## 1 Introduction
In order to alleviate the high computational costs required to describe the real topography and the rheology, which both play a key role in natural flow dynamics, thin-layer (i.e., the thickness of the flow is assumed to be small compared to its down-slope extension) depth-averaged models are  
Figure 3: Velocity profiles \(U(Z)\) at different times until the granular mass stops, measured at \(X=90\) cm from the gate, in experiments of granular collapse over an initially static granular layer of thickness \(b^{0}=5\) mm on an inclined channel of slope angle (**a**) \(\theta=19^{\circ}\); (**b**) \(\theta=22^{\circ}\); and (**c**) \(\theta=24^{\circ}\). Time \(t=0\) s corresponds to the time when the front of the flowing layer reaches the position \(X=90\) cm. These new results have been extracted from the experiments performed by [12; 13] for granular columns of initial down-slope length \(r_{0}=20\) cm, initial thickness \(h_{0}=14\) cm, and width \(W=20\) cm (i.e., volume \(V=5600\) cm\({}^{3}\)). Figure 2: Position of the static/flowing interface \(b\) as a function of time \(t\) until the granular mass stops, measured at \(X=90\) cm from the gate, from experiments of granular collapse over an initially static granular layer of thickness \(b^{0}=5\) mm on an inclined channel of slope angle \(\theta=19^{\circ}\) (green squares), \(\theta=22^{\circ}\) (blue stars), \(\theta=23^{\circ}\) (red crosses), and \(\theta=24^{\circ}\) (black vertical crosses). Time \(t=0\) s corresponds to the time when the front of the flowing layer reaches the position \(X=90\) cm. The approximate upward velocity \(b\) of the static/flowing interface is indicated in m/s for each slope angle. These new results have been extracted from the experiments performed by [12; 13] for granular columns of initial down-slope length \(r_{0}=20\) cm, initial thickness \(h_{0}=14\) cm, and width \(W=20\) cm (i.e., volume \(V=5600\) cm\({}^{3}\)). Note that the position of the free surface when the mass stops, i.e., when \(b=h\), represented by the upper point for each angle, slightly depends on the slope angle and decreases as the slope angle increases, from about 0.025 m at \(\theta=19^{\circ}\) to about 0.018 m at \(\theta=24^{\circ}\). generally used to simulate landslides [14]. Such models have been rigorously derived for arbitrary topography, but the static/flowing transition has generally been neglected, e.g., [15; 16; 17; 18]. Several attempts have been made to describe this transition in thin-layer (i.e., shallow) models, in particular by deriving an equation for the static/flowing interface position or by establishing erosion/deposition rates, e.g., [19; 20]. However, these approaches are generally based on debatable phenomenological laws and/or are too schematic to be extended to natural flows [21]. The simplifications used in some previous models lead to an inconsistent energy Equation [22]. Indeed, some of these models prescribe a given velocity profile to deduce the entrainment rate (see e.g., [23] or references in [22]). However, in transient flows, the velocity profile changes with time as shown for example in the multilayer shallow model of [24]. A better understanding of the non-averaged case is necessary before defining more physically relevant depth-averaged models including the static/flowing transition. This is why we focus here on non-averaged thin-layer models. Recent work has shown that viscoplastic flow laws with yield stress describe well granular flows and deposits in different regimes, from steady uniform flows [25; 26; 27; 28] to transient granular collapse over rigid or erodible horizontal beds [29; 30; 31] and inclined beds [24; 32; 33] or accelerating/decelerating channel flows [34]. Very good agreement with experiments is obtained even though these laws, and in particular the so-called \(\mu(I)\) rheology, where \(I\) is the inertial number [28], have been shown to be ill-posed in the quasi-static regime, i.e., for small \(I\) (and also for large \(I\)) [35].