**3. Rheology of mixtures**

268 Hydrodynamics – Natural Water Bodies

represented by very fine sand and silt sized glass beads, and cohesive particles represented by kaolin clay. Both sediments have density approximately of 2600 kg/m³. In total, 21 experiments (Fig. 4) were carried out with eight values of bulk volumetric concentration (2.5%, 5%, 10%, 15%, 20%, 25%, 30% and 35%). In addition, for each value of concentration were used three different proportions of clay in the mixture from 0% (pure

non-cohesive flows) passing to 50% (mixed) and finally, 100% (pure cohesive flows).

Fig. 4. Initial properties of the mixtures simulated and the particles properties.

length) in which the water (and flow) were drained after the experiment.

body and tail zones.

The experiments were performed in a 2D Perspex tank (4.50 m long x 0.20 m wide x 0.50 m height). A 120 litres mixture was prepared in a mixing box (full capacity of 165 litres) connected at the upstream part of the tank through a removable lock-gate (0.21 m wide and 0.70 m high). An electric-mechanical mixer was installed within that box to assure the full mixing of sediment mixture. The tank also had a dispersion zone (approximately 1.00 m

In all sets of experiments were used lock-exchange methodology characterized by the instantaneously release of the mixture (lock-gate opening) reproducing a catastrophic event on nature. As soon as the mixture entered into the channel, the dense flow was generated. In order to measure the flow properties during the experiments, a group of equipments was installed within the tank. Four UHCM's *(Ultrasonic High-Concentration Meter)* were set along the vertical profile (at 1.0; 3.2; 6.4 and 10 cm from the bottom) to acquire time-series concentration data, whilst ten UVP's (*Ultrasonic Doppler Velocity Profiler*) of 2 MHz transducers were set along vertical profile (15 cm) to register time-series of velocity data. Both equipments were located at 340 cm from the gate. With both velocity and concentration data, the hydrodynamic properties were established for all flows such as: time series of velocity and concentration, mean vertical profiles, non-dimensional parameters for the head, The rheology is the study of deformation and flow of matter and is a property of the fluid that expresses its behaviour under an applied shear stress. Through the rheological characterization of mixtures (water and sediment), it is possible to establish the relationship between shear stress and strain rate (shear rate), and consequently the coefficient of dynamic viscosity (and/or apparent) as well as the constitutive equations in terms of volumetric concentration and presence of clay.

In natural flows, the non-conservative condition of the sediment gravity flows, i.e. erosion and deposition during the movement, modifies the mechanisms of transport and deposition of particles within the flow (e.g. local concentration, size and composition of grains in suspension), which impact also their rheological behaviour.

Based on this, a rheological characterization of mixtures was carried out aiming to establish such property of the mixtures and verify its behaviour for different initial conditions. To do that, it was used a Rheometer device with two types of spindle (cone plate and parallel plate). For the tests, the mixtures were prepared following the same proportions of sediment used in the experimental work and also considering the same temperature (~ 19°C). The rheogram - output data of the Rheometer consisting in the ratio of shear stress and strain rate - was compared to typical rheological models found in literature. The simplest rheological model of imposed stress *(x)* related to strain rate *(u/z)* is the *Newtonian* model (due to the definition of Newton's law of viscosity) and it can be expressed for twodimensional flow in the x – z plane as:

$$
\pi\_\chi = \mu \stackrel{\mathfrak{\partial}u}{\underset{\partial \mathbf{z}}{\rightleftharpoons}} \tag{1}
$$

The equation (1) shows a linear relationship between the imposed shear stress and strain rate (gradient of deformation). As a consequence, the viscosity of the fluid or mixture (*coefficient of dynamic viscosity -* ) is constant for all values of shear rate. Any deviation from linearity between the stress-strain curve converts the rheological property to non-Newtonian behaviours, which can be generally divided into four more groups: *plastics* in which there is no deformation of the flow until the critical initial stress (yield strength - *0*) is overcome; *dilatant and pseudoplastic*, in which the deformation (strain rate) is expressed by a power law type (if coefficient of power law n > 1 then the fluid is *dilatant* otherwise (n < 1) is *pseudoplastic*) and; *Herschel-Bulkley* in which the fluids has a plastic behaviour (yield strength - *<sup>0</sup>*) followed by a power law behaviour. The *Herschel-Bulkley* model can be expressed for two-dimensional flow in the x – z plane as:

Sediment Gravity Flows: Study Based on Experimental Simulations 271

The threshold line represents the transition from Newtonian to non-Newtonian behaviour (plastic) and can be represented by the occurrence of yield strength. Clearly, there is not a unique value representing this change of rheological behaviour. A transition interval must be considered (dashed line around the threshold) to more accurate analysis. In addition, different composition of clay may move the position of the curve, for instance; the threshold of montmorillonite shows similar shape. However this curve of yield strength (high values for this particular type of clay) is moved into the top-left of the diagram. For the group of Herschel-Bulkley plastic mixtures (high concentration and more presence of clay - below the threshold line) the constitutive equations were empirically determined (eq. 4, 5) correlating the apparent viscosity, the clay content in the mixture, the bulk concentration of the mixtures and, the gradient of deformation (strain rate) for this group of

1 2 24 0 44 *C Clay vol* . .%

0 24 1 8 <sup>31</sup>

0 59 8 7 0 0016

It was also established an empirical relationship to yield strength in terms of the volumetric

<sup>2790</sup> 0 00104 *vol %Clay C*

The rheological characterization (rheometry) has classified the mixtures into two distinct groups as it was illustrated in Fig. 5. Based on that approach, all data and results obtained through experimental work were compared in order to establish groups with similar properties. A total of 15 parameters divided into seven categories were used to fully characterize and distinguish each group: geometry, rheology, analysis of mean vertical profiles, time-series of data, internal dynamics of the flow, depositional features and, non-

After applying this method of analysis, it was possible to identify six regions (or groups) of similar sediment gravity flows generated experimentally. Each one has typical properties and characteristics in terms of rheology, geometry, hydrodynamic and depositional processes along time and space. Moreover, the relationship with initial properties (concentration and clay content) demonstrates the cause-consequence of the experiments (from source to deposit) and the entire dynamic involved. The Fig. 7 illustrates this diagram-

Each region properties will be completely described below from non-cohesive dominated flows (regions I, II and III) to cohesive dominated flows (regions IV, V and VI). The

*u e C z*

*vol*

*<sup>u</sup> C e* 

. % .

*vol*

. %

*clay*

(5)

. *e* (6)

(4)

*C*

. .

*Clay Clay*

*z*

0

0

*clay*

concentration and the presence of clay in the mixture.

**4. Experimental results** 

dimensional parameters as seen in Fig. 6.

phase with delimited boundaries amongst the regions.

averaged vertical profiles will be discussed apart (item 4.6).

1 39

*i* 

.

*ap C*

mixtures.

where

(3)

$$
\pi\_{ap} = \pi\_0 + K \left(\frac{\partial u}{\partial z}\right)^n \tag{2}
$$

To non-Newtonian mixtures, the determination of viscosity (curve slope at the rheogram) is no longer direct, implying that for each value of gradient of deformation (strain rate) applied, there will be a different coefficient of dynamic viscosity. When this occurs, the viscosity is called *apparent viscosity of the fluid* (ap) rather than the dynamic viscosity. From the results obtained with the rheometry tests, it was defined two distinct groups for

the mixtures simulated in terms of different values of concentration and clay content: the Newtonian group of mixtures and the Herschel-Bulkley plastic group of mixtures (Fig. 5).

Fig. 5. Rheological characterization of the mixtures simulated and the constitutive equations in terms of volumetric concentration and presence of clay for each group.

For the group of Newtonian mixtures (above threshold line) it was possible to establish an empirical relationship (linear) between the values of dynamic viscosity with the volumetric concentration and clay presence, which allows properly assess the effect of viscosity on the hydrodynamic parameters for this group of mixtures (eq. 3). The coefficient values were similar to those found in literature for non-cohesive grain mixtures (e.g. Coussot, 1997; Einstein, 1906). The rheological characterization was carried out to the volumetric concentration of 35% only. Extrapolation to higher values must be handled carefully (see Coussot, 1997; Wan & Wang, 1994).

$$\frac{\mu}{\mu\_0} = 1 + C\_{vol} \left( 2.24 + 0.44 \text{ \%Clay} \right) \tag{3}$$

The threshold line represents the transition from Newtonian to non-Newtonian behaviour (plastic) and can be represented by the occurrence of yield strength. Clearly, there is not a unique value representing this change of rheological behaviour. A transition interval must be considered (dashed line around the threshold) to more accurate analysis. In addition, different composition of clay may move the position of the curve, for instance; the threshold of montmorillonite shows similar shape. However this curve of yield strength (high values for this particular type of clay) is moved into the top-left of the diagram.

For the group of Herschel-Bulkley plastic mixtures (high concentration and more presence of clay - below the threshold line) the constitutive equations were empirically determined (eq. 4, 5) correlating the apparent viscosity, the clay content in the mixture, the bulk concentration of the mixtures and, the gradient of deformation (strain rate) for this group of mixtures.

$$\frac{\mu\_{\rm up}}{\mu\_0} = \left[1.39 \left(e^{\left(31 \text{ C}\_{\rm vol}\right)} \left(\frac{\partial \mu}{\partial z}\right)^{\left(0.24 - 1.8 \text{ C}\_{\rm vol}\right)}\right] \cdot \text{C}\_{\rm clay} \tag{4}$$

where

270 Hydrodynamics – Natural Water Bodies

To non-Newtonian mixtures, the determination of viscosity (curve slope at the rheogram) is no longer direct, implying that for each value of gradient of deformation (strain rate) applied, there will be a different coefficient of dynamic viscosity. When this occurs, the

From the results obtained with the rheometry tests, it was defined two distinct groups for the mixtures simulated in terms of different values of concentration and clay content: the Newtonian group of mixtures and the Herschel-Bulkley plastic group of mixtures (Fig. 5).

Fig. 5. Rheological characterization of the mixtures simulated and the constitutive equations

For the group of Newtonian mixtures (above threshold line) it was possible to establish an empirical relationship (linear) between the values of dynamic viscosity with the volumetric concentration and clay presence, which allows properly assess the effect of viscosity on the hydrodynamic parameters for this group of mixtures (eq. 3). The coefficient values were similar to those found in literature for non-cohesive grain mixtures (e.g. Coussot, 1997; Einstein, 1906). The rheological characterization was carried out to the volumetric concentration of 35% only. Extrapolation to higher values must be handled carefully (see

in terms of volumetric concentration and presence of clay for each group.

Coussot, 1997; Wan & Wang, 1994).

*<sup>u</sup> <sup>K</sup> z*

 

*n*

(2)

0

viscosity is called *apparent viscosity of the fluid* (ap) rather than the dynamic viscosity.

*ap*

 

$$C\_{\rm clay} = 0.0016 \text{ } e^{\left(8.7 \cdot \% \text{Clay}\right)} \frac{\partial \mu^{\left(0.99 - \% \text{Clay}\right)}}{\partial z} \tag{5}$$

It was also established an empirical relationship to yield strength in terms of the volumetric concentration and the presence of clay in the mixture.

$$
\tau\_i = 0.00104 \text{ } e^{\{2790 \text{ } \%Clay \text{ C}\_{vol}\}} \tag{6}
$$
