Free surface flow characteristics of multi-phase viscoplastic fluids on inclined flumes and planes

International Journal of Multiphase Flow 78 (2016) 59–69

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Free surface flow characteristics of multi-phase viscoplastic fluids on inclined flumes and planes A.H. Azimi∗ Department of Civil Engineering, Lakehead University, Thunder Bay ON P7B 5E1, Canada

a r t i c l e

i n f o

Article history: Received 19 June 2015 Revised 21 September 2015 Accepted 27 September 2015 Available online 9 October 2015 Keywords: Foam Free-surface flow Non-Newtonian flow Sand–foam mixture Viscoplastic fluid Yield stress fluid

a b s t r a c t Laboratory experiments were conducted to understand the dynamics of gravity-driven free surface flow of multi-phase viscoplastic fluids. Foam and sand–foam mixtures were employed to represent multi-phase yield stress fluids. Two different foam–water ratios of 0.1 and 0.2 were used and a wide range of sand concentrations from 0 to 0.82 were selected. A series of flume tests were conducted to model the two-dimensional flow of sand–foam mixtures for two slopes of 10° and 15°. Effects of sand concentration co on the dynamics of free surface flow were investigated. Three-dimensional spreading tests were carried out using an inclined plane with four bed slopes of α = 10°, 12°, 15° and 18. Single and multi-discontinuity were observed in the spreading of foam mixtures. It was found that the number of discontinuities can be correlated with the bed slope and multi-discontinuity occurred for α ≥ 12°. Flow heights and frontal velocities were measured for different experimental conditions (slope angle, rheological parameter). The results of spreading tests were properly scaled using a non-dimensional time scale and Froude number. Experimental results were used to predict the rheological characteristics of sand–foam mixtures based on the Herschel–Bulkley constitutive law. Predictions were compared with independent measurements of the rheological parameter. The inclined plane test results showed that the shear stress predictions were independent of the bed slope and the uncertainty of the predictions was slightly higher than the rheometery results from the literature. © 2015 Elsevier Ltd. All rights reserved.

Introduction The flow of multi-phase viscoplastic fluids on inclined surfaces is often encountered in engineering applications such as transport of mining products, fresh concrete and sewage sludge. (Coussot et al., 1996; Chambon et al., 2009; Chambon et al., 2011). Complex flow mixtures involve in natural catastrophic events and geophysical situations such as debris flows, lavas, and submarine slides are also belong to this class of materials (Coussot and Meunier, 1996; Coussot et al., 1998; Ancey, 2007). Foam and sand–foam mixtures are also categorized as multi-phase viscoplastic fluids and their rheological characteristics can be well predicted by the Herschel–Bulkley constitutive law (Azimi, 2015a; Azimi, 2015b). Sand–foam mixtures can be made by the addition of either powders or liquid foaming agents and sand particles to water forming a uniform texture with a macroscopic continuous structure. These mixtures have many potential applications in underground mining and land stabilization (Abazari Torgabeh, 2013). A proper mixture of sand and foaming agent can be spread uniformly over unstable areas to prevent sand storm. Sand–foam mixtures can also be used

∗

Corresponding author. Tel.: +18073438560. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.09.013 0301-9322/© 2015 Elsevier Ltd. All rights reserved.

in under-balanced drilling to enhance oil recovery in porous media (Edrisi et al., 2013) and to suppress underground fires in mining industry (Bobert et al., 1997). The performance of high-expansion firefighting foam has been tested in the past to control underground coal mine fires (Chasko et al., 2003). Understanding the dynamics of multi-phase viscoplastic fluids is of particular importance to predict the flow behavior at different conditions for improving industrial processes and protect against hazard. Since conventional rheometrical tests deal with small scales, other practical techniques such as slump test, inclined flume and surface tests have been used in the past to study the dynamics of complex flows and mixtures in relatively larger scales (Clayton et al., 2003; Gawu and Fourie, 2004; Chanson et al., 2006; Cochard and Ancey, 2009). Those tests were used to identify the rheological characteristics and expansion rates of the mixtures in transient and steady state conditions. Gumati and Takahshi (2011) conducted experimental and numerical studies to evaluate the effect of foam–water ratio on the structure of foam-cuttings mixtures by measuring the pressure loss in horizontal pipes. Gawu and Fourie (2004) found that some rheological characteristics of different tailings can be measured by a slump test and the results showed a good agreement with the controlledstress rheometery. Rheological characteristics of foam mixtures to design pipeline network and pump operation were also studied using horizontal pipe

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A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

and horizontal flume tests (Briceno and Joseph, 2003). Coussot and Boyer (1995) examined the accuracy of inclined surface test under a steady state condition to determine the yield stress of clay–water mixture whose yield stress ranged from 35 to 90 Pa. Coussot and Boyer (1995) used a 1 m long flume with a slope ranging from 10° to 30°. They showed a relatively good agreement between yield stress measurements from an inclined surface test and rheological measurements if the edge effects being considered. Coussot and Proust (1996) employed a mixture of fine mud suspension at different solid concentrations to model the expansion of mudflows in an inclined plane. They used the Herschel–Bulkley model to predict the longitudinal and lateral mean velocities and fluid depths of mudflows. The Herschel–Bulkley model can be defined as

γ˙ = 0 ⇔ τ < τo

γ˙ = 0 ⇔ τ = τo + K γ˙ n

(1)

where γ˙ is the shear rate, and τ are the shear stress and τ o is the critical shear stress. K is the consistency and n is an index of the model and both of them are positive numbers. Cochard and Ancey (2009) studied the spreading of viscoplastic fluids (i.e., Carbopol Ultrez 10) on a plane surface, whose inclination ranged from α = 0° to 18°. It was found that the front position of the flow varied as a power function of time and an empirical formulation was developed to predict the front position of the flow. The proposed formulation was a function of time and plane inclination. Chanson et al. (2006) conducted a series of laboratory experiments using a tilting flume with a slope of 15° to study the flowability of Bentonite suspension. They developed an empirical formulation to predict the frontal position and velocity of the Bentonite clay flowing in the flume. Variations of the frontal position and velocity with time were found to be a function of the initial flume height Ho and the bed slope α . They reported that the proposed formulation provide reasonable accuracy for [t(gHo )1/2 ] < 6 where t is time and g is the gravitational acceleration. Recently, Chambon et al., (2014) presented a series of experimental studies on the hydraulic properties of free-surface flow of viscoplastic fluids. Two types of Herschel–Bulkley fluids (i.e., Kaolin slurries and Carbopol microgels) with different microstructure were employed. They found an excellent agreement between the results of rheometrical tests and theoretical predictions for Kaolin slurries. For Carbopol microgel, a systematic discrepancy was reported between experimental results and analytical predictions due to larger microstructure of Carbopol samples. This paper aimed at understanding the free-surface flow dynamics and spreading of multi-phase viscoplastic fluids on inclined flumes and planes. Sand–foam mixtures with different foam–water ratios and sand concentrations were selected as an example of multiphase viscoplastic fluids. The foremost objective of this study is to correlate the dynamics of multi-phase viscoplastic fluids with different fluids characteristics such as density and rheological parameters. Density and rheological characteristics of sand–foam mixtures are controlled by foam–water ratio and sand concentration. Experimental results of this study can be used to validate numerical models and improve general understanding on the dynamics of multi-phase viscoplastic fluids. Experimental studies began with two-dimensional flume tests. The flow positions and frontal velocities were measured and the results scaled based on the non-dimensional time scale and Froude number. Spreading of the flow mixtures were tested by conducting inclined plane tests. A more empirical point of view was adopted in order to describe the intermediate flow of sand–foam mixtures. The simplified one-dimensional momentum equation and the discharge equation using the Herschel–Bulkley constitutive law were employed to predict the rheological characteristics of the flow. This paper is structured as follows. Section two of this paper describes the theoretical background on the free-surface flow of complex fluids on inclined surfaces. Section three describes the experimental design and procedures. Section four is devoted to experimen-

tal results followed by proposed empirical formulations. Section five discusses about the performance of the theoretical models to predict the dynamics of sand–foam mixtures. The conclusions of the experimental studies are drawn in section six. Theory and background Flow of viscoplastic fluids on inclined flumes The averaged critical shear stress in an inclined flume can be estimated using one-dimensional force balance between the gravitational force and the resistance shear force as

ρ gho( sin α)xF = τo(xF + 2ho)

(2)

where uF is frontal velocity, xF is frontal position and ho is the asymptotic depth of the mixtures when uF → 0 (see Fig. 1). Neglecting the wall effects on the force balance, Eq. (2) can be simplified as

τo = ρ gho( sin α)

(3)

Frontal velocity of the two-dimensional granular flow on inclined flume can be predicted using dimensional analysis. Pouliquen (1999) found a linear relationship between the normalized depth and Froude number as

u

F = β gh

h hstop (θ )

(4)

where h is the flow depth, hstop (θ ) is the critical depth of flow at specific angle θ and β = 0.136. The steady-state gravitational flow of a Herschel–Bulkley fluid in an open channel and in laminar flow condition was studied by Coussot (1994). He showed that the flow can be described by the non-dimensional fluid depth G and the nondimensional shear stress Hb (i.e., Herschel–Bulkley number) as

G=

ρ gh( sin α) τo  n τo h

Hb =

K

(5)

(6)

uF

Coussot and Boyer (1995) defined the asymptotic depth ho corresponding to rest state when G drops to 1 (i.e., h = ho ). Using the discharge equation established for the Herschel–Bulkley fluids Coussot (1994) showed that the following correlation can be written if the edge effects and surface tension assume negligible.

Hb−1/n =

1

(1/n + 1)(1/n + 2)

G−2 (G − 1)(1/n+1) ((1/n + 1)G + 1) (7)

Eq. (7) shows a non-dimensional stress–strain rate relationship and it has been extensively used to predict the free surface flow of the Herschel–Bulkley fluids on inclined flumes (Chambon et al., 2014).

α Fig. 1. Image of flow of sand–foam mixture in flume experiment (Test T6-10 at t = 45 s) with the coordinate system.

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

61

Coussot (1994) simplified Eq. (7) for the Herschel–Bulkley fluids with an index value of n = 1 /3 as

G = 1 + a(Hb )−0.9 for h/xF < 1 where a is a function of h and xF



a = 1.93 − 0.43 arctan

10h xF

(8)

20  (9)

α

The correlation between the non-dimensional mixture velocity dG/dT and the non-dimensional depth G can be formed by using the mass flux balance in a control volume (Coussot, 1994) as

−1 dG (G − 1)(1/n+1) ((1/n + 1)G + 1) = dT (1/n + 1)(1/n + 2)

(10)

where T is the non-dimensional time and it can be written as

T=

 τ 1/n o

K

τo t ρ g xF ( sin α)

(11)

The non-dimensional depth G and time T can be calculated from the flume experiments data. Eq. (10) can be used to predict the fluid depth and velocity at different time steps. The expression for the velocity profile established in a layer of a Herschel–Bulkley fluid flowing down an inclined flume in a steadystate laminar flow condition can be derived analytically (Kaitna et al., 2007). For uniform flow condition, the velocity profile can be formulated as



U ( y) =

 U ( y) =

ρ f g sin α K

ρ f g sin α

n n

K

1 y1+n 1+n o

for



Ho ≥ y ≥ yo

1 1+n y1+n − (yo − y) 1+n o

for

(12)

y ≤ yo

Fig. 2. Schematic of the force balance in an inclined plane test.

inlet, three forces become dominant to determine the flow of sand– foam mixtures in an inclined plane. The horizontal component of the foam weight Wx and the pressure at the inlet FP play as the acting forces and the friction due to the bed shear stress resisting the flow motion Fr .

dM  = F = Wx + FP − Fr dt

Schematic of the forces in flow of sand–foam mixture over an inclined plane is shown in Fig. 2. Assuming that after foam spreading the mixture velocity at the inlet is negligible, the rate of change of momentum and force balance can be expressed as

1 2

¯ sin α + ρm gBh2 − τ A ρm Au2P = ρm Ahg o

yo = Ho −

τo

ρ f g sin α

(14)

Eqs. (12) and (13) show the velocity profiles in plug flow and sheared flow regimes, respectively. For the unsteady free surface slump of viscoplastic fluids in two-dimensional channel, the flow motion occurs when the shear stress exceeds the yield stress at some point within the flow (Hogg and Matson, 2009).

  ∂h ρ g cos α h tan α − > τo ∂x

(15)

The slump flow attains a final static state when the streamwise pressure gradient and the along-slope component of the gravitational acceleration are balanced by the yield stress. Hogg and Matson (2009) defined a non-dimensional velocity scale to show the status of the dynamic motion as

U∗ =

ρ g cos α Ho3 μ xo

(16)

where xo is the along-slope slump length Ho is the initial slump height. Flow of viscoplastic fluids on inclined planes A simplified one-dimensional momentum equation can be formulated to estimate the averaged bed shear stress. Assuming that the pressure force at the front of the flow is negligible compared to the

(18)

where A is the change on the bed area at each time, uP is the frontal velocity and B is the width of the inlet. Deposition of concentrated suspensions with a yield stress after free flow stoppage on inclined plane can be predicted as a function of fluid characteristics. Coussot et al. (1996) proposed a dimensionless equation to model the flow depth in any direction as a function of the distance from the edge as



(13) where U is the longitudinal velocity at different height y in a steadystate laminar flow condition and the flow height is measured from the bed in perpendicular direction. yo is the height of the sheared layer and it can be calculated as

(17)

H2

∂H 1− ∂X

2



∂H + ∂Y

2 

=1

(19)

where X, Y and H are dimensionless parameters define as

X=

ρ gx( sin i)2 τc cos i

(20)

Y =

ρ gy( sin i)2 τc cos i

(21)

H=

ρ gh( sin i) τc

(22)

Experimental setup Eight series of laboratory experiments were carried out to study the free surface flow characteristics of multi-phase viscoplastic fluids on inclined flumes and planes. Foams containing gas–liquid phases and sand–foam mixtures containing gas–liquid–solid phases were used to represent the multi-phase viscoplastic fluids. The detailed rheological characteristics of the present foam and sand–foam mixtures were investigated by Azimi (2015a; 2015b). Flow of foam and sand–foam mixtures for foam–water ratios of Gf = 0.1 and 0.2 and different sand concentrations ranging from co = 0−0.82 were studied. Recent rheological studies indicated that the shear stress of sand–foam mixtures with different sand concentrations ranging from co = 0.42 to 0.82 can be predicted by the Herschel–Bulkley model with an index of n = 1 /3 (Azimi, 2015a). Two-dimensional flume tests were carried out in a Plexiglas flume with bed slopes of 10° and 15°. Three-dimensional spreading tests were carried out using an inclined wooden plate with different bed slopes of 10°, 12°, 15° and 18°. Details of laboratory experiments and

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A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

Table 1 Details of laboratory experiments and mixture characteristics. Test number

Gf

T1 T2 T3 T4 T5 T6 T7 T8

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2

a

ms (g)

co (vol/vol)

0 400 800 1400 0 400 800 1400

0.00 0.42 0.59 0.72 0.00 0.58 0.73 0.82

ρ m (kg/m3 )

ρ mu a

φ

78.8 125.0 172.7 229.0 81.7 188.9 305.2 523.8

1011 1653 1915 2107 1023 1892 2128 2274

0.925 0.925 0.910 0.891 0.919 0.899 0.856 0.769

(kg/m3 )

Densities of unexpanded sand–foam mixtures ρ mu were calculated using Eq. 2.

mixture characteristics are shown in Table 1. Overall, 48 tests were conducted to study the effects of sand–foam parameters at different bed slopes in both flume and inclined plane tests. In all tests, the first two letters indicate the test number and the two digits after the hyphen denoting the bed slope. Twenty grams of Controlled Low-Strength Material admixture (Rheocell Rheofill, BASF Co., US) was used as a foaming agent. The foaming agent was mixed with 100 ml and 200 ml of demineralized water to form well-mixed liquid solutions with foam–water ratios of Gf = 0.2 and 0.1, respectively. Densities of both foam solutions were found to be ρ f = 1011 kg/m3 and 1023 kg/m3 for Gf = 0.1 and 0.2, respectively. Fine blasting sand particles (Sill Industrial Minerals Inc., CA; www.sil.ab.ca) with a mean diameter of D50 = 0.21 mm and density of ρ s = 2540 kg/m3 were used to form sand–foam mixtures. Sand particles were considered to be uniform in size after examining the geometric standard deviation σ g = D84 /D50 = D50 /D16 (Breusers, and Raudkivi, 1991). The temperature of the mixtures is controlled by Thermosel and all tests are made at room temperature of 20 °C. Details about the foam preparation and mixing process can be found in Azimi (2015a). Densities of sand–foam mixtures before and after mixing are shown in Table 1. The expanded sand–foam densities ρ m were directly measured (Azimi, 2015a) and the unexpanded sand–foam densities ρ mu were calculated to show the degree of expansion.

ρmu = coρs + (1 − co)ρ f

(23)

Flume tests were conducted using a 2.4 m long, 0.115 m wide and 0.3 m high Plexiglas flume which the bed was covered with fine sand

Experimental results Two-dimensional flume tests Flume tests were carried out to study the two-dimensional free surface flow of foam and sand–foam mixtures. Variations of the longitudinal free surface profiles of sand–foam mixtures with time in an inclined flume for a foam–water ratio of Gf = 0.2 are shown in Fig. 3.

(b)

1.0

0.8

0.8

0.6 0.4 0.2

10o

2s

0.2 60 40

0.0

10o

10

30

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

X (m)

0.0

1.2

(d) 1.2

1.0

1.0

0.8

0.8

Z (m)

Z (m)

0.6 0.4

1s

20 8 4 45 30 90 60

180

360

(c)

1.2

1.0

Z (m)

Z (m)

(a) 1.2

paper to minimize slippage. A removable gate was designed to block the flow of foam before the onset of experiments. Experiments were conducted with two slopes of 10° and 15° for two different foam– water ratios of Gf = 0.1 and 0.2 and different sand concentrations. The flume top surface was covered by a plastic sheet to prevent foam evaporation during the tests. Fig. 1 shows an image of the flow of sand–foam mixture with a slope of 10° (Test T6-10) at 45 s after the beginning of the test. Frontal positions of the mixtures and the surface profiles were recorded at each time. Spreading tests with an inclined plane were conducted using a thick laminated timber sheet (2.8 m x 1.6 m). The bed slopes were set at 10°, 12°, 15°, and 18°. White grid lines were drawn to form squares of 0.1 × 0.1 m2 to correct any image distortions. The total duration of each test was between 200−400 s. The foam spread and its height were simultaneously recorded by two cameras. A CCD camera (TM 1040, Pulnix America Inc.) with a resolution of 1400 × 1000 pixels and a speed of 15 frames per second was used to capture the spreading of foam and sand–foam mixtures. The camera was controlled by a computer frame grabber system (Stream 5, IO Industries Inc.) and located 1.5 m from the flume and 4 m above the inclined plane. The recorded images were used to study the development of foam and sand–foam mixtures with time and investigate the flow characteristics. Sand–foam mixtures were found to be slightly time-variant and sensitive to mixing conditions due to their large micro structural size, unstable bubble structures and bubble breakup. To study the uncertainty of foam preparation, repeatability tests were conducted for one experiment (i.e., Test T610). The uncertainties of foam and sand–foam mixtures for mixture density and viscosity were found to be ± 5% and ± 5.7%, respectively (Azimi, 2015a). Frontal positions of the sand–foam spread at different times were measured and the uncertainty of the tests was found to be ± 3.5%. Similar uncertainty ranges of ± 5−8% were reported for testing free-surface flow of viscoplastic fluids (Chambon et al., 2014).

0.6 0.4

90

0.0 0.0

60 40

20 10

0.4

0.6

0.8

15o

87

1.0

1.2

1.4

1.6

1.8

X (m)

0.6

1s 2

0.4

1s

0.2

0.2

15o

5

0.2

10

0.0 0.2

0.4

0.6

0.8

1.0

X (m)

1.2

1.4

1.6

1.8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

X (m)

Fig. 3. Variations of the longitudinal profiles of sand–foam mixtures with time. (a) Test T6-10 (α = 10°, co = 0.58). (b) Test T8-10 (α = 10°, co = 0.82). (c) Test T6-15 (α = 15°, co = 0.58). (d) Test T8-15 (α = 15°, co = 0.82).

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69 595

600 T2-10

T2-15

T3-10

T3-15

T4-10

T4-15

T6-10

T6-15

T7-10

T7-15

T8-10

T8-15

(a) 1500

400

1000

xF (mm)

t (s)

63

183

200

139

120 79

500

58

40 9

8

4.4

19

8

0

Experiments Fig. 4. Comparison of the required time for sand–foam mixtures of various sand–foam ratios and sand concentrations to reach 1 m downstream of the flume.

(b)

T6-10

T7-10

T8-10

T6-15

T7-15

T8-15

0 1

10

100

1000

10000

0.1

1

t (s)

Two different slopes of α = 10° and 15° and sand concentrations of co = 0.58 and 0.82 were tested. As can be seen in this figure, the flow height h slightly drops with time in all tests. Free surface profiles in flume tests show that the height of sand–foam mixture is almost uniform along the x axis. Fig. 3a shows the flow of sand foam mixture for test T6-10 with co = 0.58 and α = 10°. It was found that h reduces as sand concentration increases. A comparison between flow height of sand–foam mixtures 10 s after the release for co = 0.58 and 0.82 indicated that h reduces by 36% for both α = 10° and 15°. Effect of controlling parameters such as foam–water ratio, sand concentration and bed slope on the flow dynamics of sand–foam can be analyzed by measuring the frontal velocity of mixtures. The required time for each mixture to flow and reach a certain distance from the release location was recorded. Fig. 4 shows the time for sand– foam mixtures to reach one meter downstream of the flume. As can be seen, sand–foam mixtures with Gf = 0.1 (Tests T2, T3 and T4) have higher frontal velocities than the corresponding tests with Gf = 0.2 (Tests T6, T7 and T8). Similar results were observed for the radial spreading velocities of sand–foam mixtures in slump tests (Azimi, 2015a). It can be deduced from Fig. 4 that the time required for the mixture to reach one meter downstream of the flume decreased with increasing sand concentration. For example, for mixtures with Gf = 0.1 and α = 10° (Tests T2-10 and T3-10) the frontal velocity of the mixture increased by three times when the sand concentration raised by 40% (i.e., from co = 0.42 to 0.59). Effect of sand concentration on the frontal velocity became more significant for higher foam–water ratios. Comparison of tests T6-10 and T7-10 indicated that the frontal velocity of the mixture increased by 4.3 times when the sand concentration increased by 26% (i.e., from co = 0.58 to 0.73). For flume tests with a bed slope of 15° and Gf = 0.1, an increase of sand concentration by 40% resulted in increasing the frontal velocity by 8.8 times. The non-linear variations of the frontal velocities with sand concentrations may be due to the change on the density and the rheological characteristics of sand–foam mixtures (Azimi, 2015a). The horizontal component of the weight of the mixture induces the flow of sand– foam whereas for low bed slopes, the acting gravitational force becomes comparable with the resisting viscous force and the bed shear stress. Effects of bed slope and sand concentration on the flow of sand– foam mixture was investigated by measuring the frontal position xF of the mixture with time. The frontal position can be detected by basic digital image processing techniques (Chanson et al., 2006; Gonzalez, 2010). Fig. 5 shows the variation of the foam length with time for tests with a foam–water ratio of Gf = 0.2. A time scale of (g/Ho )1/2 was found to be a suitable parameter to correlate the flow of sand– foam mixture with time. Flow of sand–foam mixture in an inclined

xF / (Ho sin α)

6

4

2

0 0.0001

0.001

0.01

t

g 10 co Ho

Fig. 5. Effects of sand concentration and bed slope on the variations of the sand–foam front with time. (a) Correlation of the frontal position of the mixtures with time for sand–foam with sand concentration of co ≥ 0.58. (b) Variation of the normalized foam length in flume test with normalized time.

flume can be described as



xF = 7.7 t Ho( sin α)

g Ho

1/4 co2/5

(24)

with R2 = 0.957. Similar correlations were reported to predict the variations of the frontal position of foam mixtures (i.e., co = 0%) with time (Azimi, 2015b). Chanson et al. (2006) used similar power law formulation to predict the flow of bentonite solution on an inclined flume with α = 15°. The coefficient and the index of correlation were reported as 0.5 and 2, respectively. Effects of sand concentration on the free surface flow characteristics of sand–foam mixtures were studied with time. Frontal velocities of the mixtures were measured at various times and they were normalized with a velocity scale to form a Froude number as

F = uF /(gHo sin α)1/2

(25)

Fig. 6a shows the variation of the Froude number F with the normalized time. An empirical formulation was developed to predict the frontal velocity of the mixture at different sand concentration with R2 = 0.986 as



F = λ1 t

g Ho

λ2

coλ3

(26)

where the coefficients of λ1 , λ2 , and λ3 are 2.75, −0.8, and 1.25, respectively. For the flow of foam mixtures (i.e., co = 0), Eq. (26) simplified to as



F = λ1 t

g Ho

λ2

(27)

64

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

0.1

(a)

(a)

z

0.01

x

F

20 s

40 s

80 s 120 s 180 s 240 s 300 s 340 s 380 s 420 s

40 s

60 s 80 s 180 s 220 s 260 s 280 s 300 s

0.001

(b) 20 s

0.0001 100

1000

10000

320 s

100000

t g / Ho

uF (mm/s)

(b)

co

30

T6-10

T6-15

T7-10

T7-15

T8-10

T8-15

(c) 20 s

40 s 60 s

80 s 100 s 120 s 140 s 160 s 180 s 200 s

40 s 60 s

80 s 100 s 120 s 140 s 160 s 180 s 200 s

20

(d) 10 20 s

0 1

10

100

1000

t (s) Fig. 6. Effect of sand concentration and bed slope on the frontal velocity of sand–foam mixtures. (a) Correlations between the non-dimensional time and Froude number in log–log scales. (b) Variations of the frontal velocity of sand–foam mixture with time. Dashed curves show the performance of the proposed equation to predict the frontal velocity of sand–foam mixture for Tests T6-10 and T8-15.

Azimi (2015b) showed that the coefficient of λ1 varied with the foam–water ratio and for Gf = 0.1 and 0.2, λ1 was 0.22 and 0.11, respectively. The coefficient of λ2 was −0.6 and it was independent of Gf . Similar formulation was used for the flow of bentonite solutions in an inclined bed (i.e., α = 15°) with λ1 and λ2 equal to 1 (Chanson et al., 2006). The difference between the values of λ2 for foam and sand–foam mixture indicates that for a similar bed slope and foam– water ratio, the frontal velocity of sand–foam mixtures reduces with a slower rate than the foam mixtures. This may be due to the density difference and the changes on the structure of sand–foam mixtures. As Eq. (26) shows, effect of sand concentration on the frontal velocity is significant and positive. Fig. 6b shows the variations of the frontal velocity with time for Gf = 0.2. Depends on the bed slope and sand concentration, frontal velocities of sand foam mixtures varied between 15 and 25 mm/s and reached to almost zero velocity around 1000 s after the release. Dashed curves in Fig. 6b show the prediction of the Eq. (26). The proposed formulation was able to predict the frontal velocity with ± 8% uncertainty. Inclined plane tests Study the flow of foam mixture and the discontinuity of its structure is of importance for pumping of the foam mixture through pipes and channels. Flow development and spreading of foam mixtures on an inclined plane with a foam–water ratio of Gf = 0.2 and bed slope

Fig. 7. Flow development and spreading of foam mixtures (co = 0) with time on an inclined plane with a foam–water ratio of Gf = 0.2 (Test T5) and various bed slopes ranging from 10° to 18°. (a) α = 10°, (b) α = 12°, (c) α = 15°, (d) α = 18°. Each square is 0.1 × 0.1 m2 .

ranging from 10° to 18° are shown in Fig. 7. As can be seen, the bed slope increases the frontal velocity of the foam to an extent. By increasing the bed slope from 10° to 15°, the frontal velocity of the foam mixtures increased by almost two times (see Fig. 7a and c). It was observed that the lateral expansion of the foam mixtures was limited in all tests and the spreading was independent of the bed slope. The flow of foam mixtures over an inclined bed induces by gravity. Considering the simplified one-dimensional momentum equation described in Eq. (17), the net force between the horizontal component of the foam weight (acts as a driving force) and the bed friction (acts as a resisting force) determines the change on the momentum flux. Assuming negligible net pressure force FP , the force balance becomes Ft = Wx − Fr where Ft is the resulting tension force acting on a foam cross section perpendicular to the inclined plane. The foam discontinuity may occur once the resulting tensile stress from the momentum balance overcomes the ultimate tensile strength of the foam mixture. Foam discontinuities were observed in all spreading tests. As can be seen from Fig. 7, the initiation of the foam discontinuity and the shapes of the foam mixtures after the spreading depended on the bed slope. Since the magnitude of Wx was relatively small in tests with small bed slopes (i.e., α ≤ 12°), single discontinuity was formed in those tests (see Fig. 7a and b). The frontal parts slid over the inclined bed while the remaining part of the foam mixture was stagnant after the foam discontinuity. It was found that for small slopes (i.e., α ≤ 12°) the size of the frontal foam mixture was constant. Comparison of the foam spreading on a plane with bed slopes of 10° and 12° indicated that the foam discontinuity occurred earlier for α = 12°. The discontinuity occurred at t = 300 s for α = 10° whereas for α = 12° the discontinuity occurred earlier at t = 200 s. For bed slopes of 15° and 18°, the magnitude of Wx became larger and this resulted in

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

Variations of the spreading length xP with time for an inclined plane bed at different bed slopes are shown in Fig. 10. Two different foam water ratios of Gf = 0.1 and 0.2 were studied. As can be seen, addition of foaming agent reduces the frontal velocity of the flow. For instance, the time required reaching the spreading length of 1200 mm for Test T2-10 (Gf = 0.1) was 140 s whereas this time was 280 s for similar test with Gf = 0.2 (Test T6-10). Similar flow behavior was observed from the standard slump tests of sand–foam mixtures (Azimi, 2015a). Variations of the spreading length with time can be predicted by employing a suitable time scale (g/Ho )1/2 , bed slope (sinα ) and sand concentration co . Fig. 10c and d show the correlations of the normalized spreading length with the normalized time for both foam–water ratios of Gf = 0.1 and 0.2, respectively. The following empirical formulations can be used for prediction of sand–foam spreading over an inclined plane.

(b)

(a) II z I

x

(d)

(c)

Fig. 8. Snapshot images of the spreading of sand–foam mixtures for Test T8 (co = 0.82) 30 s after the beginning of the test for various bed slopes ranging from 10° to 18° (a) α = 10°, (b) α = 12°, (c) α = 15°, (d) α = 18°. Dashed curves show the boundaries of the direct and lateral spreading of the sand–foam mixtures. Each square is 0.1 × 0.1 m2 .



xP = λ1 t Ho( sin α) multi-discontinuity. As can be seen from Fig. 7c and d, foam mixtures held their integrity up to 100 s after the beginning of the experiments. The first discontinuity of the foam mixtures occurred around 120 s from the onset of the tests whereas the second discontinuity occurred earlier for α = 18°. For this range of bed slopes, the foam frontal velocity and the separation time was found to be constant. Comparison of the foam spreading on a plane with bed slopes of 15° and 18° indicated that the volume of the frontal foam became smaller for α = 18°. Fig. 8 shows the effects of bed slope on the spreading of sand– foam mixtures for test T8 (co = 0.82) at 30 s after the release. Experimental observations on the flow and spread of sand–foam mixtures indicated that the mixtures remained homogeneous during the plane tests and no discontinuity occurred. Spreading of sand–foam mixtures on an inclined plane was divided into two regions. Region I shows the main flow of sand–foam along the longitudinal axis and Region II shows the lateral spreading of the mixtures with an angle from the x axis. The width of sand–foam was found to be constant for high bed slopes (α ≥ 15°). Effect of bed slope on the spreading shape at different time can be analyzed by drawing contour plots of the sand–foam top surfaces. Fig. 9 shows the contour plots of the flow spreading with time for Test T6 (co = 0.58). As can be seen from Fig. 9, the spreading widths of the mixtures were constant and independent of time.

(a) 0.6

(b)

(28)

The coefficient of λ1 for Gf = 0.1 and 0.2 are 23 and 10 and the exponent of λ2 for Gf = 0.1 and 0.2 are 0.20 and 0.27, respectively. The R2 value of Eq. (28) for Gf = 0.1 and 0.2 are 0.96 and 0.98, respectively. Frontal velocities of sand–foam mixtures in inclined plane tests can be calculated by measuring the movement of the front at each time. Fig. 11a and b shows the variations of the frontal velocity with time. As can be seen, frontal velocities decrease with time for all sand concentrations and slopes. It was found that the sand concentration has a small impact of the flow of sand–foam mixture since the frontal velocity of sand–foam with higher sand concentrations was slightly larger than mixtures with low sand concentrations. Empirical correlations were found to predict frontal velocity for different foam– water ratios as



F = λ1 t

g Ho

λ2

(29)

The coefficient of λ1 for Gf = 0.1 and 0.2 are 5/4 and 2 and the exponent of λ2 for Gf = 0.1 and 0.2 are −3/4 and −4/5, respectively. The R2 value of Eq. (29) for both foam–water ratios is 0.98. In order to model the free surface flow of multi-phase viscoplastic fluids on inclined surfaces, steady-state and laminar flow conditions were assumed. Effect mixture density and the rheological characteristics of sand–foam mixtures on the longitudinal velocity 0.6

Y (m)

180

120

0.0

-0.3

-0.6

-0.6 0.0

0.5

1.0

1.5

0.0

0.5

X (m)

(d)

1.5

2.0

0.6

0.3

Y (m)

180

120

80

0.0

60

60

40

-0.3

40

20

180

120

80

20

0.0

1.0

X (m)

0.6

0.3

Y (m)

λ2

80 60 40 20

180 120 80

0.0

-0.3

(c)

g 2 c Ho o

0.3

40 20

Y (m)

0.3

65

-0.3

-0.6

-0.6 0.0

0.5

1.0

X (m)

1.5

0.0

0.5

1.0

1.5

2.0

X (m)

Fig. 9. Contour plots show the effect of bed slope on the flow spreading of sand–foam mixture with time for Test T6 (co = 0.58), (a) α = 10°, (b) α = 12°, (c) α = 15°, (d) α = 18°.

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A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

(a)

(b)

2000

1600

1600

1200

1200

x (mm)

xP (mm)

2000

800

400

T2-10

T2-12

T2-15

T2-18

T3-10

T3-12

T3-15

T3-18

T4-10

T4-12

T4-15

T4-18

800

400

0

T6-10

T6-12

T6-15

T6-18

T7-10

T7-12

T7-15

T7-18

T8-10

T8-12

T8-15

T8-18

0 0

50

100

150

200

250

0

100

200

t (s)

(c)

(d)

100

400

500

1500

2000

2500

100

80

x / do sin α

80

xP / Ho sin α

300

t (s)

60

40

60

40

20

20

0

0 0

500

1000

1500

2000

2500

0

500

1000

g 2 t co Ho

g 2 t co Ho

Fig. 10. Variations of the spreading length of sand–foam mixtures with time for different bed slopes, sand concentrations and foam–water ratios (a) Gf = 0.1, (b) Gf = 0.2. Variations of the normalized spreading length with normalized time for different foam–water ratios (c) Gf = 0.1, (d) Gf = 0.2.

100

(b) 100

10

10

x (mm)

uP (mm/s)

(a)

1

1 T2-10

T2-12

T2-15

T2-18

T6-10

T6-12

T6-15

T6-18

T3-10

T3-12

T3-15

T3-18

T7-10

T7-12

T7-15

T7-18

T4-10

T4-12

T4-15

T4-18

T8-10

T8-12

T8-15

T8-18

0.1

0.1 0

50

100

150

200

250

0

100

200

t (s)

(c)

300

400

500

1500

2000

2500

t (s)

1

(d)

F

x / do sin α

0.1

0.01

1

0.1

0.01

0.001

0.001 0

500

1000

g t Ho

1500

2000

2500

0

500

1000

g t Ho

Fig. 11. Variations of the frontal velocity of sand–foam mixtures with time for different bed slopes, sand concentrations and foam–water ratios (a) Gf = 0.1, (b) Gf = 0.2. Variations of the Froude number with normalized time for different foam–water ratios (c) Gf = 0.1, (d) Gf = 0.2.

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

(a)

(a)

0.3 Plug Flow

67

5

4

3

zP / Ho

y (m)

0.2 Sheared Flow

2

0.1

1

0

0 0

0.3

0.6

0.9

0

1

2

3

u (m/s)

(b) 0.3 (b)

Plug Flow

zP / Ho

y (m)

Sheared Flow 0.1

co =0.0 co =0.40 co =0.58

5

3

2

1

co =0.82 0 0

0.3

0.6

0.9

0

Table 2 Rheological parameters of a Herschel–Bulkley equation fit to the rheological data for foam and sand–foam mixtures (Gf = 0.2) and the height of the sheared layer for α = 10° and 18°.

a b

t g / H o = 200

t g / H o = 300

t g / H o = 600

0

Fig. 12. Predicted velocity profiles of foam and sand–foam mixtures over an inclined plane for steady state laminar flow conditions. (a) α = 10°, (b) α = 18°.

0.00 0.40 0.58 0.82

t g / H o = 100

t g / H o = 1200

u (m/s)

ρm

τ o (Pa)

K (Pa.sn )

n

yo (α = 10°) (m)

yo (α = 18°) (m)

81.7 132.2 188.9 523.8

19.8b 17a 19a 39a

6 6 7 6

1/6 1/3 1/3 1/3

0.157 0.225 0.241 0.256

0.220 0.257 0.267 0.275

(kg/m3 )

5

4

0.2

co (vol/vol)

4

x / do

Data obtained from Azimi (2015a). Data obtained from Azimi (2015b).

profiles were studied using Eqs. (12–14). Fig. 12 shows the predicted velocity profiles for foam (co = 0) and sand–foam mixtures with sand concentrations of co = 0.40, 0.58 and 0.82 and for bed slopes of α = 10° and 18°. The critical shear stresses τ o , K and n parameters of the fitted Herschel–Bulkley model for foam and sand–foam mixtures were taken from Azimi (2015a, 2015b) and those parameters were listed in Table 2. The boundary between the plug flow and the sheared flow regimes was denoted by a solid line in Fig. 12. The predicted velocity profiles indicated that the plug flow velocity in steadystate condition is higher than the frontal velocity of the foam spread by almost an order of magnitude. It was found that the thickness of the sheared flow increased with increasing sand concentration. As can be seen from Fig. 12a, for α = 10°, the sheared flow thickness of foam mixtures was around 40% less than the thickness of the sheared flow for co = 0.82. For higher bed slopes (i.e., α = 18°), the thickness of the sheared foam flow was around 80% of the thickness of sheared flow for sand–foam mixture with co = 0.82. Visual observations of the inclined plane tests showed that the front profiles of sand–foam mixtures had a smooth shape during the flow motion. Fig. 13 shows the variations of the front profiles of sand–foam mixtures at different normalized time scales of 100 < t(g/Ho )1/2 <1200 and bed slopes of α = 10° and 18°. Both longitudinal and transverse lengths were normalized with the initial height

1

2

3

4

5

xP / Ho Fig. 13. Variations of the curved profiles of the front of sand–foam mixtures in spreading test (Test T6) at different time. (a) α = 10°, (b) α = 18°. Dashed curves are the predictions from the empirical formulation. Table 3 Coefficients for prediction of the frontal profiles of sand– foam mixtures for inclined plane tests. Bed slope α

ψ1

ψ2

ψ3

ψ4

10° 18°

4e−4 4e−4

1.9 1.85

4e−4 2e−4

3/4 5/8

of the gate opening Ho . The results showed that the frontal curvature increased with time, hence the travelled distance xP from the gate for a given experiment. Similar observations were found in the frontal motion of bentonite solution with solid mass concentrations ranging from 10% to 17% (Chanson et al., 2006). Dashed curves in Fig. 13 show the prediction of empirical formulation for the time scales range of t(g/Ho )1/2 = 100 and 1200. A power law formulation was proposed to predict the frontal curvature of bentonite solutions as xP /Ho ∼ (zP /Ho )0.4 where zP is the half-width of the mixture (Chanson et al., 2006). Data fit correlation for the present experimental tests showed that a logarithmic formula can better expresses the variations of the frontal profile with time.

zP = Ho



ψ1 t

g Ho





+ ψ2 ln

x  P

Ho

+



ψ3 t

g Ho





+ ψ4

(30)

where the coefficients of ψ 1 , ψ 2 , ψ 3 , and ψ 4 for both bed slopes of 10 and 18 degrees are listed in Table 3. The regression coefficient of correlations ranging from R2 = 0.97–0.99. Discussion Employing the analytical solution for uniform flow of a Herschel– Bulkley fluid down an inclined surface (Eqs. 5–11) reveals the effects of mixture density and rheological characteristics of sand–foam mixtures. Fig. 14 shows the variations of the non-dimensional fluid depth G with the non-dimensional time T for flume tests T6 (co = 0.58)

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A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

100

3.0

2.0

τ (Pa)

G

2.5

1.5

1.0

10

0

10

1

10

2

10

3

10

4

10

5

10

6

T T6-15

T8-10

T8-15

Coussot and Boyer (1995)

Predictions

Fig. 14. Variations of the non-dimensional fluid time T with the non-dimensional depth G.

and T8 (co = 0.82) with different flume slopes of 10° and 15°. The solid curve in Fig. 14 is added for comparison and it shows the relationship between T and G for Kaolin–water mixtures (Coussot and Boyer, 1995). Dashed curves are the predictions for sand–foam mixtures based on the Gaussian profile formulations as

G = λ1 e−λ2 T

2

(31)

where λ1 and λ2 are fitting coefficients. The magnitude of the nondimensional time relates to the ratio of τ o /K. This ratio was between 2 and 3.2 for clay–water mixtures (Coussot and Boyer, 1995) whereas for sand–foam mixtures τ o /K ranged from 2.7 to 6.5. The value of T for G = 1.1 was reported to be 1000 for clay–water mixtures whereas this non-dimensional time was found to be 35,480 for Test T6. Two different flow regimes were observed for sand–foam mixtures of low and high sand concentrations with a critical sand concentration of co = 0.6. The yield stress of sand–foam mixtures significantly increases for co > 0.6 (Azimi, 2015a). As can be seen from Fig. 14, the non-dimensional time was independent of bed slope for co < 0.6 whereas for co > 0.6, T became larger for higher bed slopes. The non-dimensional velocity dG/dT can be deduced from the slope of the correlations between T and G. It can be deduced from the magnitude of the curve slope (i.e., dG/dT) that the frontal velocities of sand–foam mixtures were higher than the frontal velocity of clay– water mixture from the experimental study of Coussot and Boyer (1995). Considering the slope of correlations, indicated that the bed slope had a limited impact on the non-dimensional velocity of sand– foam mixtures with co < 0.6 whereas for Test T8 (co = 0.82), bed slope increased the dG/dT significantly. Variations of shear stress τ at different shear rates γ˙ can be calculated from the flume tests using the proposed equations for the Herschel–Bulkley fluids (Coussot, 1994). The values of τ , G and Hb are unknown at each time and they can be calculated simultaneously by solving Eqs. (5–7). Shear rate can be estimated using uF /h¯ at each time where h¯ is the average height of the sand–foam mixture. A simplified one-dimensional momentum equation (Eqs. (17–18)) can be used to estimate the variations of the average bed shear stress with time for inclined plane tests. At the beginning of the spreading test for α = 10° the maximum ratio of FP /Wx was 19.7% and as more mass introduced to the plane, the ratio of FP /Wx reduced to 9.8%. The horizontal component of the mixture weight became more significant for higher bed slopes. At the beginning of the test for α = 18° the maximum FP /Wx ratio was only 3.5% and the contribution of the pressure force reduced to 1.7% after 60 s from the onset of the spreading. The impact of momentum flux (i.e., dM/dt) on the simplified momentum equation was limited and it was noticeable only at the onset the spreading since both A and uP were high. Contribution of the momentum flux was varied from 3.4% to 13.1% at the onset of the spreading and it reduced to less than 0.1% at the end of the test.

10 0.001

0.01

0.1

1

10

100

1000

γ• (s-1) T6-10-F T8-10-F T6-10-P T6-15-P Azimi (2015a), co = 0.58

T6-15-F T8-15-F T6-12-P T6-18-P Azimi (2015a); co = 0.82

Fig. 15. Comparison between the measured shear stress-shear rate relationship using a programmable viscometer and flume/plane tests for sand–foam mixtures with Gf = 0.2. Flume tests denoted was as (F) and inclined plane test was denoted as (P). Error bars indicate the uncertainty of rheological measurements for shear stress.

Fig. 15 shows the variations of the shear stress τ prediction from the flume and plane tests at different shear rates γ˙ in log-log scale. The shear rate was approximated as γ˙ = uF /h¯ for the flume test and γ˙ = uP /h¯ for the plane test. Experimental measurements of Azimi (2015a) for sand–foam mixtures with sand concentrations of co = 0.58 and 0.82 were added for comparison. Experimental data of Azimi (2015a) were obtained using an accurate programmable viscometer (Brookfield DV-II+). The viscometer was able to measure the shear stress for wide ranges of shear rate from 0.1 s−1 to 200 s−1 . As can be seen from Fig. 15, the flume test predicted the shear stress for low shear rates ranging from 0.003 s−1 to 0.1 s−1 and the plane test predicted the shear stress for a narrow range of 0.2 s−1 < γ˙ < 3 s−1 . Comparison between the shear stress predictions of both flume and inclined plane tests indicated that the plane test results were in a good agreement with the rheological data. The inclined plane test results showed that the shear stress predictions were independent of the bed slope and the uncertainty of the prediction were slightly higher than the measurements of Azimi (2015a) using a programmable viscometer. The flume test results covered a wide range of shear rates but the results were dependent on the flume slope. 0.6 T6-10-F

T6-15-F

T8-10-F

T8-15-F

T6-10-P

T6-12-P

T6-15-P

T6-18-P

0.3

Δτ / τ

T6-10

0.0

-0.3

-0.6 0.01

0.1

1

h/x Fig. 16. Relative error on shear stress determinations using flume tests (F) and inclined ¯ plane tests (P) as a function of h/x.

A.H. Azimi / International Journal of Multiphase Flow 78 (2016) 59–69

The relative errors of the shear stress predictions for both flume and inclined plane tests were calculated and the results were plotted ¯ against the foam aspect ratio h/x. As can be seen from Fig. 16, both ¯ test provided better predictions for 0.05 ≤ h/x ≤ 0.2. It was found that the prediction of the shear stress was on average within ± 12.5% from the proposed Herschel–Bulkley equations for sand–foam mixtures (Azimi, 2015a). Chambon et al. (2014) predicted τ o and K of Carbopol microgel from free-surface flow modeling with an error of 10% and 20%, respectively. Summary and conclusions A series of laboratory experiments were conducted to investigate the free surface flow characteristics of multi-phase viscoplastic fluids on inclined flumes and planes. Foam and sand–foam mixtures were selected as multi-phase gas–liquid and gas–liquid–solid viscoplastic fluids. Effects of flow controlling parameters such as bed slope, foam– water ratio and sand concentration on the dynamics of foam and sand–foam mixtures were examined. Flume tests were conducted to investigate the free surface flow of sand–foam mixtures under twodimensional flow condition. Three-dimensional spreading of foam and sand–foam mixtures over an inclined plane was tested. Foam–water ratio was found to be important in controlling the flow of sand–foam mixture in two-dimensional flow. Surface flow of foam and sand–foam mixtures was scaled using proper velocity and time scales. Empirical power law equations were proposed to model the flow of foam and sand–foam with time with ± 8% uncertainty. It was found that the sand concentration can accelerate the flow of sand–foam mixture with the rate of co5/4 . It was found that for foam mixtures with small slopes (i.e., α ≤ 12°), the size of the frontal foam mixture was constant. Single and multi-discontinuity were observed for the flow of foam mixtures on an inclined plane. The imbalance between the horizontal component of the foam weight and the bed shear force caused different types of foam discontinuity. Single discontinuity was observed for small slopes whereas for higher bed slopes (i.e., α ≥ 12°) multi discontinuity was observed. It was found that increasing the foaming agent reduced the flow of sand–foam mixtures. Spreading of sand–foam mixtures at different times were modeled using the normalized time and Froude number as proper scales. Empirical correlations were proposed to predict the free surface flow of sand–foam spread for different foam–water ratios. Velocity profiles of the selected multi-phase viscoplastic fluids were predicted using a steady-state laminar equation with the Herschel–Bulkley constitutive law. It was found that the thickness of the sheared flow increased with increasing sand concentration. Shear stress-shear rate diagram were plotted for the selected multi-phase viscoplastic fluid using discharge and momentum equation with the Herschel–Bulkley rheological model. Effects of sand concentration and bed slope on the shear stress-shear rate were studied and the model predictions were compared with the direct rheological measurements. Comparison between the shear stress predictions of both flume and inclined plane tests with the rheological measurements from the literature indicated that the plane test results were in agreement with the rheological data. The inclined plane test results showed that the shear stress predictions were independent of the bed slope and the uncertainty of the prediction was

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slightly higher (i.e., ± 12.5%) than the accuracy of the programmable viscometer. Acknowledgments The work presented here was partially supported by the NSERCDiscovery grant No. 421785. The author wishes to thank Dr. Ehsan Abazari Torghabeh for his help on conducting a part of the present experimental work. References Ancey, C., 2007. Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142, 4–35. Azimi, A.H., 2015. Experimental investigations on the physical and rheological characteristics of sand foam mixtures. J. Non-Newtonian Fluid Mech. 221, 28–39. Azimi, A.H., 2015. Flow of foam mixtures on inclined flumes and surfaces. In: CSCE Conference. Regina, CA DPM-067-01-08. Abazari Torgabeh, E., 2013. Stabilization of Oil Sands Tailings Using Vacuum Consolidation 149 p PhD Thesis. Bobert, M., Persson, H., Person, B., 1997. Foam concentrates: viscosity and flow characteristics. Fire Technol. 33 (4), 336–355. Breusers, H.N.C. and Raudkivi, A.J. (1991). International Association for Hydraulic Research, Scouring. Balkema, Rotterdam, The Netherlands, 143 p. Briceno, M.I., Joseph, D.D., 2003. Self-lubricated transport of aqueous foams in horizontal conduits. Int. J. Multiph. Flow 29, 1817–1831. Chambon, G., Ghemmour, A., Laigle, D., 2009. Gravity-driven surges of a viscoplastic fluid: an experimental study. J. Non-Newtonian Fluid Mech. 158, 54–62. Chambon, G., Bouvarel, R., Laigle, D., Naaim, M., 2011. Numerical simulations of granular free-surface flows using smoothed particle hydrodynamics. J. Non-Newtonian Fluid Mech. 166, 698–712. Chambon, G., Ghemmour, A., Naaim, M., 2014. Experimental investigation of viscoplastic free-surface flows in a steady uniform regime. J. Fluid Mech. 754, 332–364. Chanson, H., Jarny, S., Coussot, P., 2006. Dam break wave of thixotropic fluid. J. Hydraul. Eng., ASCE 132 (3), 280–293. Chasko, L.L., Conti, R.S., Derick, R.L., Krump, M.R., Lazzara, C.P. (2003). In-mine study of high-expansion firefighting foam, US Department of Health and Human Services, Public Health Service, Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health, Pittsburgh Research Laboratory. Clayton, S., Grice, T.G., Boger, D.V., 2003. Analysis of the slump test for on-site yield stress measurement of mineral suspensions. Int. J. Miner. Process. 70, 3–21. Cochard, S., Ancey, C., 2009. Experimental investigation of the spreading of viscoplastic fluids on inclined planes. J. Non-Newtonian Fluid Mech. 158, 73–84. Coussot, P., 1994. Steady, laminar, flow of concentrated mud suspensions in open channel. J. Hydraul. Res. 32 (4), 535–559. Coussot, P., Boyer, S., 1995. Determination of yield stress fluid behavior from inclined plane test. Rheol. Acta 34, 534–543. Coussot, P., Meunier, M., 1996. Recognition, classification and mechanical description of debris flows. Earth-Sci. Rev. 40, 209–227. Coussot, P., Proust, S., 1996. Slow, unconfined spreading of a mudflow. J. Geophys. Res. 101 (B11) 25, 217-25, 229. Coussot, P., Proust, S., Ancey, C., 1996. Rheological interpretation of deposits of yield stress fluids. J. Non-Newtonian Fluid Mech. 66 (1), 55–70. Coussot, P., Laigle, D., Arattano, M., Deganutti, A., Marchi, L., 1998. Direct determination of rheological characteristics of debris flow. J. Hydraul. Eng. 124 (8), 865–868. Edrisi, A., Gajbhiye, R., Kam, S.I., 2013. Experimental study of polymer-free and polymer-added foams for underbalanced drilling: Are two foam-flow regimes still there? Soc. Petrol. Eng. J. 19 (1), 55–68. Gawu, S.K.Y., Fourie, A.B., 2004. Assessment of the modified slump test as a measure of the yield stress of high-density thickened tailings. Can. Geotech. J. 41, 39–47. Gonzalez, R.C., 2010. Digital Image Processing Using MATLAB. McGraw-Hill Education, India, p. 738. Gumati, A., Takahshi, H., 2011. Experimental study and modeling of pressure loss for foam-cuttings mixture flow in horizontal pipe. J. Hydrodyn. 23 (4), 431–438. Hogg, A.J., Matson, G.P., 2009. Slump of viscoplastic fluids on slope. J. Non-Newtonian Fluid Mech. 158, 101–112. Kaitna, R., Rickenmann, D., Schatzmann, M., 2007. Experimental study on rheologic behavior of debris flow material. Acta Geotech. 2, 71–85. Pouliquen, O., 1999. Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.