David Kibe Muchiri
email: david.muchiri@pg.canterbury.ac.nz

James N Hewett

Mathieu Sellier

Miguel Moyers Gonzalez

Jerome Monnier

Numerical simulations of dam-break flows of viscoplastic fluids via Shallow Water equations

Keywords: Shallow Water equations, viscoplastic, rheology, dam-break flows, sensitivity analysis, non-flat topography

This paper presents simulations of channelized dam-break flows of Herschel-Bulkley viscoplastic fluids over complex topographies using the Shallow Water equations (SWE). Particularly, the study aims at assessing the effects of rheological parameters: power-law index (n), consistency index (K) and yield stress (τ c ), on the flow height and velocity over different topographies. Three practical examples of dam-break flow cases are considered: a dam-break on an inclined flat surface, a dam-break over a non-flat topography, and a dam-break over a wet bed (a downstream containing a certain amount of fluid). Effects of the bed slope and depth ratios (ratio between upstream and downstream fluid levels), on the flow behaviour are also analyzed.

Numerical results are compared with experimental data from the literature, and are found to be in good agreement. Results show that for both dry and wet bed conditions, the fluid front position, the peak height, and mean velocity decrease when any of the three rheological parameters is increased. However, based on parametric sensitivity analysis, the power-law index appears to be the dominant factor in dictating the fluid behaviour. Moreover, increasing the bed slope and/or the depth ratio, the wave frontal position moves further downstream. Furthermore, the presence of an obstacle is observed to cause the formation of an upsurge that moves in the upstream direction; which increases by increasing any of the three rheological parameters. The study is useful for an in-depth understanding of the effects of rheology on catastrophic gravity-driven flows of non-Newtonian fluids (like lava or mud flows) for risk assessment and mitigation.

Introduction

Dam-break flows of non-Newtonian fluids such as lava, snow avalanches, debris, and mud flows are commonly observed in nature [START_REF] Huang | A Herschel-Bulkley model for mud flow down a slope[END_REF][START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Hogg | Slumps of viscoplastic fluids on slopes[END_REF][START_REF] Nj Balmforth | Yielding to stress: recent developments in viscoplastic fluid mechanics[END_REF][START_REF] Frigaard | Simple yield stress fluids[END_REF]. They are normally generated from the sudden collapsing of barricades or reservoirs holding fluids, which results in rapid flows downstream. Unfortunately, effects of these flows are often catastrophic. It is thus very important to understand the rheological behaviour and propagation characteristics of dam-break flows for hazard mitigation.

Previous studies have shown that the propagation of dam-break flows is influenced by various factors, among them, fluid rheological properties, bed slope, topographical variations and downstream fluid levels, see [START_REF] Griffiths | The dynamics of lava flows[END_REF][START_REF] Khoshkonesh | A comprehensive study on dam-break flow over dry and wet beds[END_REF][START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF][START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Ozmen | Dam-break flow in the presence of obstacle: experiment and CFD simulation[END_REF][START_REF] Chanson | Dam break wave of thixotropic fluid[END_REF][START_REF] Hulme | The interpretation of lava flow morphology[END_REF][START_REF] Moyers-Gonzalez | Non-isothermal thin-film flow of a viscoplastic material over topography: critical Bingham number for a partial slump[END_REF] and references therein. These factors play an important role in determining the maximum height and speed of flow downstream. However, for non-Newtonian fluids, in particular, viscoplastic fluids, the influence of these factors on dam-break flows is still an open field of research. Viscoplastaic flows are flows characterized by a yield stress (τ c ) that results in the formation of two regions within the flow: a plug (unyielded) region and a sheared (yielded) one [START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Nj Balmforth | Yielding to stress: recent developments in viscoplastic fluid mechanics[END_REF][START_REF] Bird | The rheology and flow of viscoplastic materials[END_REF][START_REF] Saramito | Progress in numerical simulation of yield stress fluid flows[END_REF]. The two regions are separated by an interface referred to as a yield surface. By deriving consistent thin-layer solutions, [START_REF] Balmforth | A consistent thin-layer theory for Bingham plastics[END_REF] demonstrated that the plug region is weakly sheared, hence the name pseudo-plug. This resolved the so-called plug paradox on the existence or non-existence of a true plug region [START_REF] Nj Balmforth | Yielding to stress: recent developments in viscoplastic fluid mechanics[END_REF]. Nevertheless, if the yield stress is below a certain threshold, viscoplastic flows become solids.

Propagation of dam-break flows for non-Newtonian fluids (in particular, viscoplastic) strongly depends on rheological properties such consistency index (K), yield stress (τ c ), and flow behaviour index (n) [START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Chanson | Dam break wave of thixotropic fluid[END_REF][START_REF] Nj Balmforth | Viscoplastic dam breaks and the Bostwick consistometer[END_REF][START_REF] Gp Matson | Two-dimensional dam break flows of Herschel-Bulkley fluids: the approach to the arrested state[END_REF].

The effects of these properties over a dry bed of a dam-break problem have been reported previously, see [START_REF] Minussi | Numerical experimental comparison of dam break flows with non-Newtonian fluids[END_REF][START_REF] Ancey | The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes[END_REF][START_REF] Cochard | Experimental investigation of the spreading of viscoplastic fluids on inclined planes[END_REF]. However, they did not consider the case of a wet bed and that of a non-flat topography.

Effects of the bed slope on dam-break flows over a dry channel have been addressed extensively, see for instance [START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF][START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Ancey | The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes[END_REF][START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF]. The fluid was observed to move further downstream when the slope is increased [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF]. This is different from the observation made recently in [START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF] for a wet bed case, where increasing the slope was reported to slow the front position. Nonetheless, more need to be done to investigate the impact of the slope on non-Newtonian flows over a wet bed. Further, downstream fluid depth has also been shown to affect the free-surface dynamics, see [START_REF] Khoshkonesh | A comprehensive study on dam-break flow over dry and wet beds[END_REF][START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Hunt | Asymptotic solution for dam-break on sloping channel[END_REF]. Experimental and numerical investigation on the effects of tailwater have been summarized in [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF] and references therein. However, this has not been reported exhaustively for the case of non-Newtonian fluids.

Bed variations due to the presence of an obstacle or an irregular topography downstream of a dam, is another factor that can influence flow dynamics. Apart from blocking the flow, an irregular topography can result in variation of the flow depth and mean velocity. The presence of an obstacle has been shown to cause the formation of a negative wave travelling in the upstream direction, see [START_REF] Ozmen | Dam-break flow in the presence of obstacle: experiment and CFD simulation[END_REF][START_REF] H Ozmen-Cagatay | Investigation of dam-break flood waves in a dry channel with a hump[END_REF]. Dam-break flows over non-flat topographies have been studied in [START_REF] Ozmen | Dam-break flow in the presence of obstacle: experiment and CFD simulation[END_REF][START_REF] Aureli | Dam-break flows: Acquisition of experimental data through an imaging technique and 2D numerical modeling[END_REF][START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF] for example, see also references therein. However, little attention has been given to investigate the impact of topographical variations on dam-break flows of non-Newtonian fluids.

Gravity-driven flows of non-Newtonian fluids are normally described by rheological models such as powerlaw, Bingham, and Herschel-Bulkley [START_REF] Huang | A Herschel-Bulkley model for mud flow down a slope[END_REF][START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Hogg | Slumps of viscoplastic fluids on slopes[END_REF][START_REF] Nj Balmforth | Viscoplastic dam breaks and the Bostwick consistometer[END_REF][START_REF] Bernabeu | Numerical modelling of non-Newtonian viscoplastic flows: Part II. viscoplastic fluids and general tridimensional topographies[END_REF][START_REF] Acary-Robert | A well-balanced finite volume-augmented lagrangian method for an integrated Herschel-Bulkley model[END_REF][START_REF] Nj Balmforth | Viscoplastic flow over an inclined surface[END_REF][START_REF] Chambon | Gravity-driven surges of a viscoplastic fluid: an experimental study[END_REF]. The Herschel-Bulkley model is commonly used because of its ability to describe many complex fluid behaviors in a non-linear and history independent manner.

In particular, dam-break flows of viscoplastic fluids can be modeled by the full Navier-Stokes equations, the Herschel-Bulkley constitutive law, and the transport equation of the free-surface dynamics. However, solving these equations simultaneously is computationally expensive and time consuming. To overcome such computational difficulties, reduced models such as the lubrication model and the Shallow Water Equations (SWE) are usually employed, see e.g. [START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Balmforth | A consistent thin-layer theory for Bingham plastics[END_REF][START_REF] Ed Fernández-Nieto | Shallow water equations for non-Newtonian fluids[END_REF][START_REF] Boutounet | Multi-regime shallow free surface laminar flow models for quasi-Newtonian fluids[END_REF][START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF][START_REF] Denisenko | A consistent three-equation shallow-flow model for Bingham fluids[END_REF].

Shallow Water (SW) type models are derived by depth integration assuming that the depth is much smaller than the characteristic length (the long-wave assumption). Only a few SWE models for Herschel-Bulkley fluids have been derived, see [START_REF] Ancey | Plasticity and geophysical flows: a review[END_REF][START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF][START_REF] Chambon | Quelques contributions à la modélisation des écoulements à surface libre de fluides complexes[END_REF] and references therein. The present study conducts a parametric analysis of dam-break flows whose numerical computations rely on the SW flow model derived (and meticulously evaluated) in [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF]. This model enables us to take into account varying conditions at the bottom and wet dry front dynamics on flat/non-flat topographies. Note that during the early stages of dam-break problems, the long-wave assumption for asymptotic analysis of shallow flows is often not satisfied. For that reason, analysis of the flow patterns at early times of dam-break, resulting from the sudden opening of the dam gate are beyond the scope of this work. In addition, it has been shown that flows of viscoplastic fluids on an incline can stop in finite time when the gravitational forces become equal to the yielding conditions, see [START_REF] Moyers-Gonzalez | Non-isothermal thin-film flow of a viscoplastic material over topography: critical Bingham number for a partial slump[END_REF][START_REF] Gp Matson | Two-dimensional dam break flows of Herschel-Bulkley fluids: the approach to the arrested state[END_REF] and references therein. The stopping behaviour leading to the arrested (stationary) state is also beyond the scope of this work. This paper focuses on the effects of rheological parameters, bed slope, downstream fluid levels, and obstacles on the velocity field and free-surface dynamics, particularly for the dam-break flows of viscopastic fluids like lava and mud flows. Numerical simulations are carried out over a dry, wet, and non-flat bed. In each case, computed results are firstly compared with experiments for validation.

The outline of the paper is as follows. In Section 2, we present the governing equations and the numerical methods. Section 3 presents numerical results on dam-break flows for the three different cases considered. In Case 1 (Subsection 3.1), dam-break flows down a dry inclined channel are analyzed. The effects of rheological parameters are discussed here in detail. In Case 2 (Subsection 3.2), we discuss the numerical results of dambreak flows over a wet channel. The influence of the bed slope and fluid depth ratio is analyzed herein.

Numerical simulations over a dry non-flat topography are discussed in Case 3 (Subsection 3.3). Conclusions are drawn in Section 4.

Mathematical formulation

We consider a two-dimensional incompressible flow of a viscoplastic fluid (like lava) down an inclined plane as shown in Fig. 1, with x being the axis of the slope at an angle θ and z, the axis normal to the slope. The flow is driven by gravity g = (gsinθ, -gcosθ), and described by its velocity u = (u, w) assuming a hydrostatic pressure field p. The fluid density is denoted by ρ, and the time-dependent fluid depth denoted by h(t, x, z) = H(t, x, z) -b(x, z) where H(t, x, z) is the fluid elevation and b(x, z) the basal topography elevation. The flow is assumed to be independent of the spanwise direction, although a 3D formulation is presented in our previous work [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF], which is followed herein. Conservation of mass:

∂ x u + ∂ z w = 0 (2.1)
Conservation of momentum:

ρ(∂ t u + u∂ x u + w∂ z u) = -∂ x p + ρg sin θ + ∂ x τ xx + ∂ z τ xz , (2.2) ρ(∂ t w + u∂ x w + w∂ z w) = -∂ z p -ρg cos θ + ∂ x τ xz + ∂ z τ zz . (2.
3)

The Herschel-Bulkley rheology law is given by

   τ xz = τ c + K (∂ z u) n if τ xz > τ c ∂ z u = 0 if τ xz ≤ τ c (2.4)
where τ xz is the shear stress, n > 0 the power-law index, K > 0 the consistency index, and τ c the yield stress.

The Herschel-Bulkley model provides a good mathematical law from which other fluid models can be obtained. For instance, when n = 1, the law reduces to the Bingham model where the consistency index K becomes the plastic viscosity η. When n ̸ = 1 and τ c = 0, we have a power-law fluid model. When n = 1 and τ c = 0, the law reduces to a Newtonian fluid model. The governing equations (2.1) -(2.4), are closed by defining appropriate boundary conditions for the free-surface and the basal topography. For the free-surface at z = H, we use the kinematic condition:

∂ t h + u∂ x H = w (2.5)
and the no stress condition:

τ = • n = 0 where the unit normal n = 1 √ 1+(∂xH) 2 -∂xH 1 . At the bottom surface z = b, a non-slip condition is used: u = w = 0.
The present study relies on the 1D version of the 2D SW model derived in [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF]. The derivation is done by depth integration of the above equations applying the long-wave assumption, see Muchiri at al. [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF] for the formal asymptotic derivation. The complete model of the SWE in one-dimension reads

∂ t h + ∂ x q = 0, ∂ t q + ∂ x q 2 h + 1 2 gh 2 cosθ = ghcosθ (tanθ -∂ x b) - 1 ρ τ b , (2.6)
with the bottom shear stress approximated as

τ b = K K ρg|S θ | 1-n n τ c ρgS θ + h c q D (2.7)
where S θ can be taken as S θ = sinθ, a zeroth-order approximation, or as S θ = sinθ -cosθ∂ x H, an improved approximation with a corrective slope term [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF]. It is worth noting that the difference between the two terms only appears in areas of sharp changes of the fluid local slopes as reported in [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF]. However, the former is not valid for θ = 0, hence not applicable for Case 2 and 3, as seen in the next section. The denominator

D is given by D = h n+1 n c n n+1 h - n 2 (n+1)(2n+1) h c , where h c (x, t) = max (0, h(x, t) -h p (x, t))
is the thickness of the sheared zone, and h p (x, t) = τc ρgS θ the plug thickness, see Fig. 1 (b). The flow rate q is given by

q = hū = H b
udz where ū is the mean velocity. We note that the basal shear stress approximation (2.7) is of zeroth order, therefore, the so-called pseudo-plug concept is not addressed herein. We also note that this approximation is a development of a similar expression presented in [START_REF] Chambon | Quelques contributions à la modélisation des écoulements à surface libre de fluides complexes[END_REF].

We also recall the Froude number, Fr = ū/ √ gh cos θ and the Reynolds number, Re = ρū 2-n h n /K from [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF], which are calculated herein from the local values, as shown in the next section. The Froude number indicates whether the flow is subcritical (Fr < 1), critical (Fr = 1), or supercritical (Fr > 1), while the Reynolds number indicates whether the flow regime is laminar (Re < 500), turbulent (Re > 2000), or in the transitional range 500 <Re < 2000 [START_REF] Hanif | Open-channel flow[END_REF].

The flow model (2.6) -(2.7) is solved by COMSOL multiphysics software version 6.0 using the "SWE interface" which employs the Finite Element method (a first order Discontinuous Galerkin scheme in space and Runge-Kutta in time) for discretization of the equations. Each term is matched with the corresponding term in the interface, with the basal shear stress implemented as the domain force. For the numerical computation, we use a uniformly spacedmesh of 2500 elements and a time-step ∆t = 0.05 s.

Results and discussion

The model is applied to three dam-break cases discussed in the following sections. In each case, the model is firstly compared with experiments for validation, and to test its applicability and reliability in simulating real flows over different geometries. Other simulation results mimicking viscoplastic lava flows are presented and analyzed thereafter. Effects of rheology on the flow depth, velocity, and free surface profile are investigated and detailed in each case.

Case 1: Dam-break flows on an inclined-flat topography

For this case, we study the influence of rheological parameters on the flow behaviour down an inclined flume. We use data of a dam-break experiment investigated in [START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF][START_REF] Ancey | The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes[END_REF]. It involves the sudden release of fixed volumes of a viscoplastic fluid down a channel inclined at some angle θ, as shown in Fig. 2. The fluid is initially locked in a reservoir set at the top of the channel before it is released suddenly by opening the lock gate to flow freely driven by gravity, on a dry flat surface. Following [START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF], we suppose that the reservoir is of length l = 0.51 m, while the channel is of length L = 6.0 m, and the inclination angle is θ = 12 o . We impose some initial fluid height defined linearly by h = h g + (x -l)tan(θ) where h g = 0.34 m is the height at the gate and wall boundary conditions on all sides. The material used for the simulations is 0.3% concentration of Carbopol ultrez 10 as indicated in Table 1 unless stated otherwise, with density ρ = 1000 kg m -3 . Data for simulations and comparison is extracted from [START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF][START_REF] Ancey | The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes[END_REF] using WebPlotDigitizer [START_REF] Rohatgi | Webplotdigitizer user manual version 3[END_REF], and figure resolution constraints correspond to an error margin of less than 5%.

The results in Fig. 3 show a good agreement between the experiments and numerical simulations. Slight deviations could be attributed to the experimental errors reported in the literature [START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF]. For instance, besides the dam-wall inertia and vibrations in the experimental facility, the entire fluid was not released instantly as the gate did not open momentarily, although the requirement for an instant dam-break is satisfied, i.e., t ≤ 2△h/g, where △h is the initial difference between the average upstream and downstream fluid levels, see [START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF] and references therein. On the other hand, for numerical simulations we assumed that on opening the gate the entire fluid column was released momentarily. Therefore, some deviations in the free surface dynamics are expected, especially at early times.

For the following simulations, the rheological properties of the fluid used are given by Eq. the power-law index is observed to reduce the Froude number. As mentioned earlier, increasing the power-law index increases the basal friction term, which in this case reduces the inertial forces over gravitational forces.

Effects of yield stress

The spreading of the fluid is observed to decrease with increasing the yield stress as shown in Fig. 9.

In particular, the free-surface front position, the mean velocity and flow rate tend to reduce with time as the yield stress increases, see Fig. The Froude numbers are also observed to decrease with time as the yield stress increases, see Fig. 10.

For smaller τ c -values, the Froude numbers are above unity, hence the flow is supercritical. However, this transcends with time to subcritical. For larger τ c -values, the flow regime is entirely subcritical. . It can also be observed that the plug thickness at the front position increases with time. This is because the stress at the front position is decreasing as the flow slows down, which indicates that a greater proportion becomes unyielded, see [START_REF] Moyers-Gonzalez | Non-isothermal thin-film flow of a viscoplastic material over topography: critical Bingham number for a partial slump[END_REF].

Effects of consistency index

The front position of the free-surface, the mean velocity, and the discharge are observed to increase with the decrease in consistency index, as shown in Fig. 12 In other words, the power-law and Herschel-Bulkley fluids appear to form a group, and Newtonian and Bingham fluids another group. This is mainly due to the flow-behaviour index which is n = 0.6, for both power-law and Herschel-Bulkley fluids, and n = 1 for both Newtonian and Bingham fluids. The difference between the two groups increases as the power-law index for power-law and Herschel-Bulkley fluids reduces, see also Fig. 6. In addition, it can be observed that the Herschel-Bulkley fluid tend to lag behind the 11 power-law fluid, and the Bingham fluid behind the Newtonian fluid. This due to the effect of yield stress which is absent in power-law and Newtonian fluids. Conclusively, from the analysis of the results discussed so far, two main types of hydrographs for dambreak flows arising from the influence of rheological parameters can be defined as shown in Fig. 16: i. Type I: for fast flows with a rapid peak; corresponding to the smaller values of rheological parameters ii. Type II: for relatively slow flows with a gradual peak; corresponding to the larger values of rheological parameters In Type I, the flow thickness increases rapidly to the peak within a very short time after which it starts decreasing rapidly with time. In Type II, the flow reaches the maximum elevation gradually over time, before decreasing slowly compared to Type I, see Fig. 16 (a). Similarly, the mean velocity for type I shoots within a short period of time before it decreases rapidly with time. This is opposite for Type II, in which the mean velocity grows gradually to the peak before it starts reducing, see Fig. 16 (b). The discharge hydrograph in Fig. 16 (c) is a reflection of the mean velocity and the height, showing similar trends: a rapid and gradual increase of the flow rate to the peak with time for Type I and II, respectively.

Another difference between the two types is on the time of arrival of the wave at the hydrograph position.

Type I arrives earlier than Type II as seen in the three state variables in Fig. 16. Smaller values of n, K, However, very near the dam gate where waves are rapid due to the sudden opening of the gate, Type I hydrographs are likely to be observed even for some largers values of rheological parameters. Similarly, far from the gate where waves are much slower, Type II hydrographs are likely to form even for the lower values of rheological parameters. Similar observations are reported in [START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF] for Newtonian flows. It is worth noting that there could be other factors that can influence the type of hydrograph including the bed slope and the volume of the released fluid.

Case 2: Dam-break flows over an inclined wet bed

In this section, we investigate the effect of rheology on the flow dynamics for a dam-break configuration over a wet bed. Numerical results are compared with experimental data from [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF]. An experimental setup described in [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF] is considered for validation purposes, where the flume is 18 m in length, 1 m in width, and with a bed slope of 0.02, see Fig. 17. A reservoir gate of thickness 1.5 cm, is located 8.37 m from the upstream wall to separate the upstream and the downstream. We assume a fixed wall at the downstream end. For the upstream, an initial fluid height of 0.2 m is used. For the downstream, three initial fluid depths As seen in Fig. 18, there is a good agreement between the experiments and simulations. It is observed that the fluid travels a longer distance on a dry surface (α = 0), as compared to cases of a wet one within the same duration. Moreover, for a wet bed, the progression of the front position is observed to decrease when the tail-fluid is increased as seen for α = 0.2 and α = 0.4 in Fig. 18 (b) and (c), respectively. As the depth ratio increases the resistance offered by the downstream fluid increases, which slows the fluid propagation, making the wave front position tend to decrease. The front position in this case is defined as the location where the free surface reaches the downstream asymptotic value.

The jet-like rising of the fluid surface at early dam-break stages reported in the literature [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF][START_REF] Turhan | Experimental and numerical investigation of shock wave propagation due to dam-break over a wet channel[END_REF], see Fig. 17 (b), is also observed numerically, though not well predicted (not shown here), as the flow from the reservoir interacts with the downstream static-fluid. The rising height for α = 0.2 is observed to be higher than that for α = 0.4, which is in agreement with experimental results.

To consider a real-like viscoplastic flow e.g., a less viscous lava [START_REF] Griffiths | The dynamics of lava flows[END_REF][START_REF] Hulme | The interpretation of lava flow morphology[END_REF][START_REF] Blatt | Petrology: igneous, sedimentary, and metamorphic[END_REF][START_REF] Mo Chevrel | Measuring the viscosity of lava in the field: a review[END_REF], a Herschel-Bulkley case is considered with the rheological parameters given by Eq. For K = 0.01 Pa s, the flows are observed to be supercritical and turbulent, at least for α = 0 and α = 0.2, see Fig. 19 (a) and (b). For α = 0.4, the Froude numbers are below unity, hence subcritical but still turbulent as seen in Fig. 19 (b). For K = 10 Pa s, the flow is supercritical for α = 0 and subcritical for α = 0.2 and α = 0.4, but all are in the laminar regime, see Fig. 19 (c) and (d), respectively. It is evident that the Froude and Reynolds numbers decrease with the increase in consistency index, as had been seen in the previous subsection.

Moreover, for the two cases, the peak values of the two numbers are observed to decrease with the increase in depth ratio. This is due to the decrease in velocity as the depth ratio increases. However, the time where the peak values are first registered is observed to decrease with the increase in depth ratio for K = 0.01 Pa s (in agreement with experiments in Fig. 18), and to increase with the increase in depth ratio for K = 10 Pa s. Increasing the viscosity (consistency index) decreases the interaction between the upstream and the downstream fluids, which lowers the impact force (due to the upstream fluid) at the interface between the two fluids, thus reducing the viscous dissipation rate. This makes the upstream fluid layer tend to speed up as the depth ratio increases for K = 10 Pa s. For all the three depth ratios, the power-law fluids are observed to spread furthest, followed by Herschel-Bulkley, Newtonian and Bingham fluids in that order, under the same conditions. This is largely due to the influence of the power-law index which is higher for Newtonian and Bingham than for power-law and Herschel-Bulkley fluids. Increasing the power-law index increases the friction term and hence slows down the fluid motion, see Eq. 2.7. Moreover, the peak thicknesses of Herschel-Bulkley and power-law fluids are observed to be higher than those of Newtonian and Bingham fluids in that order, as seen in the stage hydrographs in Fig. 20 (g) -(i). The peak height of Herschel-Bulkley is higher than that of power-law (and that of Bingham higher than that of Newtonian) fluids, due to the effect of the yield stress which is absent in power-law (and Newtonian) fluids. Moreover, the free-surface elevation increases with increasing the depth ratio, as the volume of the fluid increases. Moreover, the mean velocity of power-law fluids is observed to be higher than that of Herschel-Bulkley, Newtonian and Bingham fluids in that order, see Fig. 20 (d) -(f). Again this is due to the influence of the flow-behaviour index as explained earlier. However, the peak velocity is observed to reduce with increasing the depth ratio, for the four fluid types. Increasing the downstream fluid reduces the thickness of the moving layer which implies a decrease in the gradient of the fluid height which plays a big part in the fluid evolution.

The shape of the velocity profile at the front position for α = 0, indicates the effects of the bottom friction which appear low for other ratios. For non-zero depth ratios, the wave front positions tend to speed up over a low-friction surface, with the spreading increasing with the depth ratio.

(a) (b) (c) (d) (e) (f) (g) (h) (i)
As seen previously, it is observed that power-law and Herschel-Bulkley fluids appear to have a similar flow response which is different to that of Newtonian and Bingham fluids, see Fig. 20. Both power-law and Herschel-Bulkley fluids attain their maximum heights within a short time unlike Newtonian and Bingham fluids for which the heights tend to grow gradually with time before reaching the peak. The arrival time of the maximum height at the hydrograph position is also different for different fluid types. For the power-law and Herschel-Bulkley, the arrival time is much earlier than that of Newtonian and Bingham fluids. The difference between the two groups of fluids is observed to increase or decrease as the power-law index approaches zero or unit, respectively, see also Fig. 22. This indicates that the power-law index is the dominant factor in dictating the fluid response. This is contrary to the observations made previously in Fig. 18 (and in the literature [START_REF] Liu | Experimental investigation on the effects of bed slope and tailwater on dam-break flows[END_REF][START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF]) for Newtonian flows with a viscosity of water, K ≈ 0.001 Pa s, where the increase of the bed slope increases the fluid depth in the downstream which increases the blockage effects (resistance to motion), hence more energy dissipation, making the front position progression tend to decrease.

Effects of bed slope

Effects of rheological parameters

Effects of the three rheological parameters (K, n and τ c ) for α = 0.2 are shown in Fig. 22. It can be observed that the free-surface front position, maximum height, and mean velocity increases with the decrease in power-law index, see Fig. 22 (a) -(c). As explained earlier, reducing the power-law index reduces the apparent viscosity and the friction term hence increasing the fluid flow dynamics.

Decreasing the consistency index is also observed to increase the front position, the maximum height and the mean velocity, see Fig. Similarly, the front position and the mean velocity are observed to increase by reducing the yield stress as seen in Fig. 22 (g) and (i). Increasing the yield stress increases the basal friction term which slows down the fluid propagation. However, the fluid thickness increases with increasing the yield stress. We also note that the maximum height is reached gradually or rapidly by increasing or decreasing any of the three parameters, respectively.

Case 3: Dam-break flows over a non-flat topography

In this section, we study the effects of rheological parameters on the flow behaviour over an obstacle. We consider set-up of a dam-break flow over a bump on a horizontal channel as presented in [START_REF] Ozmen | Dam-break flow in the presence of obstacle: experiment and CFD simulation[END_REF] The upsurge is observed to grow in height towards a maximum height around t = 5 s. After attaining the peak height, it starts moving backward on the free-surface as its height reduces. This forms the first negative wave in the upstream direction. On reaching the upstream wall around t = 11 s, this wave is reflected in the downstream direction forming a second positive wave, see Fig. 25 (b). This wave is seen to propagate as it reduces in size. Due to the interference of the obstacle, a second upsurge is observed to start forming around t = 17.5 s, which develops into a second negative wave that moves again in the upstream direction, see Fig. 25 (c). At the same time, the second positive wave continues moving past the obstacle in the downstream direction on the free-surface of the first wave.

This phenomenon is observed to repeat itself forming a series of positive and negative waves behind the obstacle until no more fluid can flow past the obstacle. Correspondingly, a series of positive waves (propagating in the positive direction) that build upon each other are also observed past the obstacle, see Fig. 25 (c). This increases the fluid elevation with time on the downstream side until no more fluid is able to be transferred from the upstream, limited by the height of the obstacle. The amplitudes (heights) of the waves, measured from the free-surface, reduce one after the other as the fluid evolves. This is as the total fluid elevation reduces in the upstream direction of the obstacle and increasing downstream of the obstacle.

We also note some shocks right behind the obstacle, due to its blockage.

The mean velocity on the other hand, is observed to decrease to negative with time as the surge moves backward, as shown in Fig. 25 (d). For the reflected positive waves, the mean velocity is observed to increase towards the obstacle and decrease past the obstacle, see Fig. 25 (e) and 25 (f), respectively. Moreover, the mean velocity decreases for each wave after the other. Negative waves behind the obstacle are observed to be slower than the corresponding positives waves. This is in agreement with the theoretical conclusion drawn in [START_REF] Castro-Orgaz | Ritter's dry-bed dam-break flows: Positive and negative wave dynamics[END_REF] and references therein, where the front propagating speed of positive waves was approximated to be twice that of negative waves. Additionally, A bore shape (like a downfall) on the velocity profile, opposite of the upsurge on the elevation profile, is observed around the obstacle, which indicates a decrease in velocity around the obstacle, due to the obstacle. The mean velocity on the other hand, is highest at Point C and lowest at Point B, for the first wave, highest at Point B and lowest at Point A, for all other subsequent waves. The fluid layer downstream of the obstacle will be very thin and therefore to maintain the flow rate, the velocity will be relatively higher.

The first wave creates an overflow which leads to a high velocity at Point C downstream of the obstacle.

Subsequent waves do not overflow as much as the first wave, since they are the reflected waves which travel back and forth through Point A and C, respectively, as their velocity reduces in time due to energy loss.

For all the subsequent waves, the maximum velocities are registered at Point B. This is due to the fact that the fluid depth is lowest at this point, which means the averaged velocity will need to be highest in order to maintain the flow rate.

The Froude numbers for the first wave are above unity at the three points, hence supercritical, see Fig. Increasing any of the three rheological parameters is observed to increase the size of the surge, see Fig. • The front position of the free-surface is observed to advance faster with the decrease of any of the three rheological parameters (K, τ c and n), for both dry and wet bed conditions. This is due to the decrease of the friction term that speeds up the fluid evolution when any of these parameters is decreased. The maximum height and velocity at each position are also observed to increase with the decrease of any of these parameters. The thickness at each position, however, increases gradually for the larger values of these parameters, and rapidly within a short period of time for the smaller values. Consequently, two main types of hydrographs are defined: Type I characterizing rapid flows with an overshoot and Type II representing gradual flows. The local Froude numbers for Type I are relatively larger than for Type II. Also, the arrival time of the peak height and velocity is much earlier for Type I than Type II.

• Shear-thickening fluids are observed to be slower than shear-thinning fluids. Under the same conditions (and Re < 1000), power-law fluids appear to spread faster than Herschel-Bulkley, Newtonian, and Bingham fluids in that order. This is mainly due to the effect of the flow-behaviour index; which when reduced, the flow rate increases. In addition, the power-law and Herschel-Bulkley fluids appear to form a group with a similar flow response, as Newtonian and Bingham fluids form another group with a similar flow response. The difference between the two groups of fluids observed to increase or decrease as the power-law index approaches zero or unit, respectively. The power-law index therefore, appears to be the dominant factor in dictating the fluid response.

• Increasing the bed slope or the depth ratio, the wave frontal position moves faster downstream. This is due to the decrease of the rate of viscous dissipation as the downstream fluid level increases. Due to high viscosity, increasing the downstream fluid depth reduces the local Reynolds numbers and hence the interaction between the upstream and the downstream fluids, thus reducing the viscous dissipation effects. This is contrary to the observations made in the literature (also observed in the current results) for for high Reynolds number flows (in particular, Newtonian flows with a viscosity of water, K ≈ 0.001Pa s), where the increase of the fluid depth in the downstream area increases the blockage effects (resistance to motion) [START_REF] Zhang | Experimental study of the dam-break waves in triangular channels with a sloped wet bed[END_REF], and hence more energy dissipation, making the front positions tend to decrease.

• The presence of an obstacle is observed to impact the fluid flow speed and height due to its blockage.

It causes the formation of an upsurge that develops into a series of negative and positive waves which propagate upstream and downstream of the obstacle, respectively. Interestingly, increasing any of the rheological parameters is observed to increase the development of the surge upstream of the obstacle

Figure 1 :

 1 Figure 1: Sketch showing (a) the flow geometry with a non-flat topography, and (b) the plug and the sheared zone in the velocity profile.

Figure 2 :

 2 Figure 2: Side-view sketch of a dam-break problem on an inclined channel.

Figure 3 :

 3 Figure 3: Comparing front positions (x f ) of experiments [8, 20] and numerical simulations with time: (a) and (b) using 0.3% concentration of Carbopol on a slope of θ = 12 o and θ = 18 o , respectively, and (c) using 0.25% concentration of Carbopol on a slope of θ = 12 o .

Figure 4 :

 4 Figure 4: Time evolution of the (a) fluid height, (b) mean velocity, and (c) flow rate.

Fig. 4

 4 Fig.4shows the evolution of the state variables (h,ū,q) at various time instants. As one would expect, the height is observed to reduce in time as the fluid spreads, see Fig.4 (a). As seen in Fig.4(b) and (c), the mean velocity and discharge are maximum at early times of dam-break, due to high gradients in fluid height which reduces in time as the fluid spreads and the available hydrostatic head decreases. Consequently, due to high resistance ( e.g., the bottom surface friction) which opposes the x-component of the gravity force, the fluid is observed to decelerate with time.

Figure 5 :

 5 Figure 5: Variation of (a) Froude and (b) Reynolds numbers over distance at various time instants.

Figure 6 :

 6 Figure 6: Effects of flow-behaviour index: (a) evolution of the fluid height at t = 1 s, the corresponding (b) mean velocity and (c) flow rate at t = 1 s (broken line) and t = 2 s (solid line), and hydrographs at x = 1.1 m for the (d) fluid height, (e) mean velocity, and (f) flow rate, respectively.

Figure 7 :

 7 Figure 7: Variation of Froude numbers over distance for various values of n at (a) t = 1 s and (b) t = 2 s, respectively.

Figure 8 :Fig. 8

 88 Figure 8: Stage and Froude number hydrographs at different positions for n = 0.4, n = 0.8, and n = 1.2, respectively.

  9 (a) -(c). Increasing the yield stress increases the friction term which reduces the fluid advancement. Furthermore, the plug thickness (between the yield surface and free-surface) is observed to increase with the yield stress as the yield zone decreases, see Fig.9(a). As expected, the sheared zone reduces when the yield stress is high, see 9(a), also Fig.1 (b).

Figure 9 :Figure 10 :

 910 Figure 9: Yield stress effects: (a) time evolution of the fluid depth and yield surface (broken line) at t = 2 s, the corresponding (b) mean velocity, (c) flow rate at t = 1 s (broken line) and t = 2 s (solid line), respectively, and the respective hydrograph profiles (d) -(f) at x = 1.1 m.

Figure 11 :

 11 Figure 11: Yield stress effects: time evolution of the fluid height (solid line) and the corresponding yield surface (broken line) for τ c = 200 Pa.

Fig. 11

 11 Fig. 11 shows the fluid height profiles and their corresponding yield surfaces for a gravity induced yield stress (τ c ≃ 200 Pa), calculated from the relation τ c = ρgh p sinθ. It can be observed that as the fluid propagation increases with time, the spreading rate is reducing, due to the retardation associated with the basal friction. It can be noticed that the plug thickness for τ c = 200 Pa is much higher than for smaller τ c -values seen in Fig.9(a), as one would expect, see also Fig.1 (b). It can also be observed that the plug thickness at the front position increases with time. This is because the stress at the front position is

Figure 12 :

 12 Figure 12: Effects of consistency index: (a) the fluid depth evolution at t = 2 s, (b) the corresponding mean velocity and (c) flow rate at t = 1 s (broken line) and t = 2 s (solid line), respectively, (d) -(f) stage, velocity and discharge hydrographs, respectively, at x = 1.1 m.

Figure 13 : 8 .

 138 Figure 13: Flow response of different fluid types: (a) fluid depth profiles at t = 2 s, (b) -(c) the corresponding mean velocity and flow rate profiles at t = 1 s (broken line) and t = 2 s (solid line), respectively, (d) -(f) hydrographs at x = 1.1 m for the fluid height, mean velocity and flow rate, respectively.

Figure 14 :

 14 Figure 14: Variation of Froude numbers over distance for the four fluid types at (a) t = 1 s and (b) t = 2 s, respectively.

Figure 15 :

 15 Figure 15: Effects of the rheological parameters ( n, K, τ c ) on the fluid response: front positions (x f ) at t = 2 s.

Figure 16 :

 16 Figure 16: Types of hydrographs for the (a) surface elevation, (b) mean velocity and (c) discharge. Type I and II characterizes rapid flows and slow flows, respectively.

Figure 17 :Figure 18 :

 1718 Figure 17: Dam-break flow over a wet bed: (a) a sketch of the problem, (b) a screenshot of the experimental results at t = 0.4 s for a Newtonian fluid [9].

( 3 . 3 )

 33 and increasing the slope angle to θ = 6 o , unless stated otherwise. (K, n, τ c ) = (10 Pa s n , 0.6, 10 Pa) (3.3) Firstly, we determine the flow regime for K = 10 Pa s (Newtonian) with respect to the experimental one K = 0.01 Pa s, by plotting the Froude and Reynolds number hydrographs for various depth ratios, as shown in Fig. 19.

Figure 19 :

 19 Figure 19: Froude and Reynolds number hydrographs at x = 10 m for Newtonian flows with (a) and (b) K = 0.01 Pa s, and (c) and (d) K = 10 Pa s for the three depth ratios, respectively.

Figure 20 :

 20 Figure 20: Flow response of different fluid types: comparison of free-surface profiles (a) -(c), mean velocity (d) -(f) at t = 5 s, and stage hydrographs at x = 10 m (g) -(i), for α = 0, α = 0.2, and α = 0.4, respectively.

Figure 21 showsFigure 21 :

 2121 Figure21shows the effects of bed slopes on the fluid evolution for different depth ratios. As shown in both dry and wet bed cases in Fig21 (a) -(c), the wave front position moves further downstream with increasing bed slope. Increasing the slope increases the x-component of acceleration due to gravity. For a wet bed, the front position tends to increase with the increase of the slope and/or the depth ratio. Increasing the slope or the depth ratio, increases the fluid in the downstream which lowers the gradient of the upstream fluid, thereby minimizing the interaction between the upstream and the downstream fluids. Consequently, the spreading top-layer tend to move faster with the increase of bed slope and depth ratio. Moreover, the

  22 (d) -(f). Reducing the consistency index implies a decrease in viscosity which increases the fluid propagation. High consistency index increases the resistance to fluid motion causing a decrease in the fluid evolution.

  , and shown in Fig 23. The channel is of length 8.9 m, width 0.3 m and height 0.34 m. The reservoir located 4.65 m from the channel entrance is initially filled with water of height 0.25 m. A trapezoidal shaped obstacle of height

Figure 22 :

 22 Figure 22: Effects of parameters n, K and τ c for α = 0.2: (left) free-surface profiles at t = 5 s, (center) stage hydrographs, and (right) velocity hydrographs at x = 10 m.

Figure 23 :

 23 Figure 23: Sketch of a dam-break problem on a dry surface with a bump (not to scale).

  24 (b) and (c). The surge formation is due to the reflection of the flow wave against the obstacle[START_REF] Ozmen | Dam-break flow in the presence of obstacle: experiment and CFD simulation[END_REF]. There is a good agreement between the experimental results and numerical results of the corresponding Newtonian

Figure 24 :Figure 25 :

 2425 Figure 24: Comparison between experimental data [10] and simulations using a Newtonian fluid (K = 0.001 Pa s, n = 1, τ c = 0 Pa).

Fig. 25

 25 Fig. 25 shows the development of the surge with time into negative and positive waves. The presence of an obstacle causes the formation of an upsurge in the upstream direction of the obstacle, see Fig. 25 (a).

20

 20

Figure 26 :

 26 Figure 26: Hydrographs for the (a) surface elevation and corresponding (b) mean velocity, (c) Froude number, and (d) Reynolds number at Point A, B, and C.

Fig. 26

 26 Fig. 26 shows hydrographs for Point A, B, and C, defined in Fig 23. A series of waves passing through

  26 (c). This however, transcends to subcritical with time for other waves. Moreover, the peak values are larger at Point B compared to A and C, which indicates that the effects of inertia compared to gravity are larger at Point B than at the other two points. The Reynolds numbers, on the other hand, indicate that the flow is laminar, as seen in Fig.26 (d). These numbers decrease with time, which means that the flow speed is reducing as expected.3.3.2 Effects of rheological parameters

Figure 27 :

 27 Figure 27: Effects of rheological parameters on the development of the surge at t = 2 s (broken line) and t = 5 s (solid line).

27 . 4 Conclusion

 274 Interestingly, as the front position (and the speed) of the first wave decreases due to the increase of the rheological parameters (as seen in the previous section), the development of the surge behind the obstacle is observed to increase. Increasing any of the rheological parameters is observed to reduce the advancement (propagating speed) of the front position of the positive waves but increase the propagation of the corresponding negative waves. This is the conservation of wave energy phenomenon between the negative and positive waves. Further analysis of the effects of the rheological parameters on the flow height and speed at Points A, B, and C of the obstacle, see Fig.23, is presented in Appendix A.Numerical simulations of channelized dam-break flows for Herschel-Bulkley fluids using SWE over complex topographies are presented. Effects of rheological properties, bed slopes, and fluid depth ratios on the progression speed and height over a dry and a wet bed are analyzed. A dam-break flow over a non-flat topography is also discussed. Computed results are compared with experimental results from the literature and are in good agreement. The main conclusions are as follows:

Figure 29 :

 29 Figure 29: Effects of power-law index: hydrographs for the (a) -(c) fluid elevations, and (d) -(f) corresponding mean velocities at Point A, B, and C, respectively.

Figure 30 :

 30 Figure 30: Effects of yield stress: hydrographs for the (a) -(c) fluid elevations and (d) -(f) corresponding averaged velocities at Point A, B, and C, respectively.

Table 1 :

 1 Rheological details of Carbopol ultrez 10 used, data obtained from[START_REF] Cochard | Measurements of time-dependent free-surface viscoplastic flows down steep slopes[END_REF][START_REF] Ancey | The dam-break problem for Herschel-Bulkley viscoplastic fluids down steep flumes[END_REF].

	Concentration K (Pa s n )	n	τ c (Pa )
	0.25%	32.1	0.39	78
	0.3%	47.7	0.42	89

Funding

The Royal Society of New Zealand through the Marsden grant UOC1802 is acknowledged for providing funds to aid this research work towards the completion of the first author's PhD studies.

Availability of data and materials

Any data or material presented in this work is available from the corresponding author on request

but decrease the progression of the front position downstream of the obstacle. This is due to the conservation of momentum and energy between the negative and positive waves.

The present numerical investigations of model sensitivities, based on the SW model derived in [START_REF] Dk Muchiri | Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel-Bulkley fluids[END_REF], contribute to the understanding of geophysical flows of non-Newtonian fluids over non-flat, dry, and wet beds, for hazard assessment and management plans. The future work will be to extend this study to flow impacts of viscoplastic fluids on a downstream wall and around occlusions.

Declarations Ethical Approval

Not applicable

Competing interest

The authors have no known competing interests to declare.

Authors' contributions

The first author prepared and wrote the original draft and all the others contributed to supervision, formal analysis, review, and editing the manuscript. However, increasing any of these parameters, the mean velocity is observed to decrease at the three points.

A Appendix. Effects of rheological parameters over an obstacle

Velocity at Point A drops drastically within a short period of time, due to the blockage effects of the obstacle.

In particular, Point A records the highest fluid height and the lowest velocity while Point C records the lowest height but the highest velocity, an indication that the flow tries to maintain the flow rate.


 Les exceptions prévues par la loi (au moins en France) ont permis à certaines organisations de modifier leur modèle économique qui reposait auparavant sur la vente de ces données.

Merci

s.grellet@brgm.fr

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Elles peuvent permettre également de retrouver les articles, publications scientifiques associés.Accessible Accessible : une fois les données trouvées, l'utilisateur doit savoir comment y accéder • A1 : Données et métadonnées accessibles par leur identifiants via un protocole standard • A1.1 : Le protocole utilisé est ouvert, libre et peut-être implémenté de manière universelle • A1.2 : Le protocole peut gérer des procédures d'authentification et d'autorisation si nécessaire • A2 : Métadonnées accessibles même si les données ne sont plus disponibles  I1 : Données et métadonnées décrites dans un langage formel, accessible, partagé et permettant l'application de la rerésentation des connaissances o Représentation des connaissances = modèles de données (UML), ontologies (vocabularies) o En 2024, il existe de nombreuses normes et bonnes pratiques internationales à réutiliser. Elles doivent l'être. o Ces utilisations permettent de fournir des ressources compréhensibles pour les humains ET les machines. Ensuite, les (méta)données peuvent être échangées de manière transparente (sans aucune intervention humaine pour « câbler/coder » les éléments pour qu'ils fonctionnent)  I2 : Vocabulaires suivants les principes FAIR o = vocabulaires documentés et accessibles par un PID, possèdant des métadonnées et étant lisibles par les machines  I3 : Liens et références vers d'autres données et métadonnées o Les principes FAIR sont fortement basés sur les normes et les bonnes pratiques du Web des données o Étape par étape, cela crée un réseau mondial d'informations scientifiques reliant les données sources (brutes) à d'autres basées sur elles. o La recherche de (méta)données est alors améliorée et permet de découvrir d'autres données Interoperable Interoperable : les données sont interopérables, intégration avec d'autres données, interopérabilité avec des applications • I1 : Données et métadonnées utilisant un langage formel, accessible, partagé et largement applicable pour la représentation des connaissances • I2 : Données et métadonnées utilisant des vocabulaires suivant eux-mêmes les principes FAIR • I3 : Données et métadonnées contenant des liens vers d'autres (méta)données  R1 : Les méta)données ont une pluralité d'attributs précis et pertinents o Plus les (méta)données sont décrites avec de nombreux attributs associés plus la compréhension et la réutilisation peuvent être maximisées.  R1.1 : Licence d'utilisation des données et métadonnées claires o Permet de définir explicitement les droits d'utilisation pour les ré-utilisateurs o Le choix de la licence dépend fortement de la nature juridique de chaque donnée (leur contexte d'acquisition, etc…) o Il existe des licences gratuites adaptées à l'Open Data Creative Commons, Open Licence, Open Database License  R1.2 : Informations de provenance o Les informations de provenance, de filiation ('lineage') sont cruciales pour prouver la fiabilité des (méta)données, pour évaluer la qualité des données et permettre de reproduire leur création (ex : expérience scientifique). Celles-ci doivent être fournies lors de la déclaration des données dans une ressource consultable (comme d'autres informations : auteur, institution, dates, …)  R1.3 : Répondre aux normes communautaires pertinentes pour le domaine o Les standards communautaires permettent d'avoir des données suivant un modèle commun pour faciliter l'intégration des données au sein d'une communauté o Si aucune norme communautaire n'existe, la meilleure façon de partager les (méta)données doit être mise en place (Re : principes FAIR) Reusable Reusable : les données doivent être réutilisables pour de futures recherches • R1 : Données et métadonnées richement décrites, par une pluralité d'attributs précis • R1.1 : Fournies avec une licence d'utilisation claire • R1.2 : Associées à leur provenance • R1.3 : Suivent les standards des communautés Règles et principes d'un vocabulaire FAIR Ten simple rules for making a vocabulary FAIR (https://doi.org/10.1371/journal.pcbi.1009041) Un continuum vers un accès plus ouvert et interopérable aux données  La recherche et le domaine public doivent se conformer à ces dynamiques  Ces dynamiques s'appuient fortement (sinon uniquement) sur les normes d'interopérabilité convenues au niveau international et les bonnes pratiques établies par des communautés telles que W3C, OGC, RDA, INSPIRE.

https://force11.org/info/the-fair-data-principles/ RDA revolves around this https://www.rd-alliance.org/ and now other organizations (OGC included FAIR in its communications)

Merci à

Hélène Bressan h.bressan@brgm.fr