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Unified Formula for Critical Shear Stress for Erosion of Sand, Mud, and Sand–Mud Mixtures

August 2018
Journal of Hydraulic Engineering 144(8)

DOI:10.1061/(ASCE)HY.1943-7900.0001489

Authors:

Dake Chen

Hohai University

Bruce W. Melville

University of Auckland

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Research on erosion behavior of sediments usually treats sand and mud separately. However, natural soil often occurs as a sand- mud mixture. An expression for cohesive forces in sand-mud mixtures, which considers the effect of the sand component on the mud component, is first proposed. Then the balance of forces under the initial motion condition is analyzed, and a unified formula for the critical shear stresses of sand, mud, and sand-mud mixtures is derived. The formula reproduces well the way critical shear stress varies with mud content and is more accurate than other formulas that are widely used for sand-mud mixtures and pure cohesive sediments. The proposed formula also promises to be more convenient, being based on the dry bulk density of the mixture and the stable dry bulk density of the mud component. The effect of consolidation on the erosion thresholds of both sand-mud mixtures and pure muds is considered in the formula, which makes the formula capable of predicting the erosion thresholds of sediments in the process of consolidation.

Initial motion of sand-mud mixture.

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Force balance for a sediment particle with median size of a mixture at the surface of a bed exposed to a unidirectional flow.

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Calibration and validation of Eq. (23) in sand-mud mixtures by three groups of experimental data.

…

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Critical shear stress of uniform pure mud, as per Eq. (25). The values of 0.2, 0.4, 0.6, 0.8, and 1.0 represent the ratios of the dry bulk density of the mud to its stable dry bulk density.

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Unified Formula for Critical Shear Stress for Erosion of

Sand, Mud, and Sand–Mud Mixtures

Dake Chen1; Yigang Wang2; Bruce Melville, M.ASCE3; Huiming Huang4; and Weina Zhang5

Abstract: Research on erosion behavior of sediments usually treats sand and mud separately. However, natural soil often occurs as a sand–

mud mixture. An expression for cohesive forces in sand–mud mixtures, which considers the effect of the sand component on the mud

component, is first proposed. Then the balance of forces under the initial motion condition is analyzed, and a unified formula for the critical

shear stresses of sand, mud, and sand–mud mixtures is derived. The formula reproduces well the way critical shear stress varies with

mud content and is more accurate than other formulas that are widely used for sand–mud mixtures and pure cohesive sediments. The proposed

formula also promises to be more convenient, being based on the dry bulk density of the mixture and the stable dry bulk density of the mud

component. The effect of consolidation on the erosion thresholds of both sand–mud mixtures and pure muds is considered in the formula,

which makes the formula capable of predicting the erosion thresholds of sediments in the process of consolidation. DOI: 10.1061/(ASCE)

Author keywords: Sand–mud mixtures; Cohesive sediments; Erosion threshold; Critical shear stress for erosion; Shields theory;

Adhesive force; Cohesive force; Consolidation.

Introduction

The differences of incipient motion of noncohesive and cohesive

sediments are widely recognized (Torfs 1995;Sharif 2003;Jacobs

et al. 2011). The incipient motion of noncohesive sediments is pri-

marily dependent on the balance of weight, buoyancy, drag, and lift

forces acting on the sediment grains, which in turn are associated

with the size, density, and shape of the particles and the hydrody-

namic condition. Compared with noncohesive sediments, the in-

cipient motion of cohesive sediments is generally dominated by

cohesive forces due to electrochemical actions between particles,

rather than by gravity. Previous researchers have conducted numer-

ous detailed and in-depth studies on the erosion threshold of non-

cohesive and cohesive sediments and many significant theories and

models have been proposed and developed [see summaries by

Chien and Wan (1999), Beheshti and Ataie-Ashtiani (2008), and

Zhu (2008)]. However, in many natural environments, noncohesive

sediments and cohesive sediments are not completely separated and

often occur as sand–mud mixtures. Compared with the separate

treatments of sand and mud, the study of sand–mud mixtures at the

erosion threshold is far from mature; the influence of the compo-

sition of mixtures on erosion behavior has not been fully studied,

and the theories and models are relatively few.

After a brief review of previous work on the erosion threshold

of noncohesive sediment, cohesive sediments, and sand–mud mix-

tures, we propose an expression for cohesive forces in sand–mud

mixtures. We then apply the Shields theory, which is widely used in

noncohesive sediment transport studies, to sand–mud mixtures and

build a theoretical and unified formula for the critical shear stress

for erosion of sand, mud, and sand–mud mixtures.

Previous Work

Critical Shear Stress of Noncohesive Sediments

The critical shear stress for the incipient motion of noncohesive

sediments can be expressed in the dimensionless form

ϑsh;cr ¼τcr

ðρs−ρÞgd ¼ϑcr0ðRcÞð1Þ

where ϑsh;cr = critical Shields parameter; τcr = critical bed shear

stress; d= median particle size; g= gravitational acceleration;

ρsand ρ= sediment and water densities, respectively; ϑcr0=

function of Rc; and Rc= particle Reynolds number, defined as

Rc¼ðucdÞ=υ, where ucis the critical shear velocity and υis the

kinematic viscosity of water.

Eq. (1) was first obtained by Shields (1936) from dimensional

analysis of a single grain in a cohesionless bed, and expressed in a

graph of the critical Shields parameter against the particle Reynolds

number. This graph is the well-known Shields diagram. However,

the original Shields diagram had only a few data points, especially

for small-particle Reynolds number, hence it has been repeat-

edly updated when additional data have become available (Mantz

1977;Miller et al. 1977;Yalin and Karahan 1979;Buffington and

Montgomery 1997).

1Ph.D. Student, Key Laboratory of Coastal Disaster and Defence,

Ministry of Education, Hohai Univ., Nanjing 210098, China; College of

Harbor, Coastal and Offshore Engineering, Hohai Univ., Nanjing 210098,

China (corresponding author). ORCID: https://orcid.org/0000-0002-5160

-589X. Email: chdake@hhu.edu.cn

2Professor, College of Harbor, Coastal and Offshore Engineering,

Hohai Univ., Nanjing 210098, China. Email: ygwang@hhu.edu.cn

3Professor, Dept. of Civil and Environmental Engineering, Univ. of

Auckland, Auckland 1010, New Zealand. Email: b.melville@auckland

.ac.nz

4Associate Professor, College of Harbor, Coastal and Offshore

Engineering, Hohai Univ., Nanjing 210098, China. Email: 120133393@

qq.com

5Ph.D. Student, College of Harbor, Coastal and Offshore Engineering,

Hohai Univ., Nanjing 210098, China. Email: ovaltinewei@163.com

Note. This manuscript was submitted on November 29, 2016; ap-

proved on February 9, 2018; published online on May 29, 2018.

Discussion period open until October 29, 2018; separate discussions must

be submitted for individual papers. This paper is part of the Journal of

Hydraulic Engineering, © ASCE, ISSN 0733-9429.

J. Hydraul. Eng., 2018, 144(8): 04018046

A drawback of the Shields diagram is that the shear velocity

appears on both axes, making the Shields diagram implicit and dif-

ficult to use in practice. To avoid trial-and-error solutions, alterna-

tive parameters have been proposed by some researchers allowing

direct computation of the critical Shields parameter without

recourse to an iterative procedure (Vanoni 1964;Yalin 1972;

Soulsby and Whitehouse 1997;Cao et al. 2006). The problem

can be circumvented through the use of a dimensionless grain

diameter, defined as d¼ f½ðρs−ρÞ=ρðg=υ2Þg1=3d, which is com-

monly used in threshold curves (van Rijn 1993). Currently, many

equations are available to account for the Shields diagram

(Bonnefille 1963;van Rijn 1984;Soulsby and Whitehouse 1997;

Wu and Wang 1999;Paphitis 2001;Sheppard and Renna 2005;Cao

et al. 2006). Among them, the equation proposed by Soulsby

and Whitehouse (1997) is widely used

ϑcr0¼0.3

1þ1.2dþ0.055ð1−expð−0.02dÞÞ ð2Þ

Critical Shear Stress for Erosion of Cohesive

Sediments

With decreasing particle size, typically when the particle diameter

is of the order of 100 μm or less, the stabilizing effect of cohesive

forces cannot be neglected and plays a dominant role in controlling

the incipient motion conditions (Dade et al. 1992;Lick et al. 2004).

Current research shows that the cohesive force comes from electro-

chemical actions between particles, and is influenced by clay

mineralogy, pore water chemistry, organic content, biological ce-

mentation, and packing state, among other things (Sharif 2003;

Papanicolaou et al. 2007). Although it is difficult to describe the

cohesive force in a mathematical formulation covering all the im-

pact factors, many efforts have been made to quantify the cohesive

action by addressing some main factors, e.g., van der Waals forces

and the thin film of water (bonding water) coating cohesive

particles (Dou 1962;Han 1982;Israelachvili 1985;Dou 2000;

Zhang 2012). Deriagin and Malkin (1950) proved the existence

of cohesive forces by experiments with cross-quartz fibers, and

obtained an expression for the cohesive force between two quartz

fibers

fc¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

d1d2

pεð3Þ

where fc= cohesive force; d1and d2= diameters of the two quartz

fibers; and ε= coefficient denoting the strength of cohesive action

of the material.

Dou (1962) thought that the cohesive force not only comes from

the van der Waals force but also comes from the additional water

pressure due to the thin film of water (bonding water) coating co-

hesive particles. He considered these two causes and obtained a

unified formula for the critical near-bed velocity of coarse-grained

noncohesive sediments and fine-grained cohesive sediments. The

effect of water pressure (water depth) on the erosion threshold

of cohesive sediments has also been considered and accepted by

many other researchers (Han 1982;Dou 2000;Zhang 2012).

Tang (1963) quoted the results of Deriagin and Malkin for the

erosion threshold of cohesive sediments and developed two formu-

las, one for critical shear stress and one for critical velocity. More

notably, he noticed that the cohesive force is not only related to the

diameter, but also is greatly influenced by the compactness. Tang

therefore obtained a relation for the cohesive force between two

cohesive sediment particles based on the measured erosion thresh-

olds of cohesive sediments

fc¼εdρdc

ρsdc10

ð4Þ

where ρdc = dry bulk density; and ρsdc = stable dry bulk density of

cohesive sediments (also called the consolidated bulk density,

which is the dry bulk density of the fully consolidated sediments;

see Appendix Ifor details). The influence of the compactness on

cohesive force leads to a tendency of the erosion threshold curves

for cohesive sediments to sort according to bulk density. This ten-

dency also has been observed by many other researchers (Miller

et al. 1977;Otsubo and Muraoka 1986;Mehta et al. 1989;

Krone 1999;Roberts et al. 1998;Sanford and Maa 2001). Tang’s

approach of characterizing the compactness of sediments by using a

power function of the ratio of the dry bulk density to the stable

dry bulk density has also been followed and promoted by many

researchers including Li et al. (1995), Yang and Wang (1995), Dou

(2000), and Zhang (2012).

Han (1982) pointed out that cohesive forces are essentially van

der Waals forces and can be quantified through the intermolecular

dispersion potential energy derived from electrochemical theory.

He assumed that the cohesive force per unit area between two

points on the surface of two sediment grains is inversely propor-

tional to the cube of the distance between the two points. This

assumption led to a theoretical formula for cohesive forces between

two particles and let him establish a formula for critical near-bed

velocity. Independently and similarly, Israelachvili (1985) pro-

posed an expression for cohesive force between two spherical

particles based on van der Waals interaction

fc¼Ahd

24t2ð5Þ

where Ah= Hamaker constant; and t= particle separation, which is

defined as the smallest distance between the surfaces of two par-

ticles. Since the bulk density decreases with increasing particle sep-

aration, the particle separation involved in the formulas of Han

(1982) (Appendix II) and Israelachvili (1985) can also represent

the influence of the compactness on cohesive force. However, com-

pared with Tang’s empirical approach of using a power function of

the ratio of the dry bulk density to the stable dry bulk density to

characterize the compactness of sediments, the particle separation

is not easily obtained in practice, and this restricts the application of

their formulas.

Other attempts and efforts to determine the cohesive force and

the erosion threshold have also been made. Dade et al. (1992) pro-

posed that the apparent yield stress represents an average strength

of cohesive bonds between fine particles and derived a formula for

cohesive force by the multiplication of the apparent yield stress and

the average contact surface between adjacent particles. This al-

lowed Dade et al. to obtain the critical Shields parameter for co-

hesive sediments from force analysis. Lick et al. (2004) quoted

the result of Israelachvili (1985) and used a reference critical shear

stress to consider the effect of bulk density on erosion threshold,

and developed a formula for the critical shear stress that is suitable

for both coarse noncohesive quartz and fine cohesive quartz. Zhang

(2012) stated that the effect of fluid properties on incipient motion

should not be ignored and introduced three parameters (the kin-

ematic viscosity of water, relative density of particles submerged

in water, and acceleration of gravity) to reflect the effects of

the fluid boundary layer, temperature, and gravity features of sub-

merged sediment grains on incipient motion. He obtained an

expression for cohesive force by dimensional analysis and built

a unified formula for critical velocity for coarse noncohesive sedi-

ments and fine cohesive sediments.

J. Hydraul. Eng., 2018, 144(8): 04018046

The previously mentioned formulas for the erosion threshold

regarding the critical shear stress, critical velocity, and critical

Shields parameter are listed in Appendix II. Although these formu-

las were produced from different methods or theories, it is observed

that in all these formulas the stabilizing strength of resisting incipi-

ent motion consists of two or three parts contributed by different

actions. All these formulas (note that the critical near-bed velocity

can be transformed to the critical shear stress through the velocity

distribution) can thus be transformed into the same general expres-

sion in terms of the critical Shields parameter as follows:

ϑsh;cr ¼τcr

ðρs−ρÞgd

¼AG=d2

ðρs−ρÞgd þBFc=d2

ðρs−ρÞgd þCFh=d2

ðρs−ρÞgd ð6Þ

where G= submerged weight of the particle with median size of the

sediment; Fc= cohesive force suffered by the particle with the

median size; Fh= additional water pressure due to the water film

acting on the particle with the median size; G=d2,Fc=d2, and

Fh=d2= gravity force per unit area, cohesive force per unit area,

and additional water pressure per unit area, respectively; A,B, and

C= coefficients respectively related to the contributions of the

gravity force, the cohesive force, and the additional water pressure

to the erosion threshold; and the denominator, ðρs−ρÞgd, creates

dimensionless numbers by expressing each force relative to the

gravity force per unit area. The additional water pressure was con-

sidered by some researchers, e.g., Dou (1962,2000), Han (1982),

and Zhang (2012) as mentioned previously, but is not taken into

account in this study because most of the existing erosion tests

of cohesive sediments and sand–mud mixtures were conducted

in small-depth water flumes. The expressions for the gravity force,

the cohesive force, and the additional water pressure (if consid-

ered), along with the corresponding parameters A,B, and Cfor

each formula from transforming it into the form of Eq. (6) are also

listed in Appendix II.

Critical Shear Stress for Erosion of Sand–Mud

Mixtures

In most natural environments, soils are rarely pure noncohesive

sediments or pure cohesive sediments, but are often mixtures of

sand and mud (where mud is defined as clay and silt particles with

sizes smaller than 63 μm). Compared with the behavior of pure

sand or mud, the erosion behavior of sand–mud mixtures is poorly

understood. The study of the erosion threshold of sand–mud mix-

tures is preliminary and mainly experimental (Kamphuis and Hall

1983;Alvarez-Hernandez 1990;Nalluri and Alvarez 1992;Torfs

1995;Mitchener and Torfs 1996;Panagiotopoulos et al. 1997;

Sharif 2003;Barry et al. 2006;Kothyari and Jain 2008;Le Hir

et al. 2008;Jacobs et al. 2011;Jarrell et al. 2015).

According to these experiments of sand–mud mixtures, the criti-

cal shear stress (i.e., the erosion threshold) of a mixture is a function

of mud content. A small amount of mud added into sand could

significantly increase erosion resistance, decrease erosion rate,

and change the erosion behavior from noncohesive to cohesive.

This critical or transitional mud content seems to be in the range

of 3–15% (Alvarez-Hernandez 1990;Torfs 1995;Mitchener and

Torfs 1996;Sharif 2003;Jarrell et al. 2015). van Ledden et al.

(2004) pointed out that, compared with mud content, clay content

turns out to be more generic as a descriptor for the transition be-

tween noncohesive and cohesive erosion behavior. However, obser-

vations by Le Hir et al. (2008) showed that the relationship between

the critical shear stress and the mud volume fraction is clear, but the

correlation is not as good for the clay fraction.

Below the critical mud content, the erosion behavior is unclear

and needs more research. Different experimental results have been

observed by different researchers: erosion threshold remaining con-

stant with increasing mud content (Sharif 2003), or decreasing

slightly (Barry et al. 2006), or increasing slowly (Jarrell et al.

2015). Barry et al. (2006) attributed the decrease they found to

a lubrication effect of clay particles on sand grain erosion.

Beyond this critical mud content, the erosion threshold will in-

crease markedly with mud content up to an optimum mud content,

at which the critical erosion shear stress reaches a peak. The

optimum mud content has been confirmed by the experiments

of artificial sand–mud mixtures, and is approximately between

30 and 50% (Alvarez-Hernandez 1990;Mitchener and Torfs

1996;Sharif 2003;Jarrell et al. 2015). The network structure is

considered to be responsible for this optimum mud content because

the sand particles are completely coated with the mud particles,

resulting in an increase in the erosion strength of the sand particles

for a mud content of between 30 and 50% by weight (van Rijn

2016). The similar phenomenon that the cementation from the pres-

ence of optimal levels of sand and clay can enhance bed armoring

and the erosion resistance of the river bed has been observed in the

field (Papanicolaou et al. 2017).

When the mud content is beyond the optimum one, the critical

shear stress decreases slowly with increasing mud content up to

100% mud. Mitchener and Torfs (1996) reported that the critical

shear stress increases slightly with increasing sand content in

the region of 0–50% sand when sand is added to mud. The experi-

ments of Sharif (2003) and Jarrell et al. (2015) further confirmed

this slow decrease beyond the optimum mud content. Waeles et al.

(2007) attributed this phenomenon to the decrease in the density of

the mixture.

These experimental results of mixtures show that sand–mud

mixtures behave differently from pure sand or mud. The existing

formulas for erosion threshold of pure noncohesive and pure cohe-

sive sediments cannot be directly applied to sand–mud mixtures.

van Ledden (2003) developed a conceptual framework for the

erosion behavior of sand–mud mixtures. He introduced a critical

mud content to define the type of bed. When the mud fraction

is smaller than the critical mud content, the bed is defined as a non-

cohesive bed; otherwise it is defined as a cohesive bed. According

to van Ledden, the critical shear stress of a sand–mud mixture is a

function of the critical shear stresses of pure sand and pure mud

τcr ¼8

τcr;nð1þξcÞβ;ξc≤ξc;cr

τcr;nð1þξc;crÞβ−τcr;c

1−ξc;cr ð1−ξcÞþτcr;c;ξc>ξc;cr

ð7Þ

where ξc= mud content; τcr;nand τcr;c= critical shear stresses of

pure sand and pure mud, respectively; ξc;cr = critical mud content,

approximately 5–10%; and β= empirical coefficient, which is

generally between 0.75 and 1.25.

Ahmad et al. (2011) adopted a similar approach to van Ledden,

but, compared with the formula by van Ledden, their formula for

the critical shear stress of sand–mud mixtures is simpler and has no

critical mud or sand content in the expression

τcr ¼eζð1−1=ξnÞτcr;nþð1−ξnÞτcr;cð8Þ

where ξn= sand content (dry mass of sand divided by total dry

mass); and ζ= empirical coefficient, which is generally between

0.1 and 0.2.

J. Hydraul. Eng., 2018, 144(8): 04018046

Although the approaches of van Ledden (2003)andAhmad

et al. (2011) provided two sets of formulas for the full range

of sediment mixtures and considered the transition between co-

hesive and noncohesive behavior, their formulas are still highly

empirical. The dependence on the critical shear stresses of pure

sand and pure mud also limits the application of the formulas in

practice, as for a given sand–mud mixture the critical shear stress

of the pure mud is not easy to measure or calculate.

Analyzing erosion tests on quartz particles mixed with 2% ben-

tonite, Lick et al. (2004) extended their model for critical shear

stress of fine-grained quartz sediments by introducing an additional

adhesive force between particles due to bentonite coating; this force

is shown to be proportional to the square of particle diameter. The

expression for the adhesive force leads to an additional constant

term in the formula for critical shear stress.

Righetti and Lucarelli (2007) quoted the work of cohesive force

from Israelachvili (1985) and the work of adhesive force from Lick

et al. (2004), and developed two theoretical models for incipient

motion of cohesive and adhesive benthic sediments by moment bal-

ance, one model for single particles and one for flocs. The adhesive

force again leads to an additional constant term in both models.

None of the preceding studies on sand–mud mixtures investi-

gated the effect of consolidation on the erosion threshold. However,

Migniot (1989) observed that the dry bulk density of fine particles

in the space between sand grains was the relevant parameter for the

variation of critical shear stress of mixtures. This parameter was

called the relative mud concentration by Waeles et al. (2008),

Le Hir et al. (2011), and Mengual et al. (2017) when modeling

the erosion threshold of mixtures for sand–mud mixtures. In their

models, they used two critical mud contents to divide the mud

content range into three regimes.

In the first regime, the sand–mud mixture is presumed to be

noncohesive, but with the critical shear stress assumed to increase

with the mud fraction.

In the third regime, the mixture is presumed to be fully cohesive,

and the critical shear stress depends only on the relative mud con-

centration, which can vary due to consolidation degree. The critical

shear stress is taken as a power function of the relative mud

concentration.

There is an intermediate regime, in which the critical shear stress

depends on both the mud fraction and the relative mud concentra-

tion. The critical shear stress can be interpolated from the other

two regimes according to mud content.

Theoretical Framework of Incipient Motion for Sand,

Mud, and Sand–Mud Mixtures

Cohesive Force among Sand–Mud Mixtures

Natural sediments often occur as sand–mud mixtures. In a sand–

mud mixture, the mud particles fill the voids between the sand

particles (Fig. 1). According to the definition by Israelachvili

(1985), the cohesive forces at the molecular scale are the result

of the attractive interactions in the vacuum between contiguous par-

ticles of the same medium, while the adhesive forces are defined

as the additional binding forces between particles due to the pres-

ence of a second, interparticle medium. Based on this definition,

there are two types of bonding strength in sand–mud mixtures:

cohesive forces between mud particles, and adhesive forces be-

tween sand particles and mud particles. Although the distinction

of cohesive forces and adhesive forces is emphasized here, actually

they are essentially the same, and the cohesive force can be seen as

a particular case of the adhesive force.

Because little is known about adhesive forces between sand par-

ticles and mud particles, we make an assumption that when incipi-

ent motion begins, the motion-initiating particles (no matter what

kinds of particles, sand or mud) move together with the mud par-

ticles within a thin layer coating the individual motion-initiating

grains (Fig. 1). Since the coating layer is thin and the sizes of

the mud particles are small, the weights of the coating mud particles

are ignored. An advantage of this assumption is that consideration

of the adhesive forces between sand particles and mud particles can

be avoided. When a particle (either sand or mud) begins to move,

the adhesive forces can be ignored, leaving a focus on the weight of

the particle, the hydrodynamic forces, and the cohesive forces

between the mud particles coating the motion-initiating particle

and mud particles outside the coating layer.

As mentioned in “Previous Work,”the cohesive force between

mud particles is proportional to the grain size of sediments and is

mainly affected by the degree of consolidation, which can be para-

meterized by the bulk density of the sediment, the particles sepa-

ration, or dry bulk density, among other things. Because the mud

particles fill in the voids of sand particles in a sand–mud mixture,

the consolidation of the mud component is affected by the sand

component. In this study, following Tang (1963) and referring

to Migniot (1989), Waeles et al. (2008), and Le Hir et al.

(2011), as mentioned in “Previous Work,”we chose a power func-

tion of the ratio of the dry bulk density of the mud component in the

voids of sand particles to the stable dry bulk density of the mud

component to investigate the influence of consolidation. Thereby,

the cohesive force can be further expressed as

fc¼αcρεdcρdcm

ρsdc m

ð9Þ

where ε= coefficient denoting the cohesive strength of the mud

component (m3=s2); αc= shape coefficient; dc= median size of

the mud component of the mixture; m= exponent; ρdcm = dry bulk

density of the mud component in the voids of sand particles; and

ρsdc = stable dry bulk density of the mud component, which is

the dry bulk density of the fully consolidated mud component

(see Appendix Ifor details). The ratio of the dry bulk density of

the mud component to its stable dry bulk density represents the

degree of the consolidation of the mud component in the mixture.

The dry bulk density of the mud component in the voids of sand

particles can be calculated by

ρdcm ¼ξcρdm

nsð10Þ

where ρdm = dry bulk density of the mixture; ξc= weight fraction of

the mud component in the mixture; and ns= void ratio of the sand

component per unit volume of the mixture. The value of nscan be

calculated by

ns¼1−ξnρdm

ρsn ð11Þ

where ξn= weight fraction of the sand component in the mixture;

and ρsn = density of the sand particles.

Substituting Eqs. (10) and (11) into Eq. (9), the cohesive force in

the sand–mud mixture can be expressed as

fc¼αcρεdcξcρdm

ρsdc

ρsn

ρsn −ξnρdmm

ð12Þ

If the weight fractions of mud and sand are 100 and 0%, respec-

tively (ξc¼100%,ξn¼0%), i.e., the mixture is pure mud, then

ρdm can be substituted by ρdc. Eq. (12) can then be simplified to

J. Hydraul. Eng., 2018, 144(8): 04018046

fc¼αcρεdcρdc

ρsdcm

ð13Þ

which is consistent with the expressions of Tang (1963), Li et al.

(1995), Yang and Wang (1995), and Dou (2000) (Appendix II) for

pure cohesive sediments.

Unified Formula of Critical Shear Stress for Erosion

It is very difficult to determine the critical condition corresponding

to the incipient motion of a sediment. Here we have taken the en-

trainment of sediment particles with median size of the sediment as

the incipient motion criterion of the sediment. For the case where a

sand–mud mixture is exposed to a unidirectional flow and the sedi-

ment particles with median size of the mixture begin to move under

the actions of the gravitational force, the cohesive force, the drag

force, and the lift force, as shown in Fig. 2, the force equation for

the equilibrium condition can be expressed as

Fd¼ðGþFc−FlÞtan φð14Þ

where G= effective gravitational force; Fdand Fl= drag and lift

forces, respectively; φ= characteristic internal friction angle; and

Fc= resultant cohesive force of numerous cohesive forces between

the coating cohesive particles and cohesive particles outside the

coating layer, which represents the average cohesive effect over the

motion-initiating particle coming from the mud particles. The di-

rection of the resultant cohesive force depends on the distribution of

the surrounding mud particles and is assumed to be the same as the

direction of Gfor simplicity.

The drag and the lift forces acting on a sediment particle with

median size of the mixture can be expressed as

Fd¼1

2Cdα1d2ρu2

bð15Þ

Fl¼ηFdð16Þ

where Cd= drag coefficient; α1= particle shape factor; d= median

size of the mixture; η= ratio of the lift force and the drag force; and

ub= instantaneous velocity approaching the particle on the bed,

which can be evaluated by λu(Dou 2000), where uis the shear

velocity and λis a coefficient.

Only the submerged weight of the particle with median size of

the mixture is considered here and can be written as

G¼π

6ðρs−ρÞgd3ð17Þ

where ρs= density of the particle with median size of the mixture.

When the particle with the median size is a sand particle, then

ρs¼ρsn; when the particle is a mud particle, then ρs¼ρsc, where

ρsc is the density of the mud particles.

The resultant cohesive force can be evaluated by Fc¼kNcfc,

where fcis the cohesive force between a mud particle coating the

motion-initiating particle and another mud particle outside the coat-

ing layer; kis a coefficient; and Ncis the number of the mud par-

ticles coating the motion-initiating particle and can be evaluated by

Nc¼πð1−ηΔÞd2nc, where ηΔ¼dΔ=d,dΔis the distance from

the top surface of the motion-initiating particle to the surface of the

bed, which indicates the position of the motion-initiating particle in

the bed surface, and ncis the number of mud particles per unit area

of the surface of the motion-initiating particle and can be calculated

Fig. 1. Initial motion of sand–mud mixture.

Fig. 2. Force balance for a sediment particle with median size of a

mixture at the surface of a bed exposed to a unidirectional flow.

J. Hydraul. Eng., 2018, 144(8): 04018046

by nc¼½6=ðπd2

cÞðρdcm=ρsc Þ(Cao 1997). Consequently, the result-

ant cohesive force can be expressed as

Fc¼kNcfc

¼6kαcð1−ηΔÞρεd21

dcρsdc

ρsc  1

ρsdc

ξcρdmρsn

ρsn −ξnρdmmþ1

ð18Þ

Further substituting Eqs. (15)–(18) into Eq. (14), we can obtain

the following relation:

ρucr2

ðρs−ρÞgd ¼πtan φ

3Cdα1λ2ð1þηtan φÞ1þ36

πkαcð1−ηΔÞε

×ρ

ρs−ρ

dcρsdc

ρsc  1

ρsdc

ξcρdmρsn

ρsn −ξnρdmmþ1ð19Þ

When cohesion is negligible, Eq. (19) is consistent with Eq. (1),

yielding

ϑcr0¼πtan φ

3Cdα1λ2ð1þηtan φÞð20Þ

In general, Eq. (19) then can be rewritten as

ϑsh;cr ¼ϑcr0þϑcr1

ρs−ρ

dcρsdc

ρsc  1

ρsdc

ξcρdmρsn

ρsn −ξnρdmmþ1

ð21Þ

where ϑsh;cr = critical Shields parameter; and ϑcr1= combined

coefficient, ϑcr1¼½12kαcð1−ηΔÞεtan φ=½Cdα1λ2ð1þηtan φÞ,

whose value not only can reflect the cohesive strength of the mud

component but is also related to the characteristics of sediments in

the bed surface. Currently, evaluation of ϑcr1is unavailable because

some coefficients involved in the formula of ϑcr1, e.g., ηΔ,ε,φ,Cd,

and η, are complicated and difficult to be determined. Therefore,

ϑcr1will be treated as a presently unknown coefficient that will

be determined by the measured data of erosion thresholds.

When the mud content is 100%, i.e., the mixture is a pure mud,

then ξc¼100%,ξn¼0%, and dc,ρdm, and ρsc should be substi-

tuted by d,ρdc, and ρs, respectively. Consequently, Eq. (21)is

simplified to

ϑsh;cr ¼ϑcr0þϑcr1

ρs−ρ

d2ρsdc

ρsρdc

ρsdcmþ1

¼AG=d2

ðρs−ρÞgd þBFc=d2

ðρs−ρÞgd ð22Þ

where according to Eqs. (17) and (18), G¼ðπ=6Þðρs−ρÞgd3,

Fc¼6kαcð1−ηΔÞρεdðρsdc=ρsÞðρdc =ρsdcÞmþ1, and A¼B¼

ð6=πÞϑcr0. Eq. (22) is consistent with Eq. (6), which is the general

expression of the critical Shields parameter for cohesive sediments.

As a result, the unified formula of the critical shear stress for

erosion of sand, mud, and sand–mud mixtures can be expressed

as follows, from Eq. (21):

τcr ¼ϑcr0ðρs−ρÞgd þϑcr1ρ1

dcρsdc

ρsc  1

ρsdc

ξcρdmρsn

ρsn −ξnρdmmþ1

ð23Þ

Eq. (23) is the primary outcome of this work. Although Eq. (23)

is derived from force balance analysis of a sand–mud mixture, it

also works for pure sand and pure mud, which will be further dem-

onstrated in the next section after the unknown coefficients of the

formula are determined.

Calibration and Validation

Comparison with Experimental Studies

Data from three previous studies were used to test the validity of

our proposed model. Here we give an overview of these three stud-

ies. Torfs (1995) conducted a series of erosion tests of artificial

sand–mud mixtures. The mixed sediments were prepared by mix-

ing varying fractions of mud with fine sand ranging in size from

0.13 to 0.5 mm. Two different muds, kaolinite and bentonite, were

used in the experiments. The mud content ranged from 0 to 28%.

The bulk densities of the sand–mud mixtures were kept constant at

1,850 kg=m3during all the experiments. Sharif (2003) conducted a

series of more extensive and systematic erosion tests of sand–mud

mixtures. The mixtures were prepared by mixing well-sorted fine

sand, kaolinite, and water in an electric mixer and consolidating the

mixtures for 48 h before the experiments. Because of the consoli-

dation process, the bulk densities of mixtures at different depths

ranged from 1,200 to 1,400 kg=m3, and the mud content varied

from 0 to 100%. Jacobs et al. (2011) conducted a large number

of erosion tests on artificially generated and relatively dense

sand–mud mixtures with bulk densities ranging from 1,800 to

2,000 kg=m3. The mixtures were of sand, silt, and clay. The

median particle diameters of the sand and silt fractions were

0.17 and 0.028 mm, respectively. Two different clay materials,

kaolinite and bentonite, were used in the experiments. The mud

(silt and clay) content ranged from 0 to 80%. The relevant param-

eters of the three experiments are shown in Table 1.

The critical Shields parameter in Eq. (23) is calculated by

Eq. (2). Because the stable dry bulk densities were usually not mea-

sured in previous erosion experiments of sediments, the stable dry

bulk density in Eq. (23) is calculated by an empirical formula de-

veloped by Dou (2000) (see Appendix Ifor details), which in this

study is assumed to be valid. The calculated stable dry bulk den-

sities of the mud components in the three previously mentioned

experiments are also shown in Table 1. Half of the collected data

of the three experiments were used to determine the coefficients of

Eq. (23), and the other half were used to validate its applicability

for sand–mud mixtures. Consequently, the values of the coeffi-

cients in Eq. (23) were obtained as ϑcr1¼6.20 ×10−8m3=s2

and m¼1.55. The calibration and validation are shown in Fig. 3.

The calculated critical shear stresses agree well with the measured

critical shear stress in the case of Sharif (2003). Calculated shear

stresses are a little larger than those measured by Jacobs et al.

(2011), but a little lower than those measured by Torfs (1995).

However, in both these studies, the calculated and measured critical

shear stresses are well correlated with each other. The discrepancies

between measurements and calculations are probably caused by the

different criteria of erosion threshold adopted by Jacobs et al.

(2011) and Torfs (1995). On the whole, the formula developed

in this study can reasonably predict the critical shear stress for

erosion of sand–mud mixtures. All the mixtures used in the three

experiments are mainly sand–kaolinite mixtures and a few of sand–

bentonite mixtures. Different kinds of clay mineral may correspond

to different values of the coefficients ϑcr1and mbecause the co-

hesion of different kinds of clay mineral is different. The param-

eters obtained here need more tests and validations, especially of

different kinds of clay.

Next, we further validate the applicability of Eq. (23) in pure

sand and pure mud. When the mud content is 0%, i.e., the mixture

is pure sand, Eq. (23) is simplified to the general formula for the

critical shear stress of noncohesive sediments

τcr ¼ϑcr0ðρs−ρÞgd ð24Þ

J. Hydraul. Eng., 2018, 144(8): 04018046

Eq. (24) is widely accepted and used in noncohesive sediments

transport, and the coefficient ϑcr0can be calculated by Eq. (2),

which was verified by Soulsby and Whitehouse (1997).

When the mud content is 100%, i.e., the mixture is pure mud,

Eq. (23) is simplified as

τcr ¼ϑcr0ðρs−ρÞgd þϑcr1ρ1

dρsdc

ρsρdc

ρsdcmþ1

ð25Þ

Five groups of experimental data for mud were collected from

research in the following areas—Fodda estuary (Yang and Wang

1995), La Vilaine estuary (Yang and Wang 1995), Hangzhou Bay

(Yang and Wang 1995), Tianjin New Port (Yang and Wang 1995),

and Lianyungang (Yang and Wang 1995)—for validating the

application of Eq. (25) in pure mud. The relevant parameters of

the five experiments and the stable dry bulk densities of pure muds

in each experiment calculated by Eq. (28) are shown in Table 2.

The values of the coefficients obtained from the data sets of mixtures

were applied in the validation of pure mud, i.e., ϑcr1¼6.20 ×

10−8m3=s2and m¼1.55, and the result is shown in Fig. 4. It was

found that the calculated critical shear stresses agree well with the

measured values, demonstrating the capability of the unified formula

in pure mud.

In addition, it needs to be pointed out that the value of the stable

dry bulk density has an impact on the values of the coefficients ϑcr1

and min Eqs. (23) and (25). Therefore, the parameters obtained

here need further validation, especially using data sets where the

measured stable dry bulk densities and critical shear stresses are

available.

Comparison with Other Common Formulas

Here, the unified formula developed in this paper is compared with

some other formulas that are widely used in practice to further

evaluate its accuracy.

The unified formula [Eq. (23)], along with van Ledden’s for-

mula [Eq. (7)] and Ahmad’s formula [Eq. (8)], were applied to cal-

culate the critical shear stresses of sand–mud mixtures for different

soil depths in the case described by Sharif (2003). The calculated

and measured critical shear stresses are plotted against mud content

in Fig. 5. Figs. 5(a and b) represent the results of different soil

depths. In the calculation, the coefficients ξc;cr and βin Eq. (7)

are assigned values of 10% and 1.0, respectively; and the coeffi-

cient ζin Eq. (8) is given the value 0.15; these values were based

on Sharif’s(2003) data set.

As shown in Fig. 5, Eqs. (7) and (8) reflect only the general

trend of the critical shear stress for erosion varying with mud con-

tent, while the unified formula developed in this study [Eq. (23)]

reproduces well the effect of the mud fraction on critical shear

stress. The variation of the critical shear stress calculated by

Eq. (23) with mud content is basically consistent with the exper-

imental observations described previously. When the mud content

is less than 10%, the calculated critical shear stress increases

slightly with increasing mud content. This is a little different from

the measured values, which basically stay constant, but is consis-

tent with some other experimental observations as mentioned in

“Previous Work.”This discrepancy may result from the slight in-

crease in the critical shear stress, which can be covered by the sys-

tem errors brought by other factors, e.g., determination of the

criterion of erosion threshold by eye observation. Additionally,

it is noted that Eqs. (7) and (8) are related to the critical shear

stresses of pure sand and pure mud. However, as stated previously,

for a given sand–mud mixture, the critical shear stress of the pure

mud is not easy to measure or calculate. By contrast, the unified

formula developed in this paper depends only on the gradation

and dry bulk density of the mixture, which are easy to measure,

making it more convenient to apply than Eq. (7)or(8).

Fig. 6shows the application of the unified formula to pure

mud, along with some common formulas for pure mud compared

with measured data from the reported studies. It is obvious that the

Table 1. Erosion thresholds of sand–mud mixtures

Reference Sand size (mm) Mud size (mm) Mud type

Mud

content (%)

Bulk density

(kg=m3)

Stable dry bulk

density (kg=m3)

Torfs (1995) 0.130–0.500 Smaller than 0.063 Kaolinite and bentonite 0–28 1,850 868 (kaolinite)

780 (bentonite)

Sharif (2003) 0.180–0.212 Smaller than 0.013 Kaolinite 0–100 1,200–1,410 1,095

Jacobs et al. (2011) 0.063–0.200 Smaller than 0.063 Kaolinite and bentonite 0–80 1,800–2,000 709–1,224

Fig. 3. Calibration and validation of Eq. (23) in sand–mud mixtures by

three groups of experimental data.

Table 2. Erosion thresholds of muds

Mud source Grain size (mm) Dry bulk density (kg=m3) Stable dry bulk density (kg=m3)

Fodda estuary dc¼0.0035,dc25 ¼0.0026 170–720 1,030

La Vilaine estuary dc¼0.0031,dc25 ¼0.00031 148–365 506

Hangzhou Bay dc¼0.0104,dc25 ¼0.0023 415–640 952

Tianjin New Port dc¼0.0053,dc25 ¼0.0033 120–980 1,053

Lianyungang dc¼0.0040,dc25 ¼0.00075 170–700 767

Source: Data from Yang and Wang (1995).

J. Hydraul. Eng., 2018, 144(8): 04018046

critical shear stresses calculated by Eq. (25), which is a simplifi-

cation of the unified formula [Eq. (23)] for pure mud, are in good

agreement with the measured values. The critical shear stresses

calculated using the formulas of Li et al. (1995) and Lick et al.

(2004) (Appendix II) are far lower than the measured values.

The accuracies of the formulas of Yang and Wang (1995) and

Dou (2000) (Appendix II) are approximately the same and their

values are a little lower than the measured values. The calculated

results of the unified formula are a little larger than the mea-

sured data in some cases, especially in the case of Hangzhou

Bay and Fodda estuary, and are slightly smaller than the measured

data in the case of La Vilaine estuary. However, Fig. 6shows that

the calculated results of the formulas of Li et al. (1995), Yang and

Wang (1995), and Dou (2000) and the unified formula in this study

are all highly correlated with the measurements. With the optimized

parameters based on measured data from each site instead of the

common parameters used in the calculation of Fig. 6, all the calcu-

lated results from the four formulas agree well with the measure-

ments. Fig. 6also shows the calculated results of the unified

formula with the optimized parameters of ϑcr1and m, which agree

very well with the measured data. The discrepancies between mea-

surements and the calculated results of the unified formula of the

common parameters may be caused by the different criteria for

the erosion threshold adopted by different researchers. This also

emphasizes that when using the unified formula in this study,

the parameters could be optimized.

Discussion

General Behavior of the Unified Formula in

Sand–Mud Mixtures

In order to easily understand the behavior of the unified formula,

Eq. (23) is rewritten in the following form:

τcr ¼ϑcr0ðρs−ρÞgd þϑcr1ρ1

dcρsdc

ρsc ρdcm

ρsdc mþ1

ð26Þ

where ρdcm and ρsdc = dry bulk density and stable dry bulk density

of the mud component in the mixture. The density ratio, ρdcm=ρsdc ,

Fig. 4. Validation of Eq. (25) in pure muds by five groups of experi-

mental data.

Fig. 5. Variation of the measured and calculated critical shear stresses with mud content in the case of Sharif (2003).

J. Hydraul. Eng., 2018, 144(8): 04018046

represents the degree of consolidation of the mud component in the

mixture. The stable dry bulk density can be calculated by Eq. (28)

in Appendix Iand the dry bulk density of the mud component can

be calculated by the following formula according to Eqs. (10)

and (11):

ρdcm ¼ξcρdmρsn

ρsn −ξnρdm ð27Þ

According to Eq. (26), the erosion resistance of a mixture con-

sists of two parts: the submerged weight of the motion-initiating

particle [the first item on the right side of Eq. (26)], and the cohesive

strength coming from the mud particles around the motion-initiating

particle (the second item). The submerged weight is proportional to

the median size of the mixture, while the cohesive strength is

inversely proportional to the median size of the mud component

and is greatly influenced by the degree of consolidation of the

mud component in the mixture. As the median size of the mud com-

ponent is often in the order of 10−5or 10−6m, the coefficient of ϑcr1

related to the contribution of the cohesive strength to the erosion

threshold is in the order of 10−8m3=s2according to the calibration

result. In addition, the cohesive strength is independent of the diam-

eter of the sand component in the mixture, which shows that the

cohesive strength makes the same contribution to the erosion thresh-

olds of a large sand particle and of a small sand particle.

For a mixture with a known gradation, the stable dry bulk den-

sity of the mud component is determined according to Appendix I.

In this case, the critical shear stress of the mixture would mainly

depend on the dry bulk density of the mixture according to Eqs. (26)

and (27). However, at present little is known about the variation of

the dry bulk density of the mixture with mud content. It is therefore

not easy to understand the general behavior of the unified formula

in sand–mud mixtures. Here we will briefly investigate the behavior

of the unified formula using the experimental data from Sharif

(2003), which includes the measured dry bulk densities of mixtures

with different mud contents.

Fig. 7(a) shows the critical shear stresses of sand–mud mixtures

calculated by the unified formula [Eq. (23)] varying with mud

content in the case of Sharif (2003), and Fig. 7(b) shows the ratio

of the dry bulk densities of the mud component to its stable dry bulk

density, also varying with mud content. Different solid lines with

markers represent different soil depths, but all of the soil samples

had consolidated for the same time of 48 h before the experiments.

The dashed line in Fig. 7(a) represents the part of the critical shear

stress contributed by the effective gravity, which is calculated by

the first term on the right-hand side of Eq. (23). The gaps between

the solid lines and the dashed line therefore represent the parts of

the critical shear stress of the mixtures contributed by the cohesive

strength. It was found that the contribution to the erosion threshold

from the cohesive strength increases rapidly with increasing mud

content to a maximum at an optimum mud content, then has a slow

decrease up to 100% mud content. The part of the critical shear

stress contributed by the cohesive strength was more than 70%

when the mud content was higher than 20%, and more than

90% when the mud content was higher than 50%. It was also found

that when the mud content was higher than 30%, the critical shear

stress of a mixture was larger than that of pure sand and pure mud.

The mixture with the optimum mud content has a maximum critical

shear stress, which can be seven times the critical shear stress of

pure sand and 1.7 times the critical shear stress of pure mud.

Comparing Figs. 7(a) and 7(b) shows that the calculated critical

shear stress of a mixture varies with the mud content in the similar

way as the ratio of the dry bulk density to its stable density. The

critical shear stress and the density ratio (ρdcm =ρsdc) both first in-

crease markedly with increasing mud content to a maximum at an

optimum mud content of 30–50%. With further increase in mud

content up to 100%, there is a more gradual decrease in both

the density ratio and critical shear stress. The similarity of the

behaviors of the calculated critical shear stress and the density ratio

was not a feature of this data set but was generally predicted by

Eq. (26). This is because the critical shear stress of a cohesive

Fig. 6. Application of the unified formula and some other common formulas in pure muds.

J. Hydraul. Eng., 2018, 144(8): 04018046

mixture (mud content exceeding the critical value, usually 3–15%)

is mainly due to the cohesive strength, which in turn depends

mainly on the degree of consolidation of the mud component in

the mixture. Fig. 7(b) shows that the mud components in the mix-

tures with different mud contents have different consolidation rates.

The different consolidation rates resulted in different degrees of

consolidation of the mud components over the same time, which

further led to the variation of the critical shear stresses over mud

contents shown in Fig. 7(a). The dips of the calculated critical shear

stresses at mud content of 40% result from the lower values of the

dry bulk densities of the mud component at mud content of 40%

[Fig. 7(b), noting that the stable dry bulk density is a constant in this

case (Table 1)]. However, the dry bulk densities of the mud com-

ponent were calculated from the measured dry bulk density of the

mixtures according to Eq. (27). Therefore, the dips in this data set

may be caused by uncertainty in measurements. This uncertainty

may be better constrained when other data sets become available.

General Behavior of the Unified Formula in Pure Mud

When the mud content is specified as 100%, the mixture is pure

mud and the unified formula [Eq. (23)] can be simplified to

Eq. (25). According to Eq. (25), the critical shear stress of pure

mud is a function of the median size of the mud, the dry bulk

density, and its stable dry bulk density. Fig. 8shows the critical

shear stress of uniform pure mud varying with the median size.

The different types of dashed lines represent different ratios of

the dry bulk density to its stable dry bulk density, which represent

different degrees of consolidation. It was found that the critical

shear stress of the pure mud was far larger than that of the nonco-

hesive sediments with the same size. The degree of consolidation

has a significant effect on the critical shear stress. For pure mud

with a certain particle size, the more consolidated it is, the larger

the critical shear stress for erosion. The critical shear stress of a

fully consolidated mud (ρdc=ρsdc ¼1.0) can be several orders of

magnitude larger than that of a weakly consolidated mud.

Conclusions

Previous research of critical shear stress for erosion in pure non-

cohesive sediments, pure cohesive sediments, and sand–mud mix-

tures was reviewed. A general expression for erosion threshold of

pure mud was summarized. An expression for cohesive forces in

sand–mud mixtures, which considers the effect of the sand compo-

nent on the mud component, was proposed. Further, the balance of

forces under the initial motion condition was analyzed, leading to a

formula for critical shear stress of sand–mud mixtures. The formula

can be simplified to the general formulas for the critical shear

stresses of pure sand and pure mud.

The proposed formula has been successfully tested with three

groups of experimental data of sand–mud mixtures and five groups

of experimental data of pure mud. These tests have demonstrated

the capability of the unified formula in sand–mud mixtures and

pure mud. However, the parameters need more tests and valida-

tions, especially of different kinds of clay and using data sets in

which the measured stable dry bulk densities and critical shear

stresses are available. When relevant measurements are available,

the formula’s parameters can be optimized.

The experimental data of Sharif (2003) show that the mud com-

ponents in sand–mud mixtures with different mud contents have

different degrees of consolidation over the same time. The differing

degrees of consolidation result in the erosion threshold of sand–

mud mixtures varying with mud content. The critical shear stress

first increases markedly with increasing mud content to a maxi-

mum, then decreases slowly up to 100% mud content. The opti-

mum mud content at which the critical shear stress reaches a

peak is between 30 and 50%. The variation of the critical shear

Fig. 7. (a) Critical shear stresses calculated by the unified formula [Eq. (23)] and (b) ratios of the dry bulk densities of the mud component to its stable

dry bulk density as they vary with mud content and soil depth. The dashed line in Fig. 7(a) represents the part of the critical shear stress contributed by

the effective gravity, which is calculated by the first term on the right-hand side of Eq. (23).

Fig. 8. Critical shear stress of uniform pure mud, as per Eq. (25). The

values of 0.2, 0.4, 0.6, 0.8, and 1.0 represent the ratios of the dry bulk

density of the mud to its stable dry bulk density.

J. Hydraul. Eng., 2018, 144(8): 04018046

stress with mud content, and the optimum mud content, were well

reproduced by the proposed formula.

The effect of consolidation on the erosion threshold of pure mud

was also analyzed. As found by other researchers, the more con-

solidated the mud, the larger the erosion threshold. The critical

shear stress of a fully consolidated mud can be several orders of

magnitude larger than that of a weakly consolidated mud.

Considering the effect of consolidation on the erosion thresh-

olds of both sand–mud mixtures and pure muds makes the pro-

posed formula capable of predicting the erosion thresholds of

sediments in the process of consolidation.

The proposed formula was expressed in terms of mud content

and densities, which are reasonably easily determined from a field

sample. This makes the formula more convenient in practical

application compared with the existing formulas for sand–mud

mixtures.

Appendix I. Stable Dry Bulk Density of Cohesive

Sediments

When fine sediment first settles on a bed, it tends to have a low bulk

density due to its loose network structure and high water content.

The deposit will begin to consolidate either under its own weight or

under the external load of the layer above it. The volume will de-

crease while the bulk density will increase, and the deposit

will become more compact as the deposit drains the pore water

gradually during the consolidation. The dry bulk density will reach

its maximum when the deposit is fully consolidated, and this maxi-

mum dry bulk density is called the stable dry bulk density, which is

widely used by hydraulic researchers (e.g., Tang 1963;Yang and

Wang 1995;Dou 2000;Zhang 2012) and similar concepts are used

in agricultural research (Alberts et al. 1995).

Dou (2000) found that the stable dry bulk density is a function

of the grain size gradation. The smaller the median particle size, the

smaller the stable dry bulk density; the higher the proportion of

finer particles in the deposit, the smaller the stable dry bulk density.

Dou obtained the following empirical formula by analyzing a lot of

the experimental data he collected:

ρsdc ¼0.68ρsdc

drn

ð28Þ

where ρsdc = stable dry bulk density of cohesive sediments; ρs=

particle density; dc= median size of the sediments (mm); dr=

reference diameter with the value 1 mm; and n= coefficient reflect-

ing the amount of the relatively finer fraction. The value of nis

calculated by

n¼0.080 þ0.014dc

dc25ð29Þ

where dc25 = sediment size (mm) such that 25% of the mud by

weight is finer than this size.

Appendix II. Formulas for Erosion Thresholds of Cohesive Sediments

The formulas for the erosion thresholds of cohesive sediments, and the expressions for the effective weight, the cohesive force, and the

additional water pressure (if considered), along with the corresponding parameters for each formula from transforming it into the form

of Eq. (6), are listed.

Reference Erosion threshold Terms

Dou (1962)ub;cr ¼2.24ρs−ρ

ρgd þ0.19 εkþghδ

d1=2

G¼π

6ðρs−ρÞgd3;Fc¼π

2εd;Fh¼π

2ρghdδ;A¼6

π2.24 ucr

ub;cr2

;

B¼0.38

ρεk

ε2.24 ucr

ub;cr2

;C¼0.38

π2.24 ucr

ub;cr2

Tang (1963)τcr ¼3.2

77.5ðρs−ρÞgd þ1

3.2ρdc

ρsdc10 cg

dG¼αcðρs−ρÞgd3;Fc¼εdρdc

ρsdc10

;A¼3.2

77.5αc

;B¼1

77.5

ε;C¼0

Han (1982)

ub;cr ¼φ2

3Cx

ρs−ρ

ρgd þ2ﬃﬃﬃ

pq0δ3

Cxρd3−t

δ11

t2−1

δ2

1

þ4ﬃﬃﬃ

pK2gh

Cxd3−t

δ1ðδ1−tÞ

1=2

G¼π

6ðρs−ρÞgd3;Fc¼1

2q0πδ3

0d1

t2−1

δ2

1;

Fh¼ﬃﬃﬃ

2πK2ρghd3−t

δ1ðδ1−tÞ;A¼8

πCxφucr

ub;cr2

;

B¼4ﬃﬃﬃ

Cxπφucr

ub;cr23−t

δ1;C¼8

πCxφucr

ub;cr2

Dade et al.

(1992)

ϑsh;cr ¼πtan ϕ=b148

1þ0.1R;cr tan ϕ1þFc

GG¼π

6ðρs−ρÞgd3;Fc¼1

2b2πd2ð1−cos ϕÞτy;

A¼B¼tan ϕ=8b1

1þ0.1R;cr tan ϕ;C¼0

Yang and

Wang (1995)

τcr ¼ϑcr0ðρs−ρÞgd þ9×10−6

dS

Ss2.35

G¼π

6ðρs−ρÞgd3;Fc¼εdS

Ss2.35

;A¼6

πϑcr0;B¼9×10−6

ε;C¼0

Li et al.

(1995)

τcr ¼π

78 ðρs−ρÞgd þ6

dρdc

ρsdc3.25 G¼π

6ðρs−ρÞgd3;Fc¼εdρdc

ρsdc3.25

;A¼B¼1

13;C¼0

J. Hydraul. Eng., 2018, 144(8): 04018046

Appendix II. (Continued.)

Reference Erosion threshold Terms

Dou (2000)

τcr ¼0.0164ρΔh

Δ1=32

3.6ρs−ρ

ρgdþ

ε0þghδﬃﬃﬃﬃﬃﬃﬃﬃ

δ=d

dρdc

ρsdc2.53

G¼π

6ðρs−ρÞgd3;Fc¼π

2αcρεdρdc

ρsdc2.5

;

Fh¼ahπρghdδﬃﬃﬃﬃﬃﬃﬃﬃ

δ=d

pρdc

ρsdc2.5

;A¼0.354

πΔh

Δ1=3

;

B¼0.0328

παc

ε0

εΔh

Δ1=3

;C¼0.0164

πahΔh

Δ1=3

Lick et al.

(2004)

τcr ¼414dþ1

c4

τcrðρbc ;5Þ

τcrG¼π

6ðρs−ρÞgd3;Fc¼c4τcrðρbc ;5Þ

τcrd;A¼2,484

ðρsc −ρÞgπ;B¼414

;

C¼0

Zhang

(2012)ub;cr ¼C1

ρs−ρ

ρgd þC2

ρs−ρ

ρg0.33ρdc

ρsdc6.6υ1.34

þC3ρdc

ρsdc6.6ghδ

1=2

G¼αcðρs−ρÞgd3;Fc¼αzρdρdc

ρsdc6.6ρs−ρ

ρg0.33υ1.34 ;

Fh¼ahπρghdδρdc

ρsdc6.6

;A¼1

αcC1

ucr

ub;cr2

;B¼C2

αzC1

ucr

ub;cr2

;

C¼C3

πahC1

ucr

ub;cr2

Note: ub;cr = critical near-bed velocity; ucr = critical shear velocity; ε= coefficient representing the cohesive strength of the material;

εk= coefficient related to cohesive strength and the characteristics of sediments on the bed surface, εk¼2.56 cm3=s2;δ= parameter

of water film thickness, δ¼2.31 ×107m; ρdc and ρsdc = dry bulk density and table dry bulk density of cohesive sediments, respectively;

c= coefficient related to cohesive strength and the characteristics of sediments on the bed surface, c¼2.9×10−5kg=m; αc= shape

coefficient; φ= coefficient related to the characteristics of sediments on the bed surface; Cx= coefficient for the drag lift; q0= coefficient

denoting the cohesive strength of the material and is valued as 1.30 ×109kg=m2based on measured critical velocities; δ0¼3×10−10 m;

K2= coefficient related to the additional water pressure due to the water film, K2¼2.58 ×10−3;δ1= thickness of water film and is given the

value of 4×10−7m; h= water depth; t= particle separation; b1and b2= coefficients; ϕ= particle packing angle of sediments; τy= apparent

yield stress; R;cr = critical particle Reynolds number; Sand Ss= sediment concentration and the stable sediment concentration, respectively;

Δ¼20 mm; Δh= roughness height, Δh¼1mm for d≤0.5mm, Δh¼2dfor 0.5mm < d<10 mm, Δh¼Δfor d≥10 mm;

ah= coefficient related to the additional water pressure; c3¼πðρs−ρÞg=6;c4¼1.33 ×10−4N=m; τcr¼1.35 N=m2;ρbc = bulk density

of sediments; τcrðρbc ;5Þ= value of the critical shear stress at different values of ρbc and for d¼5μm, τcrðρbc;5Þ¼a1expðb1ρbc Þ,

a1¼7×108N=m2, and b1¼9.07 L=kg; C1¼2.1;C2¼6.59;C3¼0.0352; and αz= coefficient.

Acknowledgments

This work was supported by the National Science & Technology

Pillar Program (Grant No. 2012BAB03B01), the Fundamental

Research Funds for the Central Universities, Hohai University

(Grant No. 2014B31214), Postgraduate Research & Practice Inno-

vation Program of Jiangsu Province (KYLX_0489), and the State

Scholarship Fund of China Scholarship Council (CSC). Also, the

valuable suggestions from Mr. Graham Macky are appreciated.

Notation

The following symbols are used in this paper:

d=median size of sediments;

dc=median size of the mud component in a sand–mud

mixture;

dc25 =sediment size such that 25% of the mud

component by weight is finer than this

size;

dΔ=distance from the top surface of the

motion-initiating particle to the surface of bed;

d=dimensionless grain size,

d¼ f½ðρs−ρÞ=ρ½ðgd3Þ=υ2g1=3;

Fc=resultant cohesive force;

Fd,Fl=drag and lift forces, respectively;

fc,fa=cohesive and adhesive forces, respectively;

G=submerged weight of the particle with median size

of a sediment;

g=acceleration of gravity;

h=water depth;

m=exponent;

n=exponent;

nc=number of mud particles per unit area of the plane

of failure;

ns=void ratio of the sand component per unit volume of

a mixture;

ub=instantaneous velocity approaching the

motion-initiating particle on the bed;

ub;cr =critical near-bed velocity for the inicipient motion

of sediments;

uc=critical shear velocity;

αc=coefficient;

β=empirical coefficient;

ε=coefficient denoting strength of cohesive action of

the materials;

ζ=empirical coefficient;

ϑsh;cr =critical Shields parameter;

ξc,ξn=mud content and sand content, respectively;

ξc;cr =critical mud content;

ρ=water density;

ρdcm =dry bulk density of the mud component in the voids

of sand particles;

J. Hydraul. Eng., 2018, 144(8): 04018046

ρdm =dry density of a sand–mud mixture;

ρs=density of the particle with median size of a

sediment;

ρsdc =stable dry bulk density of mud;

ρsn,ρsc =sand particle and mud particle density, respectively;

τcr =critical shear stress;

τcr;n,τcr;c=critical shear stresses of pure sand and pure mud,

respectively; and

φ=characteristic internal friction angle.

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May 2024

The quantification of the erodibility of cohesive sediments is fundamental for an advanced understanding of estuarine sediment transport processes. In this study, the surface erosion threshold τc for cohesive sediments collected from two sites in the area of the Port of Hamburg in the River Elbe is investigated in laboratory experiments. An improved closed microcosm system (C-GEMS) is used for the erosion experiments, which allows the accumulation of suspended sediment concentration (SSC) over an experimental run. A total of 34 erosion experiments has been conducted with homogenized samples and bulk densities between 1050 kg/m³ and 1250 kg/m³. The covered range of bulk densities is seen to represent the values commonly exhibited by freshly deposited cohesive sediments. Two approaches to derive τc based on the erosion rate (ε-method) and the SSC (SSC-method) were elaborated and compared. For both approaches, only one parameter has to be set in order to facilitate transferability to other devices. The results show a better performance of the SSC-method in terms of lower uncertainties, especially at the upper application limits of the utilized C-GEMS. The application of the SSC method yields values for τc between 0.037 N/m² and 0.305 N/m², continuously increasing with bulk density. Repetition tests proved the repeatability of the experimental procedure and utilized methods to derive τc . The derived data for τc is used to fit two mathematical models: i) a highly empirical model relating τc to dry bulk density and ii) a recently proposed model relating τc to the physical properties of the sediment-mixture. While the derived parameters for the first model vary widely for the two sampling sites, the fit-parameter for the latter model is virtually independent of the investigated site, suggesting the superiority of this approach.

The incipient motion safety factor of sediment considers the coupling between the surface and subsurface flows under multi-motion modes

Article

Mar 2024
J HYDROL

The incipient movement of sediment particles in natural rivers under the action of water scour has remained a key issue in river sediment movement research. In this study, we establish a mathematical coupling model between surface and subsurface flows for investigating the interface velocity. The Navier-Stokes equation and Darcy's law are employed to describe surface and subsurface water flows, respectively. The interface velocity U between surface and subsurface flows is derived by introducing the Beavers–Joseph boundary conditions. Furthermore, based on the force equilibrium of a single particle, combined with three incipient modes, namely, sliding, rolling, and saltation, analytical solutions are derived for the critical instantaneous bottom velocity u0, and mean incoming velocity. The derived analytical solutions exhibit a higher computational accuracy than existing flume test data and classical flow velocity model results, with relative errors less than 12%. Furthermore, the incipient motion state of sediment particles can be directly assessed by defining the critical incipient safety factor for particle motion (i.e., k = U/u0). Sensitivity analysis of the safety factor for sediment particle motion reveals a nonlinear positive correlation with the slope angle, riverbed permeability, riverbed porosity, and hydraulic gradient, while indicating a linear positive correlation with the water depth. However, a nonlinear negative correlation is observed with the B-J coefficient, relative exposure degree, and sediment particle dry bulk density. The safety factor shows a rapid decrease followed by a stabilizing trend with increasing B-J coefficient. Additionally, the safety factor first increases and then decreases with increasing in sediment particle size.

On the erosion of cohesive granular soils by a submerged jet: a numerical approach

Article

Full-text available

Dec 2022
GRANUL MATTER

This paper presents an erosion interpretation of cohesive granular materials stressed by an impinging jet based on the results of a micromechanical simulation model. The numerical techniques are briefly described, relying on a two-dimensional Lattice Boltzmann Method coupled with a Discrete Element Methods including a simple model of solid intergranular cohesion. These are then used to perform a parametric study of a planar jet in the laminar regime impinging the surface of granular samples with different degrees of cohesive strength. The results show the pertinence of using a generalized form of the Shields criterion for the quantification of the erosion threshold, which is valid for cohesionless samples, through empirical calibration, and also for cohesive ones. Furthermore, the scouring kinetics are analysed here from the perspective of a self-similar expansion of the eroded crater leading to the identification of a characteristic erosion time and the quantification of the classical erosion coefficient. However, the presented results also challenge the postulate of a local erosion law including erodibility parameters as intrinsic material properties. The paper then reviews the main limitations of the simulation and current interpretation models, and discusses the potential causes for the observed discrepancies, questioning the pertinence of using time-averaged macroscopic relations to correctly describe soil erosion. The paper concludes addressing this question with a complementary study of the presented simulations re-assessed at the particle-scale. The resulting local critical shear stress of single grains reveals a very wide dispersion of the data but nevertheless appears to confirm the general macroscopic trend derived for the cohesionless samples, while the introduction of cohesion implies a significant but systematic quantitative deviation between the microscopic and macroscopic estimates. Nevertheless, the micro data still shows consistently that the critical shear stress does actually vary approximately in linear proportion of the adhesive force.

Mechanisms of the Non‐Uniform Breach Morphology Evolution of Landslide Dams Composed of Unconsolidated Sediments During Overtopping Failure

Article

Full-text available

Sep 2022

Overtopping flows in landslide dams erode and entrain materials on the dam surface resulting in erosional features that undermine the dam stability and facilitate the subsequent outburst flooding. A comprehensive understanding of dam surface evolution is therefore crucial for flood risk assessment and hazard mitigation. In this research, we study the mechanisms that influence the non‐uniform morphology evolution of landslide dam breaches (i.e., non‐linear variation of the dam surface gradient) through experiments and numerical modeling. Analog landslide dam models, constructed using unconsolidated poorly sorted soils, are exposed to different inflow discharges. We find that although the breach discharge evolves more consistently with the erosion along the sidewalls than with bed erosion, it is the erosion along the bed that controls the change in dam surface profiles. Erosion rates, expressed as a function of the difference between the flow shear stress and the apparent erosion resistance, vary at different points along the dam surface due to localized erosional features induced by scouring. The apparent erosion resistance is found to increase linearly along the dam surface. Dam failure is numerically modeled using depth‐averaged equations which assume that the complex evolution of the dam profiles is due to the coupled effects of erosion, entrainment, and channel bed collapse. Good agreement between the observed and modeled dam profiles further demonstrates that the gradual saturation of the breach flow with entrained sediment is responsible for the linear variation of the apparent erosion resistance, which in turn contributes to the formation of the surface scouring.

Numerical Simulation of Hydrodynamic and Topographical Changes Caused by an Offshore Wind Farm on a Muddy Coast

Chapter

Feb 2025

Erosion Thresholds and Rates for Sand-Mud Mixtures

Technical Report

Full-text available

Jul 2020

Differences in erosion behavior of non-cohesive and cohesive sediments are widely recognized. In many natural environments, sand and mud are not completely separated and occur as mixtures. Significantly less research has been conducted on the erosion behavior of sand-mud mixtures compared to the separate treatment of sand and mud erosion. Sedflume erosion experiments were conducted on sand-mud mixtures with varying mud content to define the relationships between mud content, critical stress for erosion (τc), and erosion rate. Sand-mud mixtures were prepared with three mud sources: (1) non-swelling clay (kaolinite), (2) swelling clay (kaolinite/bentonite), and (3) a swelling, natural mud from the Mississippi River. Test results showed that critical shear stresses of the mixed sediments departed from that of pure sand with mud fractions on the order of 2% to 10%. Peak τc was observed between 30% to 40% mud content, with swelling muds achieving a tenfold increase in τc while a five-fold increase in τc was measured for kaolinite. Additionally, this study demonstrated that the introduction of small amounts (≤5%) of mud to sand reduced erosion rates by a factor of 10 to 100. This observed abatement of erosion rate has implications for the use of dredged materials in civil and environmental engineering projects.

Modelling Fine Sediment Dynamics: Towards a Common Erosion Law for Fine Sand, Mud and Mixtures

Article

Full-text available

Jul 2017

This study describes the building of a common erosion law for fine sand and mud, mixed or not, in the case of a typical continental shelf environment, the Bay of Biscay shelf, characterized by slightly energetic conditions and a seabed mainly composed of fine sand and muddy sediments. A 3D realistic hydro-sedimentary model was used to assess the influence of the erosion law setting on sediment dynamics (turbidity, seabed evolution). A pure sand erosion law was applied when the mud fraction in the surficial sediment was lower than a first critical value, and a pure mud erosion law above a second critical value. Both sand and mud erosion laws are formulated similarly, with different parameters (erodibility parameter, critical shear stress and power of the excess shear stress). Several transition trends (linear or exponential) describing variations in these erosion-related parameters between the two critical mud fractions were tested. Suspended sediment concentrations obtained from simulations were compared to measurements taken on the Bay of Biscay shelf with an acoustic profiler over the entire water column. On the one hand, results show that defining an abrupt exponential transition improves model results regarding measurements. On the other hand, they underline the need to define a first critical mud fraction of 10 to 20%, corresponding to a critical clay content of 3-6%, below which pure sand erosion should be prescribed. Both conclusions agree with results of experimental studies reported in the literature mentioning a drastic change in erosion mode above a critical clay content of 2-10% in the mixture. Results also provide evidence for the importance of considering advection in this kind of validation with in situ observations, which is likely to considerably influence both water column and seabed sediment dynamics.

Understanding mass fluvial erosion along a bank profile: Using PEEP technology for quantifying retreat lengths and identifying event timing: Understanding mass fluvial erosion along a bank profile using PEEPs

Article

Full-text available

Mar 2017
EARTH SURF PROC LAND

This study provides fundamental examination of mass fluvial erosion along a stream bank by identifying event timing, quantifying retreat lengths, and providing ranges of incipient shear stress for hydraulically driven erosion. Mass fluvial erosion is defined here as the detachment of thin soil layers or conglomerates from the bank face under higher hydraulic shear stresses relative to surface fluvial erosion, or the entrainment of individual grains or aggregates under lower hydraulic shear stresses. We explore the relationship between the two regimes in a representative, US Midwestern stream with semi‐cohesive bank soils, namely Clear Creek, IA. Photo‐Electronic Erosion Pins (PEEPs) provide, for the first time, in situ measurements of mass fluvial erosion retreat lengths during a season. The PEEPs were installed at identical locations where surface fluvial erosion measurements exist for identifying the transition point between the two regimes. This transition is postulated to occur when the applied shear stress surpasses a second threshold, namely the critical shear stress for mass fluvial erosion. We hypothesize that the regimes are intricately related and surface fluvial erosion can facilitate mass fluvial erosion. Selective entrainment of unbound/exposed, mostly silt‐sized particles at low shear stresses over sand‐sized sediment can armor the bank surface, limiting the removal of the underlying soil. The armoring here is enhanced by cementation from the presence of optimal levels of sand and clay. Select studies show that fluvial erosion strength can increase several‐fold when appropriate amounts of sand and clay are mixed and cement together. Hence, soil layers or conglomerates are entrained with higher flows. The critical shear stress for mass fluvial erosion was found to be an order of magnitude higher than that of surface fluvial erosion, and proceeded with higher (approximately 2–4 times) erodibility. The results were well represented by a mechanistic detachment model that captures the two regimes. Copyright © 2017 John Wiley & Sons, Ltd.

Estimation of resuspension rate of cohesive sediments by currents.

Article

Nov 1986

Essais de synthese des lois du début d'entrainement des sédiments sous l'action d'un courant en regime continu

Article

Jan 1963

R. Bonnefille

Soil component. USDA-Water Erosion Prediction Project: Hillslope Profile and Watershed Model Documentation

Article

Jan 1995

Threshold of sediment motion in coastal environments

Article

Intermolecular and Surface Forces, 3rd Edition

Book

Jun 2011

J. N. Israelachvili

This reference describes the role of various intermolecular and interparticle forces in determining the properties of simple systems such as gases, liquids and solids, with a special focus on more complex colloidal, polymeric and biological systems. The book provides a thorough foundation in theories and concepts of intermolecular forces, allowing researchers and students to recognize which forces are important in any particular system, as well as how to control these forces. This third edition is expanded into three sections and contains five new chapters over the previous edition.

A unified formula for incipient velocity of sediment

Article

Dec 2012

H.-W. Zhang

Based on analysis of previous sediment incipient velocity formulae, this paper established an expression form for the adhesive force among the sediment particles and additional downforce caused by the water depth through theoretical discussion, introduced Zhang Hongwu's velocity distribution formula based on mixing length model with eddy mode, and then the undetermined coefficient in formula has been determined in view of the same important influence on incipient motion by both adhesive force and additional downforce caused by the water depth and the eigen value of natural river sediment movement, The unified formula for incipient velocity of sediment theoretically established, which can not only summarize the thickness of the sand, but also reflect the effect of bed resistance and sediment concentration and water temperature on sediment incipient motion, and it is applicable in light sand.

Mechanics of Sediment Transport

Article

Jan 1977

M.S. YALIN

CONTENTS: FUNDAMENTALS, MECHANICAL PROPERTIES OF THE FLOW, DIMENSIONLESS EXPRESSION OF THE TWO-PHASE PHENOMENON, THE BEGINNING OF SEDIMENT TRANSPORT, SEDIMENT TRANSPORT RATE, DISTRIBUTION OF SUSPENDED LOAD, SAND WAVES, FRICTION FACTOR.

Unified Formula for Critical Shear Stress for Erosion of Sand, Mud, and Sand–Mud Mixtures

Abstract and Figures

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