Most of the computational procedures used so far, uncoupled to water flow, are unable to handle this fundamental environmental factor. Beliche is not an isolated case in this respect. Justo & Durand (2000) showed the close relationship between rainfall intensity and the acceleration of settlements experienced by different points inside the rockfill embankment of Martı´n Gonzalo Dam. Settlement plates reacted to strong rainfall events during a given period—within the first three years after the end of construction. Later, rainfall was not able to induce additional settlements. Similar behaviour was observed in dams such as Beliche (Fig. 3) or El Infiernillo (Marsal et al., 1976), and in large railway rockfill embankments (Soriano & Sa´nchez, 1999). In this paper, a coupled flow–deformation analysis of the history of construction, impoundment and the operational phase of Beliche Dam is presented. Due consideration will be given to rainfall, during both construction and subsequent phases. Model calculations will be compared with field measurements. The main source of experimental and field information was provided by Naylor et al. (1986, 1997). The paper is organised as follows. First, the elastoplastic models used for the description of dam materials will be introduced, paying special attention to the rockfill model and to the physical nature of model parameters. The results of oedometer and triaxial tests performed at the Laboratorio Nacional d’Engenheria Civil of Lisbon (LNEC) will be back-analysed in order to derive model parameters. Hydraulic parameters have been approximated on the basis of known identification data. Details of the coupled flow-mech- anical analysis performed will then be given. Spatial distributions of some measured variables at some specific times (vertical and horizontal displacements, stresses) will be obtained and compared with measurements. Also, time records of settlements and horizontal displacements will be obtained and compared with actual measurements. Calculated stress paths at some specific points within the dam will be given, with the purpose of understanding the deformation phenomena observed in the structure. An analysis was first conducted using the parameters derived directly from laboratory tests. This case represents the base case. Additional cases, modifying both the rockfill compressibility, to account for scale effects, and the rockfill permeability were also performed. Finally, the obtained results are discussed. CONSTITUTIVE MODELS Two different models were used to describe the behaviour of the dam core, a low-plasticity clay, and the rockfill shoulders. For the clay core, the model proposed by Alonso et al. (1990) (BBM: ‘Barcelona Basic Model’) is adopted. A summary of the basic relationships of this model is given in Table 1. The model for the rockfill shoulders (RM) is an extension of the compressibility model developed by Oldecop & Alonso (2001), and it will be described in more detail here. Rockfill model for triaxial conditions Volumetric compressibility of rockfill is assumed to have two components, whose origins are a particle rearrangement A REVIEW OF BELICHE DAM 269 and a particle breakage phenomenon. The second is highly dependent on the prevailing relative humidity (or, alternatively, the total suction, s) at the rockfill particles. The second mechanism is active beyond a threshold mean stress, py. Isotropic compressibility is described by the following equations: dv¼ºidpif p<py dv¼ºidpþºd(s)d pif p.py (1a) (1b) where dvis the incremented volumetric strain, pis the mean total stress, and ºiand ºd(s) are compressibility parameters. ºd(s), which is made dependent on total suction, describes a deformation mechanism based on particle breakage, and it accounts for some macroscopic phenomena observed in rockfill testing: the compressibility of rockfill increases as the material is wetted, and the rockfill collapses, at constant confining stress, when the ambient relative humidity is increased. The superscript ‘d’ in ºd(s) alludes to the delayed nature of the particle breakage mechanisms. An appropriate function for the clastic compressibility parameter ºd(s) was found to be ºd(s)¼ºd 0Æsln sþpatm patm  (2) where ºd ois the volumetric compressibility for saturated conditions and Æsis a material parameter. The atmospheric pressure, patm, is introduced to avoid indeterminacies for saturated conditions (s¼0). ºd(s) is restricted by ºd(s)>0. Rockfill elastic behaviour is characterised by two compressibility coefficients kand ks, and a Poisson ratio, . The compressibility coefficients kand ksare defined as follows: de v¼kdp¼dp E3(1 2) (3a) des v¼ks ds sþpatm (3b) where the volumetric elastic strains de vand des vare induced by total stress and suction changes. Note that the elastic bulk modulus defined in equation 3(a) does not depend on the mean stress. The swelling stiffness coefficient ksis, in general, very small, except for expansive claystones and shales. As no evidence of rockfill expansion has been reported in the case of Beliche, a value of ks¼0was adopted. For isotropic conditions, the yield stress, p0, was defined in Oldecop & Alonso (2001) by the expression p0(s)¼p 0if p 0<py p0s ðÞ¼pyþ ºik ðÞ p 0py  ºiþºds ðÞk if p 0.py (4a) (4b) where p 0defines the position of the yield curve, and it was Table 1. Basic relationships for constitutive models used in the analysis of Beliche Dam* Barcelona Basic Model (BBM) (Alonso et al., 1990) Rockfill Model (RM) (Compressibility part described in Oldecop & Alonso, 2001) Isotropic elastoplastic volumetric deformation† dv¼ºs ðÞ 1þe dp p For p<py)dv¼di v¼ºidp For p.py)dv¼di vþdd v¼ ¼ºiþºds ðÞ  dp Volumetric compressibility index† ºs ðÞ¼º0 ðÞ 1r ðÞ exp s ðÞ þr  ºiþºds ðÞ ºds ðÞ¼ºd 0Æsln sþpatm patm  Hardening law d p 0¼1þe ðÞ p 0 º0 ðÞ k dp vdp 0¼dp v ºik Loading–collapse curve (LC) p0s ðÞ¼pcp 0 pc ! º(0)k º(s)kFor p 0<py)p0s ðÞ¼p 0 For p 0.py)p0s ðÞ¼pyþºik ðÞ p 0py  ºiþºds ðÞk Shear strength criticalstate parameter Ms ðÞ¼MMs ðÞ¼Mdry Mdry Msat ðÞ Msat Mdry  s 10 patm Tensile strength parameter pss ðÞ¼kss Yield surface(triaxial) Fp,q,s ðÞ ¼q2M2pþpss ðÞ½ p0s ðÞp ½ ¼0 Plastic potential (triaxial) Gp,q,s ðÞ ¼q2ÆM2pþpss ðÞ½ p0s ðÞp ½ ¼0 Creep strain – dv dt¼1 v pdd dt¼1 3d q 1 v¼ t1cln sþpatm patm  d¼0:1v * A common notation was used for equivalent parameters. Material constants are different for the soil and rockfill models. † Suction srefers to matric suction, in the case of BBM, and to total suction, in the case of RM. 270 ALONSO, OLIVELLA AND PINYOL identified as the yield stress for a very dry rockfill. A simple volumetric hardening was proposed to follow the evolution of p 0: dp 0¼dp v ºik (5) where the plastic volumetric strains, dp v¼dvde v,may be obtained through consistency conditions. The model will now be extended to triaxial stress condition (stress variables p¼(1+23)3;q¼13)by means of a family of elliptical yield curves. The yield surface, F, in a space (p,q,s) is now defined by Fp,q,s ðÞ ¼q2M2s ðÞpþpss ðÞ½ p0s ðÞp ½ ¼0 (6) This is a simple choice, although more elaborated constitutive models for rockfill have recently been proposed (Cha´vez & Alonso, 2003). It will be shown later, by comparing model calculations with large-scale triaxial test results on compacted rockfill specimens, that the constitutive model based on the elliptical yield function provides satisfactory results. The effect of suction on rockfill strength is introduced through parameters ps(s) and M(s).Parameter ps(s) provides an apparent cohesive intercept that serves to approximate the real non-linear strength envelopes found in testing. The dependence of the critical-state slope Mon s also helps to introduce suction effects on rockfill limiting shear conditions. ps(s) is simply defined as a linear function of suction: pss ðÞ¼kss(7) where ksis a material parameter. As the available experimental data on rockfill often consist of tests performed on dry or flooded specimens, it was found convenient to define M(s) in terms of two extreme values: Mdry and Msat.Often Mdry corresponds to the relative humidity existing at the laboratory and Msat corresponds to saturated conditions. The transition from Msat to Mdry takes place for an increase in suction from a zero value, when the specimen is saturated, to a very high value, typically a few tens or even hundreds of MPa for the relative humidity prevailing in laboratories. The following expression is proposed to model this variation: Ms ðÞ¼Mdry Mdry Msat ðÞ Msat Mdry  s=10 patm (8) Rockfill behaves in a non-associated manner. Maintaining the model’s simplicity, the modification of the yield function suggested for BBM will be adopted here, and a potential function Gis defined as follows: Gp,q,s ðÞ ¼q2ÆM2s ðÞpþpss ðÞ½ p0s ðÞp ½ ¼0 (9) where Æis a material parameter. The relationships described have been collected in Table 1. The table helps to emphasise the formal similarities and differences between the RM and the BBM. Unlike the BBM, compressibility relations for the rockfill are described by linear functions relating strain and stress. The threshold stress pyhas no equivalent in the case of unsaturated soils. Note also that the hardening parameter for the case of rockfill, p 0, is defined for a very dry state, a concept that is not applicable to soils. An additional difference, important from a fundamental perspective, is that the suction term, s, has a definite mechanical meaning in the case of soils. It is generally accepted that, for low-plasticity unsaturated soils and silts, sshould be identified with the capillary or matric component of suction. The role of suction in the case of rockfill is different. Following the ideas of Oldecop & Alonso (2001), suction controls the velocity of crack propagation in rockfill particles. Therefore no direct mechanical action should be assigned to the suction term in this case. It is a state variable externally controlled by boundary conditions and flow phenomena. The crack propagation velocity depends on the total energy of the water on propagating cracks, and therefore it is measured by the relative humidity or, alternatively, by the total suction. The distinction is probably not very significant in some applications such as dams, but it may prove significant in other situations in which the chemical composition of the water, which controls the osmotic component of total suction, is a relevant variable. In this paper osmotic effects are not considered, and total and matric suction become identical. Generalisation: the viscoplastic approach Equations (6) and (9) for the yield and potential functions may be generalised to a three-dimensional stress state by writing them in terms of the second invariant of the deviatoric stress tensor, J2D, and the first invariant of the stress tensor, J1, as follows: FJ 1,J2D,sðÞ ¼3J2D 1 9M2s ðÞJ1þ3kss ðÞ 3p0s ðÞJ1 ½ ¼0 (10) GJ 1,J2D,sðÞ ¼3J2D 1 9ÆM2s ðÞJ1þ3kss ðÞ 3p0s ðÞJ1 ½ ¼0 (11) A viscoplastic formulation offers computational advantages (Zienkiewicz & Cormeau, 1974; Zienkiewicz & Taylor, 2000), which proved to be of considerable help in the dam analysis reported in this paper. This is especially interesting in geotechnical problems because the viscoplastic approach provides a unique solution, and it is used as a regularising procedure. Collapse behaviour can be viewed as a softening process (the triaxial tests included in this paper are a good illustration in this regard), and this behaviour introduces numerical difficulties. In some cases instability occurs because collapse concentrates at isolated elements, while others in the vicinity experience an induced stress relaxation, and they revert to elastic conditions. This phenomenon complicates the modelling of generalised collapse. The viscoplastic approach helps in this regard because it homogenises the spatial distribution of collapse strains. It must be added that one can always ensure a solution as close as necessary to the ‘true’ elastoplastic solution by increasing sufficiently the fluidity parameter of the viscoplastic formulation (see below). The preceding set of constitutive relations was cast in the form proposed by Perzyna (1966). Total strain rates (_ ) are calculated as a sum of elastic (_ e) and viscoplastic components (_ vp): _  ¼_ eþ_ vp (12) The viscoplastic component is computed through _ vp ¼ˆF ðÞ  @G @(13) where ˆis a fluidity parameter and k(F)lis a flow function defined as F ðÞ  ¼F ðÞ if F.0 (14a) F ðÞ  ¼0ifF<0 (14a) A REVIEW OF BELICHE DAM 271 A usual expression for the flow function is (Perzyna, 1966) F ðÞ ¼F F0  N (15) where F0is a reference stress to normalise F. The viscoplastic parameters ˆ¼100 s1and N¼5 have been chosen in order to obtain a solution as close as possible to plasticity: that is, deformations are not delayed in time. Furthermore, the viscoplastic contribution is fast compared with the creep contribution described below. The time dependence appears in the viscoplastic contribution as a result of suction or stress changes in time only, but not of delayed deformations. Another advantage of the viscoplastic approach is that an implicit scheme of integration can be easily used. Plastic strains are calculated as the integral of the viscoplastic strain rates in a time interval. Total strains are computed as ˜¼˜eþ˜p¼˜eþ˜t_ vp ¼D1 e˜þ˜tˆF ðÞ  @G @  kþŁ ¼D1 e˜þ¸@G @  kþŁ (16) where ˜t¼tkþ1tkis the time increment in which the strain increment is calculated, and the superscript k+Ł indicates that the viscoplastic strain rate is calculated at an intermediate time defined as tkþŁ¼tk(1 Ł)+tkþ1Ł. The fully implicit approach is used here, and this implies Ł¼1. Note that the term ˜t[ˆh(F)i] is formally equivalent to a plastic multiplier, ¸. Creep behaviour of rockfill Long-term deformation records of rockfill structures (Sowers et al., 1965; Sherard & Cooke, 1987) show the significant creep behaviour of rockfill. The underlying mechanism of subcritical crack propagation in stressed particles provides a physical explanation for the observed macroscopic creep behaviour. Oldecop & Alonso (2002) analysed long-term records of compacted gravel specimens of quartzitic shale tested under oedometric conditions and varying relative humidity. The strain–time records were interpreted in terms of a creep coefficient, ºt, defined in a classical manner: ºt¼dv dlnt ðÞ (17) It was found that ºtdepends on confining stress and on the prevailing total suction, as shown in Fig. 4. For an extended range of stress and suction values, ºtdepends linearly on confining stress and on the logarithm of the applied suction. The second dependence is illustrated in Fig. 5, where the creep coefficient ºt(represented in Fig. 4) is plotted in terms of ln(s) for a vertical confining stress v¼1 MPa. The creep rate decreases with increasing suction. Presumably a zero creep rate may be obtained for an essentially dry environment, but this limiting condition was not found in the experiments performed. Based on these findings, the following strain rate equation is proposed for the volumetric component of the long-term creep of rockfill: _ c v¼dc v dt¼ºtp,s ðÞ t¼1 vs,t ðÞ p(18) where the stress dependence is assumed to be linear, and v is a viscosity coefficient for volumetric creep given by 1 vs,t ðÞ ¼ t1cln sþpatm patm (19) where and care constitutive parameters. The values ¼ 0.0012 MPa1and c¼0.083 provide an accurate representation of the creep data given in Fig. 4 for v,1 MPa. These values were used in the analysis of Beliche Dam. No equivalent experimental information was found for the deviatoric creep. A similar expression for the deviatoric creep is tentatively proposed here: _ c d¼dc d dt¼1 3ds,t ðÞ q(20) where dis probably related to v. In fact, it is proposed that dwill be a fraction of v: d¼av(21) in order to maintain simplicity in the model. A small value a¼0.1 was adopted in the calculations reported below. 0 0·5 1·0 1·5 2·0 2·5 3·0 Vertical stress, ó: MPa s 5 0 MPa s 5 5·7 MPa s 5 97 MPa s 5 255 MPa ët 5 då/d(ln t) 3 1022 0 0·02 0·04 0·06 0·08 0·10 0·12 0·14 0·16 0·18 Total suction, s 0 MPa 5·7 MPa 97·0 MPa 255·0 MPa Fig. 4. Time-dependent compressibility index, ºt, measured in oedometer tests performed on compacted gravel of a quartzitic shale (after Oldecop & Alonso, 2002) 0 0·05 0·1 0·5 Total suction, s: MPa 0·4 1 10 100 1000 Creep coefficient, ët Fig. 5. Variation of ºtwith suction for vertical stress v 1 MPa (plot based on data represented in Fig. 4) 272 ALONSO, OLIVELLA AND PINYOL Under general stress conditions, creep rates will be calculated through a general viscoelastic equation: _ c¼1 2d pI ðÞ þ1 3v pI(22) where volumetric and deviatoric viscosities have already been defined, and Iis the identity matrix. In the generalised constitutive model rockfill, strain rates are calculated as the sum of three contributions—elastic, viscoplastic and long-term (or secondary)—as follows: _  ¼_ eþ_ vp þ_ c(23) Expressions for the three contributions have already been given. Material parameters have been collected in Table 2. FINITE ELEMENT MODEL The analysis of the dam was performed with the finite element code CODE_BRIGHT, developed at the Department of Geotechnical Engineering of UPC. The code solves simultaneously, in a monolithic manner, the balance equations for heat transfer, water flow and air flow, as well as the mechanical equilibrium equations. It is formulated in terms of two independent stress variables: the net stress (excess of total stress over air pressure), and suction or difference between air and water pressures. The formulation of the code and its numerical implementation are described in Olivella et al. (1994, 1996). A detailed description of the code is given in DIT-UPC (2002). Available in the current version of the code are a few mechanical constitutive models (linear and non-linear elasticity, viscoelasticity and viscoplasticity for saline and granular materials, BBM for unsaturated soils, and others). The viscoelastoplastic model for rockfill materials described before was also introduced in CODE_BRIGHT, in order to analyse rockfill structures. The basic transport and deformation phenomena mentioned can be activated independently when a given problem is solved. For the analysis of Beliche Dam, only the coupled flow and deformation relationships were simultaneously solved. Material parameters were approximated by means of a back-analysis of the large-diameter tests performed at the LNEC. Test results were interpreted through CODE_ BRIGHT, as reported in the next section. Laboratory results on foundation materials were also interpreted, to find the corresponding constitutive parameters. Flow-related parameters were not reported in the available data, but they were approximated on the basis of indirect information (grain size distributions, plasticity) and past experience. DETERMINATION OF MATERIAL PARAMETERS Rockfill materials for the inner and outer shell were tested in a large-diameter oedometer (dia. ¼500 mm) and triaxial (dia. ¼300 mm) cells. The gradation of the material tested was scaled down by making the specimen cumulative grain size distributions parallel to the gradation curve of the in situ rockfill material. Tests were performed on a dry material (presumably equilibrated with the relative humidity prevailing in the laboratory). At some stage during the triaxial tests, specimens were flooded, although the applied vertical strain rate was maintained during the sample inundation and subsequent straining. In order to interpret these tests, specimens were treated as a boundary value problem. Stress, strain and flow conditions were applied at the specimen’s boundaries as required. Flowrelated properties are therefore required to perform calculations. A common water retention curve for the two types of rockfill (inner and outer shell) was used. It is shown in Fig. 6. The curve is similar to the retention curve measured in compacted specimens of schist reported in Oldecop & Alonso (2001). It shows two distinct parts, which reflect the pore structure of a rockfill material. The relatively large voids between rock particles, which provide most of the pore volume, become empty for a very small suction (0.0001– 0.001 MPa in the figure). All the remaining water is then stored within the rock particles, which exhibit a strong resistance to becoming unsaturated, as suction increases, because of the small size of the rock pores. In this way, a bimodal shape characterises the water retention behaviour of rockfill. For the low-plasticity clay used in the core, the water retention curve shown in Fig. 6 was adopted in calculations. An air entry value close to 0.1 MPa was assumed in this case. Hydraulic conductivities of all dam materials are described by the expression K¼krwg w KrSr ðÞ (24) where kis the intrinsic permeability (m2); wis the Table 2. Mechanical parameters for rockfill Definition of parameter Symbol Units Base case for Beliche Dam Inner shell Outer shell Elastic behaviour Elastic modulus EMPa 150 180 Poisson’s ratio –0 .30 .3 Plastic behaviour Plastic virgin instantaneous compressibility (ºik)– 0 .025 0.01 Virgin clastic compressibility for saturated conditions ºd 0–0 .028 0.01 Parameter to describe the rate of change of clastic compressibility with total suction Æs–0 .010 0.003 Slope of critical-state strength envelope for dry conditions Mdry –1.75 1.90 Slope of critical-state strength envelope for saturated conditions Msat –1.30 1.80 Parameter that controls the increase in cohesion with suction ks–00 Threshold yield mean stress for the onset of clastic phenomena pyMPa 0.01 0.01 Parameter that defines the non-associativeness of plastic potential Æ–0 .30 .3 Creep behaviour Creep coefficient for saturated conditions MPa10.0012 0.0012 Parameter that controls the influence of suction on creep rate c–0 .083 0.083 Initial state for dam model Initial suction s0MPa 20 20 Initial mean yield stress p 0MPa 0.02 0.02 A REVIEW OF BELICHE DAM 273 viscosity of the fluid (for water, w¼0.001 Pa s); rwis the density of the fluid (for water, rw¼1000 kg/m3); and gis the acceleration due to gravity. The intrinsic permeability depends on the porosity and fabric of the porous material. The selected values are included in Table 3. Parameter Kr(Sr), the relative permeability, introduces the effect of partial saturation. Krvaries between 0 (dry soil) and 1 (saturated soil), and it is usually expressed as a non-linear function of the degree of saturation (Sr). The functions used are represented in Fig. 7. A relative permeability defined as a power law of the degree of saturation has been considered. Exponent values of 3 for the clay core and foundation soil and 10 for the rockfill are used. In the case of the rockfill, the relative permeability starts to decrease below 1 when the degree of saturation becomes smaller than 0.3. It was assumed that, for degrees of saturation in excess of 0.3, water can circulate freely through the rockfill macropores. Mualem (1976) proposed a ‘consistent’ methodology to derive the relative permeability function on the basis of the water retention curve. The method proposed by Mualem, verified for sandy soils, leads to the theoretical curve shown in Fig. 7. The power law adopted was close to this theoretical law, and it is considered more suitable for rockfill conditions. Oedometer tests were simulated by means of a onedimensional mesh. A constant displacement rate was applied to the upper boundary, and stresses were calculated. Triaxial specimens were simulated by means of an axisymmetric mesh, defined by 32 linear quadrilateral elements. The confining stress was applied against the lateral boundaries, and a constant vertical displacement rate was applied to the upper boundary. Stresses were computed at this boundary. For the dry stage of tests, a constant high suction, which represents the laboratory conditions, was maintained in the specimen, and the boundaries were assumed to be impervious. Flooding was simulated by converting the lower boundary into a zero-suction boundary. Inner schist shell Measured and simulated responses for the oedometer and triaxial tests are shown in Figs 8 and 9. Compressibility parameters (instantaneous, clastic) (as well as an approximation of the elastic modulus) could be derived from the oedometer results. A comparison of the calculated and measured responses of the simulated triaxial test is represented in Fig. 9. The stress–suction paths actually experienced by the specimens are not particularly simple. This is illustrated in Fig. 10, in a (p,q,s) space. The compacted specimen is initially loaded isotropically under very dry conditions (point A). The specimen is then loaded vertically at constant Degree of saturation Suction: MPa Clay core Rockfill 0 0·2 0·4 0·6 0·8 1·0 1025 1024 1023 1022 1021 100 101 102 Fig. 6. Water retention curves for rockfill material and clay core Table 3. Initial void ratio and intrinsic permeability adopted in calculations Material Initial void ratio, e0 Intrinsic permeability, k:m 2 Rockfill shells 0.538 2 31011 Clay core 0.786 8 31015 Foundation soil 0.428 3 31012 Degree of saturation Relative permeability Clay core (power law) Rockfill (power law) Rockfill (theoretical) 100 1022 1024 1026 1028 10210 10212 10214 0 0·2 0·4 0·6 0·8 1·0 Fig. 7. Relative permeability functions for rockfill and clay core 2.5 2·0 1.5 1·0 0.5 0 543210 Volumetric strain: % Vertical stress: MPa Model Experimental Fig. 8. Oedometer test on compacted schists from inner shell 274 ALONSO, OLIVELLA AND PINYOL suction until the limiting condition (point B) is reached. Then it was flooded, while the constant rate of vertical strain is maintained. The volumetric collapse induces a sudden loss of vertical stress, and, at the same time, the suction is reduced to zero. Point C is reached. The position of point C depends on the rate of applied vertical displacement and on the specimen’s permeability: it cannot be determined on the basis of the constitutive equation alone. Once at zero suction, the yield envelope experiences a reduction in size. As the vertical strain rate is maintained, the specimen starts resisting again along the loading path and eventually reaches a new yield condition at point D, and a final limiting state at point E. These triaxial tests provide direct information on the strength envelope for dry and saturated conditions (Mdry, Mwet), and on the elastic parameters. The volumetric behaviour, reproduced in Fig. 9(b), provides an indication of the value of the Æparameter defining the plastic potential. For low confining stresses, dilatancy is underestimated by the model. This is a feature of the constitutive model used, which is not well suited to reproducing the observed dilatancy behaviour in samples exhibiting a ductile response in shear. Rockfill specimens for laboratory tests were prepared by compacting the material in thin layers to obtain uniform samples (Naylor et al., 1986). In order to simulate the oedometer and triaxial tests, a significant preconsolidation effective mean stress, associated with compaction, was used in the model ( p 0¼0.9 MPa for both inner and outer shell). A non-linear elastic model would facilitate the modelling of the tests, and would require a lower apparent preconsolidation stress. However, for the dam model, yielding was assumed to be active at a low stress ( p 0¼0.02 MPa for both shells). It is believed that yield conditions in the field, which include particle breakage, especially for the lightly compacted inner shell in 1 m thick layers, start immediately after compaction. The parameters used to reproduce the tests have been collected in Table 2. They will be considered as the base case. Outer greywacke shell The results of the simulations performed and a comparison with actual test measurements are shown in Figs 11 and 12. The greywacke rockfill is substantially stiffer than the lightly compacted schist, as the derived elastic and plastic compressibility parameters collected in Table 2 indicate. Strength is also higher for the greywacke. The model captures reasonably well the measured stress–strain curves under dry conditions, the transient loss of deviatoric stress associated with flooding, and the final recovery of strength under wet conditions. Compression volumetric strains are overpredicted (Fig. 12(b)), however, especially for low confining stresses when dilatancy effects seem to dominate. The identified clastic compressibility of the outer material (ºd 0¼ 0.01) is small compared with the value found for the inner schist. This implies that relative humidity changes will have a limited effect on computed collapse in this material. The analysed experimental data are not sufficient to discriminate accurately among the different model parameters. In particular, oedometer tests, which would include a suction reduction stage (ideally under suction control), are required for a precise determination of the parameters (ºi k), ºd 0and Æs. Suction control also helps in the interpretation of triaxial testing. Nevertheless, the set of large-diameter tests performed at LNEC provided sufficient data to derive model parameters with an acceptable degree of confidence. 0 0·5 1·0 1·5 2·0 Axial strain: % Deviatoric stress: MPa Model Experimental Confining pressure 5 0·3 MPa Confining pressure 5 0·1 MPa 25 0 5 10 Volumetric strain: % Model Experimental Confining pressure 5 0·3 MPa Confining pressure 5 0·1 MPa 0246810121416 (a) Axial strain: % (b) 0246810121416 Fig. 9. Triaxial tests on compacted schists from inner shell: (a) deviatoric stress–strain curves; (b) volumetric–axial strain curves q E D C A p LC B 1 M(s) s Fig. 10. Sketch showing stress–suction path applied in triaxial tests initially dry and later flooded when limiting conditions are reached 22·5 22·0 21·5 21·0 20·5 0 3·02·52·01·51·00·50 Volumetric strain: % Vertical stress: MPa Model Experimental Fig. 11. Oedometer test on compacted greywacke from outer shell A REVIEW OF BELICHE DAM 275 permeability and, more generally, to hydraulic properties. Rockfill permeability was inferred from available grain size distribution curves. The actual field value is, however, uncertain. There was an interest in exploring the effect of changing some of these material properties on the computed response of the dam. A change in rockfill permeability implies a change in the suction field for a given rainfall/evaporation regime. An increase in permeability results in lower water potential gradients and a more limited reduction of initial suction. Therefore a reduction of rockfill collapse should be expected in this case. Fig. 26 shows the effect of a tenfold increase in the rockfill permeability (for both inner and outer shells) on the vertical settlements of marks M54, J54 and J40 and the evolution of representative suctions. Suctions are now higher, and a reduction of vertical displacements is computed. Heavier rains will be required in this case to develop the full collapse potential of the rockfill. (d) 0·32 0·33 0·34 0·35 0·36 Mean net stress: MPa Porosity Rockfill downstream (collapse due to rain cycles) Rockfill upstream (collapse due to impounding) 1 12 2 (a) 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 Mean net stress: MPa 0 0·5 1·0 1·5 2·0 Suction: MPa LC curve during construction LC curve during collapse 1 12 2 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 (b) 0·41 0·42 0·43 0·44 Mean net stress: MPa Porosity Clay core upstream side (collapse due to impounding) Clay core downstream side (collapse due to rain cycles) 1 1 2 2 (c) 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·45 0 0·5 1·0 1·5 2·0 Mean net stress: MPa Suction: MPa LC curve during construction LC curve during collapse 1 1 2 2 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 Fig. 25. Stress–liquid pressure and stress–porosity plots for points indicated of (a, b) inner rockfill and (c, d) clay core: base case M54 21·0 20·8 20·6 20·4 20·2 0 0·2 Time: days Vertical displacements: m Base case J54 21·0 20·8 20·6 20·4 20·2 0 0·2 Time: days Vertical displacements: m Base case J40 21·0 20·8 20·6 20·4 20·2 0 0·2 Time: days Vertical displacements: m Base case 21·2 21·0 20·8 20·6 20·4 20·2 0 Time: days Water pressure: MPa Base case 0 360 720 1080 1440 1800 2160 2520 2880 3240 3600 0 360 720 1080 1440 1800 2160 2520 2880 3240 3600 0 360 720 1080 1440 1800 2160 2520 2880 3240 3600 0 360 720 1080 1440 1800 2160 2520 2880 3240 3600 Fig. 26. Evolution of vertical displacements of surface markers M54, J54 and J40: comparison of measured and calculated values when the rockfill permeability of shells is increased 10 times with respect to base case. Also indicated is the calculated evolution of water pressure (suction) at a reference point within inner downstream rockfill 282 ALONSO, OLIVELLA AND PINYOL The effect of particle size on rockfill compressibility was recently investigated by Montobbio (2002). Suction control oedometer tests were performed on specimens of crushed quartzitic shale of scaled grain-size distribution (cumulative distribution curves were kept parallel). The test results were interpreted through the elastoplastic compressibility model described by Oldecop & Alonso (2001). Fig. 27 shows the measured values of the saturated virgin compressibility index ºd 0plotted in terms of the D50 value of the grain size distribution. Coarser specimens are more deformable, and ºd 0 increases linearly with logD50. Naylor et al. (1986) reported the gradation curves of rockfill in situ and the curves used in different testing devices. For the 500 mm diameter oedometer, a value of D50 ¼18 mm was found. In situ material is characterised by D50 ¼100 mm. For these two values, Fig. 27 suggests that the compressibility index of the in situ rockfill may be 50% higher than the laboratory-determined value. The results of increasing the elastoplastic compressibility index (ºiþºi 0) of the rockfill shells by 50% with respect to the base case are shown in Figs 28 and 29. A significant improvement of the calculated settlements is now found. The measured downstream settlements (I6) remain higher than the model calculations, and this suggests that a more compressible material than expected was placed in situ. The evolution in time of vertical displacements is plotted in Fig. 29. Water pressure in the rockfill does not change, because changes in porosity have a negligible effect on rockfill permeability. Model calculations are now closer to measured surface settlements. Actual settlements indicate a more marked creep deformation than the effect provided by the creep coefficients initially selected. However, within the time interval used to make the comparisons (10 years), creep effects are of minor significance if compared with suction-induced phenomena. CONCLUSIONS Recent developments in rockfill constitutive behaviour provide new tools for the coupled flow and deformation analysis of rockfill embankments, which offer definite advantages over previously reported modelling approaches. Zoned earth dams integrate fine compacted soils in the impervious core and rockfill shoulders into a single structure. In both types of material the role of suction or, alternatively, the prevailing relative humidity is a fundamental variable to explain the behaviour of the dam during construction, impounding and operation. Numerical models developed within the framework of effective stress analysis require some ad hoc computational devices to handle the development of strains as a consequence of suction or relative humidity changes, both in the core and in the rockfill shoulders. They may be useful to reproduce stress and deformation states at a given stage of dam construction and operation, even if the representation of material behaviour is inconsistent with physical phenomena. However, they lack the possibility of handling environmental factors, such as a weather regime. The Beliche Dam case highlighted the relevant effects of climate, which are directly responsible for the behaviour of the downstream shoulder. Ideally, the constitutive models used require suctioncontrolled testing for the determination of material parameters. It has been shown, however, that the performance of two series of oedometer and triaxial tests under dry and saturated conditions allows the identification of model parameters. Despite the limitations of a methodology for parameter determination that relies on a limited number of tests on compacted specimens, the computed dam response for the set of parameters derived in this manner proved to be quite satisfactory. It has been argued that size effects tend to reduce the compressibility of coarse materials as determined in laboratory experiments. A corrected compressibility, which incorporates this effect, is better able to reproduce the field data. One of the main advantages of the proposed modelling approach is the consistent integration of water transfer and deformation mechanisms. It has been shown that the defor- 500 100 110 D50: mm Saturated compressibility index, ëd 0 0·08 0·06 0·04 0·02 0 Fig. 27. Variation of virgin saturated compressibility index of scaled gravel specimens of quartzitic shale with grain size distribution (Montobbio, 2002) 0 5 10 15 20 25 30 35 40 45 50 55 Y coordinate: m Computed Measured A B C DE 0 5 10 15 20 25 30 35 40 45 50 55 21·421·221·020·820·620·420·20 Vertical displacement: m Y coordinate: m A B C D E 0 5 10 15 20 25 30 35 40 45 50 55 21·421·221·020·820·620·420·20 Vertical displacement: m Y coordinate: m A B CD E CD E B Computed Measured Computed Measured 21·421·221·020·820·620·420·20 Vertical displacement: m (c)(b)(a) Fig. 28. Calculated and measured vertical displacements at position of extensometers (a) I1, (b) I3 and (c) I6 during construction and impoundment for stages [A] to [E]. Elevation 0 corresponds to bottom of clay core. Virgin compressibility index of rockfill increased by 50% A REVIEW OF BELICHE DAM 283 mations of the dam, immediately after construction, are highly dependent on the weather regime. The analysis helps to distinguish deformations associated with suction changes from pure creep effects. The former may be relevant for a number of years following dam completion. In fact, it is the joint consideration of the rainfall pattern and the rockfill permeability that dictates the time period in which ‘waterrelated effects’ may deform the structure in time. These effects, which have been associated with the breakage of particles, tend to vanish when the rockfill reaches, for the first time, a 100% relative humidity. Flooding of the rockfill is not necessary to reach this condition. At later stages, pure creep effects are held responsible for rockfill deformation. ACKNOWLEDGEMENTS The authors would like to acknowledge and express their thanks for the careful review of the original manuscript made by Professor Maranha das Neves from the Instituto Superior Te´cnico of Lisbon. NOTATION (RM refers to Rockfill Model and BBM to the Barcelona Basic Model for unsaturated soil.) afraction of volumetric creep coefficient that provides deviatoric creep coefficient (RM) Dematrix of elastic coefficients Eelastic modulus evoid ratio e0initial void ratio Fyield function (RM and BBM) F0reference stress to normalise F(RM and BBM) Gplastic or viscoplastic potential (RM and BBM) ggravity Iidentity matrix J1first invariant of stress tensor (¼1+2+3) J2D second invariant of deviatoric stress tensor, (¼1 6[(12)2þ(13)2þ(23)2] þ2 12 þ2 13 þ2 23) Khydraulic conductivity Krrelative permeability kintrinsic permeability ksparameter describing increase in cohesion with suction (RM and BBM) Mslope of critical-state line (RM and BBM) Mdry slope of critical-state line of rockfill material for a very dry state (RM) Mwet slope of critical-state line of rockfill material for a saturated state (RM) pmean total stress (RM); mean net stress (BBM) patm atmospheric pressure pcreference stress (BBM) psapparent yield stress in tension (RM and BBM) pythreshold yield mean stress for onset of clastic phenomena of rockfill material (RM) pwwater pressure p0preconsolidation stress (RM and BBM) p 0yield stress for a very dry rockfill (RM); preconsolidation stress for saturated conditions (BBM) qdeviatoric stress (¼13) rparameter defining maximum soil stiffness (BBM) Srdegree of saturation ssuction s0initial suction ua,uwair and water pressures Æparameter defining non-associativeness of plastic potential (RM and BBM) Æsparameter describing rate of change of compressibility with total suction (RM) parameter controlling rate of increase of soil stiffness with suction (BBM) cparameter describing effect of suction on viscosity coefficient for volumetric creep (RM) ˆfluidity parameter (RM) ,e,p,vp total, elastic, plastic and viscoplastic strain tensors respectively ddeviatoric strain (RM) vvolumetric strain (RM and BBM) d vvolumetric strain component associated with clastic deformation mechanism (RM) M54 21·0 20·8 20·6 20·4 20·2 0 0·2 Time: days Vertical displacements: m Base case J54 21·0 20·8 20·6 20·4 20·2 0 0·2 Vertical displacements: m Base case J40 Base case 21·2 21·0 20·8 20·6 20·4 20·2 0 Water pressure: MPa 0 360 720 1080 1440 1800 2160 2550 2880 3240 3600 Time: days 0 360 720 1080 1440 1800 2160 2550 2880 3240 3600 Time: days 0 360 720 1080 1440 1800 2160 2550 2880 3240 3600 Time: days 0 360 720 1080 1440 1800 2160 2550 2880 3240 3600 21·0 20·8 20·6 20·4 20·2 0 0·2 Vertical displacements: m Base case and present case (2 curves) Fig. 29. Evolution of vertical displacements of surface markers M54, J54 and J40. Comparison of measured and calculated values when the rockfill virgin compressibility index is increased 50% with respect to the base case. Also indicated is the calculated evolution of water pressure (suction) in a reference point within the inner downstream rockfill 284 ALONSO, OLIVELLA AND PINYOL i vvolumetric strain component associated with particles rearrangement (RM) _  strain rate tensor ddeviatoric viscosity coefficient (RM) vvolumetric viscosity coefficient (RM) kelastic stiffness coefficient for changes in mean stress (RM and BBM) ksElastic stiffness coefficient for changes in suction (RM and BBM) ¸plastic multiplier (RM) ºvirgin compressibility (BBM) º(0) virgin compressibility for saturated conditions (BBM) ºdcompressibility parameter for virgin states associated with clastic deformation mechanism (RM) ºd 0virgin clastic compressibility for saturated conditions (RM) ºicompressibility parameter for virgin states associated with particle rearrangement (RM) ºtcreep coefficient (RM) volumetric creep parameter for saturated conditions (RM) wviscosity of fluid (water) Poisson’s ratio rwdensity of fluid (water) vvertical confining stress 1,2,3total principal stresses (F) flow function (RM) REFERENCES Alonso, E. 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