2.2.1. *μ*(*I*) Rheology

The pressure induced by collisions and frictional interactions is modeled following Chauchat et al. (2017) [21]:

$$p^s = \left(\frac{B\_\phi \,\phi}{\phi\_{\text{max}} - \phi}\right)^2 \rho^s d^2 \parallel S^s \parallel^2 \tag{10}$$

where *Bφ* = 1/3 is a parameter of the dilatancy law [22] and || *Ss* || is the norm of the deviatoric part of the solid phase strain rate tensor *Ssij* defined as || *Ss* ||= <sup>2</sup>*SsijSsij*.

Following Jop et al. (2006) [23], the particle shear stress is proportional to the particle pressure following a frictional law:

$$
\pi\_{ij}^s = \mu(l)(p^s + p^{ff})\frac{S\_{ij}^s}{||\:S^s\:\mid\: \vert}.\tag{11}
$$

According to GDRmidi(2004) [13], the friction coefficient *μ*(*I*) is given by:

$$
\mu(I) = \mu\_s + \frac{\mu\_2 - \mu\_s}{I\_0/I + 1},
\tag{12}
$$

where *I* =|| *Ss* || *d ρ<sup>s</sup>*/*p*˜*<sup>s</sup>* is the inertial number, *μs* = 0.63 is the static friction coefficient for sand, and *μ*2 = 1.13 and *I*0 = 0.6 are empirical coefficients.

A frictional shear viscosity is introduced to be consistent with the fluid phase momentum equation, and the particle shear stress is written as *τsij* = *<sup>ν</sup>sFrSsij*, with the frictional shear viscosity *νsFr* written as:

$$\nu\_{Fr}^{s} = \min\left(\frac{\mu(I)(p^s + p^{ff})}{\rho^s \left(||\ \mathcal{S}^s||^2 + D\_{small}^2\right)^{1/2}}, \nu\_{max}\right). \tag{13}$$

*Dsmall* is regularization parameter from Chauchat and Médale (2014) [24], and *νmax* is a viscosity limiter set to *νmax* = 10

#### 2.2.2. Kinetic Theory for Granular Flows

The model adopted was suggested by Ding and Gidaspow (1990) [25]. The particulate pressure is a function of the particle velocity fluctuations represented by the granular temperature Θ following:

$$p^s = \rho^s \phi[1 + 2(1+\varepsilon)\phi \mathbf{g}\_{s0}] \Theta\_{\prime} \tag{14}$$

with *e* the coefficient of restitution during the collision and *gs*0 = (2 − *φ*)/2(1 − *φ*)<sup>3</sup> a radial distribution function from Carnahan and Starling (1969) [26] introduced to describe the crowdedness of particles.

The particle shear stress *τsij* is decomposed into the sum of a frictional and a collisional stress component:

$$
\pi\_{ij}^s = \pi\_{ij}^{ff} + \vec{\pi}\_{ij}^s. \tag{15}
$$

The frictional components allow reproducing the immobile sediment bed behavior and are defined as *τ f f ij* = <sup>2</sup>*ρ<sup>s</sup>νsFrSsij*, with *νsFr* calculated as:

$$\nu\_{Fr}^{s} = \frac{p^{ff}\sin(\theta\_f)}{\rho^s \left(||\,^{S^s}\,||^2 + D\_{small}^2\right)^{1/2}},\tag{16}$$

using a constant friction angle *θ f* = 32◦.

> The particle collisional stress is calculated as:

$$
\pi\_{ij}^s = 2\mu^s S\_{ij}^s + \lambda \frac{\partial u\_k^s}{\partial x\_k} \delta\_{ij}. \tag{17}
$$

The particle shear viscosity *μs* and bulk viscosity *λ* are functions of the granular temperature and the radial distribution function following:

$$
\mu^s = \rho^s d\sqrt{\Theta} \left[ \frac{4\phi^2 g\_{s0} (1+\varepsilon)}{5\sqrt{\pi}} + \frac{\sqrt{\pi} g\_{s0} (1+\varepsilon)(3\varepsilon - 1)\phi^2}{15(3-\varepsilon)} + \frac{\sqrt{\pi}\phi}{6(3-\varepsilon)} \right] \tag{18}
$$

and:

$$
\lambda = \frac{4}{3} \phi^2 \rho^s d\varrho\_{s0} (1+\varepsilon) \sqrt{\frac{\Theta}{\pi}}.\tag{19}
$$

The balance equation for the granular temperature is written as:

$$\frac{3}{2} \left[ \frac{\partial \phi \rho^s \Theta}{\partial t} + \frac{\partial \phi \rho^s u\_j^s \Theta}{\partial x\_j} \right] = (-p^s \delta\_{ij} + \mathfrak{r}\_{ij}^s) \frac{\partial u\_i^s}{\partial x\_j} - \frac{\partial q\_j}{\partial x\_j} - \gamma + f\_{\text{int}}.\tag{20}$$

where the granular temperature flux *qj* is modeled following Fourier's law of conduction:

$$q\_{\dot{j}} = -D\_{\Theta} \frac{\partial \Theta}{\partial x\_{\dot{j}}} \tag{21}$$

with *D*Θ the conductivity calculated as:

$$D\_{\Theta} = \rho^s d\sqrt{\Theta} \left[ \frac{2\phi^2 g\_{s0} (1+\varepsilon)}{\sqrt{\pi}} + \frac{9\sqrt{\pi} g\_{s0} (1+\varepsilon)^2 (2\varepsilon - 1) \phi^2}{2(49 - 33\varepsilon)} + \frac{5\sqrt{\pi} \phi}{2(49 - 33\varepsilon)} \right]. \tag{22}$$

The expression of the dissipation rate of granular temperature *γ* is modeled following Ding and Gidaspow (1990) [25]:

$$\gamma = 3(1 - \varepsilon^2) \phi^2 \rho^s g\_{s0} \Theta \left[ \frac{4}{d} \sqrt{\frac{\Theta}{\pi}} - \frac{\partial u\_j^s}{\partial x\_j} \right]. \tag{23}$$

Finally, the fluid particle interaction term *Jint* is expressed as:

$$J\_{int} = \phi K (2ak - 3\Theta),\tag{24}$$

where *α* characterizes the degree of correlation between particles and fluid velocity fluctuations following the expression *α* = *<sup>e</sup>*<sup>−</sup>*BSt*, where *B* is an empirical coefficient and *St* is the Stokes number defined as the ratio between the particle response time *tp* = *ρ<sup>s</sup>*/(<sup>1</sup> − *φ*)*K* and the characteristic time scale of the most energetic eddies *tl* = *k*/(<sup>6</sup>*ε*), with *ε* the dissipation rate of TKE.
