tim fitton, atc williams - self formed channels in tailings disposal

Fitton – Self-formed channels in tailings disposal

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Third Annual Slurry Pipelines Conference, Perth 2013

Self-formed channels in tailings disposal

Tim Fitton - ATC Williams

Abstract

Self-formed channel occur in nature, wherever the flow of a fluid shapes the open channel

in which it is travelling. Rivers, mudflows, lava flows and glaciers are examples of this.

Tailings slurry (mine waste mixed with water) is typically hydraulically discharged into large

impoundments for its disposal, though in recent decades new technology has been applied

to thicken the slurry to form deposits with gently sloping “beaches” (surfaces), allowing the

tailings to be stored more efficiently.

The discharge of a tailings slurry into a tailings disposal area will result in the slurry being

transported along self-formed channels across the beach. Such channels are normally

turbulent, but occasionally they are laminar with a plug flow mechanism. It has been

observed that such channels can be grouped into three regimes; segregating turbulent flow,

non-segregating turbulent flow, and non-segregating plug flow.

This paper presents three different methods of predicting the headloss in a self-formed

channel on a tailings beach, to cover each of the three tailings deposition regimes that are

encountered. Each of these three methods has been found to be of use in the prediction of

tailings beach slopes.

1 Introduction

A self-formed channel is a gravity-flow open channel with its cross section and course

shaped by the fluid that flows along it. Such channels are common in nature, with the most

obvious example being rivers. Glaciers, mudslides and lava flows are some other examples.

The water flowing in a river will erode the river bed in some areas and deposit sediments in

other areas. Most of the erosion takes place in the upper reaches of the river, where the

gradient of the stream is steep enough to induce high velocity flow. Conversely, deposition

of sediments mainly occurs in the river delta at the lower reaches of the river, where the

velocity diminishes rapidly with the chaotic mixing of the river outflow with the stiller

waters of the sea. In the mid-section of a river, there is a more even balance between

erosion and deposition. In this paper, such conditions are labelled as “equilibrium flow”. At

all points along a river, the cross section is being shaped by the flowing water, which in turn

causes the river to gradually alter its course over time. Due to the low viscosity of water,

rivers always flow in the turbulent regime. The turbulent eddies carry the sediment

particles, and often sweep up others.


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The mechanics of a lava flow are very different to that of a river, despite both natural

processes featuring a fluid that forms its own channel shape and course. In the case of a

lava flow, the fluid is very viscous, and it also develops a yield stress as it cools. Lava flows

do not erode the underlying material to the same degree as a river, and typically a lava flow

channel is beset by stationary lava that stopped only hours or days beforehand.

Tailings is waste from the mining industry. It is typically sand and silt that has been

artificially created through the process of digging or blasting rock from the earth, and then

mechanically crushing and grinding it to the optimum size for the extraction of the target

mineral. Depending on the type of mine and mineral extraction process in use, the tailings

may contain introduced chemicals. Occasionally tailings is radioactive. Depending on the

chemistry of the original rock, the tailings may also generate acid if it is susceptible to

oxidation upon contact with the air. For all these reasons, it is generally accepted that

tailings should be disposed of in environmentally responsible manner. The quantity of

tailings that is produced by a small mine could be 500,000 tonnes per year, whilst a large

mine might produce as much as 40 million tonnes per year. There are a few huge mines

that produce more than 100 million tonnes per year, and it is expected that the number and

sizes of mines will continue to grow in time.

The disposal of such large quantities of tailings is a significant concern. Typically this has

been achieved by constructing a dam to form an impoundment of some sort, and then filling

up the impoundment with hydraulically transported tailings (tailings and water slurry

pumped through a pipe or gravity fed along an open channel). The tailings particles settle

out in the impoundment to form a deposit, the surface of which is called a beach.

Segregation of the tailings particles is typical in such impoundments, with the coarse

particles depositing nearer to the discharge point. In more recent years Eli Robinsky’s

concept of thickened tailings disposal (Robinsky 1975) has been progressively taken up as an

alternative to the traditional impoundment disposal method (Williams et al. 2008).

Robinsky’s concept involves the discharge of a tailings slurry that is thickened beyond the

point of segregation. Such non-segregating slurries form beaches with steeper overall

slopes, which can be exploited to use much smaller embankments for containing the same

volume of tailings. Robinsky’s concept also provides other benefits such as reduced water

usage, reduced seepage of contaminants into the ground, and greater stability of the tailings

deposit, but such matters are beyond the focus of this paper, and have been well

documented elsewhere (Fourie 2012).

On thickened and conventional tailings deposits alike, it is typical to see self-formed

channels of tailings slurry flowing across a tailings beach. These channels form naturally as

the slurry adopts a path of least resistance towards its active deposition area. At the

downstream end of the channel, the slurry will spread out in a sheet in the deposition area.

Over time the downstream end of the channel will gradually advance across the sheet,

making the channel grow in length. This will continue until such time as the sheet depth


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builds up too much, at which time it will present a greater resistance to the incoming

channel flow. Almost instantly, the tailings slurry will “break out” of its self-formed channel

at some point upstream and form sheet flow elsewhere, and in the following minutes a new

channel will start advancing across that new sheet. This process sees the channel course

change periodically. An inspection of a tailings beach will reveal hundreds of dry “extinct”

channels that were left abandoned by the slurry as it broke out at some point upstream. It

is evident from this process that it is the channel slope that dictates the overall slope of the

tailings beach. Such channels on tailings beaches are typically flowing under equilibrium

flow conditions, such that deposition and erosion are effectively cancelling one another out.

The slope of a channel with equilibrium flow conditions is labelled the “equilibrium slope”.

A photograph of a self-formed channel on a tailings beach is presented as Figure 1.

Figure 1 A self-formed channel on a tailings beach (Tailings Info website)

Prior to the introduction of Robinsky’s thickened tailings concept there was little interest in

tailings beach slopes, since tailings beaches typically exhibited overall slopes of about 0.1 to

0.3%, allowing engineers to design storage impoundments without too much fuss. Since the

adoption of Robinsky’s thickened tailings concept however, the mining industry has been far

more interested in predicting the slope of the beach, since the resultant overall beach


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slopes have varied from 0.5% to 5.0% from one mine site to another. An inaccurate

prediction of a beach slope could cost a mining company millions of dollars in the

construction of the dams and pumping infrastructure that do not suit the particular

thickened tailings disposal. It was this mining industry interest in beach slope prediction

that led us to equilibrium slopes of self-formed channels on tailings beaches.

2 Segregating slurries

A self-formed channel on a tailings beach is not unlike a river, particularly in situations

where segregating tailings slurry is being discharged. Turbulent eddies in the channel carry

the tailings particles along, as is seen in a river. However, the segregating slurry may be

concentrated enough to exhibit non-Newtonian viscous characteristics, so it is necessary to

take the slurry rheology into account if one is attempting to model its behaviour.

Fitton (2007) presented an a-priori model that enabled the prediction of the channel

equilibrium slope for segregating slurries. (“A priori” is a Latin term meaning “from the

earlier”. It is used to describe an equation that is based on previously published equations,

without introducing any new empirical factors.) The model featured equations from the

fields of non-Newtonian open channel flow, sediment transport and rheology:

The slope of an open channel with uniform flow can be calculated with the Darcy-Weisbach

equation:

(1)

where S0 is the slope of the channel bed under uniform flow conditions, RH is the hydraulic

radius, V is the average velocity of flow, g is the acceleration due to gravity and f is defined

as the Darcy friction factor.

The open channel form of the Colebrook-White equation will be used to calculate the Darcy

friction factor:

(2)

Of key interest in this work is the ks parameter, which represents the roughness of the pipe

or channel walls (expressed in meters). The value of 2 x d90 has been adopted for estimating

ks on the basis of work done by Ikeda et al. (1988).

The Re parameter in the Colebrook-White equation is the Reynolds Number. Haldenwang

et al. (2004) presented the following equation for calculating the Reynolds number for Non-

+−=

fR

k

f H

s

Re

51.2

8.14log2

110

HgR

VfS8.

2

0 =


5

Newtonian turbulent open channel flow, with the rheological properties of a fluid of density

ρ expressed in terms of the three Herschel-Bulkley fitting parameters τy, K and n:

(3)

From numerous experimental studies into the transport of sediments in turbulent flows,

there are many equations presented in the literature that enable the prediction of the

minimum average velocity required to keep particles in suspension in a pipe (VC). This

particular velocity appears in the literature under several names, with the most common

ones being “minimum transport velocity”, “deposition velocity” and the “critical velocity”.

Fitton (2007) investigated the performance of 16 minimum transport velocity equations

from the literature. It was found that the Wasp et al. (1977) equation performed quite well

for segregating slurries, which is expressed below in an open channel form:

(4)

where d is the median particle diameter, CV is the slurry concentration by volume, ρs is the

density of the solid particles and ρl the density of the carrier fluid.

In order to determine an equilibrium slope the channel cross-sectional shape needs to be

defined in such a way that the hydraulic radius can be calculated as a function of the depth.

Fitton recommended a parabola with a width 5.5 times the depth, based on some

experimentally measured data.

To predict an equilibrium slope an iterative process was used to arrive at a unique channel

depth that produced a transport velocity that was equal to the average velocity in the

channel. At this point, a slope could be calculated.

3 Non-segregating slurries without plug flow

Fitton (2007) also presented an empirical equation for the prediction of the minimum

transport velocity of a non-segregating slurry, based on two sets of field data that he had

generated in flume experiments at the Sunrise Dam gold mine in Western Australia and the

Peak gold mine in New South Wales:

n

H

y

HB

R

VK

V

+

=2

8Re

2

τ

ρ

2/16/1

4/1 )(8

48.3

−

=

l

lsH

H

VC

gR

R

dCV

ρρρ


6

(5)

In equation 5, the KBP parameter is the plastic viscosity of the slurry.

The same prediction method could be applied as for segregating slurries, except with the

use of equation 5 instead of equation 4.

4 Non-segregating slurries exhibiting plug flow

In 2011 a pilot plant trial was carried out at the Chuquicamata copper mine in Chile to

investigate the characteristics of the local copper tailings at higher levels of thickening.

(Engels et al 2012, Pirouz et al 2013) The pilot plant featured a tall pilot scale thickener that

could be used to produce batches of high concentration slurry.

During those trials it was discovered that a non-segregating slurry could be thickened to

such a degree that its self-formed channels behaved more like a lava flow than that of a

river, since the viscosity and yield stress of some batches of slurry were high enough to

induce plug flow conditions in the channel. (Fitton and Slatter, 2013) Plug flow occurs

when a portion of the slurry in the channel remains unsheared. This is illustrated in

Figure 2.

Figure 2 Open channel plug flow observed in tailings

Fitton and Slatter (2013) presented a method for analysing such channels. They adopted a

half ellipse instead of the parabola that had previously been assumed by Fitton. It was

proposed that half an ellipse is a more realistic cross sectional shape for a self-formed

channel, particularly at the banks of the channel, where the sharp points of the parabola are

deemed to be unrealistic compared to the vertical up-turns of the ellipse.

Diagrams are presented in Figure 3 to show the distribution of shear stress and velocity in

pipe and channel centrelines in laminar flow, highlighting the impact of yield stress fluids.

=−

BP

HgsegregatinCNon

K

VRV

ρln145.0


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Figure 3 Distribution of shear stress and velocity in laminar pipe flow (top) and

laminar open channel flow (bottom)

A number of assumptions were proposed for the open channel plug flow model:

• Flows in the sheared zone are laminar, with parabolic velocity profiles

• The channel cross section is a half ellipse

• The plug is also a half ellipse, with the same width to depth ratio as the greater

channel

In developing their plug flow model, Fitton and Slatter adapted pipe flow equations

presented by Slatter (1997) to cater for open channel flow. To calculate the depth of the

plug, the following relationship was presented:

(6)

where dP is the depth of the plug at the channel centreline and d is the total depth at the

centreline of the channel, as shown in Figure 2.

ddW

YP

=

ττ


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The Re3 Reynolds number was modified using the laminar pipe/channel conversion D = 4RH:

(7)

Where VSZ is the average velocity in the sheared zone and RHSZ is the effective hydraulic

radius of the sheared zone. To calculate RHSZ, the following equation was presented:

(8)

RH is the hydraulic radius of the channel, and RHP is the hydraulic radius of the plug. VSZ is

calculated:

(9)

Where QSZ is the flow rate in the sheared zone and ASZ is the cross sectional area of the

sheared zone. QSZ is calculated as follows:

(10)

where QP is the flow rate in the plug, equal to

(11)

with uP as the velocity of the plug and AP as the cross sectional area of the plug.

To determine the plug velocity a geometric approach was taken, as illustrated in Figure 4.

n

HSZ

SZ

Y

SZ

R

VK

V

+

=2

8Re

2

3

τ

ρ

HPHHSZ RRR −=

SZ

SZ

SZA

QV =

PSZ QQQ −=

PPP AuQ .=


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Figure 4 Velocity profiles for a plug flow scenario; the green line shows the actual

velocity profile, while the blue line shows the average velocity. The area contained inside

both shapes is identical by definition.

The two velocity profiles shown in Figure 4 contain the same area. By this paradigm the

plug velocity (uP) can be determined by equating the areas inside the two shapes as follows:

(12)

Strictly speaking, equation 12 is only accurate for an infinitely wide channel. To address

this, the respective hydraulic radii were substituted for the centreline depths to make

equation 12 accurate for a three-dimensional channel. Making uP the subject yields:

(13)

The wall shear stress is required for the determination of the plug depth (equation 6). In

channel flow this is typically calculated using DuBoys equation, but in this particular

situation that is not helpful because the channel slope is required as an input parameter,

and that is what is ultimately sought to predict. Instead the problem is approached by

applying with the fundamental relationship between the shear stress (τ), apparent viscosity

(η) and shear rate (γ ̇):

(14)

)(.32

PPPP ddududV −+=

)(

.

32

HPHHP

HP

RRR

RVu

−+=

γητ &.=


10

The shear rates in the channel can be calculated by deriving the quadratic equation for

theoretical velocity distribution profile in the sheared zone (note that this parabolic velocity

distribution is valid for a Bingham plastic, and will be approximate for a Herschel-Bulkley).

The velocity profile can be described as follows:

(15)

The derivative of equation 15 is the shear rate function. Deriving equation 15 and making

the depth equal to the maximum depth (d) yields the shear rate at the channel bed (γ ̇W):

(16)

Substituting the respective hydraulic radii for the centreline depths to make equation 16

accurate for a three-dimensional channel gives:

(17)

The apparent viscosity is then calculated using the rheological parameters and the shear

rate:

(18)

This equation completed the channel plug flow model. However, it will become

immediately evident in the attempted application of this new model that the plug depth is

required as an input parameter for determining the wall shear stress, and the wall shear

stress is required for calculating the plug depth. Therefore the plug flow model is implicit as

presented.

This problem arises from open channel flow possessing an extra dimension over pipe flow in

terms of its variable depth, which makes its modelling more complex than that of pipes. An

increase in the flow rate in a pipe simply results in an increase in the velocity, for example.

In an open channel, an increase in the flow rate will yield an increase in the depth as well as

2

2)(

)(P

P

PP ddepth

dd

uuu −

−−=

)(

2

P

PW

dd

u

−=γ&

)(

2

HPH

PW

RR

u

−=γ&

γγτ

η&

&n

Y K.+=


11

the velocity, particularly where non-Newtonian fluids are involved. As a result of this added

variable, an iterative solution is required.

It was found that initially estimating the wall shear stress to be equal to 1.2 times the yield

stress enabled a convergent iterative solution to four significant figures to be successfully

reached after six iterations. With the use of a computer spread sheet program, this iterative

approach was easily applied.

4.1 Channel aspect ratio

The channel plug flow model presented here will enable the modelling of yield stress fluids

in open channels of any cross-sectional geometry.

However, in the case of self-formed tailings channels, there exists the question of the aspect

ratio of the channels. Fitton (2007) carried out a numerical analysis of different width-to-

depth (w/d) ratios for a limited number of such channels, and after comparing the predicted

results against experimentally measured data, recommended 5.5:1. Fitton and Slatter

(2013) attempted to back-calculate the width-to-depth ratio (w/d) for each of the self-

formed channels that were observed in the Chuquicamata flume trials (Pirouz et al 2013).

It was not possible to measure the depth of the self-formed channels during that

experimental work, since any disturbance to the channel created by the immersion of a

probe would result in the destruction of the fragile channel bed and banks. However, flow

rates, plug velocities and channel widths were measured, so it was possible to back-

calculate the channel depth. This deduction of channel depths was undertaken, and it was

found that the aspect ratios for the 17 measured self-formed channels varied from 2.5 to

11.

Due to the considerable range of aspect ratios that were thus calculated for the

Chuquicamata flume trials, it was decided that the plug flow model should include an aspect

ratio prediction model if any reasonably strong trends could be discovered between the

deduced aspect ratio and some of the measureable characteristics of the slurry, rather than

simply assuming a constant ratio.

Analysis was carried out to identify trends relating the slurry characteristics to the width-to-

depth ratios calculated for the Chuquicamata data. It was found that a modest trend was

observed when the w/d values were plotted against the Bingham yield stress of the slurry.

This plot is presented as Figure 5. It is noted that the Bingham fit to each rheogram was

tangent to the 100 s-1

point on the curve.


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Figure 5 Plot of Bingham yield stress vs w/d ratio for the self-formed channels in the

Chuquicamata flume trials

The equation describing the trend curve that is shown on the Figure 5 plot is as follows:

(19)

This equation was adopted for predicting channel aspect ratios for self-formed channels in

the plug flow model.

4.2 Elliptical geometric equations

In order to apply the plug flow channel model to an elliptical channel shape it is necessary to

calculate the cross sectional area and the wetted perimeter of a half ellipse such as that

shown in Figure 2. The following equation can be used to calculate the area of the half

ellipse:

(20)

yB

dwτ

25/ =

4

dwA

π=


13

Ramanujan (1914) presented an excellent approximation for calculating the circumference

of an ellipse. It has been modified here for calculating the wetted perimeter of the half

ellipse:

(21)

It is noted that equations 20 and 21 can also be used to calculate the area and wetted

perimeter of the plug (AP and PP) if the plug depth and plug width are used instead of the

overall channel depth and width respectively.

4.3 Predicting a beach slope with the plug flow model

The overall approach is similar to the a priori beach slope model previously presented, but

the plug flow model adds some more steps to the process. The application of the model

remains the same – the channel depth is adjusted until the channel velocity is equal to the

predicted minimum transport velocity. The Fitton 2007 non-segregating minimum transport

velocity equation (equation 5) is recommended for use in the plug flow beach slope model.

5 Conclusion

The discharge of a tailings slurry into a tailings storage facility will result in the slurry forming

its own channels across the beach. Such channels can be laminar or turbulent. It has been

proposed that these channels can be grouped into three regimes; segregating turbulent

flow, non-segregating turbulent flow, and non-segregating plug flow.

This paper has presented a method for predicting the equilibrium slope of a self-formed

channel on a tailings beach for each of these three regimes. It has been found that each of

the three methods has its use in the prediction of tailings beach slopes.

References

Fitton, T.G. (2007) Tailings beach slope prediction, PhD thesis, RMIT University, Australia (Since

published as a book by VDM Verlag, Saarbrucken, Germany in 2010).

Fourie, A.B. 2012, Perceived and realised benefits of paste and thickened tailings for surface

deposition, paper presented to Paste 2012, Sun City, South Africa, 2012.

Pirouz, B., Seddon, K.D., Williams, M.P.A., Echevarria, J. & Pavissich, C. 2013, Tilt flume testing for

beach slope evaluation at Chuquicamata Mine CODELCO-Chile, paper presented to Paste 2013, Belo

Horizonte, Brazil

( )

+−

−+

+−

++=2

2

2/

2/3410

2/

2/3

1.2/2

dw

dw

dw

dw

dwPπ


14

Ramanujan, S., 1914, Modular equations and approximations to Pi. Quart. J.Pure. Appl. Math. 45

(1913-1914), 350{372.

Robinsky, E.I. 1975, 'Thickened discharge - A new approach to tailings disposal', CIM Bulletin, vol. 68,

pp. 47-53.

Slatter, P.T. 1997, The effect of the yield stress on the laminar/turbulent transition, 9th Int. Conf.

Transport and Sedimentation of Solid Particles, Crakow, Poland, 1997.

Tailings Info website, accessed 11 Nov 2013. http://www.tailings.info/disposal/paste.htm

Wasp, E.J., Kenny, J.P. & Gandhi, R.L. 1977, Solid-liquid flow slurry pipeline transportation, First edn,

Trans Tech Publications, Germany.

Williams, M.P.A., Seddon, K.D., & Fitton, T.G. 2008, “Surface Disposal of Paste and Thickened Tailings

- a brief history and current confronting issues”, paper presented to Paste 2008, Kasane, Botswana,

6-9 May, 2008.