Research PaperReliability analysis of rapid drawdown of an earth dam using direct coupling
Introduction
The stability of earth dams for long-term conditions depends on its geometry, material properties, and the forces to which the dam is subjected. The forces due to water are internal (pore-water pressures and seepage forces) and external (hydrostatic and hydrodynamic effects [1], [2]). During water discharge of an earth dam, the reduction of water level has two principal effects: modification of the internal Pore Water Pressure (PWP), and reduction of the external stabilizing hydrostatic pressure due to unloading. Therefore, safe operation of earth dams requires studies of seepage and stability for the upstream slope of the dam.
The level of water in a reservoir is affected by hydrological events and by user demand, in case of water supply dams. The partial or total emptying of the reservoir is part of a dam operation procedures. Partial or total emptying is required during maintenance and in case extreme events occur, such as earthquakes (stress changes due to earthquakes [3]), landslide into the reservoir or extreme rainfall. Dam equilibrium during the emptying procedure depends directly on the discharge velocity, with rapid discharges known to be critical. This leads to the problem known as Rapid DrawDown (RDD), with the discharge velocity (VRDD) being a critical parameter. The discharge velocity depends on characteristics of the hydraulic installations, such as the capacity of spillways.
The usual approach to RDD analysis is a transient seepage and limit equilibrium analyses using mean soil properties, to find deterministic factors of safety (FoS) [2], [4], [5]. However, the equilibrium depends on loading conditions and soil parameters, which can be modelled as random variables. In the last four decades, significant progress has been made into quantifying the variability of soil properties [6], [7], [8], and in performing reliability analysis of slopes and embankments in long term steady-state conditions [5], [9], [10], [11], [12], [13], [14], [15], [16], [17]. However, to the best of our knowledge and based on a comprehensive literature review, no studies were found addressing reliability of earth dams in transient conditions of rapid drawdown.
Different approaches are possible for probabilistic slope and dam stability analyses. Point estimate methods (PEM) are very efficient and popular in geotechnical engineering [18], [19], [20], [21] but were shown to be inaccurate for many problems by [22]. Accuracy can be increased by using higher-order moment methods [23], [24], but efficiency is lost. Approximation methods such as the first-order reliability method (FORM) are competitive with PEM in terms of computational cost, but this one is are more accurate and suitable for evaluating small (failure) probabilities at distribution tails. The FORM loses accuracy to Monte Carlo Simulation (MCS) only if the limit state functions are excessively nonlinear, in which case PEM also fails.
The equilibrium analysis of earth dams is based on locating the critical slip surface, i.e., the surface with the smallest safety factor. In general, only a limited number of slip surfaces is checked [25], [26], [27], [28]. The critical slip surface, in a probabilistic analysis, is generally believed to correspond to the minimum safety factor surface in the deterministic approach [29]. Likewise, one could assume that the critical time, in a probabilistic transient analysis, would be the same critical time of a deterministic analysis. These ideas are challenged herein the present paper.
This paper addresses the reliability analysis of an example earth dam during the rapid drawdown. The numerical study is conducted by the so-called Direct Coupling (DC) [30], [31], [32], [33], [34], [35], [36] between a deterministic software for slope stability (GeoStudio 2018 [37], [38]), and a structural reliability software (StRAnD [39]). The Direct Coupling method is not usual in geotechnical literature because, due to historical trends, point estimate and response surface surrogate methods became more popular [36]. However, direct coupling is the rule when it comes to (super-) structure reliability analysis using numerical models. Direct coupling allows the full features of deterministic numerical software to be explored in probabilistic or reliability analyses. In direct coupling, the probabilistic solver treats the numerical software as a black-box: it defines specific values (outcomes) of the random variables for which the numerical solution needs to be computed.
In this paper, the First-Order Reliability Method (FORM) is employed for reliability analysis, although many other algorithms are available in the StRAnD software [39]. FORM is known to be very efficient, as it computes failure probabilities with a limited number of calls to the deterministic solver (GeoStudio). The FORM method is based on an interactive search for the so-called design point, at which the limit state function is linearized. FORM yields good results when the problem is not excessively non-linear, as is the case for the transient dam equilibrium problem addressed herein. Searching for the design point requires gradients of the limit state function, with respect to random variables. These are evaluated by finite differences in the current implementation [40].
The remainder of this paper is organized as follows. The problem setting is presented in Section 2, which also briefly describes the performance function and reliability analysis techniques employed herein. The complete direct coupling technique for transient analysis is described in Section 3. Results of the transient numerical dam equilibrium analysis are presented and discussed in Section 4. Concluding remarks are presented in Section 5.
Section snippets
Review of current drawdown standards
There are different criteria for addressing the risk of Rapid DrawDown (RDD) in earth dams. In different codes, the discharge is referenced as: (i) a proportion of dam height; (ii) a proportion of dam volume; and (iii) a function of height per day [41], [42].
The RDD criteria as a function of dam height proportion (i) recommend drawdown analyses varying between 10% and 90% of the height, within 7 days and 4 months [43], [44]. The velocity of RDD as proportion of dam height is divided into hazard
Steps of the probabilistic modeling
Deterministic and probabilistic calculations were implemented by the coupling GeoStudio 2018 [37], [38] and StRAnD 1.07 [39] software. Details of the coupling approach are presented and discussed in Siacara et al. [40]. The algorithms implemented in Siacara et al. [40] were modified herein to evaluate earth dams in different long-term steady-state and transient conditions. The factors of safety (FoS) are evaluated using different limit equilibrium methods (e.g. Morgenstern and Price). The
Reliability Analyses and Results
Reliability analyses were performed for the same scenarios described in the mean value (deterministic) analysis, including four values of VRDD (VA = 2.0, VB = 1.5, VC = 1.0 and VD = 0.5 m/day). Reliability indexes (β) and probabilities of failure (Pf) where evaluated by the Direct Coupling GeoStudio and StRAnD software packages, as described in 2 Problem Setting, 3 Methodology. The limit state gradients were estimated by finite differences. The search for the DP was performed with the HLRF
Concluding Remarks
This paper addressed the transient reliability analysis of earth dams during rapid drawdown operations. Seepage and equilibrium analyses were performed numerically using the deterministic GeoStudio 2018 software. Reliability analyses were performed by the first-order reliability method in StRAnD software. A direct coupling technique was employed, in which the probabilistic software calls the deterministic software as a black box. An example earth dam was studied, and four different discharge
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors thank the financial support by the Coordination for the Improvement of Higher Education Personnel (CAPES) for research funding (Grant number n° 88882.145758/2017-01) and the Brazilian National Council of Scientific and Technological Development (CNPq).
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