Introduction

The solid fraction is commonly used to characterize different types of flow-type mass movements (Pierson and Scott 1985; Dasgupta 2003; Pierson 2005; Hungr et al. 2014). Each flow type falls within a continuous spectrum of solid fraction. As the solid fraction increases, stream flows (or flash floods, Si et al. 2022) transition to hyperconcentrated flows (or debris floods, Church and Jakob 2020), and as the solid fraction further increases, debris flows develop. Although the thresholds for defining each flow type are transitional, the reported solid fraction for characterizing stream flow is generally less than 0.1 (Pierson 2005) and the reported volumetric solid fraction for characterizing debris flows is generally greater than 0.4 (Iverson and George 2014). The most important role of the solid fraction is that it governs the rheology of the flow. More specifically, as the solid fraction increases, the yield strength and shear resistance of the flow increase accordingly (Pierson 2005). For the impact problem, high shear resistance indicates that the flow is less likely to deform, thereby generating a sharp impulse load upon impacting a barrier. Conversely, high shear resistance also means that enduring grain contacts more efficiently dissipate flow kinetic energy (Song et al. 2019), which ultimately attenuates the impact load. Despite the importance of the solid fraction in characterizing the flow type, this static index cannot holistically characterize the dynamics of flow-type mass movements.

Various impact models have been proposed and adopted for estimating the impact loads for the design of barriers to resist flow-type mass movements. In hydraulic engineering, the impact load on structures is estimated based on the conservation of momentum (Lobovský et al. 2014; Ng et al. 2021; Prieto et al. 2018; Song et al. 2021; Zhang and Huang 2022). Correspondingly, the design of countermeasures against debris flows follows the same principle, i.e., the hydrodynamic approach (Eq. 2). However, additional coefficients are necessary to compensate and capture the complex effects of solid–fluid interaction during the dynamic impact process (Hübl et al. 2009; Leonardi et al. 2016; Song et al. 2018). Hübl et al. (2009) and Cui et al. (2015) summarized the coefficients from the field and physical model tests in the literature (Fig. 1). Evidently, a wide range of values exist and a new framework is necessary to characterize and unify the impact behavior of a wide spectrum of flow types. Such a framework should capture the impact mechanisms for even the most extreme flow types.

Fig. 1
figure 1

Effects of flow inertia on dynamic pressure coefficient

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In this study, the parameters that characterize the impact behavior of flow-type mass movements are presented. A new framework is proposed to characterize and unify the impact behavior for a wide range of flow-type mass movements. Finally, the framework is validated with existing centrifuge model test data in the literature (Song et al. 2017). Limitations of the proposed framework and further work are also discussed.

Characterization of flow-type mass movements

Flow inertia

One of the principal features that governs the impact behavior of a flow-type mass movement is the state of inertia before impact. The force resulting from inertia, normalized by the force imposed by the earth’s gravitational field, is known as the Froude number (in squared form):

$${Fr}^{2}=\frac{{v}^{2}}{gh{\text{cos}}\theta }=\frac{{\rho v}^{2}}{\rho ghcos\theta }=\frac{\mathrm{Inertial\;force}}{\mathrm{Gravitational\; force}}$$
(1)

where v is the velocity of the flow (m/s); h is the incoming flow depth (m); g is the acceleration due to earth’s gravity (m/s2); \(\rho\) is the density of the flow (kg/m3), and \(\theta\) is the slope inclination (°). Whether an impact scenario is predominantly dynamic or static is governed by the inertial and gravitational forces, as characterized by Fr2 before impact (Ancey and Bain 2015; Faug 2015).

In engineering guidelines for designing protection barriers (VanDine 1996; Kwan 2012; NILIM 2016a, b; CAGHP 2018), the hydrodynamic equation is most commonly used to estimate the impact force \(F\) exerted on a barrier because of its simplicity. The hydrodynamic equation is given as follows (Hungr et al. 1984; VanDine 1996; Hübl et al. 2009; Kwan 2012):

$$F=\alpha \rho {v}^{2}hw$$
(2)

where \(\alpha\) is the dynamic pressure coefficient; and hw is the impact area (m2), with w the width of flow (m). With obvious simplifications in the hydrodynamic equation, the required parameters are easily obtainable for design. In Fig. 1, at low Fr, \(\alpha\) tends to infinity and covers a wide range of values.

Natural flow-type mass movements, especially debris flows, are characterized as elongated with low aspect ratio (depth/length) or multiple surges. As the subsequent flow approaching the barrier, the kinetic energy of the forgoing front of the elongated flow gradually dissipates out, and the frontal part approaches a static state. Given the importance of static deposition, the impact force can be represented as a combination of both static and dynamic loading (Kwan 2012; Faug 2015; Song et al. 2022)

$$F=\frac{1}{2}k\rho g{\left(\beta h\right)}^{2}w+\rho {v}^{2}hw$$
(3)

where k is the earth pressure coefficient of the static deposition, \(\beta\) is the ratio between the deposited thickness H to the incoming flow depth h impacting the barrier (Fig. 4b). The dynamic pressure coefficient \(\alpha\) in Eq. 1 can be rearranged as follows:

$$\alpha =\frac{k{\beta }^{2}}{2}\frac{1}{{Fr}^{2}}+1$$
(4)

The overall value of \(\alpha\) becomes unity when Fr tends towards infinity, which indicates that the impact process is predominantly dynamic. As Fr diminishes, \(\alpha\) increases towards infinity, indicating that static loading is important. The trends of \(\alpha\)–Fr relationship in Fig. 1 are generally consistent with the prediction with consideration of static loading.

Solid–fluid stresses

Boyer et al. (2011) reported that the complex rheology of saturated particle suspensions can be unified using a dimensionless viscous number Iv, which characterizes stresses induced by viscous shearing and grain contact shearing:

$${I}_{{\text{v}}}=\frac{\mu \dot{\gamma }}{{P}^{p}}$$
(5)

where \(\dot{\gamma }\) is the shear rate (1/s), \(\mu\) is the dynamic viscosity of pore fluid (Pa∙s), and Pp is the uniform confining pressure (Pa). The friction number Nfric proposed by Iverson and LaHusen (1993) and Iverson (1997) is the inverse of the viscous number Iv:

$$\frac{1}{{I}_{{\text{v}}}}={N}_{{\text{fric}}}=\frac{{\upsilon }_{{\text{s}}}}{1-{\upsilon }_{{\text{s}}}}\frac{\left({\rho }_{{\text{s}}}-{\rho }_{{\text{f}}}\right)gh}{\mu \dot{\gamma }}$$
(6)

where \(\rho\)s is the density of solid grains (kg/m3), \(\rho\)f is the density of pore fluid (kg/m3), and \(\upsilon\)s is the volumetric solid fraction. The numerator \({\upsilon }_{{\text{s}}}\left({\rho }_{{\text{s}}}-{\rho }_{{\text{f}}}\right)gh\) represents the confining pressure Pp. The Nfric is the ratio of sustained grain-contact stress to fluid-viscous stress. This dimensionless number reflects the efficiency of the rate of frictional work relative to that of viscous shearing (Iverson and LaHusen 1993).

Other relevant dimensionless numbers for the solid–fluid and solid–solid interactions are summarized in Table 1. Aside from characterizing the flow state, the dimensionless numbers also ensure that model flows are relevant to those observed in prototype.

Table 1 Key dimensionless numbers
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Centrifuge modelling of flow-type mass movement impact

A robust framework for characterizing the impact force exerted by flow-type mass movements is crucial for the design of safe and cost-effective protection barriers. The framework must be principally deduced from laws of physics and the framework must be verified using data taken from controlled and repeatable physical experiments. The data should not be taken from the field given the challenges of dealing with natural settings and natural materials (Iverson 2015). Furthermore, carrying out small-scale physical experiments may appear to be a cost-effective solution. However, flow-type mass movements are scale-dependent phenomena and small-scale models entail scaling disproportionalities related to the changes in the pore fluid pressure of a flow-type mass movement (Iverson 2015). To this end, the centrifuge model tests carried out by Song et al. (2017) are suitable for evaluating the proposed impact framework in this study. Details of these physical model tests are described with much brevity below.

Scaling principle

The regulation of the pore pressure is an important feature in governing the macroscopic dynamics of flow-type mass movements (McArdell et al. 2007; Iverson and George 2014). For a granular assembly subjected to shear, contraction will generate excess pore water pressures and dilation will generate negative pore water pressures, thereby influencing the mobility of a flow-type mass movement. To properly model changes in pore pressure, the absolute stress state of the granular assembly must be properly captured. Correspondingly, the geotechnical centrifuge provides an ideal means to capture the correct stress state in a granular assembly.

Two levels of scaling are required to model flow-type mass movements in the geotechnical centrifuge. The first level of scaling requires that the absolute stress state of the granular assembly is correct. Additionally, by subjecting the model to an elevated gravitational field N times that of earth’s gravitational acceleration, linear dimensions are reduced by N times, inertial velocity has a scale factor of unity, impact pressure is the same as that in prototype, and impact force is scaled 1/N2 times (Ng et al. 2016). Relevant scaling laws are summarized in Table 2.

Table 2 Summary of relevant scaling laws (Ng et al. 2016)
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In the second level of scaling, the relative stress ratios between the solid and fluid phases in the model flow are matched with those estimated in the field using the dimensionless numbers proposed by Iverson (2015). A summary of relevant dimensionless numbers is given in Table 1. To match all the relevant mesoscopic dimensionless numbers, the particle size and fluid dynamic viscosity were varied according to the Hagen-Poiseulle relationship between the intrinsic permeability k (in m2) and the particle diameter \(\delta\). Two-level scaling offers a rigorous and systematic approach to investigate the behavior of flow-type mass movements in the geotechnical centrifuge (Song et al. 2017).

Model setup

A 400-g-ton beam centrifuge with an arm diameter of 8.4 m was used to carry out the model tests in Song et al. (2017). A centrifuge model container with dimensions of 1245 mm in length, 350 mm in width, and 850 mm in height was placed on the platform (Fig. 2). A model slope was inclined at 25° and had a width and inclined length of 233 mm and 1000 mm, respectively. The model slope was installed between the Perspex window of the container and a partition to form a channel. At the top of the model container above the upstream end of the slope, a hopper was installed to hold the debris material. The debris was retained by a hinged gate, which was controlled using a hydraulic actuator. Inside the hopper, a helical ribbon mixer is installed to continuously agitate the solid–fluid mixtures. A model rigid barrier was installed perpendicular to the slope. The model barrier consists of a cantilevered steel plate with width, height, and thickness of 233 mm, 200 mm, and 10 mm, respectively.

Fig. 2
figure 2

Centrifuge model for studying the impact force exerted by flow-type mass movements on a rigid barrier (all dimensions in mm)

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A high-speed camera that captures the impact kinematics at a resolution of 1300 × 1600 pixels and at a sampling rate 640 fps was installed at the side of the model container (Fig. 2). Images were analyzed using particle image velocimetry to deduce the velocity fields for each test. A total of five miniature load cells were installed in the model rigid barrier at denser intervals near the bottom of the barrier (8 mm) and more sparsely towards the top of the barrier (60 mm). A data logger with a sampling rate of 20 kHz was used to obtain measurements. In this study, the numerator \({\upsilon }_{{\text{s}}}({\rho }_{{\text{s}}}-{\rho }_{{\text{f}}})gh\) of friction number Nfric (see Eq. 6) characterizes the effective grain-contact stress of the solid phase.

Test program and materials

Two types of tests were carried out, open channel tests and impact tests. Open channel tests were used to characterize the flow velocity and flow depth at the location where the barrier would be installed. Impact tests were then conducted to measure the impact force exerted on the barrier under the same test settings. The prototype flows in this study were simplified as idealized two-phase flows with a viscous pore fluid and uniform solid grains. The fluid phase represents a water and fine grain mixture that flows freely in the modelled grains solid phase. A prototype fluid phase dynamic viscosity of 0.5 Pa·s was adopted in this study. Leighton Buzzard (LB) fraction C silica sand comprising fairly uniform and rounded grains with diameters of about 0.6 mm and an internal friction angle 31° was adopted for the solid phase.

To study the effects of the solid fraction of the flow on the impact behavior, the solid fraction was varied from 0 to 0.5. Dry sand and air mixture with solid fraction of 0.58 is used to model the impact of dry debris avalanche. A summary of the test program, along with the characteristic velocities and flow depths for each test, is given in Table 3.

Table 3 Test program. All dimensions in prototype
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Test procedures

For each test, 0.03 m3 of debris (170 m3 in prototype) was prepared in the hopper to reach the target bulk density (Table 3). The helical ribbon mixer was activated to continuously agitate the solid–fluid mixtures, so as to prevent consolidation. The model was then subjected to an elevated gravitational acceleration of 22.4 g. The hinged gate was released by activating the hydraulic actuator. The debris transitioned onto the slope and impacted the model rigid barrier. Load measurements were taken and impact kinematics were captured by the high-speed camera.

Results and interpretation

Influence of solid fraction on Froude number Fr and friction number N fric

Figure 3a shows a sudden drop in Fr2 between the pure liquid flow (test L) and the two-phase flow with a solid fraction of 0.2 (test SL20). The abrupt change in Fr2 reflects the significance of solid fraction on the mobility of the flow. A solid fraction higher than 0.2 leads to the progressive decrease in the Fr2, implying a lower sensitivity to changes in solid fraction (Song et al. 2017). The relationship between solid fraction and friction number Nfric shows a monotonically increasing trend (Fig. 3b). The Nfric values in the literature (Table 1) are further compared here. Results show that test SL50 in this study, USGS flume experiments, centrifuge tests by Bowman et al. (2010), and natural debris and pyroclastic flows (Iverson 1997; Iverson and Denlinger 2001) lie right above the threshold Nfric = 2000 (Iverson 1997). This observation indicates that the state of these flows, although scattered, is in the frictional regime.

Fig. 3
figure 3

Influence of solid fraction \(\upsilon\)s on a square of Froude number Fr2; b friction number Nfric

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The flow dynamics change with increasing channel inclination and different boundary conditions (Choi et al. 2015a). This makes physical test data under different conditions difficult to compare. Hence, there is unlikely a universally applicable relationship between the solid fraction \({\upsilon }_{{\text{s}}}\), Nfric. Instead, it is quite apparent that the impact response is directly influenced by the effects of gravity, the inertial effect, and solid–fluid interaction. All of these factors can be well-characterized using both Fr2 and Nfric, which will be discussed later.

Effects of solid fraction on dead zone

To highlight the effects of solid fraction on the impact load exerted on a rigid barrier, PIV analysis was carried out to analyze the geometry of the dead zone when the peak force occurs for each impact test (Fig. 4). For viscous fluid flow, the impact process, when the peak load occurs, is predominantly dynamic since there are no solids to form a dead zone at the base of the barrier (Fig. 4a). At a slightly higher solid fraction (\(\upsilon\)s= 0.2), deposits still cannot be clearly observed at the base of the barrier. At solid fractions of 0.4 and 0.5, more pronounced dead zones in the shape of a ramp are observed at the base of the barrier (Fig. 4b). The PIV analysis clearly shows that the impact process is both dynamic and static when the peak force occurs. This observation further corroborates the physical meaning of Eq. 4. Not only does the dead zone impose static loading on the barrier, but some of the vectors are redirected along the barrier-parallel direction as a result of the dead zone. Finally, and as expected, the largest dead zone is observed for the dry debris avalanche (Fig. 4c). Correspondingly, the peak force exerted by the dry debris avalanche impact is governed predominantly by the static load.

Fig. 4
figure 4

Comparison of PIV analysis and dead zone geometry at peak impact force: a \(\upsilon\)s= 0, viscous liquid, showing runup mechanism, b \(\upsilon\)s= 0.5, showing runup mechanism, and c dry debris, showing pile-up mechanism. Linear dimensions in model scale

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The formation of a dead zone is directly attributed to grain contact stresses under the influence of gravity. These dead zones are responsible for exerting a static load on the barrier (Kong et al. 2021). Flows with a higher solid fraction will have more grain contacts and therefore more pronounced dead zones.

Effects of flow inertia on dynamic pressure coefficient \(\alpha\)

The total impact force on the whole barrier can be deduced by summing the pressure along the height of the barrier. The impact force time histories of each test, specifically tests L, SL20, SL40, SL50, and S are shown in Fig. 5. The impact force time histories for tests L, SL20, SL40, and SL50 are generally quite similar. Nevertheless, the momentum (velocity and density) carried by each flow is not the same. Without normalizing the loads, the peak impact loads are not directly comparable.

Fig. 5
figure 5

Total impact force (in prototype) time histories for flows with different solid fraction

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Figure 6 shows the effects of inertia on the back-calculated dynamic pressure coefficient \(\alpha\) based on Eq. 2. The dynamic pressure coefficient \(\alpha\) could be regarded as a normalized impact load resulting from different types of flows. The theoretical prediction by Eq. 4, with the ratio between deposited thickness to incoming flow depth \(\beta\) = 4 (estimated from the PIV analysis, Fig. 4) and earth pressure coefficient k = 1 (applicable for flowing granular flows, Silbert et al. 2001), is also plotted in Fig. 6. The trend of test L, SL20, SL40, and SL50 generally agrees with the theoretical prediction, however, with disparity. As Fr approaches infinity, \(\alpha\) tends towards unity. Results show that as the solid fraction decreases, \(\alpha\) also decreases. A flow with a solid fraction of 0.2 does not exceed unity for \(\alpha\), implying that grain contact stresses are insignificant for low solid fraction flows. Most of the grains in low solid fraction flows are likely suspended and do not contribute to increasing the shear resistance of the flow. Moreover, much air bubbles are entrained in the viscous liquid (aerated flow, see Fig. 4a). The compressibility partially explains the \(\alpha\) value lower than unity.

Fig. 6
figure 6

Influence of flow inertia on dynamic pressure coefficient, Eq. 4 plotted with average \(\beta\) = 4 and k = 1

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For a solid fraction of 0.4, the corresponding \(\alpha\) is greater than 1.5. Higher solid fraction leads to more pronounced dead zones, which impose static loading on the barrier. For the flow with a solid fraction of 0.5, back-calculated \(\alpha\) exceeds the value of 2. Again, this value is only possible if static loading plays a significant role. In the design of a rigid reinforced concrete barrier in Hong Kong, GEO (2020) recommends \(\alpha\) = 1.5 for estimating the impact force of a single surge impact. Based on the test results, \(\alpha\) = 1.5 seems to be a non-conservative design value. This is because the static load needs to be calculated separately, as defined by the guideline (Kwan 2012). For the case of the dry sand test, \(\alpha\) is less than unity. Although the solid fraction is the highest among the flow compositions tested, the back-calculated \(\alpha\) value is the lowest (Fig. 6) because of a pile-up impact mechanism and the higher degree of grain-contact stresses.

More interestingly, Fig. 6 shows that flows with a similar state of inertia (Fr2) before impact have remarkably different back-calculated \(\alpha\) values. Specifically, tests SL40, SL50, and S do not exhibit a unique Fr2 -\(\alpha\) relationship as expected from Eq. 4. This in turn introduces a large degree of ambiguity and uncertainty when the impact force is required for the design of rigid barriers. Clearly, the centrifuge test results highlight the need to consider the composition of the flow in order to advance the current state of impact characterization. A continuum-based approach that only considers the flow inertia, in the Fr2 and \(\alpha\) space, cannot provide a complete story. Also, results indicate that solid–fluid interaction governs the impact mechanism (Choi et al. 2015a) and needs to be captured in order to successfully characterize the impact behavior for a wide range of flow-type mass movements.

New framework for impact load estimation

Characterization of solid and fluid stresses using friction number N fric

The back-calculated \(\alpha\) from the centrifuge experiments are shown in the Nfric and \(\alpha\) space in Fig. 7. The relationship between Nfric and \(\alpha\) does not show a monotonically increasing or decreasing trend as flows transition from a viscous to a frictional-dominated regime. A specific Nfric corresponds to a single \(\alpha\) value in the Nfric-\(\alpha\) space. An increase in \(\alpha\) from test L to test SL50 denotes an increasing static load, which results from a larger dead zone at the base of the barrier (Fig. 4). As the impact mechanism transitions into a pile-up mechanism where the deposition propagates back upstream as a granular bore, \(\alpha\) drops.

Fig. 7
figure 7

Influence of friction number (solid–fluid interaction) on dynamic pressure coefficient

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The effect of the pore fluid relative to the grain stresses is an important consideration when characterizing and unifying the impact behavior of the entire spectrum of flow-type mass movements. The fluid phase generates excess pore pressure among grains, which has a significant effect in reducing grain contact stress during the flow and impact process. As previously discussed, the interstitial fluid for dry debris avalanche is air, which has a negligible dynamic viscosity and does not generate excess pore pressures between grains (Iverson et al. 2004). Instead, the grain-contact stresses are dominant in such flows, i.e., high Nfric, implying that the dissipation of flow kinetic energy is much more significant (Choi et al. 2015b; Ng et al. 2017). As a result of the contribution from the solid and fluid phases, the resulting impact mechanisms (Fig. 4) and impact force (\(\alpha\)) between dry sand flow and solid–liquid mixtures are quite different.

The Fr 2-N fric-\(\alpha\) relationship

As revealed above, the Fr2-\(\alpha\) relationship presents multi-value feature in estimating the impact force of a wide spectrum of mass movements (Fig. 6). To address this drawbacks (Fig. 6), a new framework, a Fr2-Nfric-\(\alpha\) relationship, is proposed in Fig. 8. The multi-dimensional relationship further highlights the complexity of the impact behavior of flow-type mass movements. The newly-proposed framework characterizes the type of flow using Nfric (Fig. 3), which captures the contributions between contact stress among grains and the viscous stress from the fluid. The projection of data on the Fr2-\(\alpha\) space and the Nfric-\(\alpha\) space results in a unique 3D surface. This proposed framework serves as a robust tool for engineers to estimate the impact force or to develop upper bound design scenarios.

Fig. 8
figure 8

Newly proposed 3D framework to characterize and unify impact for flow-type mass movements

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The parameters required to establish a unique 3D surface for the proposed framework are not overwhelming. Only Nfric needs to be further estimated. The characteristic particle diameter can be obtained by gathering data from past mass movements in a particular catchment. Furthermore, the typical dynamic viscosity for the flow-type mass movement generally falls within only two orders of magnitude (from 0.001 to 0.5 Pa∙s). The density of a flow-type mass movement can be conservatively estimated. Finally, the shear rate from the velocity and flow depth can be approximated using well-calibrated numerical tools (Savage and Hutter 1989).

Characterization of static loads using friction number N fric

Before the deposited material reaches a static state, the flow-type mass movements undergo either a runup or pile-up impact mechanism. These mechanisms strongly influence the stress history of the deposited material. Owing to their different deposition angles and deposition heights behind the rigid barrier, the state of deposited material cannot be directly compared using the absolute value of the static load. Instead, Fig. 9a shows a comparison of the measured static loads behind the rigid barrier against the loads exerted by the deposited material at the active state based on the Coulomb earth pressure theory. Since the state of debris material only refers to the solid phase, the static load of the liquid phase is excluded by assuming a hydrostatic liquid pressure along the barrier. Data points lying around the same diagonal line from the origin indicate that the deposits have a similar state. With an increasing solid fraction, a rotation of the diagonal line towards the active failure state is observed.

Fig. 9
figure 9

Static state of deposited material: a relationship between the measured static load and the loads at active failure mode; b quantification of the state of static debris using the Nfric

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Figure 9b further quantifies the state of static deposited material using Nfric. The ordinate shows the ratio between the measured static load and the theoretical active load. Correspondingly, a value of unity denotes the active failure state of the deposited material. Flows with low solid fractions, particularly test SL20 with Nfric = 3.4 × 102, are least affected by the interaction process with the barrier. By contrast, the dry debris avalanche has a much higher Nfric = 4.9 × 106 compared to the threshold value of 2000 (Iverson 1997). This implies that the static deposit of the dry debris avalanche is strongly influenced by flow-structure interaction. The extent to which flow-structure interaction influences the static deposits is evident by examining how close the state is to the active failure state, even without any barrier displacement. Clearly, Nfric is an effective index for quantifying the impact behavior of two-phase flows under both dynamic (Fig. 7) and static (Fig. 9b) conditions. Furthermore, the progressive transition from viscous to frictional states in the observed impact mechanism cannot be holistically captured using a conventional threshold of Nfric = 2000. There may not be a clear threshold for viscous and frictional-dominated flows.

We further adopt the schematics in Fig. 10 to demonstrate the significance of solid fraction in regulating the final state of the static deposits. With the regulation of the fluid phase, the solid grains in the flows with low solid fraction (low Nfric) tend to settle along the vertical direction. Therefore, negligible shear stress is inherited in the final deposited material (Fig. 10a). This scenario corresponds to the state at rest. Flows with high solid fraction (high Nfric) exhibit more grain contacts between grains and between the grains and channel bed. Correspondingly, boundary friction stresses are better preserved within the final deposited material (Fig. 10b). With the friction stress pointing upstream, the lateral load on the barrier is less than that of state at rest. This scenario corresponds to the active state of deposited debris. As a result, static deposits with higher Nfric are closer to the active state, even without any deflection of the barrier. These observations confirm that Nfric is an appropriate parameter in quantifying the state of a flow-type mass movement. The active state could also be achieved through barrier downward deflection (Fig. 10c), e.g., the deflection of a flexible barrier for debris-flow hazard mitigation (Ng et al. 2016). Despite the observed features in this study, it is worthwhile to note that a passive state could also be achieved for flows with high solid fraction (high Nfric). As shown in Fig. 10d, given high impact inertia (Froude number), the deposited debris could be further compressed against the barrier. This scenario is analogous with the upward movement of the barrier (Fig. 10e). The experimental measurements of Song et al. (2023) have confirmed the passive state of compressed static deposition behind a rigid barrier.

Fig. 10
figure 10

Schematics of the relative movement between a block and barrier, showing the state of static deposits with different stress history (deposition process). a State at rest through vertical settlement; b, c active state by downward movement of either debris or barrier; d, e passive state by compression of downward moving debris or upward movement of barrier. F is lateral load, mg is gravity, f is shear resistance, and \(\theta\) is slope inclination

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Discussion and conclusions

Stream flow, hyperconcentrated flow, debris flow, and dry debris avalanche have been widely regarded as distinct phenomena. Accordingly, various load models have been established by hydraulic engineers, geotechnical engineers, and geologists to estimate the impact load against protection barriers. The use of the square of Froude number Fr2 as a sole parameter for characterizing the impact behavior of a wide spectrum of flow-type mass movements against a protection barrier is incomplete. Rather than relying on the empirical design coefficients, the proposed framework (Fig. 8) in this study unifies the impact behavior for a wide spectrum of flow-type mass movements using the Fr2-Nfric-\(\alpha\) space in a scientifically-based manner. By adding the friction number Nfric to the existing \(\alpha\)-Fr2 space, reliable estimates of the dynamic and static loads for flow-type mass movements can be estimated for a particular geo-material.

Under the same solid fraction and stress state, poorly graded soil tends to dilate; while well-graded soil may contract, since the fine particles would fall into the void formed by coarse particles. From critical-state soil mechanics, it is the distance from the initial void ratio to the critical-state void ratio, rather than the absolute void ratio, that controls the behavior of soils (Been and Jefferies 1985). Kostynick et al. (2022) further demonstrated that the shear viscosity and yield stress are controlled by the distance from the initial solid fraction to the jamming solid fraction. These indicate that the initial solid fraction is insufficient to describe the soil behavior, especially the pore-pressure response. Therefore the distance from the initial solid fraction to the critical-state solid fraction is a rational state parameter. Further work considering the critical-state parameters is warranted to modify the proposed framework, especially on the granular dilatancy and pore-pressure feedback (Song et al. 2023).

With the negligible contribution of interstitial air to the viscous shear stress, the dry debris avalanche in this study can be characterized using a friction number Nfric of about 107. Conventionally, friction-dominated flows are characterized as Nfric > 2000 (Iverson 1997). However, with higher Nfric, the static state of deposited materials progressively transitions towards an active failure state. This finding indicates that there may not be a clear threshold for distinguishing between viscous and frictional flows.

Finally, there is an obvious disparity between the Nfric values for tests SL50 (Nfric = 2.5 × 103) and S (Nfric = 4.9 × 106) as shown in Fig. 7. Due to the limitation in mixing two-phase flows under elevated gravitational acceleration, simulating two-phase flows with solid fraction around 0.60 was not realized in the centrifuge tests. It is acknowledged that more high-quality scaled data (e.g., Bugnion et al. 2012) will help shed light on the impact behavior for flows with Nfric ranging from 103 to 107. The additional data is instrumental to strengthening the proposed framework. With sufficient data, a clearer representation of the proposed 3D surface can be characterized and simplified mathematically.