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Dimensionless Erosion Laws for Cohesive Sediment ARTICLE in JOURNAL OF HYDRAULIC ENGINEERING · AUGUST 2015 Impact Factor: 1.62 · DOI: 10.1061/(ASCE)HY.1943-7900.0001068
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Dimensionless Erosion Laws for Cohesive Sediment
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Joseph S. Walder 1
Abstract: A method of achieving a dimensionless collapse of erosion-rate data for cohesive sediments is proposed and shown to work well for data collected in flume-erosion tests on mixtures of sand and mud (silt plus clay sized particles) for a wide range of mud fraction. The data collapse corresponds to a dimensional erosion law of the form E ∼ ðτ − τ c Þm , where E is erosion rate, τ is shear stress, τ c is the threshold shear stress for erosion to occur, and m ≈ 7=4. This result contrasts with the commonly assumed linear erosion law E ¼ kd ðτ − τ c Þ, where kd is a measure of how easily sediment is eroded. The data collapse prompts a re-examination of the way that results of the hole-erosion test (HET) and jet-erosion test (JET) are customarily analyzed, and also calls into question the meaningfulness not only of proposed empirical relationships between kd and τ c , but also of the erodibility parameter kd itself. Fuller comparison of flume-erosion data with hole-erosion and jet-erosion data will require revised analyses of the HET and JET that drop the assumption m ¼ 1 and, in the case of the JET, certain simplifying assumptions about the mechanics of jet scour. DOI: 10.1061/(ASCE)HY.1943-7900.0001068. © 2015 American Society of Civil Engineers.
Introduction In perusing the pages of the ASCE monograph Sedimentation Engineering (ASCE 2008), it is impossible not to be struck by the divergent treatments of cohesive and non-cohesive (or cohesionless) sediments. Broadly speaking, erosion and transport of cohesionless sediments are discussed in terms of dimensionless measures of grain size and shear stress. The power of using these dimensionless variables is that diverse data sets can be collapsed and described by simple empiricisms that are valid for a broad range of the variable values. In the literature on cohesive sediments, however, there appears to have been only a single attempt at a dimensionless collapse of erosion data (Partheniades 1992, 2009); more will be written about this later. This situation is perhaps unsurprising when one considers the complexity of just one aspect of the problem: sediment mobilization at the base of flowing water. With either purely cohesive sediment or mixtures of cohesive- and cohesionless grains, the forces resisting particle movement include not just frictional resistance and weight, but also surface forces. Those surface forces depend upon mineralogy, chemical composition, and the arrangement of grains (e.g., Ternat et al. 2008). Furthermore, even if one can estimate the threshold shear stress required to mobilize a single grain at the bed surface, that threshold is not necessarily the most important one, because other modes of sediment mobilization occur, such as the entrainment of clumps or even layers (e.g., Kothyari and Jain 2008). The goal of this paper is to break from previous usage and show that at least for a certain class of cohesive sediments–mixtures of sand, silt, and clay–it is in fact possible to achieve a collapse of flume-derived erosion-rate data in terms of dimensionless measures of shear stress and erosion rate. The rest of this paper is structured as follows: First, a brief review of experimental methods for measuring the erosion resistance 1
Hydrologist, U.S. Geological Survey, Cascades Volcano Observatory, 1300 SE Cardinal Court, Building 10, Suite 100, Vancouver, WA 98683. E-mail:
[email protected] Note. This manuscript was submitted on September 22, 2014; approved on June 16, 2015; published online on August 26, 2015. Discussion period open until January 26, 2016; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydraulic Engineering, © ASCE, ISSN 0733-9429/04015047(13)/$25.00. © ASCE
of sediment is presented, followed by a discussion of the sorts of erosion laws that have been used to date for cohesive sediment. Next, by drawing an analogy to established approaches for achieving non-dimensional collapse of erosion data for cohesionless sediment, a prospective method for collapsing erosion data for certain cohesive sediments—mixtures of sand, silt, and clay–is proposed and tested against several sets of flume-erosion data, with reasonably good success. For many of these data sets, dimensionless erosion rate Φ~ varies approximately as T 7=4 , where T is a dimensionless measure of shear stress. (The variables will be defined below.) Finally, the implications of such a dimensionless erosion law for interpretation of other sorts of erosion tests are discussed.
Methods of Measuring Erosional Characteristics of Sediment Present practice in engineering and geoscience includes three primary methods for quantifying the erosional characteristics of sediments: flumes; the jet erosion test, or JET (Hanson and Cook 2004); and the hole erosion test, or HET (Wan and Fell 2004). A common type of flume erosion test (e.g., Briaud et al. 2001; Roberts et al. 2003) involves a sediment core inserted into the bed of a flume and slowly raised as erosion proceeds [Fig. 1(a)]. The sediment core is located far enough from the flume entrance that flow is well-developed and standard methods may be used for calculating the shear stress applied to the eroding core. The shear stress may be varied by changing the fluid flux or the slope of the flume. As a practical matter, however, it is hard to achieve values of shear stress as large as might be of interest in geomorphic processes. This has led some investigators (e.g., Damgaard et al. 1997; Roberts et al. 2003) to develop a modified apparatus in which the flow is within a closed duct rather than having a free surface. A quite different sort of flume has been developed for in situ erosion testing in rivers and harbors: a sort of water conduit placed at the sediment/water interface, with flow provided by pumping ambient water (Ravens and Gschwend 1999). For the HET, which was developed for purposes of characterizing piping erosion, sediment is packed into a mold and then a central hole is drilled out [Figs. 1(b and c)]. A hydraulic gradient is imposed from one end of the sample to the other, and the resulting flow erodes and enlarges the hole. The HET has only been used in laboratory settings. The JET
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(Hanson 1990b; Hanson and Cook 2004), in contrast, is intended for field use. The JET device is placed on the sediment surface and a jet of water is directed normal to that surface [Fig. 1(d)]. The impinging jet is deflected radially outward, scouring sediment in the process.
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Mathematical Representation of Erosion Rate Data There is a long tradition of writing empirical laws for erosion rate and bedload flux in terms of the time-averaged shear stress τ . Although it has more recently become clear that sediment entrainment and transport are driven by coherent turbulent structures [see, for example, Nearing (1991), Best (1992)], there is no conflict with using τ -based empiricisms as long as one is dealing with uniform, steady surface-parallel flow, in which case turbulent measures
(such as Reynolds stresses), when extrapolated to the sediment surface, give a shear stress equal to the value derived from the mean flow However, when the flow is no longer surface-parallel, the shear stress based on considerations of the mean flow diverges from the value derived from turbulent measures. This discrepancy is especially great for the JET configuration: at the stagnation point where the jet impinges on the sediment surface, τ vanishes, but the turbulent stresses do not (e.g., Haehnel et al. 2008; Haehnel and Dade 2010). The erosion laws considered below are therefore appropriate for flume studies, but potentially problematic for analyzing HET data if the eroded hole deviates substantially from cylindrical (for example, Wahl et al. 2008, p. 12), and definitely problematic for analyzing JET data (Ghaneeizad et al. 2014, 2015). Workers in coastal processes commonly assume an erosion rule for cohesive sediments of the form (Winterwerp et al. 2012)
(a)
(b)
(c)
(d)
Fig. 1. (Color) Schematic drawings and photographs to illustrate types of erosion tests discussed in this paper: (a) erosion flume. A sediment core is pushed upward by a piston, with the top of the core maintained level with the flume bed. Some erosion flumes operate as enclosed, pressurized channels; (b) photograph of a sample used for the hole-erosion test (reprinted with permission from Wahl et al. 2008); (c) the sample of Fig. 1(b) inserted into the hole-erosion test device (reprinted with permission from Wahl et al. 2008); (d) submerged jet erosion test. Ambient water is entrained into the free jet that emerges from the nozzle, initially at a distance H 0 from the sediment surface. As the free jet impinges on the sediment surface, it is strongly decelerated and deflected, and re-forms as a wall jet along the sediment surface. A scour hole of depth η forms and evolves over time © ASCE
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ð1Þ ð2Þ
The m ¼ 1 assumption in the form of Eq. (4) has the useful consequence that kd values of different sediments may be meaningfully compared. An appropriate generalization of Eq. (4) is E ¼ kd ðτ − τ c Þm
for τ > τ c
ð6Þ
where M = mass of sediment eroded per unit area and unit time; τ = shear stress; τ c = threshold shear stress for erosion; and m and M 0 = empirically determined constants. The nondimensional measure of shear stress, τ =τ c − 1, used in Eq. (2) is sometimes called the “transport stage” and denoted by T (van Rijn 1984a, b). Alternatively, one could express erosion rate in terms of sediment thickness per unit time, denoted E m τ −1 for τ > τ c ð3Þ E ¼ E0 τc
Unfortunately, there is no straightforward way to compare values of the nonlinear “erodibility” for sediments that are characterized by different values of m, as the physical dimensions of kd depend upon m. To remedy this situation, Eq. (6) has to be replaced by an expression that includes an “erodibility” coefficient with physical dimensions independent of m. Such a replacement expression turns out to be most easily understood by first recasting the erosion rule in a dimensionless form.
with E0 empirically determined. In geomorphology and engineering, a common expression is (e.g., Hanson 1990a)
Dimensionless Erosion Laws
E ¼ kd ðτ − τ c Þ
for τ > τ c
ð4Þ
Dimensionless erosion rules for cohesionless sediment have generally been written in the form (García 2008, pp. 116–118)
E is related to M by
Φ ¼ FðD ; σg ; ss ; τ ; τ c Þ
M E¼ ρd
ð5Þ
with ρd the dry bulk density. Eq. (4) incorporates the assumption of a linear erosion law, that is, m ¼ 1, in which case the coefficient kd is commonly called the “erodibility”. This assumption seems to be nearly universal, yet locating sources that clearly explain the basis for this assumption has proven to be something of a quixotic quest. Instead, one reads, for example, that “[n]ormally, [m] is assumed equal to unity” (Knapen et al. 2007, p. 77), that m “is generally assumed to be equivalent to 1” (Hanson and Cook 1997), that the linear law is “[a] commonly used relationship in describing soil erosion” (Hanson 1990a), without attribution. In fact, there are many examples of sediments that display nonlinear erosion behavior. Two such examples are presented in Fig. 2 for sand/mud mixtures (see Table 1 for descriptions) containing a range of grain sizes and consolidated to ρd ≈ 1,330–1,500 kg=m3 . [The term “mud” is used to denote all sediment finer than sand (Mitchener and Torfs 1996)]. Data have been fit to Eq. (3), with the result m ¼ 1.60 for the Canaveral Harbor sediment [Fig. 2(a)], m ¼ 1.72 for the Detroit River sediment [Fig. 2(b)]. Even larger values of m are found for poorly consolidated near-shore sediments (for example, MacIntyre et al. 1990).
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 0
(a)
5
10
Transport stage
15
ð7Þ
The dimensionless erosion rate Φ is defined [following van Rijn (1984a)] as Φ¼
M pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρs ðss − 1ÞgD50
ð8Þ
and the other symbols introduced above are median grain size, D50 ; σg ¼ D84 =D16 , a dimensionless measure of the spread in the grainsize distribution; grain specific gravity, ss ¼ ρs =ρw , where ρw is the density of water; acceleration of gravity, g; dimensionless shear stress, also called the Shields parameter, τ ¼ τ =ðρs − ρw ÞgD50 ; critical Shields parameter τ c ¼ τ c =ðρs − ρw ÞgD50 ; and D ¼ ½ðss − 1Þg=ν 2 1=3 D50 , a dimensionless measure of mean grain size, where ν is kinematic viscosity of water. The function F must be determined experimentally. A well-known example of an empirical erosion rule in the form of Eq. (7) is from van Rijn (1984a), who found 1.5 Φ ¼ 0.00033D0.3 T
ð9Þ
for loose sand ð0.13 mm ≤ D50 ≤ 1.5 mmÞ. The general validity of this nonlinear erosion rule for cohesionless sediment remains uncertain, as other investigators [e.g., Dey and Debnath (2001)
Erosion rate in meters per second
Erosion rate in meters per second
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M ¼ 0 for τ ≤ τ c m τ M ¼ M0 −1 for τ > τ c τc
0.0008
0.0006
0.0004
0.0002
0.0000
20
0
(b)
5
10
15
20
Transport stage
Fig. 2. Graphs of erosion rate as a function of transport stage for A sediment sample CDS-1 from Canaveral Harbor and B sediment from the bed of the Detroit River. Information about these sediments and the data sources is given in Table 1. Both graphs show a nonlinear relationship between erosion rate and transport stage © ASCE
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Table 1. Properties of Sediments Whose Erosional Characteristics Were Investigated Using a Flume
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Sediment name Quartz Quartz Quartz Boston Harbor midchannel Boston Harbor “open cell” Canaveral CDS-1 Canaveral CDS-2 Canaveral CH-EC-S-1 Canaveral CHB-2 Detroit River Mobile Bay Synthetic mixture Synthetic mixture Synthetic mixture
Range of dry bulk specific gravity (sd )
Mean grain size, D50 , in μm
Sand-sized fraction (percent)
Silt-sized fraction (percent)
Clay-sized fraction (percent)
Mud-sized fraction (percent)
Organic content, weight percent
1.84–1.91 1.75–1.95 1.64–1.89 1.38–1.50 1.45–1.58 1.33–1.44 1.55–1.65 1.20–1.25 1.21–1.27 1.38–1.50 1.44–1.54 1.58–1.93 1.59–1.78 1.58–1.69
1,350 75 15 36 100 52 92 23 27 12 21 350 130 130
100 62 3 20 45 30 65 13 13 22 — 78 60 70
0 35 85 38 39 42 24 62 56 52 — 13 30 20
0 3 12 42 16 28 11 25 31 26 — 9 10 10
0 38 97 80 55 70 35 87 87 78 — 22 40 30
0 0 0 2.2 3.0 3.2 2.2 4.4 3.6 3.3 1.0 0 0 0
Source a a a b b c c c c d e f g g
Note: Sand-sized fraction consists of all grains larger than 64 μm. Silt-size fraction consists of all grains in the range 4–64 μm. Clay-sized fraction consists of all grains smaller than 4 μm. Size fractions are approximate for sources a to d. No size-fraction data were given in source e. a Roberts et al. (1998). b Roberts et al. (2001). c Jepsen et al. (2001). d Jepsen et al. (1997). e Gailani et al. (2001). f Fujisawa et al. (2008). g Fujisawa et al. (2011).
and Le Hir et al. (2008, p. 147)] have found different values for the exponent of T in Eq. (9). Substituting Eq. (5) in Eq. (8) gives ρd E s E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ Φ¼ ρs ss ðss − 1ÞgD50 ðss − 1ÞgD50 where sd ¼ ρd =ρw is dry bulk specific gravity. Formally, one could use the definition of τ c to rewrite Eq. (10) as pffiffiffiffiffi ð11Þ Φ ¼ τ c Φ~ where Φ~ ¼
sd E M pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ss τ c =ρw ρs τ c =ρw
ð13Þ
The simplest plausible non-dimensional data collapse of the form of Eq. (13) is Φ~ ¼ αT m © ASCE
E¼α
ð14Þ
1=2 1=2 ss τc s 1 Tm ¼ α s ðτ − τ c Þm sd ρw sd ρw τ 2m−1 c ð15Þ
Note that no “erodibility” parameter appears in Eq. (15). As long as a non-dimensional data collapse exists in the form of Eq. (14), the dimensional erosion law is fully specified by α, m, τ c , ρs , ρd , and ρw . A generalized erodibility parameter may of course be defined by comparing Eqs. (6) and (15), namely
ð12Þ
Eq. (11) would be a needlessly cumbersome way to define Φ if one were actually dealing with sand or gravel. Consider, however, the case of cohesive sediment:τ c can be determined, or at least estimated, from experimental data, so that calculating Φ~ is straightforward, whereas τ c , now interpreted as a generalized critical Shields parameter for cohesive sediment, would typically have to be estimated by analogy to other pffiffiffiffiffi studies of the erosion threshold for cohesive sediments. With τ c uncertain to within a factor of about 2–3 (see Appendix I), the fractional error would be consid~ It therefore seems reasonable to seek a erably more for Φ than for Φ. ~ data collapse in terms of Φ instead of Φ, with the general form ~ Φ~ ¼ FðT; sediment propertiesÞ
which has three parameters: α, m and (implicitly) τ c . The dimensional equivalent is
kd
1=2 ss 1 ¼α sd ρw τ 2m−1 c
ð16Þ
but it must be emphasized that this kd is simply notational shorthand: Eq. (16) is not an empirical relationship between an independently determined kd and the other parameters. This result may appear perplexing and will be elucidated later. To the best of my knowledge, the only other attempt as a dimensionless collapse of erosion data for cohesive sediment is that of Partheniades (1992, 2009) for a small set of measurements on San Francisco Bay mud. Partheniades’ dimensionless measure of shear stress is, to within a constant, τ =τ c , while his definition of dimensionless erosion rate follows from certain assumptions about the micromechanics of erosion of mud aggregates. No such assumptions are made in what follows. Ultimately, the merits of using Φ~ to collapse erosion data must be assessed empirically: does Eq. (13) in fact provides a framework for collapsing data on erosion of cohesive sediments? I have reanalyzed several published sets of flume-erosion data to see whether such a data collapse indeed exists. The results presented next are encouraging.
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Re-Analysis of Some Flume Data The flume data picked for reanalysis come from experiments done on both natural and synthetic mixtures of sand and mud. Sediments were mixed into homogeneous composites, remolded, and allowed to consolidate for varying amounts of time, thereby achieving a range of bulk densities. Table 1 provides a synopsis of sediment properties as well as data sources. Mineralogical descriptions of the sediments were sketchy at best and in some cases non-existent. The raw data used for reanalysis, as well as computed values of Φ~ and T, are presented in tabulated form in the supplemental materials. The sediments from Boston Harbor, Detroit River, Canaveral Harbor, and Mobile Bay, as well as the quartz particles, were all studied using the SEDFLUME apparatus (McNeil et al. 1996). Data from many studies done with SEDFLUME are particularly attractive for present purposes because values of dry bulk specific gravity sd are typically reported to within a factor of 0.01, meaning that the influence of sd on erosional characteristics can be characterized accurately. The SEDFLUME investigators expressed their results in the form E ¼ Aτ p sqd
ð17Þ
where E was given in cm=s, τ was in Pa, and A, p, and q were obtained by multivariate regression. To estimate τ c , which commonly depends sensitively on sd , I used the method described in Roberts et al. (1998, p. 1263): namely, Eq. (17) was rearranged to solve for τ c as a function of sd and E, with E assigned a small but non-zero value Ec 1=p Ec τc ¼ s−q=p ð18Þ d A Following Roberts et al., Ec was chosen as 10−4 cm=s, or approximately 1 mm erosion in 15 min, a value judged to be about the smallest that could be accurately measured. Picking a different value would slightly change the various coefficients and exponents in the regression relations given below, but would not change the overall conclusion. For each sediment type, log-transformed data for T and Φ~ were fitted to the equation log Φ~ ¼ β 0 þ n0 log T
ð19Þ
which corresponds to the power-law of Eq. (14) if α ¼ 10β0 and m ¼ n0 . Graphs of the various data sets are shown in Figs. 3(a–e), while Fig. 3(f) is a composite. The proposed data collapse does a good job in every case except possibly for the Mobile Bay sediment. The two Boston Harbor data sets [Fig. 3(a)] overlap despite the grain size difference between the samples. For Canaveral Harbor [Fig. 3(b)], data for sample CDS-2 nearly all fall below the trend of the other three samples. I suggest this reflects the fact that the slower-eroding CDS-2 had a significantly smaller mud fraction than the other samples. Behavior of the synthetic sand/mud mixtures [Fig. 3(c)] reflects the influence of both mud fraction and compaction. The mixtures with only 22% mud displayed a strong influence of compaction, with the densest samples considerably less erodible than the samples at the two lowest values of sd . The drop in erodibility as sd increases from 1.78 to 1.87 may reflect a transition in the sand matrix from being relatively easily sheared to being considerably less so. The comparison between Detroit River sediment and Mobile Bay sediment shown in Fig. 3(d) is a bit confusing at first. The Mobile Bay data nearly all plot above the Detroit River data, although a look at the raw data (in the © ASCE
supplemental material) shows that the Mobile Bay sediment eroded more slowly. The reason for the appearance of Fig. 3(d) is that the Mobile Bay sediment displayed a significantly larger τ c than the Detroit River sediments: for a given value of shear stress, the value of T is significantly less for the Mobile Bay sediment than the Detroit River sediment. The Mobile Bay data are therefore, in a sense, translated leftward on Fig. 3(d) compared to where these data would plot if the sediment had a smaller τ c. The separation between the coarse-sand-sized quartz particles and the fine-grained particles shown in Fig. 3(e) probably reflects the role of cohesive forces for the fine-grained samples. Roberts et al. (1998), whose data are shown in Fig. 3(e), noted that for the fine-grained particles, erosion rate fell off sharply with increasing bulk density. It seems likely that cohesive forces would become progressively more important as bulk density increases. It is striking that the different grain sizes all show a trend in log T − log Φ~ space with a slope close to 1.5—the same value found by van Rijn (1984a) in his sediment pickup experiments. One interpretation of this observation is that though cohesive forces strengthened the fine-grained samples, the mechanics of erosion remained essentially unchanged. Finally, Fig. 3(f) is a composite of all the data sets discussed above. There is a considerable degree of overlap in the reanalyzed ~ space. I have indicated in data sets when plotted in ðlog T; log ΦÞ Fig. 3(f) a trend line that passes through the middle of the aggregated data sets and is described by Φ~ ¼ 6.3 × 10−6 T 1.75
ð20Þ
The Mobile Bay data do not fit this trend well at small values of T, where errors in τ c would have the greatest effect. The Mobile Bay sediments were also affected by the formation of chemical precipitates (Gailani et al. 2001). The synthetic sand/mud mixtures constitute the major departure from the trend of the other data sets. The discrepancy is greatest at small T values, that is, when τ =τ c is only slightly greater than one. One difference between the synthetic sediments and the natural ones is that the former were compacted artificially, while the natural sediments underwent self-weight consolidation for varying periods of time. It is not obvious, however, that this difference would account for the difference in erosion response. The dimensional erosion rate E corresponding to Eq. (20) is 1=2 s τc τ − τ c 7=4 E ≈ 6.3 × 10−6 s sd ρw τc s 1 ðτ − τ c Þ7=4 ≈ 6.3 × 10−6 s ð21Þ 1=2 sd τ 5=4 ρw c The Mobile Bay sediment, which contained about 8% bentonite, exhibits a much greater τ c than the sediments without clay minerals, and also erodes much more slowly than the clay-free sediments. The body of evidence about cohesive-sediment erosion (e.g., Knapen et al. 2007) suggests that the sediment properties influencing Φ~ may include sd , D , clay fraction f c or mud fraction fm , clay mineralogy, and possibly other dimensionless variables characterizing the state of the sediment. For the re-analyzed data sets, only sd was systematically varied. The possible dependence of Φ~ on sd for each data set was evaluated by fitting data to the equation log Φ~ ¼ β 1 þ β s sd þ n1 log T
ð22Þ
Eqs. (19) and (22) comprise a nested pair of statistical models, so it is straightforward to apply an F-test (Appendix II) to evaluate
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-3.0
-3.5
-3.5
-4.0
-4.0
-4.5
-4.5
log
log
-3.0
-5.0 -5.5
-5.5 mid-channel open cell
-6.5 -1.0
-0.5
0.0
(a)
0.5
CDS-1 CDS-2 CHB-2 CH-EC-S-1
-6.0 -6.5 -0.5
1.0
0.0
0.5
(b)
log T
-3.5
1.0
-3.5 -4.0 -4.5
log
log -4.5
-5.0 -5.5 -6.0
-5.0
Detroit Mobile
-6.5 -5.5 -1.5
1.5
log T -3.0
40% mud 30% mud 22% mud (1.68 or 1.78) 22% mud (1.87) 22% mud (1.93)
-4.0
-1.0
-0.5
(c)
0.0
0.5
-7.0 -2.5
1.0
-2.0
-1.5
(d)
log T
-1.0
-0.5
0.0
0.5
1.0
1.5
log T -3
-2.5 -3.0
-4
-3.5
log
-4.0
log
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-6.0
-3.0
-5.0
-4.5 -5.0
-6.0 -1.5
(e)
Boston Canaveral Detroit Mobile synthetic 40% mud synthetic 30% mud trend
-6
15 m 75 m 1.35 mm
-5.5
-5
-7 -1.0
-0.5
0.0
0.5
1.0
1.5
-2
(f)
log T
-1
0
1
2
log T
Fig. 3. (Color) Data from several re-analyzed sets of flume-erosion data, plotted in terms of dimensionless erosion rate Φ~ and transport stage T. Two of the data sets shown here were also shown, plotted in terms of different variables, in Fig. 2. See Table 1 for descriptions of the various sediments and for literature sources: (a) erosion data for two sediment samples from Boston Harbor; (b) erosion data for four sediment samples from Canaveral Harbor; (c) erosion data for three synthetic sediments with varying fractions of mud; sd is given in parentheses for some samples; (d) erosion data for samples from the Detroit River and Mobile Bay; (e) erosion data for three samples of quartz particles with different median grain sizes; (f) a composite of the data shown in Figs. 3(a–d). The trend line has a slope of 7=4 in terms of the logarithmic variables of the graph. None of the data sets is well described by a linear erosion law
whether Eq. (22) improves on Eq. (19) in a meaningful way. Results of all the regressions, both univariate and bivariate, are given in Table 2. In some but not all cases, the regression fit was slightly improved by including sd as an explanatory variable. The only case in which sd accounted for a large fraction of the variance in the data was the synthetic mixture of Fujisawa et al. (2008), for which compaction to sd > 1.69 caused a sharp reduction in erosion rate [Fig. 3(c)]. © ASCE
Implications for Interpretation of Hole Erosion Test Bonelli and Brivois (2008) developed a mathematical model of the HET by starting with mass- and momentum conservation equations and applying boundary conditions along with the assumption m ¼ 1. By assuming that ε, defined as hole radius R divided by sample length L, is ≪1, they were able to find an analytical solution for R as a function of time t. Using empirical pipe-flow
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Table 2. Results of Univariate and Bivariate Regressions for Dimensionless Erosion Rate
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Sediment Boston midchannel Boston open cell Canaveral CDS-1 Canaveral CDS-2 Canaveral CHB-2 Canaveral CH-EC-S-1 Detroit River Mobile Bay Synthetic 78% sand Synthetic 70% sand Synthetic 60% sand Quartz 1.35 mm Quartz 75 μm Quartz 15 μm
β0
n0
r20
β1
βs
n1
r21
N
−5.14 −5.00 −5.327 −5.756 −5.321 −5.279 −5.485 −4.868 −4.30 −4.079 −4.446 −3.945 −5.145 −5.117
2.28 1.76 1.614 1.786 1.767 1.63 1.715 1.329 0.813 1.03 0.835 1.532 1.498 1.471
0.878 0.889 0.923 0.775 0.953 0.884 0.921 0.662 0.339 0.968 0.829 0.985 0.962 0.888
0.314 — −0.414 1.372 — 8.186 −0.155 — 1.617 — −1.765 — −3.161 −2.27
−3.844 — −3.598 −4.464 — −11.11 −3.71 — −3.309 — −1.589 — −1.059 −1.564
2.18 — 1.552 1.744 — 1.769 1.716 — 1.068 — 0.815 — 1.531 1.500
0.900 — 0.954 0.828 — 0.939 0.952 — 0.88 — 0.907 — 0.968 0.915
32 35 37 25 29 21 138 82 24 14 19 24 88 89
Note: The simple statistical model fit to each data set is log Φ~ ¼ β 0 þ n0 log T, with correlation coefficient r20 . The alternative statistical model is log Φ~ ¼ β 1 þ β s sd þ n1 log T, with correlation coefficient r21 . Coefficients for the alternative model are given only if it fits data better than the simple model at the 95% confidence level. N is number of samples.
equations relating R to discharge Q, their analytical solution could be expressed in terms of QðtÞ, which is the relevant quantity that is actually measured. The parameters τ c and kd are then determined by fitting the model QðtÞ to data. Bonelli and Brivois (2008, p. 1590) argued that m > 1 is impossible, at least for the geometry and constant-pressure boundary conditions of the HET, because their analytical solution has the property that RðtÞ → ∞ in finite time if m > 1. However, RðtÞ → ∞ is equivalent to ε → ∞, and any solution with this property contradicts the key assumption (ε ≪ 1) underlying the model. The Bonelli and Brivois model therefore provides no basis for claiming that m must be ≤1. I suggest that the Bonelli and Brivois method for determining erosion-law parameters from HET data needs to be revised so that all erosion-law parameters, including m, are determined by some sort of optimization procedure. Development of such a revised method of data inversion would involve a substantial amount of computational work and is beyond the scope of this paper.
Implications for Interpretation of the Jet Erosion Test
dH ¼ dt
m 1=2 2 ~ ss τc H α − 1 sd ρw H2
ð23Þ
~ the ultimate depth of erosion in the limit t → ∞, is defined in H, Appendix III. Eq. (23) is integrable to give an implicit solution for t as a function of H, with closed-form solutions possible if m is an integer multiple of 1=2. I applied those solutions (Fig. 4) to inverting JET data for experiments done by Mazurek (2001) using a manufactured product containing 40% clay, 53% silt, and 7% fine sand. (Relevant time-series data may be found in tabulated form in the supplemental materials.) No strict optimization procedure was applied; the curves fits were rather done simply by visual inspection. The informality of this process notwithstanding, it seems clear that the erosion data are poorly described by a linear model and that m ≈ 5=2 provides a reasonable fit.
Use and Misuse of the Concept of Erodibility
The JET is widely used by agricultural- and civil engineers as well as geomorphologists in North America, and more recently has been used by investigators in Europe and Asia (e.g., Chang et al. 2011; Pinettes et al. 2011). The JET is easily set up in the field, unlike either flume measurements or the HET. The popularity of the JET also reflects the fact that Hanson and Cook (2004), the developers of the JET device, have made their data-inversion method freely available. The JET device contains a probe that allows the operator to measure the changing distance HðtÞ between the nozzle and the eroding sediment surface directly beneath the nozzle. Hanson and Cook (1997, 2004) assumed that this time series for scour depth at a single point could be used to infer erosion-law parameters. The only other parameters of the JET device that factor into the Hanson and Cook analysis are the nozzle diameter, d, the initial distance of the nozzle from the bed, H 0 , and the mean flow speed of the jet as it emerges from the nozzle, U 0 . Using several other assumptions about the erosion process (see Appendix III for a critical discussion), Hanson and Cook derived an evolution equation for HðtÞ. The solution to this first-order differential equation was then fit to the measured time series to infer erosion-law parameters. © ASCE
An alternative evolution equation is derived in Appendix III after relaxing the m ¼ 1 assumption
For convenience, Eq. (16), which defines the generalized erodibility parameter kd in terms of parameters appearing in the dimensionless erosion law, is restated here 1=2 ss 1 ð24Þ kd ¼ α sd ρw τ 2m−1 c Note that the dimensions of kd depend upon m. An alternate way to define an erodibility parameter—denoted κd , and with invariant dimensions—is as the rate of change of erosion rate E with respect to the excess shear stress τ − τ c . Using Eq. (15) and the definition of T leads to the result, after some algebra 1=2 ∂E s 1 ðτ − τ c Þm−1 ¼ mα s κd ¼ sd ∂ðτ − τ c Þ ρw τ 2m−1 c ¼ mkd ðτ − τ c Þm−1
ð25Þ
The dimensions of κd are L2 TM−1 , and its value depends upon the magnitude of the applied shear stress unless m ¼ 1. It is useful to compare κd to kd ¼ E=ðτ − τ c Þ, which one might plausibly take as a measure of erodibility if one has only a single measurement
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1.08
1.6 1.5 1.4 1.3 1.2 1.1
m=1, =0.00001 m=2.5, =0.000025 data
1.0
Dimensionless distance from nozzle to bottom of scour hole
Dimensionless distance from nozzle to bottom of scour hole
1.7
(a)
1.04
1.02 m=1, =0.005 m=2.5, =0.00075 data
1.00
0.98 101
102
103
104
105
106
107
108
Dimensionless time
(b)
101
102
103
104
105
106
107
Dimensionless time
Fig. 4. Comparison of time-series data for sediment scour by an impinging jet with theoretical predictions of scour history. The assumption of a linear erosion law implicit in the Hanson and Cook (1997, 2004) model of the JET test was relaxed. Dimensionless scour depth and dimensionless time are defined in the main text: (a) data from Test 8/8.1/8.4/1 of Mazurek (2001). The material was a manufactured product used for pottery with size distribution 7% sand, 50% silt, 43% clay, and bulk dry specific gravity of 1.54. Type curves calculated for α ¼ 1.62; (b) data from Test 8/14.5/9.0/4 of Mazurek (2001) using the same material as in a. Type curves calculated for α ¼ 1.07
7 ss 1 ðτ − τ c Þ3=4 κd ≈ α 5=4 sd 4 ρ1=2 w τc
(Fig. 5). These two measures of erodibility are equal only if m ¼ 1, in which case s 1 1=2 ð26Þ κd ðm ¼ 1Þ ¼ kd ¼ α s sd ρw τ c To reiterate a point made previously, Eqs. (24)–(26) must be understood as definitions of an erodibility parameter—a sort of notational shorthand. The erosion law is completely specified by α, m and τ c [plus a fourth parameter—the regression coefficient β s of Eq. (23)—if Φ~ does depend on sd ]. Suppose that one has a set of flume-erosion data for which there ~ log T ¼ m, as for the various data sets is actually a trend d log Φ=d in Fig. 3. In this case, Eq. (25)—or Eq. (26) if m ¼ 1—gives a sort of “best fit erodibility” for the entire data set; furthermore, one may think of Eqs. (25) and (26) as expressing a sort of average relationship between τ c and κd for that data set. For the general trend identified in Fig. 3(f), for example 30
Erosion rate, arbitrary units
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0.9
1.06
25
20
15
10
5
0 0
1
2
3
4
5
Shear stress in excess of erosion threshold, arbitrary units
Fig. 5. Geometrical interpretation of the difference between the traditional definition of the erodibility coefficient kd ¼ E=ðτ − τ c Þ and the erodibility as defined here, namely κd ¼ ∂E=∂ðτ − τ c Þ. The solid line shows erosion rate as a function of excess shear stress. The slope of the dotted line is κd , while the slope of the dashed line, which is the tangent to the plot of erosion rate versus excess shear stress, is κd © ASCE
ð27Þ
In contrast, suppose that one has inverted data from a single jet-erosion test in terms of α, m, and τ c . One can use those parameter values to calculate κd , but that κd now relates only to that single JET measurement, using a specific sediment and a specific set of experimental conditions, and provides no information at all about any possible correlation between κd and τ c for a suite of experiments. To assess such a correlation, one might do a suite of experiments with a specific sediment compacted to different values of sd . Furthermore, the possibility that m ≠ 1 indicates that one should also run sets of experiments at different values of τ . A few flume-erosion studies have taken this approach (e.g., Ghebreiyessus et al. 1994; Roberts et al. 1998, 2001), but systematic development of the function κd ðsd ; τ Þ does not seem to have been done using the JET. A number of authors have proposed empirical relationships of the general form kd ∼ 1=τ δc for suites of sediments by aggregating parameter values inferred from individual JET measurements under the assumption that m ¼ 1. (The symbols kd and τ c are used here for the linear erodibility coefficient and critical shear stress, respectively, to connote a regression relationship based on aggregated data rather than for a single sediment.) Hanson and Simon (2001) were the first to propose such a relationship: kd ¼ 0.2 × 10−6 =τ 0.5 c , when SI units are used, for loess-derived streambed sediments commonly containing 50–80% silt. The exponent of τ c in the Hanson and Simon regression is what one might expect for m ¼ 1 sediments [Eq. (26)] as long as there is not too much variability in the value of α. Subsequent investigators, including Simon et al. (2010) and Daly et al. (2013), have published other regression relationships between kd and τ c , with diverse values of the exponent δ and predicted kd values for a given value of τ c commonly varying by a factor of 100–1,000. This variability may have several causes, including diverse sediment properties, different methods of inverting experimental JET data (e.g., Mazurek et al. 2010; Simon et al. 2010; Cossette et al. 2012; Daly et al. 2013), and incorrectly treating a sediment as having m ¼ 1 behavior. It is noteworthy that Knapen et al. (2007) concluded that κd and τ c are in fact uncorrelated for a large suite of both laboratoryand field tests on soils.
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It seems reasonable to pose a rhetorical question: is erodibility even a useful concept? Bonelli and Brivois (2008, p. 1520) stated that τ c and kd “are of obvious significance” to the mechanics of surface erosion, as opposed to m, which is “just a matter of fitting the numerical and experimental results”. This claim bears closer examination. First consider τ c. For cohesionless grains, τ c can be determined with reasonable accuracy by considering hydrodynamic- and frictional forces acting at the grain scale (e.g., García 2008). The same can be done with aggregates of cohesive grains by accounting for other forces (e.g., van der Waals forces) acting at the grain scale (e.g., Ternat et al. 2008). However, there are other modes of erosion of cohesive sediment that involve detachment of lumps or flakes consisting of both cohesive and noncohesive grains; the τ c characterizing these erosion modes have not been predicted, although they can be expressed in terms of empirical formulas that involve geotechnical parameters like unconfined compressive strength and plasticity index (Winterwerp and van Kesteren 2004). The situation is rather different when it comes to erodibility, however. I am unaware of any attempt to predict erodibility on the basis of microphysical models. Moreover, as noted by Moody et al. (2005), erodibility reflects “both the initiation of motion process, which is soil property-dependent, and the transport process, which is not a fundamental property of the soil, but rather depends on the flow characteristics of the transporting fluid.” The same is true for any sediment, soil or otherwise. It is interesting to get some idea of the size of the error in κd if a sediment is incorrectly treated as linear. Suppose that one is dealing with an erosive process with a shear stress of typical magnitude τ 0 , varying over a range Δτ ≪ τ 0 − τ c . The generalized erodibility κd may then be written, to leading order, as 1=2 s 1 ðτ 0 − τ c Þm−1 κd ≈ mα s sd ρw τ 2m−1 c m−1 s 1 1=2 τ 0 ¼ mα s −1 ð28Þ sd τc ρw τ c and treated as a constant. If the sediment is incorrectly treated as obeying a linear erosion law, the relative error will be 1−m κd ðapparentÞ 1 τ 0 −1 ð29Þ ¼ κd ðactualÞ m τc For the datasets that were reanalyzed above, taking m ¼ 7=4, the error is −3=4 κd ðapparentÞ 4 τ 0 −1 ð30Þ ¼ κd ðactualÞ 7 τc The mismatch becomes worse as τ 0 =τ c increases. For example, if τ 0 =τ c ¼ 10, κd ðapparentÞ=κd ðactualÞ ≈ 0.11. The error increases as τ 0 =τ c and m increase. The only way to keep the error small is to work at a shear stress not much greater than τ c .
Appendix I. Estimates of τ c for Cohesive Sediment The form of Eq. (11) necessarily begs the question of how to estimate τ c —which may be thought of as a generalized critical Shields parameter—for cohesive sediments. A number of investigators have approached this problem. Righetti and Lucarelli (2007) used a physically based model that accounts for cohesive effects (such as van der Waals forces between similar grains) and adhesive effects (such as interaction between sand and clay grains) to calculate theoretical values of τ c and compared those values to the usual critical Shields parameter. The Shields curve was extrapolated to small values of D by using van Rijn (1984b, p. 1434, Fig. 1) mathematical expression for τ c ðD Þ. Righetti and Lucarelli compared their predictions to published data and found reasonable agreement. The predicted increase in apparent τ c due to surface forces is as much as factor of 4 for D ≥ 0.4, corresponding to D50 ≥ 15 μm. The increase may be even more—perhaps a factor of 10–for clay and clay-sized grains (Ternat et al. 2008). The corresponding pffiffiffiffiffi difference between the “standard” and “corrected” values of τ c is thus a factor of about 2–3. This may be taken as the likely error in any value of τ c that would be assigned to a muddy sediment. There are other methods of estimating τ c in terms of macroscopic, geotechnical properties such as plasticity index or compressive (e.g., Jacobs et al. 2011); again, the uncerpffiffiffiffistrength ffi tainty in τ c is likely to be a factor of 2–3 (Kothyari and Jain 2008, Fig. 10).
Appendix II. Testing the Relative Merits of Statistical Models The dimensionless erosion rate Φ~ will plausibly depend upon a dimensionless measure of shear stress as well as on some measure of the sediment’s compactness. As noted in the main text, these measures may be chosen as the transport stage T and the bulk dry specific gravity, sd . The most restrictive statistical model is then log Φ ¼ β 0 þ n log T þ β s sd
ð31Þ
while the least restrictive model is
Conclusions A method for achieving a dimensionless collapse of erosion data for cohesive sediments was proposed by drawing an analogy to the data collapse used by van Rijn (1984a) for cohesionless sediment. The data collapse works well for mixtures of sand and mud (that is, silt and clay size particles) with a wide range of mud fraction (22–87%) and dry bulk specific gravity (1.2–1.65), and indicates that for those sediments, erosion rate varies as the 7=4 power of excess shear stress. The nonlinearity of the erosion law contrasts with the commonly assumed linear erosion law © ASCE
and prompts a reassessment of diverse methods used for characterizing sediment erosion. Established methods of analyzing both the hole-erosion test and jet-erosion test need to be modified to drop the assumption of a linear erosion law. In the case of the jet-erosion test, dropping the linear erosion assumption is in addition to other modifications required to correctly characterize stresses applied by the jet to the sediment surface. Incorrectly treating the erosion law as linear can lead to large errors in the estimated apparent erodibility. Nonetheless, apparent erodibility may be a useful index for comparing one sediment to another. In any case, the concept of erodibility as a material property is probably best discarded.
log Φ ¼ β 0 þ n log T
ð32Þ
Eq. (31) will be taken as the null hypothesis (denoted NH) and Eq. (32) as the alternative hypotheses (AH). For each sediment under consideration, F statistic is then calculated as (Weisberg 1985, pp. 95–97)
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ðRSSNH − RSSAH Þ=ðdf NH − df AH Þ RSSAH =df AH
ð33Þ
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where RSS denotes the residual sum of squares and df denotes the degrees of freedom. RSSNH − RSSAH is necessarily ≥0; df NH will be N − 2, while df AH ¼ N − 3, where N is the number of samples in the data set. One then compares the calculated value of F to tabulated values of the function Fðγ; ν 1 ; ν 2 Þ, where ν 1 ¼ df NH − df AH ¼ 1, ν 2 ¼ df AH , and γ ≪ 1. If F > Fðγ; ν 1 ; ν 2 Þ, one may reject the null hypothesis at the 1 − γ confidence level: that is, at the same confidence level, Φ~ depends upon both T and sd . For present purposes, γ ¼ 0.05.
Appendix III. Derivation and Solution of Equation for Jet Scour The kinematics of scour holes formed by normally impinging jets has been investigated not only by hydraulic engineers concerned with, say, scour downstream of hydraulic structures (e.g., Aderibigbe and Rajaratnam 1996) but also in entirely different contexts, such as by investigators trying to understand formation of craters by rocket exhaust (e.g., Haehnel et al. 2008; Metzger et al. 2009). The data collected in such experiments has revealed the way that scourhole geometry is influenced by sediment properties (grain size, water content, degree of compaction) and characteristics of the jet (for example, nozzle diameter and flow rate). Modeling the mechanics of scour-hole formation as a problem in unsteady two-phase flow has recently been undertaken by Kuang et al. (2013) for the case of a gas jet impinging on the surface of a cohesionless granular material. Kuang et al. used a computationalfluid-dynamics approach to solve for the gas motion and coupled those results to a discrete-particle simulation of the granular material. It seems likely that the same method could be applied to cratering in cohesive materials by changing the description of inter-particle forces (e.g., Yao and Anandarajah 2003). The goal here, however, is to try to describe scour-hole formation in the context of macroscopic sediment properties such as might be encountered in geotechnical engineering, so a continuum-mechanics approach seems appropriate. An example of such a model for a closely related problem is that of Bonelli and Brivois (2008) for flow through an erodible “pipe” (the so-called HET, or hole erosion test). This approach entails solving coupled equations for mass- and momentum conservation, subject to Rankine-Hugoniot jump conditions and an erosion “law” at the solid/fluid boundary; for completeness, transport of eroded sediment would also have to be considered. It turns out that commonly erosion is so slow relative to the fluid velocity that fluid and sediment are effectively decoupled, and a model for evolution of the sediment surface reduces to a kinematic model, with the fluid simply being the medium that applies a shear stress to the sediment surface. The Hanson and Cook (1997, 2004) kinematic model (hereafter HCKM) will next be written in a form that relaxes the linear erosion law assumption. The assumed geometry of the model is that of the JET apparatus (Fig. 1), which has a cylindrical chamber of radius R and contains a probe for measuring η at the center of hole (r ¼ 0). The HCKM is predicated on three key assumptions: first, that dη=dt for an impinging jet flow can be described by an erosion rule developed for surface-parallel flows; second, that ηðr ¼ 0; tÞ may be treated as a proxy for the evolving geometry of the entire scour hole; and third, that dη=dtjr¼0 may be expressed in terms of shear stress imposed by the fluid and some sort of measures of sediment erodibility. These assumptions are neither trivial nor obvious. The first assumption amounts to a statement that the normal-stress gradient associated with impinging flow has nothing to do with erosion. For cohesionless sediment, at least, this seems to be incorrect (Kuang et al. 2013). The second assumption is more or © ASCE
less equivalent to assuming that the evolving scour hole has a selfsimilar shape, in line with experimental evidence (e.g., Pagliara et al. 2008). The third assumption is less easy to rationalize: after all, the time-averaged shear stress at r ¼ 0 vanishes by symmetry, so an erosion law that assumes E depends upon the local value of τ would lead to E ¼ 0 at r ¼ 0. Empirically this is of course incorrect [see also discussion in Mercier et al. (2014) and Ghaneeizad et al. (2014)]. A physically sound resolution would probably depend upon a recognition that in detail, erosion depends upon turbulent stress fluctuations at the sediment surface (e.g., Nearing 1991; Best 1992), and that those turbulent stress fluctuations can be measured. Ghaneeizad et al. (2014) have in fact done such measurements and showed that unlike the mean shear stress, the turbulent shear stress does not vanish at r ¼ 0. Unfortunately there is as yet no understanding of how to write a macroscopic erosion law in terms of measures of fluid turbulence. The resolution proposed in the HCKM is to assume that dη=dtjr¼0 may be written in terms of τ max , the maximum time-averaged shear stress exerted by the impinging jet in 0 ≤ r < R. The validity of this assumption has not been tested experimentally. Adopting the assumptions about the scour process discussed above, and using Eq. (15), one finds dη ¼ dt
1=2 1=2 ss τc ss τc τ max − τ c m m αT max ¼ α sd ρw sd ρw τc ð34Þ
on a fictitious horizontal plane at a distance H ¼ H0 þ ηðtÞ from the nozzle. The final key assumption of the HCKM is to suppose that Eq. (34) correctly describes scour rate even as a scour hole develops and the sediment surface departs from being horizontal. Experimental evidence about normally impinging jets is then used to to specify τ max. I will assume that the nozzle is placed at an initial distance H 0 of at least 8.3d from the soil surface, in which case the impinging flow is “fully developed” (Beltaos and Rajaratnam 1977; Phares et al. 2000) and the time-averaged shear stress τ ðrÞ on the bed at a radial distance r from the center of impingement is given by (Phares et al. 2000) τ ðrÞ ¼
ρw U 20 d2
f 1=2
H2 Rej
r H
ð35Þ
where Rej ¼ U 0 d=ν is the jet Reynolds number, ν is kinematic viscosity, and the function fðr=HÞ is established experimentally. If the fluid is unbounded, the maximum shear stress, τ max , occurs at r=H ¼ 0.09, where fð0.09Þ ¼ C~ ≈ 44.6. For a bounded fluid, however—such as for the JET, with a boundary at some finite radius–τ max is increased by a factor of about 2.4 and occurs at a smaller value of r=H (Ghaneeizad et al. 2014). Using Eq. (35) in Eq. (34) and recognizing that dη=dt ¼ dH=dt, the evolution equation for HðtÞ becomes dH s 1 α½C 0 Re−1=2 ρw U 20 ðd=HÞ2 − τ c m ð36Þ ¼ s j m−1=2 sd dt ρ1=2 w τc ~ as where C 0 ¼ 2.4C~ ¼ 107. I now define H ~ ¼ H
C 01=2 Re−1=4 U0 d j pffiffiffiffiffiffiffiffiffiffiffiffi τ c =ρw
ð37Þ
and then rewrite Eq. (36) as
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dH ¼ dt
m 1=2 2 ~ ss τc H α − 1 sd ρw H2
ð38Þ
~ is the maximum value of H, approached asymptotically in time. H Eq. (37) is the desired evolution equation for HðtÞ and is used in the main text. In applying the evolution equation for HðtÞ and especially when representing data and model graphically, the difference between my initial condition and the one used by Hanson and Cook (1997, 2004) should be noted. The HCKM factors in the erosion of a fictitious layer of sediment situated between the nozzle and the actual, initial sediment surface. In other words, in the HCKM, t ¼ 0 corresponds to the jet nozzle just barely touching the sediment surface. In computing “erosion” of the fictitious layer between the nozzle and the actual sediment surface at t ¼ 0, one has to deal with a jet that is not in the self-similar domain, and thus Eq. (38) does not apply. I have not factored the fictitious-layer erosion into my mathematical development because I consider only cases in which the jet nozzle is initially far enough from the sediment surface that the jet’s velocity field is fully established when it impinges on that surface. Nearly all uses of the JET in fact satisfy the fully-established condition, and in particular, both of the examples considered in Fig. 4. I now recast Eq. (38) into dimensionless form by defining dimensionless variables H ¼ H=½H, t ¼ t=½t. Hanson and Cook ~ and ½t ¼ H=k ~ d τ c . Their choice of ½t (1997, 2004) chose ½H ¼ H obviously embeds the assumption m ¼ 1. I make a different choice of scales that involves only measureable quantities ½H ¼ H0
ð39Þ
H0 ffi ½t ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ½τ =ρw
ð40Þ
solutions for all m ¼ i=2, where i is a positive integer, but I have not attempted a formal proof of that proposition. Using the shorthand IðH Þ for the integral on the right-hand side of Eq. (44), one finds For m ¼ 1=2 IðH Þ ¼ ðα2 − 1Þ1=2 − ðα2 − H 2 Þ1=2 For m ¼ 1 IðH Þ ¼ −H þ 1 þ For m ¼ 3=2 IðH Þ
ð41Þ
which follows from the discussion in Phares et al. (2000) about the maximum shear stress associated with an impinging jet. Using Eq. (41) in Eq. (40), one finds ½t ¼
H20 Re1=4 j
C 01=2 U 0 d
ð42Þ
Using Eqs. (39) and (42), the dimensionless evolution equation for scour depth becomes, after some algebra 2 m dH ss 1 ζ α ¼ − 1 ð43Þ dt sd ζ H2 ~ where ζ ¼ H=H 0 . Eq. (43) is separable, giving an implicit solution t ðH Þ m Z s ζ H H 2 dH ð44Þ t ¼ d ss α 1 ζ 2 − H2 ~ after applying the initial condition H ð0Þ ¼ 1. Since H=H 0∼ [Hanson and Cook 1997, Eq. (9)], we see that 1=τ 1=2 c ζ ∼ 1=τ 1=2 c ; thus all three parameters of the erosion rule appear on the right-hand side of Eq. (44). For a given value of m, Eq. (44) defines a “type curve” in ðt ; H Þ space. Changing the value of α stretches the curve in the t direction; changing the value of ζ causes a stretch in both the t and H directions. The integral in Eq. (44) has closed-form solutions for m ¼ 1=2, 1, 3=2, 2, and 5=2; it seems likely that there are in fact closed-form © ASCE
α α þ H αþ1 − ln ln α − H 2 α−1
ð46Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi α2 − H 2 α2 − 1 α − pffiffiffiffiffiffiffiffiffiffiffiffiffi − α α α2 − 1
ð47Þ
For m ¼ 2 2 2 αH 3 H α 3 1 − − − IðH Þ ¼ α α α2 − H 2 2 α2 − 1 2 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αþH α þ 1 3 − ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − ln pffiffiffiffiffiffiffiffiffiffiffi ð48Þ 2 α−1 α−H For m ¼ 5=2
α3 1 1 − IðH Þ ¼ 3 ðα2 − H2 Þ3=2 ðα2 − 1Þ3=2 1 1 − 2α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − pffiffiffiffiffiffiffiffiffiffiffiffiffi α2 − H2 α2 − 1 i h p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − α2 − H2 − α2 − 1 α
where ½τ is a scale for shear stress, chosen as ½τ ¼ C 0 Re−1=2 ρw U 20 ðd=H0 Þ2 j
α ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 α − H 2
ð45Þ
ð49Þ
Acknowledgments Jesse Roberts (Sandia National Laboratories) provided erosionrate data for quartz particles and for Canaveral Harbor sediments in tabulated form. I had useful correspondence about the jet erosion test with Kerry Mazurek (University of Saskatchewan, Canada), Fabienne Mercier (IRSTEA, France), Andrew Simon (Cardno ENTRIX), and Tony Wahl (U.S. Bureau of Reclamation). Wahl provided the photographs of the HET device shown in Fig. 1. In working with other investigators’ data, all errors of interpretation are my own. Mention of trade names is for identification purposes only and does not constitute endorsement by the U.S. Geological Survey.
Notation The following symbols are used in this paper: A = regression coefficient used by Roberts et al. (1998); ~ C 0 = numerical coefficients; C, D16 , D50 , D84 = grain sizes; subscript indicates percent smaller; D = a dimensionless measure of grain size; d = nozzle diameter in jet erosion test; E = erosion rate expressed as thickness per unit time; Ec = 10−6 m=s; E0 = parameter in empirical erosion law; f c , fm = respectively mass fractions of clay and mud; g = acceleration of gravity;
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~ = respectively, height of JET nozzle above sediment H, H 0 , H surface; initial value of H; and asymptotic value of H as t → ∞; kd , kd = respectively, linear and nonlinear measures of erodibility; L = length of sample in hole erosion test; M = erosion rate expressed as mass/area-time; M 0 = parameter in empirical erosion law; m = exponent in erosion law; N = number of samples; n0 , n1 = coefficients in regression equations; p, q = exponents in regression equation of Roberts et al. (1998); Q = discharge; R = hole radius in hole erosion test; Rej = jet Reynolds number; r = radial distance from stagnation point in jet erosion test; sd , ss = respectively, specific gravities of dry bulk sediment and of grains; T = ðτ =τ c Þ − 1, the transport stage; t = time; U 0 = velocity of jet fluid at the nozzle; α = numerical coefficient in dimensionless erosion law; β 0 , β 1 , β s = regression coefficients; γ = F-test parameter; confidence level is 1 − γ; δ = exponent in putative regression relationship between kd and τ c ; ε = R=L; ~ ζ = H=H 0; η = scour distance in jet erosion test; κd = generalized erodibility measure for nonlinear erosion law; ν = kinematic viscosity of water; ν 1 , ν 2 = parameters in F-test; ρd , ρs , ρw = respectively, densities of bulk dry sediment, grains, and water; σg = D84 =D16 ; τ , τ c , τ 0 , Δτ = respectively, shear stress, critical shear stress, typical value of τ , and τ 0 − τ c ; Φ, Φ~ = dimensionless measures of erosion rate; = as a superscript, denotes a dimensionless variable; ½ = denotes characteristic value of variable within brackets; and x = underscore denotes values of parameters derived from regression on aggregated data rather than a single sediment; x here is a place holder.
Supplemental Data Tables S1–S12, along with an explanatory text file, are available online in the ASCE Library (http://www.ascelibrary.org).
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Errata for “Dimensionless Erosion Laws for Cohesive Sediment” by Joseph S. Walder
2 3 4
Joseph S. Walder
5
U.S. Geological Survey
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Cascades Volcano Observatory
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1300 SE Cardinal Court, Bldg. 10, Suite 100
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Vancouver, Washington 98683
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E-mail:
[email protected]
10
11
The following corrections should be made to the original paper:
12 13
On page 04015047-8, the caption of Figure 4 should read as follows:
14
Fig. 4. Comparison of time-series data for sediment scour by an impinging jet with theoretical
15
predictions of scour history. The assumption of a linear erosion law implicit in the Hanson and
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Cook (1997, 2004) model of the JET test was relaxed. Dimensionless scour depth and
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dimensionless time are defined in the main text: (a) data from test 8/8.1/8.4/1 of Mazurek (2001).
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The material was a manufactured product used for pottery with size distribution 7% sand, 50%
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silt, 43% clay, and bulk dry specific gravity of 1.54. Type curves calculated for 1.62 ; (b)
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data from test 8/14.5/9.0/4 of Mazurek (2001) using the same material as in a. Type curves
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calculated for 1.07 .
22
On page 04015047-11, Eq. (45) should read
24
27
28 29 30 31
1/ 2
2 H *2
1/ 2
Eq. (46) should read
1 H* ln I H * H * 1 ln * 2 H 1
25
26
I H* 2 1
23
Eq. (47) should read I H *
2 H *2
2 H *2 2 1 2 1
32
Eq. (48) should read 2 2 H* 3 H* 3 1 I H 2 2 *2 H 2 1 2 *
33
34
3 ln 2
36 37
H*
ln
1 1
Eq. (49) should read I H
35
H*
*
3
1
1 1 2 3/2 2 1 2 H *2 2 1 1
3 2 H *2 3/2 1 2 H *2 2 1
