Rheology

As stated above, continuum mechanical models are associated with a rheology model. By integrating the rheological relationship of the equivalent single phase fluid, the basal resisting force is obtained as a function of the flows mean velocity, density and depth (Hungr, 1995; Pirulli, 2005). Rheological models used in debris flows adopt the concept of fluid flows to use on granular masses (Yifru, 2014).

Viscosity is defined as the ratio of applied shear stress to the rate of shear strain (Pierson

& Costa, 1987). Based on the viscosity some definitions of the fluids can be made, these are also given in Figure 11. The resistance of forces associated with the rheology are acting inside the flow and in the contact surface between the flow and surface of the bed (Quan, 2012). The basal flow resistance T assumes a linear increasing shear stress with depth to give a relation between the resistance and the other parameters of the flow (Hungr, 1995).

A Newtonian fluid is described as a fluid that flow independently of increasing or decreasing shear stress for a constant temperature. Viscosity is constant. (Pierson &

Costa, 1987; Yifru, 2014). When viscosity is decreased as a function of increasing shear strain, the flow exhibits a shear-thinning behavior and is called pseudoplastic (Pierson &

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Costa, 1987; Yifru, 2014). The opposite behavior is shown for the dilatant fluid, which exhibits a shear-thickening behavior as viscosity increases with the rate of shear strain.

In some cases, there is no flow happening until a certain threshold value is reached given by the upper three curves of Figure 11. This may happen for some naturally occurring materials (Pierson & Costa, 1987). First when this yield strength is exceeded, the material can flow. In a Bingham fluid, the viscosity is constant after hitting the yield stress, making the rate of shear strain proportional to the shear stress (Pierson & Costa, 1987; Yifru, 2014). There are also versions of the previously described dilatant fluid and pseudo-plastic fluid which exhibit a certain yield strength.

Figure 11 Flow curves of different rheologic behavior (Pierson & Costa, 1987).

Debris flows are multi-phase and consist in general of phases of solid and water. The distribution of the solid and fluid components together with the grain size distribution and physical and chemical properties of the solid phase, will affect how the flow responds to shear stress (Pierson & Costa, 1987). Pierson and Costa (1987) propose that the

rheologic response of a mixture of sediment and fluid is given primarily by the concentration of sediments. The two other effects on the response to shear stress become less important compared to this (Pierson & Costa, 1987). Figure 12 shows the variation in rheology and classification of the flow based on velocity and solid

concentration. Granular flows are at the higher end of the sediment concentration, giving a non-Newtonian behavior. When particles start to interact, the flowing fluid obtains a yield strength introducing this behavior (Pierson & Costa, 1987).

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Figure 12 Classification scheme based on rheological behavior and existing nomenclature (Pierson

& Costa, 1987).

The basal resistance of the flow called T, is dependent on which rheology that is used for the material (Hungr, 1995). 𝐴_𝑖 is the base area of the boundary block and is given as the product of length of the base 𝑑_𝑠 and the width 𝐵_𝑖.

According to Pierson and Costa (1987) the term debris flow is the most appropriate for both the behavior of a viscous slurry flow and an inertial slurry flow. A viscous slurry flow is characterized by viscous forces controlling the flow. Viscous forces can control the flow when the values of shear rates, mean grain diameter, grain density and water content are low (Bagnold, 1954; Pierson & Costa, 1987). Bingham model is stated to be

appropriate in the kinds of flow where the intergranular fluid consists of clay, silt and water (Major & Pierson, 1992). This model has been used for numerous debris flows (Takahashi, 2007, p. 40). The resistance of this model is given by solving the third degree equation for the mean flow velocity as given in Hungr (1995):

44 𝑣_𝑖= 𝐻𝑖

6𝜇_𝑣(2𝑇

𝐴_𝑖− 3𝜏 +𝜏³𝐴𝑖2

𝑇² )

The resting force is hence given dependent on the flow depth, velocity, the constant yield strength 𝜏, and the Bingham viscosity 𝜇_𝑣.

However, when the pore fluid has low viscosity and the shear rate, grain diameter and grain density are high, the opposite of the beforementioned case, the dominating forces are different. For this case, the inertial forces dominate the flow behavior and hence, momentum transfers as particles collide (Pierson & Costa, 1987). These inertial slurry flows can be modelled by a dilatant fluid model with a yield strength (Takahashi, 2007).

Takahashi (2007, p. 42) uses the dilatant model on what he calls stony type of debris flows.

Another rheologic model given in Hungr (1995) is the Voellmy rheology. This model was originally introduced for snow avalanches (Voellmy, 1955). The model concerns a friction term and a coefficient concerning the turbulence of the flow given the Greek letter 𝜉.

Hungr (1995) justified the use of this rheological model by showing to the results of (Bagnold, 1954). As the granular material is sheared the strength of the material will increase with a function of the squared strain rate (Bagnold, 1954; Hungr, 1995). The friction resistance of a landslide would hence increase proportionally the velocity squared (Salm, 1993). The resistance force of the Voellmy rheology is as given in Hungr (1995):

𝑇 = 𝐴𝑖[𝛾𝐻𝑖(cos(𝛼) +𝑎_𝑐

𝑔) tan(𝜙) +𝛾𝑣_𝑖² 𝜉

The slope angle is expressed as 𝛼 in this case. Additionally, 𝑎_𝑐 defines the centrifugal acceleration. The Voellmy rheology is discussed more in the next section.