## 1. Introduction

[2] The drawdown condition is a classical scenario in slope stability, which arises when totally or partially submerged slopes experience a reduction of the external water level. This is a common situation in riverbanks, subjected to changing river levels. Flooding conditions are critical in this case because river levels reach peak values and the velocity of decreasing water level tends to reach maximum values also.

[3] Operation of dams requires changes in water level, which modify the safety factor against sliding of the upstream slope of earth dams. When the reservoir level is high, hydrostatic pressures help to stabilize the slope. A reduction of water level has two effects: a reduction of the stabilizing external hydrostatic pressure and a modification of the internal pore water pressures. The second effect has traditionally received considerable attention in dam design because it may lead to critical conditions of the slope. The subject has been approached from different perspectives, which have been largely dictated by current advances in soil mechanics.

[4] *Sherard et al.* [1963] discuss the practical implications of rapid drawdown and a number of case histories associated with total or partial failure of the upstream slope. *International Committee on Large Dams* [1980] and *Lawrence Von Thun* [1985] provide further information on drawdown-induced failures.

[5] Current approaches to analyze drawdown are classified into two different groups: Flow methods, which should be applied in relatively pervious slopes and undrained methods, which find applications in impervious soil slopes. Methods from the first group concentrate on the solution of the flow problem in a situation that involves changes in boundary conditions and a modification of the initial free surface. These methods implicitly assume that the soil skeleton is rigid and therefore they do not consider any modification of the initial water pressure because of the change in total boundary stresses imposed by the drawdown. Methods developed to handle this problem include flow net analysis [*Reinius*, 1954; *Cedergren*, 1967]; methods based on ad hoc hypothesis (typically Dupuit-type of assumptions) [*Brahma and Harr*, 1962; *Stephenson*, 1978]; finite element analysis of flow in saturated soil [*Desai*, 1972, 1977; *Cividini and Gioda*, 1984] and finite element analysis for saturated-unsaturated flow [*Neumann*, 1973; *Hromadka and Guymon*, 1980; *Pauls et al.*, 1999].

[6] The second group considers only the instantaneous change in pore pressure induced by an instantaneous drawdown. This is the undrained case in which flow is not considered. Key early references for this approach are works by *Skempton* [1954], *Bishop* [1954] and *Morgenstern* [1963] and more recent work has been published by *Lowe and Karafiath* [1980], *Baker et al.* [1993] and *Lane and Griffiths* [2000]. In a recent contribution, *Berilgen* [2007] uses two commercial programs for transient/flow and deformation analysis respectively and reports a sensitivity analysis involving simple slope geometry.

[7] In dam engineering practice neither one of the two mentioned approaches can reliably approximate the field situation because compacted soils are far from being rigid and pure undrained conditions, even in the case of fairly impervious soils, are too conservative for common drawdown rates, which fall in the range 0.1 to 1 m/d.

[8] In this paper the term “coupled” analysis refers to the joint consideration of flow and stress deformation analysis. In the general formulation applied in this paper balance equations of fluid and gas and the equilibrium equations are solved simultaneously. However, when only flow problem is solved the mechanical equations are not considered. We refer to this case as the “uncoupled” analysis. The soil is now assumed rigid. The undrained case is solved by means of the fully coupled formulation.

[9] As an introduction to the remaining of the paper, consider, in qualitative terms, the nature of the drawdown problem in connection with Figures 1a and 1b.

[10] The position of the water level MO (height *H*) provides the initial conditions of the slope CBO. Pore water pressures in the slope are positive below a zero pressure line (*p*_{w} = 0). Above this line, pore water pressures are negative and suction is defined as *s* = −*p*_{w}. A drawdown of intensity *H*_{D} takes the free water to a new level M′ N′ O′ during a time interval *t*_{DD}.

[11] This change in level implies a change in total stress conditions against the slope. Initial hydrostatic stresses (OAB against the slope surface; M N B C against the horizontal lower surface) change to O′ A′ B and M′ N′ B C. The stress difference is plotted in Figure 1b. The slope OB is subjected to a stress relaxation of constant intensity (Δ*σ* = *H*_{D}*γ*_{w}) in the lower part (BO′) and a linearly varying stress distribution in its upper part (O′O). The bottom horizontal surface CB experiences a uniform decrease of stress of intensity, *H*_{D}*γ*_{w}. In addition, there is a change in hydraulic boundary conditions. In its new state, water pressures against the slope are given by the hydrostatic distribution O′ A′ B on the slope face and by the uniform water pressure value *p*_{w} = (*H* − *H*_{D})*γ*_{w} on the horizontal lower surface.

[12] The change in boundary total stresses result in a new stress distribution within the slope. This stress change will induce, in general, a change in pore pressure. The sign and intensity of these pore pressures depend on the constitutive (stress-strain) behavior of the soil skeleton. An elastic soil skeleton will result in a change of pore pressure equal to the change in mean (octahedral) stress. If dilatancy (of positive or negative sign) is present, shear effects will generate additional pore water pressures. Changes in total stress-induced pore pressures are, in fact, simultaneous with the dissipation process owing to the new unbalanced hydraulic boundary conditions. A transient flow will establish. In this case it is necessary to apply a fully coupled hydromechanical approach to take into account the simultaneous stress and flow phenomena. However, if the soil permeability is large enough, pore pressures may dissipate fast enough so that the effect of stress-induced pore pressures apparently disappears. Otherwise, in a pure “undrained” condition (high-speed of water level changes or very low permeability) changes in pore pressure will be exclusively induced by total stress changes.

[13] It is sometimes stated that in cases of rigid materials the flow-based analysis is sufficiently accurate, implying that no stress-related changes in pore pressures are generated. It is clear that this is never the case in practice since it is required that the effective soil volumetric modulus becomes significantly higher than the water modulus. Only if the “rigid” material happens to be pervious and for a different reason, the stress coupling seems to be absent.

[14] Consider three representative points (P_{1}, P_{2} and P_{3}) of the slope sketched in Figure 1 and their expected evolution of pore pressures in qualitative terms in Figure 2. A given time, *t*_{DD}, in the *t* axis marks the end of the drawdown operation.

[15] A point P_{1}, close to the upper part of the slope, will experience a limited change in stress due to the unloading represented in Figure 1. Therefore, no major differences should be found when comparing coupled or uncoupled analysis, even if the soil is impervious. In a pervious case, it has already been argued, no differences in practice will be found. The upper points in the slope may develop negative pore water pressures (suction).

[16] At the other extreme of the slope, point P_{3}, the slope face BO is far away. Because of the one-dimensional nature of this situation, it is well-known that pore pressures in the soil, at any depth, will follow the changing water level. However, in order to reproduce this elementary result with a computational tool, it is necessary to use a fully coupled hydromechanical approach or an “undrained” analysis. Otherwise, a change in water level will trigger a transient flow condition because no information on the instantaneous change in pore water pressure is available in an uncoupled model.

[17] Predicting the behavior of point P_{2}, near the toe of the slope is more difficult. Mean and shear stresses are high and they experience significant gradients. New pore pressures generated after unloading are far from being in equilibrium among them and with respect to the new hydraulic head imposed at the boundary. In fact, in a fully coupled approach, the transient process of pore pressure dissipation has several origins. They are: the rate of water lowering (this is a boundary condition), the heterogeneous distribution of “instantaneous” pore water pressures after drawdown and the “source” or “storage” terms provided by both, the changing saturation in some parts of the domain and the deformation of the soil skeleton. Figure 2b shows that the response of point P_{2} in a coupled analysis will depend on the permeability of the soil. The problem has, however, an additional difficulty because soil stiffness, which controls the storage term associated with changes in effective stress, will also dictate the rate of the process.

[18] Difficulties for the development of consistent, fully coupled hydromechanical codes for saturated/unsaturated soils, hampered by the issue of the effective stress principle and the development of consistent constitutive equations for unsaturated conditions, have probably prevented a more advanced and realistic analysis of the classical drawdown problem. This paper relies on one of the existing complete formulations in this regard. The solved cases use the finite element program CODE_BRIGHT [*Department of Geotechnical Engineering and Geosciences*, 2002] developed at the Department of Geotechnical Engineering and Geosciences of UPC. The code solves in a fully coupled manner thermal, mechanical and flow (air and water) problems in porous media. It may handle a variety of mechanical constitutive laws but the results presented here correspond either to elastic conditions or to elastoplastic constitutive models (BBM [*Alonso et al.*, 1990]; Rockfill model [*Oldecop and Alonso*, 2001]). These types of models go beyond previous known attempts to analyze drawdown effects. Some relevant aspects of the formulation used in CODE_BRIGHT are briefly described in Appendix A.

[19] Some of the qualitative descriptions offered above will be made more precise by solving the drawdown problem in a simple slope. The cases of instantaneous and progressive drawdown will be compared. Then a review of some existing rules to estimate drawdown effects on slopes will be performed. Despite the long list of developments and publications associated with drawdown analysis, almost no comparison between field measurements and calculations exists. For this reason, it was appropriate to perform an analysis of an interesting published field case (the response of Glen Shira Dam against a very rapid drawdown). Model results and measurements will be compared.