Hideo Aochi
email: h.aochi@brgm.fr

Thomas Ulrich

Ariane Ducellier

Fabrice Dupros

David Michea

Finite difference simulations of seismic wave propagation for understanding earthquake physics and predicting ground motions: Advances and challenges

Seismic waves radiated from an earthquake propagate in the Earth and the ground shaking is felt and recorded at (or near) the ground surface. Understanding the wave propagation with respect to the Earth's structure and the earthquake mechanisms is one of the main objectives of seismology, and predicting the strong ground shaking for moderate and large earthquakes is essential for quantitative seismic hazard assessment. The finite difference scheme for solving the wave propagation problem in elastic (sometimes anelastic) media has been more widely used since the 1970s than any other numerical methods, because of its simple formulation and implementation, and its easy scalability to large computations. This paper briefly overviews the advances in finite difference simulations, focusing particularly on earthquake mechanics and the resultant wave radiation in the near field. As the finite difference formulation is simple (interpolation is smooth), an easy coupling with other approaches is one of its advantages. A coupling with a boundary integral equation method (BIEM) allows us to simulate complex earthquake source processes.

Overview of finite difference approach in seismology

Seismic waves radiated from an earthquake propagate in the Earth, which is often considered as an elastic medium although the waves attenuate due to some anelasticity. The theory of wave propagation in an elastic medium was well established before the 20 th century. In the early years of seismology, researchers tackled the relevant equations to obtain analytical (or semi-analytical) solutions [START_REF] Aki | Quantitative seismology[END_REF]. These solutions are usually limited to some simple structures (e.g. 1D-layered model without any lateral medium heterogeneity). These solutions, however, are still useful as seismologists need a simple solution to rapidly analyse earthquakes. The advances in computing environments allowed the direct evaluation of the differential equation in the 1970s [START_REF] Boore | Finite difference methods for seismic wave propagation in heterogeneous materials[END_REF]. Since the mid-1990s, seismologists have been able to routinely carry out huge computations for various applications [START_REF] Olsen | Three-dimensional simulation of a magnitude 7.75 earthquake on the San Andreas fault[END_REF].

Let us consider a 3D isotropic elastic medium, characterized by two elastic (Lamé) coefficients λ and µ and density ρ. We denote displacement, velocity and stress as u, v, and τ, respectively. Taking a Cartesian coordinate (x, y, z), the equations of motion are written as

2 2 i ij i j u F t x ρ τ ∂ ∂ = + ∂ ∂
where indices represent each component or direction in the coordinates, and the summation convention about the same index is assumed. F is a body force (usually zero except for a particular source). The strain is defined as 1 2

j i ij j i u u x x ε   ∂ ∂ = +     ∂ ∂   .
The constitutive relation between the stress and the strain is where δ ij = 1 if i = j, otherwise 0. The differential equations of the seismic wave propagation consist of

x xx xy xz x v F t x y z ρ τ τ τ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂
and two other equations. and 2

xx x y z x v v v v t x y z x τ λ µ   ∂ ∂ ∂ ∂ ∂ = + + +   ∂ ∂ ∂ ∂ ∂  
and five other equations. This is called the stress-velocity formulation, for which it is easy to discretize the differential terms.

For reasons of space, we have shown only one equation for each component of velocity and stress.

As for many other differential equations, various approximations of these equations are possible. However, the staggered-grid discretization in space and time is common for good stability [START_REF] Virieux | Dynamic faulting studied by a finite difference method[END_REF]. Figure 1 shows the spatial definition of each component of velocity and stress, noting that each component is separated from the others by half a grid step (except for the diagonal component of stress). Stress and velocity are evaluated at different time steps (by half a time step). Recently partially-staggered grids were proposed [START_REF] Saenger | Modeling the propagation of elastic wave waves using a modified finite-difference grid[END_REF] to combine all the stress components at one point and all the velocity components at another point separated by half a grid step. Such a modified method gives a better estimation of the boundary conditions, but additional operations are necessary. In many cases, the 4 th order approximation on the spatial differential operator is adopted because of its efficiency;

1 9 1 3 3 ( ) ( ) ( ) ( ) ( ) 8 2 2 24 2 2 h h f x f x f x f x h f x h x h ∂       = + - - - + + -       ∂       .
Note that a time step is taken as max / t c s V ∆ ≤ ⋅ ∆ and the upper frequency limit is conventionally f max = V min /(5 ∆s), where V max and V min are the maximum and minimum wave velocities in the modeled structure, respectively and ∆t and ∆s are the time step and grid spacing. The constant c is often taken to be about 0.5. Seismologists often treat a finite volume extracted from the Earth for a regional problem (e.g. a cube of 100 x 100 x 50 km from a sphere of radius 6 400 km). Therefore, we should not have any artificial reflections from the given model boundaries, which should continue propagating outwards. For this purpose, an absorbing condition is required as a boundary to eliminate the reflected waves or a boundary zone absorbing the seismic waves [START_REF] Clayton | Absorbing boundary conditions for acoustic and elastic wave equations[END_REF]. Appropriate absorbing boundary conditions over a wide range of frequencies allow proper calculation not only of the wave propagation phase but also the residual displacement field. This feature becomes important when seismologists are interested in geodetic deformation of very long periods as well as short-period ground motions.

Figure 2 shows a simulation example of the wave propagation during the M9 2011 Tohoku-oki, Japan, earthquake, which was extremely large (a few hundred kilometers of fault length and more than a hundred seconds of rupture duration). High-sampling (1 Hz) continuous GPS observed the deformation process in displacement. We aim to reproduce such deformations without losing information on the residual absolute displacement, which is difficult to obtain from accelerometric data because of sensor response and low-frequency noise. In Figure 2, we show two finite difference simulations low-pass filtered (cut-off at 50 s); one is properly calibrated to use an absorbing boundary condition (FDM1), while the other, using the same code, is not (FDM2). We compare with the reference solution synthetically calculated with the wave-number method in the same 1D structure. We observe that FDM1 provides a reasonable solution compared to the reference solution including the residual displacement, while FDM2 has some bias. We note that both finite difference simulations are consistent in a band-pass filter range (1 to 10 s), so that the difference comes from the response at long periods. In fact, FDM1 is calibrated in its absorbing condition so as to work better at around a period of 100 s, while FDM2 is our conventional simulation of wave propagation adopted for a period of 1 s. This discrepancy is important to understand when applying finite difference schemes for longperiod deformation processes. The simulations do not perfectly match the observations because of the use of a simple source model with very few parameters, but these simulations are good enough as a first-order approximation. Other aspects, such as implementation of earthquake sources, treatment of the ground surface and material contrast are found in various studies [START_REF] Graves | Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences[END_REF]. The extensions to visco-elastic medium are given in [START_REF] Day | Memory-efficient simulation of anelastic wave propagation[END_REF]. The finite difference scheme has been generally developed for equally-spaced structural grids, so that it is rather easy to manage memory allocation in parallel computing. However it is possible to mix different grids or even combine with finite element grids [START_REF] Aoi | 3-D finite-difference method using discontinuous grids[END_REF]. From their early era of development, finite difference methods are used to apply a Message Passing Interface (MPI). This is necessary to reduce the unbalances between CPUs. New approaches using graphic card as well as the usual CPU (GPU-CPU) have been recently developed [START_REF] Dupros | Exploiting intensive multithreading for efficient simulaton of seismic wave propagation[END_REF].

Earthquake physics and ground motion

Coupling with a boundary integral equation method

The finite difference scheme always has difficulty in treating dislocations (earthquake faulting) in the medium, although some improvement is possible [START_REF] Cruz-Atienza | 3D finite-difference dynamic-rupture modeling along nonplanar faults[END_REF]. The reason why this is difficult is that the dislocation problem is a discontinuous phenomenon in the continuum medium, governed by nonlinearity (fracture mechanics). Nevertheless, there is no problem in calculating wave propagation for any known sources within the proper frequency range. The convenient and practical solution is to combine it with other methods, in particular with a boundary integral equation (boundary element) method specialized for the dislocation [START_REF] Aochi | The 1999 Izmit, Turkey, earthquake: Nonplanar fault structure, dynamic rupture process, and strong ground motion[END_REF]. Figure 3 shows the schematic illustration of the simulation procedure. In the following session, we present a study on the rupture dynamics and wave propagation in the very-near-field of an earthquake. 2.2. Very-near-field ground motion 2.2.1. Unresolved scientific problems. Nowadays, seismologists are able to describe the earthquake rupture phenomena by kinematics in order to explain the regional or teleseismic data set (kinematic inversion). In other words, the regional seismic wave propagation (say at more than 10 km in a low frequency range below 0.5 -1 Hz) is reasonably comprehensible. However, some observations very close to the causal fault (those leading to significant structural damage) can be somewhat unexpected.

Although it is considered that these mechanisms are related to the dynamic rupture process of the causal fault, each case is not yet well resolved. The 2008 Mw6.9 Iwate-Miyagi, Japan, earthquake is remembered for the maximum recorded acceleration of 3800 cm/s 2 at the surface sensors of the station IWTH25 (Kik-net, NIED) located above the ruptured thrust fault (Figure 4). An acceleration of more than 1000 cm/s 2 is observed even at the borehole sensor (280 m depth) at the same station. The PGV also reaches about 100 cm/s. We focus on the fact that an extreme ground motion is found even at depth, and this should have originated from the source process. In particular, the PGV is found at the very beginning of the ground motions, about 2 seconds after the arrival of S wave, as the first main pulse.

Kinematic and dynamic source process and wave radiation.

We compute the wave propagation from the published kinematic source models using a finite difference method. The 1D structure is assumed. We aim to calculate up to 2 Hz. We first model the ground motion based on the rupture scenario obtained from the kinematic inversion [START_REF] Asano | Kinematic source rupture process of the 2008 Iwate-Miyagi Nairiku earthquake, a Mw6.9 thrust earthquake in northeast Japan, using Strong Motion Data[END_REF] (Fault strike = N201°E, dip = 51°, focal depth = 7.77 km). As already pointed out [START_REF] Suzuki | Rupture process of the 2008 Iwate-Miyagi Nairiku, Japan, earthquake derived from near-source strong-motion records[END_REF], mainly because of the resolution limit of the inversion model (subfaults of 2 km), the strong first pulse is hardly reproduced in terms of its huge amplitude and short duration. This infers that a heterogeneity equivalent or smaller than this scale may be important. Then we try to simulate the beginning of the rupture process (about the first 3 s). We investigate kinematically the effect of a small circular patch (radius = 1 km) on the ground motion at IWTH25 by shifting its location. For the given geometrical situation, it is found that a patch not located beneath the station, but slightly shallower (2 km away from the hypocentral depth) can give a relatively significant fault parallel (e.g. significant vertical) component in the ground motion, but no model leads to a perfect match to the observations. We then try to model the rupture process dynamically supposing not the kinematic scenario but the physical parameters (stress field and rupture criterion). We assume a uniform shear stress field (5 MPa) except for a patch representing heterogeneity (if any) as demonstrated in Figure 2. A simple slip-weakening law is assumed (peak strength = 10 MPa, residual stress = 0 MPa, critical slip displacement = 5cm). Introducing a patch with higher initial stress generates heterogeneity in the rupture process. On the patch, the rupture passes locally faster than in other locations and leads to larger slip. We calculate the ground motion from this source model again, as shown in Figure 5, in which, for reference, the case of uniform dynamic rupture (without the patch) and the kinematic source model are also shown. The synthetic seismograms still have a smoother waveform regardless of the heterogeneity. The ground motion is a convolution over the fault plane, so the rupture propagation on the rest of the fault plane other than the given small patch affects it. We encounter two difficulties. The station is unfavorably located with respect to the S-wave radiation from the assumed fault plane, and it is difficult to give the dynamic model a strong heterogeneity at the beginning without significantly affecting the later rupture process. As recent geodetic studies imply [START_REF] Takada | Post-seismic surface deformation following the 2008 Iwate-Miyagi Nairiku earthquake detected by ALOS/PALSAR[END_REF], the true fault geometry may not be as simple as a single fault.

2.2.3.

Discussion -do we really know the earthquake process? In fact, we have been trying to simulate the recent earthquakes according to the known parameters (source parameters and structure) [START_REF] Aochi | Influence of super-shear earthquake rupture models on simulated near-source ground motion from the 1999 Izmit (Turkey) earthquake[END_REF]. We think that such reproduction process is important, because predicting the ground motion in any given situation is the objective of the quantitative seismic hazard assessment. From our several experiences, we find that the "known" parameters are not always satisfactory to explain the phenomena. This is because the earthquake process is too complex to determine all the numerous parameters necessary to construct the models. Any parameter is obtained under some hypothesis, e.g., fixing the other parameters. This is also why inversion analyses sometimes provide different models even using the same data set.

The seismology (and also the other domains of geoscience) is always in a knowledge gap; we try to model more precisely the phenomena using any numerical sophisticated method and intensive calculation (forward modeling), while the model parameters are not acquired precisely enough regardless of the increase of high-quality various data (inverse modeling). The sustainable efforts to fill in this gap are required, since this cannot be done easily. The intensive numerical simulation will be able to be a powerful tool towards this objective.

Perspectives and Conclusion

Nowadays, when the accuracy of the calculation is aimed, higher-order finite element methods (spectral element or discontinuous Galerkin) could be better regardless of their costs [START_REF] Komatitsch | The spectral-element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF]. Each method always has its merits and limits so that it is waited that various methods are developed and utilized. Such numerical codes have become also popular for engineering/industrial purpose such as the quantitative seismic hazard/risk assessments [START_REF] Douglas | A survey of techniques for predicting earthquake ground motions for engineering purposes[END_REF], however guaranteeing the numerical performance on parallel computers and obtaining the physically reasonable solutions are difficult tasks. Since 2000, various international benchmark and cross-checking tests have been also performed [START_REF] Day | Tests of 3D elastodynamic odes: final report for lifelines project 1A01[END_REF]. This is necessary for us to advance beyond.

The finite difference formulation for the seismic wave propagation is the most primary scheme. However, this always shows a good (computing) performance and this is often satisfactory for tackling the unsolved physical problems, as shown briefly in this paper. The basic formulation of finite difference is very well established and we can relatively easily handle the intensive calculation. Thus making full use of it, we are now able to explore the earthquake dynamics and the consequent ground motion. Coupling with a boundary integral equation is a practical option to take advantage of both methods. At the end, we precise that the finite difference scheme is classical, but the formulation always evolves for improving solutions and developing for other utilities [START_REF] Kozdon | Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods[END_REF].

Figure 1 :

 1 Figure 1: Illustration of standard staggered grid and partially-staggered-grid. The denoted components are defined half a grid apart from the others. For simplicity, equal grid size is taken for all the directions.

Figure 2 :

 2 Figure 2: Finite difference simulation of wave propagation for the 2011 Tohoku-oki earthquake. A simple source model (a huge ellipse with Gaussian slip distribution and a constant rupture speed) is assumed (Madariaga, pers. comm. 2012). The waveforms (deformation) applied with a low-pass filter at 50 s are shown at four GPS stations. FDM1: Absorbing boundary condition is adapted for a period of 100 s, FDM2: the same code in the same condition except for the absorbing boundary condition calibrated for a conventional seismological frequency range of 1 s. Syn. Ref.: Synthetic reference solution calculated with a discrete wavenumber method in the same 1D layered structure. Observation: displacement record observed by high-sampling (1Hz) continuous GPS network of the Geographical Survey of Japan.

AFigure 3 :

 3 Figure 3: simulation strategy of earthquake dynamics to wave propagation starting from the physical and geological initial condition (A), using a boundary integral equation (B) and a finite difference method (C). The shown example is the case of the Mw7.3 1992 Landers earthquake [13].

Figure 4 :

 4 Figure 4: Map of the fault model (courtesy of Dr. Asano [14], slip distribution in color with arrows) of the 2008 Miyagi-Iwate earthquake in the left panel. Epicenter is shown as a star. A seismograph station, IWTH25, is located above the fault plane and close to the estimated epicenter, and this station recorded the very strong ground motions (100 cm/s) as shown in the right panel. The three components recorded at borehole (260 m depth) and the trajectory between 2 and 4 seconds, the arrival of the strong pulse, are shown. Very strong vertical motion is recorded at around 3.5 second.

Figure 5 :

 5 Figure 5: Comparison of seismograms (low-pass filtered at 1 Hz) at IWTH25 at depth for various source models. The calculation is made by the finite difference method using a 1D structure model beneath IWTH25. From top to down, dynamic rupture model presented with heterogeneity, dynamic rupture model under the homogeneous condition, kinematic rupture model of Asano and Iwata (2008), and the observation. For the three synthetic models, the origin time is set at t=0. Because of uncertainty in the real origin time, the observation may be shifted in time. The dynamic rupture model provides a shaper pulse, but no model can explain the pulse of the first two seconds.

Acknowledgments

This paper is a contribution to the French national project ANR DYNTOHOKU. Most of the recent numerical simulations are carried out at the French national computing center GENCI-CINES (grant 2012-46700), but some other results realised in previous projects are also included (BRGM-ANR-CARNOT). The data recorded by National Institute of Earth Science and Disaster Prevention, Japan (strong ground motion) and Geographical Institute of Japan (GPS) are used.


F Rosquoët

A Alexis

A Khelidj

A Phelipot

Experimental study of cement grout: rheological behavior and sedimentation

Keywords: Cement paste, Rheology, Hydration, Bleeding

Three basic elements (cement, water and admixture) usually make up injectable cement grouts used for prestressed cable coating, repair and consolidation of masonry, soil grouting, etc… The present study was divided into two parts. First, in order to characterize rheologically fresh cement paste with W/C ratios (water/cement ratio) varying between 0.35 and 1, an experimental study was carried out and has revealed that the cement past behaves like a shear-thinning material whatever W/C. second, to study the time evolution of their density, a gammadensitometer bench was used. Relying on the water content and the density measured, we demonstrate that the computation of the degree of hydration of cement is possible. The cement/geotechnics interdisciplinary approach proposed here has made it possible to obtain a large range of original results useful to improve our understanding of the sedimentation processes for cement pastes with different W/C ratios.

Introduction

Injectable cement grouts are of great use in many construction domains: prestressed cable coating, repair and consolidation of masonry, soil grouting, etc… Their coating and consolidation functions are characterized by a set of physico-chemical properties of the material in-place. We have, for instance: -Level of compactness -Porosity -Mechanical properties (failure stress, etc.…) -Durability Beside those, the material has to be both injectable and stable at a given instant. Its consistency (liquid or plastic) must, allow for placing and filling of the voids. The material, moreover, must be homogeneous, i.e., showing no sedimentation or bleeding phenomena. In the absence of chemical admixtures, cement grouts are compared to cement pastes used to prepare some specific concrete types and, therefore, play a preponderant part in the rheological behavior of concrete (Self compacting concrete, in particular). The study of the rheology and the sedimentation of these mixtures becomes then essential. Both are discussed here. We have divided our study into two parts. First, the rheological behavior and the sedimentation of cement pastes with W/C ratios in the range 0.35-1 have been experimentally studied. Second, using the gammadensimetric measurement of the density and the values of free water achieved by drying the specimen at 105°C, we demonstrate that the theoretical computation of the degree of hydration of a cement paste is possible. The multidisciplinary approach presented here is innovative because combining theories applies to both geotechnics and cement [START_REF] Mckinley | A geotechnical description of fresh cement grout -Filtration and consolidation behaviour[END_REF].

Material studied

Field of use

The information about the various mixtures used to produce cement grouts found in the literature [START_REF] Woodward | Grouting Post-tensioned concrete Bridges : The prevention of voids[END_REF][START_REF] Miltiadou | Etude des coulis hydrauliques pour la réparation et le renforcement des structures et des monuments historiques en maçonnerie[END_REF][START_REF] Benhamou | Comportement rhéologique des coulis de liants hydrauliques ultrafins destinés à l'injection[END_REF][START_REF] Danot | Grout injecttion in the laboratory[END_REF], has been used to select a range of W/C values for our study. Considering this range and in order to obtain homogeneous pastes, a particular mixing procedure has been developed here.

Cement pastes (or cement grouts) designate a fluid mixture of cement, water and, possibly, admixture. The fields of use for cement pastes are many: filling, sealing, consolidation, ... For the coating of prestressed cables, the W/C ratios found in the literature range between 0.35 and 0.42. [START_REF] Woodward | Grouting Post-tensioned concrete Bridges : The prevention of voids[END_REF]. Hydraulic cement grout for the repair and consolidation of masonry structures have W/C ratios in the range 0.5-1.5 [START_REF] Miltiadou | Etude des coulis hydrauliques pour la réparation et le renforcement des structures et des monuments historiques en maçonnerie[END_REF]. Cement grouts for soil or rock injection are very fluid hydraulic binders with W/C ratios between 1 and 2 [START_REF] Benhamou | Comportement rhéologique des coulis de liants hydrauliques ultrafins destinés à l'injection[END_REF]. Regarding sealing cement grouts, W/C ratios are similar to those used for the repair and consolidation of masonry and range between 0.5 and 1 [START_REF] Danot | Grout injecttion in the laboratory[END_REF]. Figure 1 schematically presents the different fields of use. Many parameters are susceptible to affect cement paste characteristics and, the W/C ratio in particular. In order to address the influence of this ratio on cement paste characteristics, only neat cement, i.e., a fluid mixture consisting of cement and water, has been considered here. Because of the above-mentioned fields of use, seven water/cement ratio factors have been considered here: W/C = 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 1.

Composition of the material

All cements used in this study are Portland cement CEM I 52.5 PM ES CP2 from Teil in France.

Production

The procedure of the mixing cement paste (laboratory mixer with paddle) is sumarized in table 2.

Table 2: Mixing procedure for tests.

For convenience, we agree that, during the whole study, time t 0 corresponds to the time 30 second after mixing.

Setting time test (Standard NF P18-362)

The setting time test is used to determine when the skeletal structure forms. Initial and final set times are measured using the Vicat apparatus in conformity with standard NF P18-362. The results are displayed in Figure 2. We notice, that, whatever the value of the W/C ratio, the initial set time increases almost twofold when W/C is doubled while the time between initial and final set remains more or less constant (approximately two hours).

Study of the rheological behavior

Measurement of the rheological characteristics

In order to determine all the properties of their rheological behavior, all seven water/cement ratio are tested using a rheometer [START_REF] Chiara F & Ferraris | Measurement of the Rheological Properties of high performance concrete : State of the art report[END_REF]. All measurement have been carried out using a type Rhéomat 115 rheometer with a MS 145 type coaxial cylinder spindle. A thermal control system is used to maintain the temperature constant. All measurements are carried out at t 0 (30 sec. after mixing) and at the constant temperature of 20°C +/-1°C. The volume of the preparation is 50 ml. The 15 ml required for measurement are sampled in the center of the receptacle take the cement paste. For each W/C ratio, the measurements are carried out by varying progressively the shear rate from 23 to 1200 s -1 . For each shear rate, the value of shear stress measured is the value after a minimum of 30 seconds without fluctuation. The trace of the tangent for each curve is used to determine the apparent viscosity (η a ) and an apparent yield shear stress (τ a ) of the paste tested [START_REF] Chappuis | A new model for a better understanding of the cohesion of hardened hydraulic materials collords and surfaces[END_REF]. Fig. 4 : Evolution of apparent viscosity and apparent yield stress as a function of the W/C ratio.

Result post-processing

Figures 3 and4 show that both the apparent viscosity and the apparent yield increase as W/C decreases. For a W/C ratio of 1, viscosity and apparent yield are both low, whereas the apparent viscosity is much higher for a W/C ratio of 0.4. For a better understanding consideration of the shear thinning behavior of the material, it is possible to determine the evolution of the viscosity as a function of the shear rate by calculating the slopes of each time interval [n, n+1].

• + • + • γ - γ τ - τ = γ ∆ τ ∆ = η n 1 n n 1 n (1)
The viscosity values thus determined are presented in Figure 5. The short duration of the test and shearing does not allow the bleeding. We notice here that the lowest viscosity value (for W/C = 1) is thirty times higher than that of water (10 -3 Pa.s). Moreover, the increase in viscosity is sharp for shear rates lower than 300 s -1 and varies depending on the W/C ratio studied. The lower the W/C ratio is, the higher the viscosity increases (range: 0.03-0.3 Pa.s). Generally, the behavior of the cement grout can be compared to a shear thinning fluid (viscosity decreases as shear rate increases). A bibliographic study is conducted to describe various rheological behavior models for cement pastes. A model with a yield shear stress of Herschel-Buckley type appears as:

• + = n Aγ τ τ 0 (2)
This model, however, does not suit satisfactorily rheological behaviors of our tested pastes, which do not show any clear yield stress (Figure 3).

A comparison between measurements (Figure 3) and literature [START_REF] Papo | Rheological models for cement pastes[END_REF] finally allows for the selection of a simple model of the type « Power Law » written as:

n A • = γ τ (3)
By assessing the reference shear rate • γ 0 (1000 s -1 was chosen for commodity) coefficient A takes a stress dimension.

n A           = • • 0 γ γ τ (4)
From the shear stress vs. shear rate curves (Figure 3), it is possible to determine the coefficients A and n of the « Power law » model equation (Table 3).

Table 3: Coefficients A and n for the different W/C ratios.

From these values, it is possible to work out a simple model to determine the evolution of the shear stress as a function of the W/C ratio and of the shear rate. Because of the marked disagreement of the values achieved with a W/C of 1 (bleeding in excess), this ratio is not considered here. In a first approximation, coefficient n is assumed as the algebraic mean rounded to the first denary, namely 0.6. For coefficient A, a linear regression of the points according to the W/C ratio is performed (figure 6). The model is then written in the form:

6 . 0 0 137 C W 175           γ γ       + - = τ • • (5)

Study of the sedimentation

The objective of this research is to examine the sedimentation of a cement paste from the end of mixing to the complete setting by following the evolution in time of the specific density of a column of fresh cement grout for different W/C ratios using a gammadensitometer bench [START_REF] Bras | Etude expérimentale du début de consolidation de sédiments fins[END_REF].

Description of the experimental bench and measurement principle

The experimental density measurement bench [START_REF] Gallois | Settling colums parametric tests applied to coastal sediment consolidation[END_REF][START_REF] Bras | Sedimentation and consolidation of a high water content soil[END_REF][START_REF] Bras | Consolidation of high water content soils in an estuarine environment[END_REF] consists of a radioactive source (gamma ray), a scintillation counter and a computer. The counter, placed on a carriage, moves vertically by means of a back-geared motor unit combined with a speed regulator and a position transducer. A power station provides controls and data acquisition. All tests are performed in specific protected room, which is used at same time for soil testing (requiring a constant temperature of 10 °C ± 1°C) (Figure 7). Density is measured using gamma rays from a radioactive source. The scintillation counter is fixed to a moving carriage. A linear position transducer is used position the device very accurately ( ± 1 µm) up to 2000 mm. The data unit controls, on one hand, the movement of the carriage and, on the other hand, stores the data collected by the density calculator. A computer collects and analyses all data. The calibration of the gammadensitometer bench is necessary to achieve density measurements for a given type of material. The three-point calibration chosen here is completed using different samples with W/C ratios ranging between 0.3 and 0.4 (knowing that absorption coefficient is constant for a given composition of grains).

Methodology of tests

Tests are carried out inside 1500-mm high PVC tubes with an internal diameter of 94 mm. Because all test tubes are new, no special preparation is required. The test tube is tightly closed at bottom with a specific stand which is first coated with a demoulding agent. The test tube is then fastened to the wall in two places using clamping rings. The frame is positioned and the carriage is moved upwards to check the correctness of the alignment of the column body and of the carriage trajectory. The frame is finally fastened to the wall to prevent it from shifting during the tests. The initial height of the scintillation counter in the lowest position is adjusted using a calibrated height setting plate. The material, prepared in accordance with the procedure described in Table 1, is poured as quickly as possible at the end of the mixing phase into the column up to 1400 mm. An extension (a short tube closed by a plastic film) is then added on top of the column, which guarantees that the clamping system of its higher part is out of the trajectory of the scintillation counter. The test tube is closed on top, ensuring the tightness to avoid losses due to evaporation. Each test lasts 24 hours, during which the moving carriage executes many cycles. One cycle can be split up into one descending half-cycle following one ascending half-cycle. For all points, the central station records simultaneously the time, the position of the carriage in relation to a preset reference and the density. Measurements are carried out every 50 mm for the 1400 mm of the total height. One cycle lasts eight minutes (Figure 8). For each height, the specific density values retained correspond to the mean density measured during both ascending and descending half cycles. These mean density values are associated with the starting time of each cycle. Density measurements achieved with the gammadensitometer bench are accurate to approximately ± 0.02.

Results and comments

For all the different W/C ratios studied (0.35 ; 0.4 ; 0.45 ; 0.5 ; 0.6 ; 0.7 ; 1), density measurements are carried out during 24 hours. The results obtained are used to plot density profiles. Initial specific gravities are presented in table 4 for all the W/C ratios studied. Figure 8 presents the evolution of the density for a cement paste with a W/C ratio of 1 throughout the height of the sample. Density curves are plotted every 64 minutes (that is to say, every four cycles). After the 55 cycle (7hr12min), the density does not vary anymore, which corresponds to the beginning of the formation of the rigid skeleton. In the case of Figure 8, we also notice the increase of the specific density in the lower part and a decrease in the upper part. Cement grains fall to the bottom of the column. The fall of the grains and the rise of the water particles compose what is called the sedimentation process [START_REF] Gallois | Modélisation du processus de sédimentationconsolidation des vses estuariennes : approche continue unidimensionnelle[END_REF]. Sedimentation increases the density from a maximum value located at the bottom of the column to the value of the initial density located higher in the column. This interface point can vary with time. The longer the time between t 0 and the studied cycle is, the higher the interface point, when initial density is reached. Density variations stop after seven hours. The continuous measurement using the gammadensitometer bench of cement pastes during 24 hours accounts for density history in time. Figure 9 presents the density curves at t 0 and at t = 24hr for all the pastes studied. In all cases, we observe that density remains constant throughout the whole height of the column at t 0 , which proves that the cement paste is homogeneous. For cement pastes with W/C ratios in the range 1-0.5, density increases sharply with time at the bottom of the column, while it decreases at the top of the column because of the sedimentation of solid cement grains. For W/C ratios ranging from 0.4 to 0.35, density does not vary much with time, the cement pastes remaining homogeneous. For the W/C ratio equal to 0.45, a sharp decrease of the density is observed locally at a height of 450 mm. A close examination of the column at the end of the test reveals that this peculiarity is produced by the appearance of a water pocket at this point.

Evolution of the degree of hydration

A theoretical approach from the geotechnic domain can be applied here to draw a parallel between density measurements using the gammadensitometer bench, the free water content (measured by drying) and the theoretical quantity of water necessary for the hydration given by Bogue [START_REF] Bras | Etude expérimentale du début de consolidation de sédiments fins[END_REF]. For all the columns used for the measurements of the density, measurements of the quantity of free water are carried out by drying at 105°C the specimen. The free water was previously planned to be deduced at 28 days, as a reference time. In order to reduce the duration, we adopt a t = 10 days reference time.

Evolution of the phases

The decomposition of the cement paste using two simplified phase diagrams and neglecting the voids (or entrained air) is possible (Figure 10).

First of all, before the occurrence of the chemical reaction between cement and water, the solids are composed of anhydrous cement. The water content of the cement paste is equal to the W/C ratio. When water is in contact with cement the hydration begins and water consists of part free water and part chemical bonded water (water of hydration) [START_REF] Baroghel-Bouny | Caractérisation des pâtes de ciment et des bétons, Méthodes, analyse, interprétations[END_REF]. Two types of water content are then distinguished:

The free water content:

S wf f W W w = (7)
The bonded water content:

c wl l W W w = (8)
The bonded water content is the amount of water reacting with anhydrous cement. For a given cement composition and after full hydration (infinite time), the bonded water content assumes a specific value (w B ) given by Bogue's equation [START_REF] Bogue | La chimie du ciment portland[END_REF]. For the cement composition studied here, w B = 22.93%.

Conventionally, the degree of hydration (α) is defined by:

B l w w = α (9)
The measurement of the density γ given by the gammadensitometric bench:

s wf s wf V V W W V W + + = = γ (10)
Introducing the free water content, w f we found:

s w f f 1 w 1 w γ + γ + = γ (11)
w f f s w 1 w 1 ence h γ - γ + = γ (12)
Equation 12 can then be used to calculate the mass density of the solid phase, as a function of the drying-measured free water content and of the density measured using the gammadensitometer bench. Moreover, the solids formed by the reaction of hydration combine bonded water with anhydrous cement:

c wl S W W W + = (13)
During the tests, the observed volume reductions were less than 2%. The Le Chatelier contraction can therefore be neglected:

c wl S V V V + = (14)
Solid mass density is then expressed with the bonded water content: (For a specific gravity of 3150 kg/m 3 for cement and of 1000 kg/m 3 for water) which can also be written as: By combining Equations ( 12) with [START_REF] Bogue | La chimie du ciment portland[END_REF], we obtain the expression of the bonded water content as a function of mass density (measured using the gammadensitometer bench) and free water content (measured by drying at 105°C):

wl c wl c s s s V V W W V W + + = = γ (15)
            - - γ γ γ γ - + = f w c f l w 1 1 w 1 1 w (19)

Analysis of the results

From Equation (19) we calculate the degree of hydration α as a function of the water content and the density measured using the gammadensitometer bench after ten days. To obtain a value representative of the whole column, the mean degree of hydration is calculated as: The degree of hydration, of pastes with W/C ratios varying between 0.35 and 0.5, is approximately constant throughout the height of the column. For W/C ratios in the range 0.6-1, on the other hand, the upper part of the column reveals some degrees of hydration near 100%. This difference can be accounted for by the non consideration of the physical phenomena occurring during the hydration of cement pastes and linked to sedimentation. Following these measurements, we observe that the sedimentation initiates the upward migration of ion-loaded water. After the beginning of sedimentation, the upper part of the column could therefore present more water and above all more chemical constituents than the lower part. As a consequence the local chemical composition of cement is not verticaly homogeneous within the column anymore. This composition is now different from the initial and global composition of mixing cement. For this reason, the quantity of water necessary for a complete hydration of mixing cement w B (determined using Bogue's equation and global cement composition), does not fit its local value after sedimentation when cement set. The data available at the laboratory about the follow-up of cement paste hydration have revealed that, for a W/C ratio of 0.35, hydration reached 67% after ten days. This value agrees satisfactorily with our computations (63 % for the same W/C) of the degree of hydration as a function of both water content and density measurement.

Conclusions and perspectives

Injectable cement grouts usually consist of three basic elements (cement, water and admixtures). For the convenience for this first approach, we have considered here neat cement pastes only, i.e., not addition of admixture. In order to examine the impact of the ratio of water mass to that of cement (W/C), the parameter has been varied in the range 0.35-1. After a considerable measurement series, data processing has revealed that the cement paste studied here behaves like a shear-thinning material. This information is very significant to improve our understanding of the material behavior, which stands at the limit of the traditional domain. In less than twelve hours, the material changes from a fluid to the paste and finally to the solid phase. For this reason, the studies conducted to examine the material include several scientific domains (fluid, soil and solid mechanics). Then, in order to address the variation of the specific density of a cement paste with time, the study conducted by the author using a gammadensitometer bench is described. The device is ussually used, at the Civil Engineering Laboratory of Nantes Saint-Nazaire, to study mud sedimentation history. Because of the close similarities between muds and cement pastes (presence of suspensions in water in both cases), the gammadensitometer bench has required no special adjustment. In view of the results, two different groups of cement pastes have been distinguished. For high W/C ratios (within the range 0.5-1), the material is heterogeneous and we verify the presence of a large quantity of bleeding water. At t 0 , however, the material is very fluid. Conversely, for low W/C ratios (within the range 0.35-0.45), the material is verified as homogeneous and with no bleeding water. The much higher viscosity, however, can prove a drawback for injectability. The present study demonstrates the presence of phenomena closely linked to sedimentation and reveals the interaction between sedimentation (occuring with high water content pastes) and hydration. Subsequent developments of this research could begin with the improvement of our understanding of sedimentation evolution using isodensimetric profile modeling and of the material final state thanks to a sharp and local analysis of the material (thermogravimetric analysis) and to the measurement of pore spaces. And then, in order to confirm the relevance of the present research, a study to examine real cement grouts (with admixture addition) could be conducted.

Fig. 1 :

 1 Fig. 1 : Summary scheme of the different fields of use of cement grouts.

Fig. 2 :

 2 Fig. 2: Evolution of initial and final set times as determined by Vicat test for different W/C ratios.

Figure 3 Fig. 3 :

 33 Figure 3 presents the evolution of the measured shear stress as a function of the imposed shear rate for different W/C values. All tests are repeated twice, with a new batch of cement paste. The tests are repetitive. The evolution of shear stress as fonction of shear rate for each W/C ratio is a average of two tests

Fig. 5 :

 5 Fig. 5: Evolution of the viscosity as a function of the shear rate.

Fig. 6 :

 6 Fig. 6: Evolution of the coefficient A and n as a function of the W/C.

Fig. 7 :

 7 Fig. 7: Schematic diagram of the gamma-densitometer bench.

Fig. 8 :

 8 Fig.8: Density profiles for a cement paste with a W/C ratio of 1 during 7hr12min.

Fig. 9 :

 9 Fig. 9: 24-hour density history for W/C ratios from 1 to 0.35. t0 ; t = 24hr

Fig. 10 :

 10 Fig. 10 : Simplified phase diagram.

  16) we find that Bogue's bonded water content (w B ) corresponds to a solid mass density (γ SB )

Table 5 :

 5 Degree of hydration.

Table 1 :

 1 Bogue composition of the Portland cement CEM I 52.5 PM ES CP2 from Teil (France).

	The Bogue composition of the Portland cement used during our tests, is detailed in table 1.
	According to Bogue formula and this table, the mass of water necessary to compete hydration
	per unit of anhydrous cement is 22.93%.

Table 4 :

 4 Initial density.

Acknowledgments

The assistance of Dr. Agnès MONTILLET, assistant professor at the Saint-Nazaire Technical Institute (L.G.P., C.R.T.T.), which put the rheometer at our disposal, is sincerely appreciated. Special thanks to Mr. Vincent PICANDET, who has been a great help for the power station programming. With our grateful thanks for Mr. Roger COUE, technical collaborator, for his assistance during bench setting and column tests.