Impact of Cohesive and Seepage Forces on Riverbank Erosion: A Truncated Pyramid Model Study

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Abstract

The riverbank erosion mechanism is extremely complicated and sediment particles on the bank are subjected to variety of forces. Cohesive force between the particles and the force due to seepage of water play an important role among those forces. In this paper the ‘‘Truncated Pyramid Model’’ for the arrangement of particles has been used and generalized equations for calculating escape velocity considering both cohesive and seepage forces have been proposed. The escape velocity is the fundamental pre-requisite to evaluate other parameters directly associated with the riverbank erosion, for example, entrainment rate and volumetric rate of the river bank erosion.

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The effect of seepage force along with the cohesive force has been studied for different degrees of exposure (fully exposed, half exposed and fully submerged) and compared with the previous study where only the force of cohesion has been studied. The change in escape velocity due to seepage force for different degrees of exposure have been determined in this paper for three different particle sizes (0,6mm, 0,8mm and 1mm diameter) along with different inter-particle distances.

Introduction

River bank failure is connected to bank erosion. Erosion of a bank causes loss of bank lands and threatens flood defence. It is a widely known fact that bank-derived sediment plays an important role within the evolution of stream and floodplain morphology. Bank erosion may be a phenomenon that has its root within the formation of the productive floodplain. Every river system has eroding bank, though the rate of erosion may vary. The river bank erosion is a function of multiple factors.

Among those, cohesive force between the particles and seepage force due to entrapment of water between the particles play a huge role. Lots of studies have been done by the previous researcher in macroscopic level to quantify the bank failure. However the microscopic approach to quantify the bank failure is yet to be explored to study the hydrodynamic effect of cohesive force between the particles and seepage force simultaneously.

In this paper microscopic view of approach has been considered to analyze the different forces acting on the particles for the determination of the escape velocity of the particles. Basal bank erosion is correlated to the bank failure process by considering hydrodynamic static pressure and pore-water pressure. An analytical approach to estimate the basal erosion rate is suggested by. The rate of bank erosion is calculated through analytical model by considering dynamic equilibrium of the sediment particles by. Distinct Element Method (DEM) is used for simulation of the mechanical behaviour of the cohesive soil by. Capillary cohesion between two particles is quantified by. They suggested the equation for calculating the force acting between the particles which is a function of the radii of two unequal particles and the other geometric parameters.

The proposed equation is:

Fc=πσR1R2[c+exp(a(D/R)+b)]

Where Fc is the capillary cohesion force between the particles of two different radii of R1 and R2 having inter-particle distance of D, R is the greater of the two radii and σ is the surface tension. The coefficients a, b and c can be expressed in terms of the volume of liquid bridge (V) and contact angle (∅) as follows:

a=-1.1(V/R^3 )^(-0.53) (2a)

b=(-0.148ln(V/R^3 )-0.96) ∅^2-0.0082ln(V/R^3 )+0.48 (2b)

c=0.0018ln(V/R^3 )+0.078 (2c)

The effects of the liquid volume and the separation distance on the static liquid bridge is studied by.

An analytical model named “Truncated Pyramid Model” is used to quantify the effects of inter-particle distance on riverbank erosion under cohesion by considering the dynamic equilibrium.

The incipient motion of the cohesionless sediment particles is studied by. They introduce an expression for the seepage force with the different degrees of exposure of the particle to the stream. The expression proposed by is:

FS=ρg{1−f(e)}(1−c)d3

This equation can be resolved into two components along x and y-direction as:

Fsx=ρg{1−f(e)}(1−c)d3cosα

And

Fsy=ρg{1−f(e)}(1−c)d3sinα

Where, Fs is the seepage force, e is the relative degree of exposure, α is the angle of slope, d is the particle diameter, c is the shape factor relative to the particle volume and f(e) is the exposure function which depends on particle shape and is given by the expression of:

f(e)=e^2 (3-2e) (5)

For the completely hidden particle degree of exposure e=0 (strong presence of mostly cohesive force), and the value of f=0, and for the completely exposed particle e=1 and f=1; whereas for the half exposed particle e=0.5 and f=0.5. In the present study this expression is used to determine the effect of seepage force with three different types of exposure along with the expression proposed by for capillary cohesion for the “Truncated Pyramid Model” which was proposed by.

Framework of Model

Truncated Pyramid Model proposed by is utilized in this present work to calculate the escape velocity of the particle on the riverbank. This model has been designed based on the fact that each spherical shaped particle rests on top of two particles with a small volume of waterbridge between them which creates the pyramid structure of the bank. The particle size is assumed to be increased in such a manner that the heavier and bulkier particles will lie at the lowest and away from the river. Particles arrangement in the Truncated Pyramid Model with the magnified view of the two neighboring particles has been shown in fig. 1. Free body diagram for the calculation of the dynamic force acting on the particles, where, suffices k and l indicate spatial location of the particle in the two-dimensional coordinate.

Assumptions

The assumptions for Truncated Pyramid Model to analyze forces acting on the particles have been discussed below:

Particles are spheres made of isotropic material. The size of the particles increases gradually row-wise and column-wise. The position of the particle is denoted by suffices k and l, where k denotes the row number and l denotes the column number.
Force of friction between the water and the particle is neglected. Angles are manifested by approximate trigonometric relations. Angles are functions of the radii of the three adjacent particles.

The threshold condition of the particle separation from the riverbank defines the impending acceleration. This acceleration can be determined by considering the dynamic equilibrium condition of the particle in x and y-direction. This acceleration leads to the escape velocity of the particle. The mass of the spherical particle is given by the expression((4πR_kl^3 ρ_s)⁄3), where ρ_s indicates the density of the sediment particle material. The general equations for the calculation of impending acceleration in x-direction can be written as follows:

x ̈_(k,l)=(3/4πR_(k,l)^3 ρ_s )(F_c1x-F_c2x-F_c3x-F_c4x+F_c5x+F_c6x+F_sx )

Where,

Fc1x= Force between the particles k-1,l and k,l in x-direction

=πσ√(R_(k,l) R_(k-1,l) ) [c_(k,l)+exp{a_(k,l) (D/R_(k,l) )+b_(k,l) } ][{(R_(k,l)+R_(k-1,l) )^2+(R_(k,l)+R_(k,l+1) )^2-(R_(k-1,l)+R_(k,l+1) )^2 }/ {2(R_(k,l)+〖 R〗_(k-1,l) )(R_(k,l)+R_(k,l+1) ) } ]

F_c2x=Force between the particles k-1,l-1 and k,l in x-direction

= πσ√(R_(k,l) R_(k-1,l-1) ) [c_(k,l)+exp{a_(k,l) (D/R_(k,l) )+b_(k,l) } ][{(R_(k,l)+R_(k-1,l-1) )^2+(R_(k,l)+R_(k,l-1) )^2-(R_(k-1,l-1)+R_(k,l-1) )^2 }/{2(R_(k,l)+R_(k-1,l-1) )(R_(k,l)+R_(k,l-1) ) } ]

F_c3x=Force between the particles k, l-1 and k,l in x-direction

= πσ√(R_(k,l) R_(k,l-1) ) [c_(k,l)+exp{a_(k,l) (D/R_(k,l) )+b_(k,l) } ]

F_c4x=Force between the particles k,l and k+1,l in x-direction

= πσ√(R_(k,l) R_(k+1,l) ) [c_(k+1,l)+exp{a_(k+1,l) (D/R_(k+1,l) )+b_(k+1,l) } ][{(R_(k,l)+R_(k,l-1) )^2+(R_(k,l)+R_(k+1,l) )^2-(R_(k,l-1)+R_(k+1,l) )^2 }/ {2(R_(k,l)+〖 R〗_(k,l-1) )(R_(k,l)+R_(k+1,l) ) } ]

F_c5x=Force between the particles k,l and k+1,l+1in x-direction

=πσ√(R_(k,l) R_(k+1,l+1) ) [c_(k+1,l+1)+exp{a_(k+1,l+1) (D/R_(k+1,l+1) )+b_(k+1,l+1) } ][{(R_(k,l)+R_(k,l+1) )^2+(R_(k,l)+R_(k+1,l+1) )^2-(R_(k,l+1)+R_(k+1,l+1) )^2 }/ {2(R_(k,l)+〖 R〗_(k,l+1) )(R_(k,l)+〖 R〗_(k+1,l+1) ) } ]

F_c6x=Force between the particles k,l and k,l+1 in x-direction

= πσ√(R_(k,l) R_(k,l+1) ) [c_(k,l+1)+exp{a_(k,l+1) (D/R_(k,l+1) )+b_(k,l+1) } ]

F_sx= Seepage force along x-direction

=ρg{1-f(e) }(1-c) d^3 cos⁡α

And the general equations for the calculation of impending acceleration in y-direction can be written as follows:

y ̈_(k,l)=(3/4πR_(k,l)^3 ρ_s )(-F_c1y-F_c2y+F_c4y+F_c5y+F_sy )+g(1-ρ/ρ_s )

Where,

F_c1y=Force between the particles k-1,l and k,l in y-direction

=πσ√(R_(k,l) R_(k-1,l) ) [c_(k,l)+exp{a_(k,l) (D/R_(k,l) )+b_(k,l) } ] [1-[{(R_(k,l)+R_(k-1,l) )^2+(R_(k,l)+R_(k,l+1) )^2-(R_(k-1,l)+R_(k,l+1) )^2 }/{2(R_(k,l)+〖 R〗_(k-1,l) )(R_(k,l)+R_(k,l+1) ) } ] ]^0.5

F_c2y=Force between the particles k-1,l-1 and k,l in y-direction

= πσ√(R_(k,l) R_(k-1,l-1) ) [c_(k,l)+exp{a_(k,l) (D/R_(k,l) )+b_(k,l) } ] [1- [{(R_(k,l)+R_(k-1,l-1) )^2+(R_(k,l)+R_(k,l-1) )^2-(R_(k-1,l-1)+R_(k,l-1) )^2 }/{2(R_(k,l)+R_(k-1,l-1) )(R_(k,l)+R_(k,l-1) ) } ] ]^0.5

F_c4y=Force between the particles k,l and k+1,l in y-direction

= πσ√(R_(k,l) R_(k+1,l) ) [c_(k+1,l)+exp{a_(k+1,l) (D/R_(k+1,l) )+b_(k+1,l) } ] [1-[{(R_(k,l)+R_(k,l-1) )^2+(R_(k,l)+R_(k+1,l) )^2-(R_(k,l-1)+R_(k+1,l) )^2 }/{2(R_(k,l)+〖 R〗_(k,l-1) )(R_(k,l)+R_(k+1,l) ) } ] ]^0.5

F_c5y=Force between the particles k,l and k+1,l+1in y-direction

=πσ√(R_(k,l) R_(k+1,l+1) ) [c_(k+1,l+1)+ exp{a_(k+1,l+1) (D/R_(k+1,l+1) )+b_(k+1,l+1) } ] [1-[{(R_(k,l)+R_(k,l+1) )^2+(R_(k,l)+R_(k+1,l+1) )^2-(R_(k,l+1)+R_(k+1,l+1) )^2 }/{2(R_(k,l)+〖 R〗_(k,l+1) )(R_(k,l)+〖 R〗_(k+1,l+1) ) } ] ]^0.5

F_sy=Seepage force along y-direction

=ρg{1-f(e) }(1-c) d^3 sin⁡α (9e)

The resultant of the impending acceleration (in m/s2) of the particle k,l is

f_(k,l)=(x ̈_(k,l)^2+y ̈_(k,l)^2 )^0.5

And the equation of the escape velocity (in m/s) proposed by [4] for the particle k,l would be

V_(escape k,l)=(2R_(k,l) f_(k,l) )^0.5

Where, R_(k,l)is expressed in mm.

Calculation of Escape Velocity

Variation of escape velocity with the variation of particle size and different degrees of exposure with different inter-particle distance is calculated in the present work. In this paper the left-most particle (k=1) has been taken under consideration being the most vulnerable one. Three cases of three different radii are considered. Three different radii of the particles are 0.3 mm, 0.4 mm and 0.5 mm. In each set of calculation pertaining to a specific particle radius, α = 30 for obvious reason. The influence of mean particle radius variation on escape velocity of the particle with three different degrees of exposure is calculated and the various escape velocities with for three different size of particle and three different degrees of exposure are plotted against different inter-particle distances.

Following are the properties which have been considered in the present calculation

Density of water = 1000 kg/m3;

Contact angle = 0;

Surface tension = 0.073 N/m;

Volume of water bridge = 20 nl;

Above values have been considered to analyze the nature of variation. But these values may vary case to case and this model has the ability to integrate different kind of variations.

Results and Discussion

It is clear that as the inter-particle distance increases escape velocity decreases. It means less velocity required for the particle to escape from the bank surface for greater inter-particle distance. For the higher value of inter-particle distance force between the particles is low, so the requirement of velocity to escape is less. From the three figures it has been seen that as the mean particle radius increases escape velocity decreases. It means larger particle requires less velocity to separate from the bank surface. Again, all the three figures suggest that as the degree of exposure increases escape velocity decreases and the effect of variation is more in case of larger particles. It means for the larger particle effect of the seepage force is less compared to the smaller particle. Water plays a significant role in binding the particles together.

Discussions

Following conclusions may be drawn from the present study:

The escape velocity decreases as the inter-particle distance increases. A higher value of inter-particle distance makes the force between the particles lower and, therefore, the requirement of velocity to escape gets reduced.
The escape velocity decreases as the mean particle radius goes up. This accounts for the vulnerability of larger particles compared to the smaller ones in terms of separation from the riverbank.
With the rise in the degree of exposure the escape velocity drops and the effect of variation is more pronounced in case of bigger particles. In case of the larger particle the effect of the seepage force is smaller compared to the particles with less diameters. The presence of water between the particles acts as the binding agent.