# Towards a better forecast for reservoir slope sliding

Didier Marot, Alain Alexis, Pierre Thomas

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Didier Marot, Alain Alexis, Pierre Thomas. Towards a better forecast for reservoir slope sliding. International Journal of Sediment Research, 2003, 18 (4), pp.278-289. hal-01007390

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# TOWARDS A BETTER FORECAST FOR RESERVOIR SLOPE SLIDING

Didier MAROT<fnref n="1" />, Alain ALEXIS² and Pierre THOMAS³

The siltation of reservoir causes many problems of management and environmental disturbances that are important to minimize. Solving these problems requires in particular a good knowledge of mud bank instabilities. In partnership with the Laboratoire National d'Hydraulique (LNH), a subsidiary of Electricité de France (E.D.F.), authors realize a study of soft soils sliding entailed by reservoir emptying. Using a geotechnical flat sliding mechanism, the theoretical formulation of the safety factor can be established when the sediment mass is submerged and emergent. The measurement campaign carried out makes it possible to obtain bathymetric and topographic surveys and a geotechnical characterization of the soil for the whole reservoir of Grangent dam, located on the Loire (France). The proposed model appears to agree satisfactorily with the instability of the zones emerged because of the emptying and with the stability of the profile after emptying. The use of this model within the framework of forecasts makes it possible to calculate significant minimal displaced heights.

**Key Words:** Soft soils, Mud, Bank, Sliding, Dam reservoir, Emptying

## 1 INTRODUCTION

Hydroelectric dams and their storage reservoirs allow for the production of electricity, the creation of water reserves for consumption or irrigation, and the regulation of river flow rate. The long-term future of these installations requires the dredging (often non-viable economically) or the emptying of storage reservoirs. This release of stored water is carried out with two objectives:

- periodic inspection of the structure and the carrying out of maintenance work,

- expulsion (flushing) of sediments accumulated as upstream slopes of the dam.

The improvement of the sedimentary control and the development of operational solutions require increased knowledge of the impacts of emptying a storage reservoir on the slope stability of its silt deposition. Bouchard (1995), Paquier and Khodashenas (2002) presented numerical codes which simulate erosion and deposition in a storage reservoir at times of high waters. However, emptying is also accompanied by a considerable amount of slide of the upstream slopes. Zhou and Lin (1998) arbitrarily set a maximum slope gradient for the sediments, but the difficulty lies in the evaluation of this maximum slope as this is influenced by several factors:

- the thickness of the sediment mass

- the slope of the bedrock

- the soils characteristics

The objective of this study is thus to propose a criteria for upstream slope slide during emptying of storage reservoirs of dams, and to then validate it on a real site: the Grangent dam (Loire-France). An initial criteria is developed with a rectangular-shaped slice (Alexis et al., 2001). It supposes that the shear plane and the water-sediment interface are parallel. In order to deal with a case more typical of the natural configuration while freeing itself from this simplistic assumption, we develop a sliding criteria for a slice of sediment with a trapezoidal shape. Moreover to improve the validation of proposed criteria, the topographic profiles carried out before the construction of the dam are exploited.

<footnote n="1">¹ Doctor, ² Doctor-Engineer, ³ Professor, IUT de Saint Nazaire, Laboratoire Génie Civil Nantes-Saint Nazaire, BP 420, 44606 SAINT NAZAIRE cedex, FRANCE, Tel.: 00 33 2 40 17 81 50, Fax: 00 33 2 40 17 81 60,</footnote>
## 2 BIBLIOGRAPHICAL ASPECTS

Landslides can be classified into two great rheological families (Filliat, 1981): sliding in the traditional sense of the word and mud flows. The move to the second type may in particular be obtained by increases in the water content. We therefore need to model the falling of upstream slopes in accordance with these two approaches.

### 2.1 Hydraulic Flow Criteria (Mud Flows)

The low consistency of the mud makes it possible to consider it as a fluid. Its behaviour can be predicted by various models, including viscoelastic models (Feng and Mehta, 1995) and viscoplastic models (Coussot and Piau, 1994). The plasticity of material results in the need to reach a threshold stress value in order to start the movement. Coussot (1995) specified that in order for flow to take place, it is necessary that the thickness of the fluid $h_e$ should satisfy the following inequality:

$$h_e \geq \frac{\tau_s}{\gamma_{sat} \sin \alpha} \quad (1)$$

$\tau_s$: threshold shear stress; $\gamma_{sat}$: bulk density of the saturated fluid; $\alpha$: inclination of the slope.

This model supposes that the shear plane and the water-sediment interface are parallel.

When the shearing of the fluid gets underway, the velocity profile depends on the behavioural law selected (Suhayda and Prior, 1978; Périgaud, 1983; De Kee et al., 1990).

### 2.2 Geotechnical Slip Criteria

Although of low consistency, the mud can also be regarded as a soil. The study of the stability of the sediment massif is then carried out according to two approaches: the finite element method, which is a numerical method of calculation of the variational type (Zou et al. 1995), and the limit failure state design. The limit failure state design involves studying the stability of the sediment mass at the boundary equilibrium, i.e. when the failure is on the point of occurring. This study is carried out by the introduction of the safety factor (F) which is the ratio of the shear strength to the shear stress. The unstable volume can be divided into a certain number of vertical slices and the boundary equilibrium of each slice is studied. Two types of calculation are then possible. The first is based on the formulation of the safety factor for an arbitrarily defined sliding surface. This study can be undertaken using two dimensional models (Okagbue and Ifedigbo, 1995) and can also take account of the evolution of the pore pressure which influences the stability of the sediment mass (Fredlund et al., 1993). Lam and Fredlund (1993) have for their part carried out a three-dimensional analysis. The second type of calculation by the slice method is based on researching the critical sliding surface, obtained by minimizing the safety factor. This safety factor and the corresponding critical sliding slope gradient can then be found by various numerical methods, in two dimensions (Chen, 1992) or even in three dimensions (Yamagami and Jiang, 1997).

The two dimensional general expression of the equilibrium of each slice leads to four equations:

- 2 equations of force equilibrium,

- 1 equation of moment equilibrium,

- 1 equation of the Coulomb criterion.

Thus for a sediment mass which comprises n slices, 4n equations are obtained (the three dimensional study results from a generalisation of the equations obtained).

The resolution of the equilibrium of all the slices leads to (6n - 2) unknown factors. There are thus (2n - 2) equations missing to solve the problem, and we must introduce various simplifying assumptions.

An assumption is generally made: the normal force applied to the lower surface of the slice is assumed to apply to the centre of this surface. The number of unknown factors is thus reduced to (5n - 2). Consequently (n - 2) assumptions are necessary to the determination of all unknown factors. These assumptions generally relate to the shape of the sliding surface, the application point of the inter-slice forces or to the expression of these inter-slice forces. In particular, we can mention Bishop (1954), who supposes that the sliding line is circular and that the vertical component of the inter-slice forces applied to a slice cancels itself out.
### 3 CRITERIA OF THE TRAPEZOID FLAT SLIDE

Soft soils depositing within barrage reservoirs are characterized by their low strength and their high water content. These characteristics imply the development of specific slip criteria and the justification of used assumptions. The observation campaigns carried out during several emptying (Marot, 1998) enabled us to underline the presence of flat sliding that follows the level of water along the banks of the river. We did not observe discontinuity in the transverse direction. The two dimensional model of flat sliding therefore appears to be adequate to describe in-situ slides.

Thus, we develop a flat-sliding criteria for a slice of soil with a trapezoidal shape. Fig. 1 represents the slice studied, with the forces applied to it.

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Schematic representation of the trapezoid flat slide</figcaption></figure>

The notations used are the following:

$H_i$ height of water above the side surface i

$h_i$ height of the side surface separating slice i from the slice i-1

$W_i$ weight of slice i

$P_{wi}$ force resulting from the pressure of the water applied to the upper surface of slice i

$T_{ti}$ and $\Sigma_{ti}$ tangential and normal components of the force applied to the lower surface of the slice i

$\Phi_i$ force applied by slice i-1 to slice i

$f_i$ and $v_i$ horizontal and vertical components of the inter-slice force $\Phi_i$

$\alpha_i$ inclination of the lower surface of slice i

$\theta_i$ inclination of the upper surface of slice i

$b$ width of the slice

Assumptions: the material is assumed to be

a1) homogeneous,

a2) saturated,

a3) normally consolidated (overpressure is zero compared to the surrounding pressure).

Assumption b): the failure occurs in undrained conditions.

Justification of assumption a1: the profiles of total stress and undrained shear strength obtained by different researchers (Mekerta, 1995; Alexis et al., 2001) for this type of soils allow us to suppose the sediment massif to be homogeneous.
Justification of assumption a2: given the low incidence of non-saturation on the effective stress (Marot et al. 1998) we can assume that the sediment mass is saturated.

Justification of assumption a3: sedimentation and sediment consolidation background storage reservoir is often regular. We can thus suppose that the sediments are normally consolidated prior to emptying. The consolidation phenomenon is often caused by a loading, which creates a variation in the volume of the soil. There is thus a change from a state of hydrostatic balance to a secondary state of hydrostatic balance. For this to happen, there is a dissipation of pore pressure that is accompanied by an increase in the effective stress.

On the other hand we have seen that the mud can be considered as a saturated fluid, thus the lowering of the water level is instantaneously transmitted to pore pressure (incompressible fluid) which remains hydrostatic. Consequently the lowering does not bring about a variation of volume within the material, and does not create pore overpressure. The effective stress thus does not undergo any modification and the material remains normally consolidated.

Justification of assumption b: to carry out laboratory drained shear tests on fine soils samples of low thickness, the shear velocity must be very low (lower than 25 µm per min or 3,6 cm per day, AFNOR NF P 94-071-2). However in-situ shearing involves great thicknesses of sediments, and the speed of the drawdown of the storage reservoir, which heavily affects the shear velocity, is generally about several tens of centimetres per day (with a maximum of 10 cm per hour). We can therefore suppose that shearing appearing at the time of the slope slide occurs in undrained conditions.

### 3.1 Expressing Sliding Criteria as Equations

For convenience and to simplify, we consider slice number 1.

In accordance with the assumptions of Bishop (1954), we suppose that the vertical components of the inter-slice forces are equal: $\Delta v = v_1 - v_2 = 0$. In a sediment mass, the effective horizontal ($\sigma'_h$) and vertical ($\sigma'_v$) stresses are linked by the earth pressure coefficient ($K_0$):

$$ \sigma'_{h} = K_{0} \sigma'_{v} \qquad (2) $$

The interstitial fluid is considered saturated, its pressure is noted as $u_w$.

Therefore $\sigma_h = K_0 \sigma'_v + u_w$

$$ \frac{\sigma_h}{\sigma_v} = K_0 \left( 1 - \frac{\gamma_w}{\gamma} \right) + \frac{\gamma_w}{\gamma} \qquad (3) $$

$\gamma_w$: density of water (9.81 kN/m³).

Whatever the value of $K_0$, the total stress ratio in case of soft soil is around the unit. The values of $K_0$ for the mud being very difficult to measure but around the unit (Schlosser, 1988), we can suppose that, as in a fluid, $\sigma_h = \sigma_v$.

The force $f_l$ can then be expressed by:

$$ f_l = \int_{0}^{h_1} \gamma z dz + \gamma_w H_1 h_1 L \qquad (4) $$

i.e.:

$$ f_l = \frac{1}{2} \gamma L h_1^2 + \gamma_w H_1 h_1 \qquad (5) $$

$$ \text{thus} \qquad \Delta f = \frac{1}{2} \gamma L (h_1^2 - h_2^2) + \gamma_w L (h_1 H_1 - h_2 H_2) \qquad (6) $$

With the assessments of the vertical and horizontal forces, we obtain the expression of the safety factor :

$$ F = \frac{c_u L \frac{b}{\cos \alpha}}{W \sin \alpha - \Delta f \cos \alpha + P_w \sin (\alpha - \theta)} \qquad (7) $$

The study of the stability of the slice is carried out according to two states of loading:

- with the slice completely submerged (high water level)

- with the slice completely emergent (low water level)
### Sediment massif completely submerged:

We combine Equation (6) and:

$$P_w = \gamma_w \left( \frac{H_1 + H_2}{2} \right) b \frac{L}{\cos \theta} \quad (8)$$

$$W = \frac{1}{2} \gamma (h_1 + h_2) b L \quad (9)$$

to obtain:

$$F = \frac{2 c_u \frac{b}{\cos \alpha}}{D_1 + \gamma_w [(H_1 + H_2)b(\sin \alpha - \tan \theta \cos \alpha) - 2(h_1 H_1 - h_2 H_2)\cos \alpha]} \quad (10)$$

with,

$$D_1 = \gamma [b(h_1 + h_2)\sin \alpha - (h_1^2 - h_2^2)\cos \alpha] \quad (11)$$

### Sediment massif completely emergent:

In this case,

$$\Delta f = \frac{1}{2} \gamma L (h_1^2 - h_2^2) \quad (12)$$

and
yield to:

$$P_w = 0$$

$$F = \frac{2 c_u \frac{b}{\cos \alpha}}{D_1} \quad (13)$$

with $D_1$ defined by Equation (11).

Both approaches, mud flow and flat slide with rectangular slice in the undrained conditions (Alexis et al., 2001), thus lead to the same expression of the sliding condition. The mud flow model has the advantage of giving the profile of the movement rates of the fluid, but the water-sediment interface and the shear plane are supposed parallel. The trapezoid criteria is more in conformity with the different configurations that can exist in nature.

## 3.2 Method of Criteria Validation

With the aim to check the validity of the sliding criteria, we have to compare its results with the changes in the thickness of sediment layer during emptying.

This comparison may be carried out in three steps:

- by considering the sediments displaced during emptying,

- by considering the sediments not displaced,

- by using the criterion within the framework of a displaced height forecast.

The three kinds of surveys are thus referenced (level N, inclination α):

- top level of soft soil before emptying: $N_0, \alpha_0$

- top level of soft soil after emptying: $N_1, \alpha_1$

- base of soft soil: $N_T, \alpha_T$

- height of the displaced layer: $h_d = N_0 - N_1$

- height of the soft soil layer: $h_T = N_0 - N_T$

Fig. 2 summarizes these notations.

To facilitate calculations, we geometrically represented the trapezoidal slice studied (Fig. 3). We propose an overall (spatiotemporal) indexing (n), corresponding to a configuration that may be (according to the case in question), temporal (instant t) or spatial (top level of soft soil).

$Z_1^0$ and $Z_2^0$: elevations of the top of the slice in the initial configuration, $Z_1^n$ and $Z_2^n$: elevations of the top of the slice in configuration n, $Z_e^n$: elevations of the storage reservoir in configuration n, $T_1$ and $T_2$: elevations of the shear plane.

Geometrically, we obtained the following expressions:

$$h_1 = Z_1^n - T_1$$

$$h_2 = Z_2^n - T_2$$
<figure><img src="image_3.png" /><figcaption>Fig. 2 Summary of the notations</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3 Geometrical representation of the trapezoid</figcaption></figure>

$$ \cos \alpha = \frac{b}{\sqrt{b^2 + (T_2 - T_1)^2}} $$

$$ \sin \alpha = \frac{T_2 - T_1}{\sqrt{b^2 + (T_2 - T_1)^2}} $$

$$ \cos \theta = \frac{b}{\sqrt{b^2 + (Z_2^n - Z_1^n)^2}} $$

$$ \sin \theta = \frac{Z_2^n - Z_1^n}{\sqrt{b^2 + (Z_2^n - Z_1^n)^2}} $$

The heights of water above the top of the trapezoid at the n-th instant are:

$$ H_1^n = Z_e^n - Z_1^n $$
$$H_2^n = Z_e^n - Z_2^n$$

We should also emphasize that shear strength in the undrained condition along the potential line of sliding defined by the elevations $T_1$ and $T_2$, can be expressed by:

$$c_u = c_{u0} + a \frac{(Z_2^0 - T_2 + Z_1^0 - T_1)}{2} \quad (14)$$

### - Submerged sediment massif:

Using the preceding expressions and Equation (10), we can express the safety factor of the fully submerged sediment massif by:

$$F^n = \frac{2 \left[ \frac{a}{2} (Z_2^0 - T_2 + Z_1^0 - T_1) + c_{u0} \right] [b^2 + (T_2 - T_1)^2]}{D_1 + D_2} \quad (15)$$

with,

$$D_1 = \gamma b \left[ (Z_2^n - T_2 + Z_1^n - T_1)(T_2 - T_1) + (Z_2^n - T_2)^2 - (Z_1^n - T_1)^2 \right] \quad (16)$$

$$D_2 = 2b \gamma_w \left[ (Z_e^n - Z_2^n)(Z_2^n - T_2) - (Z_e^n - Z_1^n)(Z_1^n - T_1) + \left(Z_e^n - \frac{Z_2^n + Z_1^n}{2}\right)(T_2 - T_1 - Z_2^n + Z_1^n) \right] \quad (17)$$

### - Emergent sediment massif:

Equation (13) enables us to express the safety factor in the form:

$$F^n = \frac{2 \left[ \frac{a}{2} (Z_2^0 - T_2 + Z_1^0 - T_1) + c_{u0} \right] [b^2 + (T_2 - T_1)^2]}{D_1} \quad (18)$$

$D_1$ is given by Equation (16).

#### (a) Calculation of the safety factor of the displaced sediment

For this study, we need to characterize the sediments located between the level $N_0$ (top level of soft soil before emptying) and the level $N_1$ (top level of soft soil after emptying) (Fig. 2). Consequently we assumed that in configuration $n$, no sliding has occurred yet, i.e. that:

$$Z_1^n = Z_1^0 \quad \text{and} \quad Z_2^n = Z_2^0$$

These heights correspond to the level $N_0$.

The potential line of sliding is given by the level $N_1$, the heights $T_1$ and $T_2$ thus correspond to this level.

Before emptying, the safety factor is calculated using Equation (15) (submerged sediment mass) and the preceding conditions. The values for the safety factor thus obtained enabled us to check the stability of the submerged sediment mass.

After emptying, the same conditions were applied to the Equation (18).

We obtained the safety factor noted $F_A$, making it possible to identify the unstable zones of the exposed sediment mass ($F_A < 1$).

#### (b) Calculation of the safety factor of the sediments remaining after emptying

The soil remaining is likely to slide on the surface corresponding to the base of soft soil, and its water-sediment interface is provided by post-emptying level ($N_1$). The heights $T_1$ and $T_2$ are thus represented by level $N_T$, the heights $Z_1^n = Z_1^0$ and $Z_2^n = Z_2^0$ correspond to the level $N_1$, and the initial top soft soil level $N_0$ is used to obtain the values of the heights $Z_1^0$ and $Z_2^0$.

The safety factor thus obtained is referred to as $F_B$ and the values that are lower than the unit make it possible to locate the unstable zones of the remaining sediments.
(c) Calculation of forecast minimal displaced heights

For the unstable slices, the remaining layer must be stable and thus conform to the expression of the safety factor of the emergent sediment mass (Eq. 18), we obtain, then, the quadratic equation:

$$ (Z_1^n)^2 - Z_1^n(T_2 + T_1) - Z_2^n(Z_2^n - T_2 - T_1) + \frac{2}{F^n \gamma b} \left[ a \frac{Z_2^n - T_2 + Z_1^n - T_1}{2} + c_{u0} \right] [b^2 + (T_2 - T_1)^2] = 0 \quad (19) $$

that we can solve, by adopting only the physically acceptable solution:

$$ Z_n^i = \frac{1}{2} \left[ T_2 + T_1 - \sqrt{(T_2 + T_1)^2 + 4 Z_2^n (Z_2^n - T_2 - T_1) - \frac{8}{F^n \gamma b} c_u (b^2 + (T_2 - T_1)^2)} \right] \quad (20) $$

with, $c_u = c_{u0} + a \frac{Z_2^n - T_2 + Z_1^n - T_1}{2}$

The heights $T_1$ and $T_2$ are given to us by the base level of soft soil and the heights $Z_1^n$ and $Z_2^n$ correspond to the top level of soft soil before emptying ($N_0$). The safety factor at limit state is set to the unit. Equation (20) thus makes it possible to calculate the elevation of the low point $Z_1^n$, knowing the elevation of the high point $Z_2^n$.

In order to forecast the final profile, this $Z_2^n$ elevation is unknown and depends on the evolution of the whole profile. To break free from the interaction of the slices, we can make a local assumption on $Z_2^n$.

We chose to consider a particular configuration where, for a given slice, the slide results in the lowering of the low point, the high point remaining fixed. The minimal displaced height ($h_{dmin}$) taken into account is the lowering of the mid point of the slice, that is to say:

$$ Z_2^n = Z_2^0 \quad \text{and} \quad h_{dmin} = \frac{1}{2}(Z_1^n - Z_1^0) \quad (21) $$

# 4 CONFRONTING THE CRITERIA WITH BATHYMETRIC CHANGES

## 4.1 Description of the Selected Site

The storage reservoir studied is that of the Grangent dam which is located on the Loire (France) close to Saint-Étienne. Built between 1955 and 1957, the structure is a cylindrical concrete vault with a height of 55 m and a crown length of 200 m.

The storage reservoir has a volume of 57.4 Mm³, a length of 23 km and the surface area of the total catchment is 3850 km². Into this storage reservoir, the Loire receives three tributaries, these being the Semène, the Ondaine and the Lizeron. The study zone extends upstream and downstream from the confluence of the Ondaine with the Loire, for approximately 4 km. We distinguish four zones: upstream, Ondaine, post-confluence and downstream (Fig. 4).

<figure><img src="image_5.png" /><figcaption>Fig. 4 Positioning of the sampling and bathymetric profile stations</figcaption></figure>
## 4.2 Topographic Profiles

Fifty land surveys were carried out every 400 m in 1952, i.e. before the construction of the dam. These make it possible to know the elevation of the base of the soft soil ($N_T$).

## 4.3 Bathymetric Profiles

During the emptying between November 1995 and May 1996, the storage reservoir was brought back to a height of 404 m NGF, this being 16 m below its nominal elevation. This lowering of the water table reduced by half the length of the storage reservoir, whose tail then reached the confluence with the Ondaine. An objective being to highlight the influence of the lowering of the water level on the geomorphology of the upstream slopes, two bathymetric survey campaigns using echo sounder were carried out by the Laboratoire National d'Hydraulique (EDF). The first batch of surveys were taken before partial emptying (level $N_0$) and the second batch upon completion (level $N_I$). Positioning was achieved by satellite (GPSD) in order to improve the accuracy of a comparative study of the profiles.

Over half of the surface area of the sediments underwent displacement due to the flow of the river or slide on the banks. About the quarter of the sediments present in the storage reservoir moved in this way, to settle in the post-confluence area. The thickness variation of the silt deposition is very marked :

- upstream area (22 profiles): from -0.24 m to -2.40 m and -1m on average,

- post-confluence area and downstream (10 profiles): up to +3.40 m and +1.70 m on average.

- The weakness of the suspended solids concentrations in water downstream of the dam shows that this deposition occurs totally before reaching the structure. Most of the sediment is then concentrated in the post-confluence area and downstream.

## 4.4 Samples and Associated Measurements

With the aim of characterizing all of the sediments submerged before emptying, a sampling campaign associated with laboratory measurements is carried out. Fourteen core samples of a diameter of 6 cm and an average length of 2.5 m were taken equitably over the whole surface of the pre-defined study area to detect possible heterogeneities of the sediments. Several samples were taken at each sampling point in order to eliminate singularities non-representative of the true state of the soil.

We carry out a multivariate analysis in order to define all variables according to three factors and to thus underline the presence of correlation relationships between the various values (Marot et al., 1998). The correlation analysis shows that the profile representative of undrained shear strength may be estimated by:

$$ 
\begin{array}{l}
c_u = c_{u0} + a h \\
c_{u0} = 0.16 \text{ kPa} \\
a = 1.47 \text{ kPa/m} \quad (r = 0.858; n = 48; \xi = 8.9 \times 10^{-11})
\end{array} 
\quad (22) 
$$

with $r$: correlation coefficient; $n$: number of individuals; $\xi$: risk of error

The calculation of the total ($\sigma$) and effective stresses ($\sigma'$) according to the vertical elevation in the sediment ($h$) leads to:

$$ \sigma [\text{kPa}] = 10.17 \text{ h [m]} \quad (r = 0.994, n = 177, \xi < 1.10^{-11}) \quad (23) $$

$$ \sigma' [\text{kPa}] = 2.71 \text{ h [m]} \quad (r = 0.927, n = 177, \xi = 1.10^{-11}) \quad (24) $$

## 4.5 Locating the Unstable Zones and Calculation of Minimal Displaced Height

The eleven profiles for which all of the bathymetric and topographic data are accessible, and at the level of which the mud mass became exposed during emptying are studied.

The values of the safety factor of the submerged sediments (Eq. 15) being all higher than the unit, the model confirms the stability of the sediment masses submerged before emptying. The values lower than the unit of the safety factor ($F_A$) of the displaced sediments indicate the location of the unstable zones created during emptying.

Fig. 5 shows these unstable zones for one of the profiles studied (P32). These zones, which characterize the sediments before emptying, are thus located on the slopes of the profile $N_0$. We note that the central part of the profile, only slightly sloping, is stable with the slide but undergoes major erosion, (not taken into account in this criteria). This same remark can also be made for the profiles located far upstream in the storage reservoir, for the slopes are very gentle there. On level $N_I$, we can see the zones
of instability of the remaining sediments, characterized by a value for the safety factor $F_B$ lower than the unit. We can thus note that sliding has made it possible for the massif to recover its stability.

<figure><img src="image_6.png" /><figcaption>**Fig. 5** Localization of the zones of instability, before and after emptying (profile P32)</figcaption></figure>

Fig. 6 makes it possible to compare all the measured displaced heights ($h_d$) and minimal heights calculated by the criteria, for all of the studied profiles.

<figure><img src="image_7.png" /><figcaption>**Fig. 6** Measured displaced height vs calculated minimum displaced height</figcaption></figure>

On this graph, all of the points with a x-value equal to zero correspond to the central parts of the profiles, which are the places of slight sloping and of major erosion. The other points, corresponding to the embankment, show a calculated minimal height slightly lower than the measured height. This graph thus presents a satisfactory estimate of the heights displaced by sliding. The trapezoidal shape allows to obtain better estimations than with a rectangular shaped slice.
## 5 CONCLUSION

The aim of this study is to simulate mud bank sliding during a dam reservoir emptying process. A flat-sliding geotechnical criteria is developed for a slice of soil with a trapezoidal shape. This sliding criteria is based on the theoretical formulation of the safety factor when the massif is submerged and emergent. The data necessary to the criteria are available before emptying.

This criteria first of all made it possible to check the stability of the submerged sediment masses and to locate the unstable zones which appear as the mud is exposed. Then we check that the slide allow for the remaining sediments to recover their stability.

Lastly, with prediction in mind, we estimate the mud thicknesses which would be displaced during emersion in order for the remaining sediments to remain stable.

These estimates are shown to agree with the measurements carried out on the selected site. In the zones subjected to heavy erosion, the thicknesses displaced are underestimated.

However, the simplicity of this criterion means that it can be coupled with a model of erosion-deposition in order to better allow for the representation of the true mechanisms of the soft soil movement caused by the emptying.

## REFERENCES

* Afnor NF P 94-071-2. 1994, Essai de cisaillement rectiligne à la boite. Association française de normalisation (in French).

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