Technical Note Estimating Equilibrium Scour Depth at Cylindrical Piers in Experimental Studies Gonzalo Simarro 1 ; Cristina M. S. Fael 2 ; and António H. Cardoso 3 Abstract: Using data from six very long experiments on local scour at single cylindrical piers, some of the existing equilibrium criteria found in the literature are assessed. It is found that common criteria, which consider scour depth increments in 24 h or the observation of horizontal plateaux in records of scour depth time evolution, can incur important errors on the equilibrium scour depth. Using some of the expressions for time evolution found in the literature to fit the experimental data and infer equilibrium scour depth through extrapolation to infinite time, it is found that the equilibrium scour depth cannot be specified, in general, for experiments shorter than one to two weeks. DOI: 10.1061/ (ASCE)HY.1943-7900.0000410. © 2011 American Society of Civil Engineers. CE Database subject headings: Scour; Equilibrium; Bridge foundations; Bridge failures; Piers; Experimentation. Author keywords: Scour; Equilibrium scour; Bridge foundation; Bridge failure; Rivers. Introduction Since the early stages of research on scouring, e.g., Chabert and Engeldinger (1956), it is well known that, in clear-water scour, the equilibrium scour depth is approached very slowly in time. Chabert and Engeldinger (1956) assumed equilibrium to occur when the scour depth does not change “appreciably” with time. Ettema (1980) has identified three phases of the scour process; the third phase is the equilibrium phase, where the scour depth “practically” does not increase anymore. Coleman et al. (2003) state that an equilibrium scour hole may continue to deepen at a “relatively slow rate. ” Each author has a different interpretation of the meaning of concepts like “practically, ”“appreciably, ” or “relatively slow rate. ” Some investigators state that equilibrium cannot be achieved in finite time (Franzetti et al. 1982), or even that the scour hole never stops developing (Oliveto and Hager 2002). The reported subjectivity has important implications on the design of scour experiments. Assuming that equilibrium scour exists but that it is not reached in a finite time, the question is “how long should experiments be until the scouring rate becomes insig- nificant or practically null and scour depth is close enough to its ultimate value?” In an attempt to answer the previous question, Melville and Chiew (1999) defined time to equilibrium as the time when the rate of scour reduces to 5% of the pier diameter in a 24-h period. Coleman et al. (2003) defined the equilibrium time as the time at which the rate of scour reduces to 5% of the smaller of the foundation length (pier diameter or abutment length) or the flow depth in the succeeding 24-h period. Grimaldi (2005) suggested a more restrictive criterion, namely, the reduction of scour rate to less than 0:05d p =3 in 24 h (d p = pier diameter). The value of 5% is obviously arbitrary; if the variation is reduced to, say, 2%—which is arbitrary as well—the time needed to reach equilibrium may be significantly longer. Similarly, Fael et al. (2006) have suggested using 2d 50 , again arbritary, as the limiting increment in 24 h. Furthermore, the 24 h usually considered are also arbitrary. Adopting a suggestion by Ettema (1980), Cardoso and Bettess (1999) assessed the onset of the equilibrium phase as the time where the slope of plots of the scour depth, d s , versus the logarithm of time, t , changes and tends to zero, in an attempt to mitigate arbitrariness. Radice et al. (2002) claim that this approach may also fail, since scouring can be triggered again, after the observation of a long-lasting quasi-horizontal plateau. Beside the preceding “equilibrium criteria, ” a number of expres- sions to represent the time evolution of the scour can be found in the literature (e.g., Breusers et al. 1977; Franzetti et al. 1982; Bertoldi and Jones 1998; Melville and Chiew 1999; Oliveto and Hager 2002; Kothyari et al. 2007). Some of these expressions do not even consider the existence of equilibrium scour depth. Assuming that equilibrium scour exists, probably reached in an infinite time, the focus of this work will be on expressions that do consider the existence of equilibrium. Defining in this paper ℓ i and τ i as the characteristic lengths and times, the expression by Bertoldi and Jones (1998) reads: B 4 ðtÞ¼ ℓ 1 t=τ 1 1 þ t =τ 1 þ ℓ 2 t=τ 2 1 þ t=τ 2 ð1Þ and has been extended by Lança et al. (2010) to B 6 ðt Þ¼ ℓ 1 t =τ 1 1 þ t =τ 1 þ ℓ 2 t =τ 2 1 þ t =τ 2 þ ℓ 3 t =τ 3 1 þ t =τ 3 ð2Þ In the preceding equations and throughout this paper, indexes on the left hand side indicate the number of free parameters of each expression (e.g., B 4 considers ℓ 1 , τ 1 , ℓ 2 , and τ 2 ). Defining c i as dimensionless coefficients, the expression by Franzetti et al. (1982) is of the form: 1 Researcher, Dept. of Marine Geology, Instituto de Ciencias del Mar (CSIC), Barcelona, Spain; formerly, Dept. of Civil Engineering, Universi- dad de Castilla-La Mancha, Ciudad Real, Spain (corresponding author). E-mail: gonzalo.simarro@uclm.es, simarro@icm.csic.es 2 Associate Professor, Dept. of Civil Engineering and Architecture, Universidade da Beira Interior, Covilha, Portugal. E-mail: cfael@ubi.pt 3 Professor, Dept. of Civil Engineering and Architecture, Instituto Superior Tecnico, Lisbon, Portugal. E-mail: ahc@civil.ist.utl.pt Note. This manuscript was submitted on October 19, 2010; approved on February 22, 2011; published online on February 24, 2011. Discussion per- iod open until February 1, 2012; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Hydraulic Engineering, Vol. 137, No. 9, September 1, 2011. ©ASCE, ISSN 0733- 9429/2011/9-1089–1093/$25.00. JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2011 / 1089 Downloaded 20 Sep 2011 to 150.214.217.169. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org