A bilinear flume is adopted to model the block movement under the action of debris flow (Fig. 2a). The upstream section is 4 m long and inclined at 25°. The top 1 m acts as a reservoir that is isolated by an uplift gate for releasing debris material. The downstream section has a length of 6 m and is inclined at 5° which is a typical slope for the deposition area. The width of the flume is 0.3 m, and the sidewall is transparent for observing the movement of the block. Spherical glass beads (0.6 mm) are used to roughen the flume bed, which is also used as the solid phase of debris flow. A flat aluminium block is positioned 0.75 m downstream from the smooth transition zone (Fig. 2a).To ensure that the block only moves by sliding rather than rolling and saltation, the block is designed as a flat shape, and the edges are rounded. The dimension of the block is 40 mm × 40 mm × 10 mm (Fig. 2b). The debris flow accelerated after being released upstream and began to decelerate (deposit) after reaching the 5° section.

The flume bed contains 4 basal sensing modules (Fig. 2a), and each module is equipped with a triaxial load cell (LH-SZ-02, 50N, ±0.1 % BSL) located at the center of the force plate (Fig. 2b). These load cells are used to measure normal and shear stresses. Additionally, each module has a pore-water pressure transducer (PPT, OMEGA PX409, 6.9 kPa/34.5 kPa, ±0.08 % BSL) upstream of the force plate (Fig. 2b) to measure pore-water pressure. Above each basal sensing module, there is an ultrasonic sensor (BANNER U-GAGE T30UXUA, 0.1–1.0 m, resolution 0.1 % of distance) to measure the flow depth. The whole data acquisition system (National Instruments) is set to a sampling rate of 500 Hz. To derive the frontal velocity prior to contact, a high-speed camera (PHONTRON FASTCAM Mini WX50) with resolution of 1280 × 1024 pixels is placed at the sidewall of the flume. The frame rate of the high-speed camera is set at 250 fps. Three video cameras (DJI Osmo Action 4, resolution 3648 × 2736 pixels, 120 fps) are used to capture the movement kinematics.

Owing to the incomplete transparency of the modeled debris flow, the block movement is not easily observable by eye. Therefore, employing a micro inertial measurement unit (IMU) is imperative for analyzing its movement mode (Curley et al., 2021; Maniatis, 2021). A commercial IMU (WITMOTION, WT901SDCL) is embedded into the block. It has an acceleration range of ±16 g with accuracy of 0.0005 g per LSB (least significant bit), and an angular velocity range of ±2000° s⁻¹ with resolution of 0.061 (° s⁻¹) per LSB. The sampling rate of the IMU is 200 Hz. The IMU can be switched on and off manually before and after the experiments. Through the images captured by the high-speed camera, the response of IMU can be manually synchronized with the flow depth and stress measurements.

For the basal sensing modules, the raw data from the load cells and the pore-water pressure transducer exhibit substantial noise. To facilitate data visualization and comparison, a moving average filtering (with interval of 0.02 s) is employed to smooth these datasets. The raw data from the IMU are retained without filtering.

In this study, glass beads with diameters of 0.6 mm and densities of 2540 kg m⁻³ are used as the solid phase of debris flows. A solution of glycerol and water is used as the fluid phase. The blocks have a density (2700 kg m⁻³) close to that of reinforced concrete.

To investigate the mechanism of debris flows displacing building blocks under different flow conditions, debris flows are modeled with volumetric solid concentrations of 45 %, 50 %, and 53 % (Table 1). The fluid viscosity is kept at 0.01 Pa s, i.e., ten times that of water. The solid particles and fluid phase are thoroughly mixed using a mixer before release. The debris-flow volume is 50 L. A steady flow with relatively constant height is generated by adjusting the opening of the gate. The test program of this study is summarized in Table 1.

The leading-edge model developed by Takahashi and Yoshida (1979) is introduced. When the debris flow (with velocity v0 and depth h0) enters into deposition area (slope θ), the velocity vd slows down, and the flow depth H thickens. The debris-flow deposition is regarded as a cohesive whole. Compared with the original model, we further consider the influence of the degree of liquefaction on basal friction resistance.

According to the conservation of mass and momentum in the flow area and accumulation zone, the governing equations (conservation of mass and momentum) are expressed: 1HS=h0v0t2ddtρdHvS=ρdgHSsin⁡θGravitational driving force-ρdgHScos⁡θ1-λμdFrictional resistance+ρdh0v0v0cos⁡θ0-θMomentum flux of upstream+12kρdgh02cos⁡θ0cos⁡θ0-θEarth pressure of upstream where ρdHvS is the momentum of deposition, and its time derivative is the force on the deposition. On the right-hand side of Eq. (2), all the forces acting upon the deposition are summed up. The first row is gravitational driving force, and the component of the gravity of deposition along the flow direction; the second row is the frictional force generated by self-weight considering the degree of liquefaction λ and friction coefficient between flow and bed μd; the third row is the momentum flux of the incoming flow, where v0, h0, and θ0 are velocity, depth, and slope of upstream; the last row is the earth pressure from the upstream to the downstream, and k is the earth pressure coefficient. Substitute Eq. (1) into Eq. (2): 3ddtvdt=gsin⁡θ-cos⁡θ1-λμdt+v0cos⁡θ0-θ1+k2Fr02=-Gdt+v1 where Gd = gμdcos⁡θ(1-λ)-sin⁡θ is the acceleration of the deposition downslope, v1 = v0cos⁡θ0-θ1+k2Fr02 is the equivalent upstream inflow velocity, Fr0 = v0gh0cos⁡θ0 is the Froude number of upstream incoming flow. We further solve Eq. (3) and obtain the velocity of debris flow: 4vd=-12gcos⁡θ1-λμd-sin⁡θt+v0cos⁡θ0-θ1+k2Fr02=-Gd2t+v1

From the solution, the debris flow in the deposition area is in uniform deceleration motion, and the deceleration is Gd2. The equivalent upstream inflow velocity v1 is not used in the next sections, because the velocity of the debris flow at the downstream can be directly measured in the experiment, and v0 is used to represent the initial frontal velocity of debris flow at the downstream start. The deposition length of debris flow can be obtained by integrating the velocity: 5S=v1t-14Gdt2=2v12Gd

Based on the aforementioned leading-edge model, we developed a model where the kinematic behavior of building blocks is exclusively governed by the debris flow dynamics, with negligible feedback effects on the flow regime. This hypothesis generally requires the ratio between mass of debris flow (discharge) and the mass of block to be higher than 10.

As a preliminary study focusing on mechanisms, the proposed model in this study only considers the movement of one single building fragment. This means the complicated destruction process of buildings is not covered. Without a deep understanding on the mechanisms of a simplified scenario, it is pessimistic to further forward our understanding into the complicated real-world cases. When the density of building fragment is higher than that of debris flow, the fragment sinks and contacts the bed (Fig. 8). The motion of a fragment sinking to the bed can be influenced by flow conditions and local terrain, leading to various forms of movement, such as sliding, rolling, and saltation (Imamura et al., 2008; Nandasena and Tanaka, 2013). The movement of a fragment can also be affected by factors such as its shape, size, and density, as well as the forces it experiences. Since the fragments of destroyed buildings are mostly flat, we consider that the movement mode of the fragment (block) is sliding.

Based on Takahashi and Yoshida's model, a flat block sinks at the bed under a decelerating incoming flow condition and is displaced a certain distance from its initial position. During the movement, we assume that the forces acting on the block can be described as the sum of its own gravity, buoyancy, friction resistance, dynamic drag force, and active/passive earth pressures (Fig. 8). Compared to the drag force, the fluid viscous force is negligible (with the Friction number; Iverson, 2015; higher than 100), hence it is not considered in the model. The governing equation of block movement can be expressed: 6ρbVbdvbdt=CdρdAvdn-vb2Sgnvdn-vbDynamic drag force+12kp-ka1-λρgA2H-hcos⁡θSgnvdn-vbEarth pressure force+ρb-ρdgVbsin⁡θGravity-bouyancy driving force-ρb-ρdgVbμbcos⁡θ1-λFrictonal resistance

The left hand-side of Eq. (6) is the derivative of block momentum with respect to time, where ρb and Vb are the density and volume of the block, respectively. On the right-hand side, all the forces acting upon the block are summed. The first row represents the dynamic drag force, which is proportional to the square of the velocity difference and the block's frontal area A, and Cd is the drag coefficient. The ratio of basal sliding velocity to frontal flow velocity defined as 1/n, and the basal sliding velocity is expressed as vd/n. The second row represents the coupled active and passive earth pressures, where ka and kp are the active and passive earth pressure coefficients, respectively (Iverson and Denlinger, 2001), and H and h are the heights of debris flow and block. The third row represents the component of the block's gravity down slope, which excludes the buoyancy. The last row represents the frictional resistance, where μb is the friction coefficient between the block and bed, the frictional resistance is influenced by the degree of liquefaction, when the degree of liquefaction is 0, the model is applicable to dry granular flows, and when the degree of liquefaction is unity, the model is applicable to pure fluid flows.

The directions of drag force, active and passive earth pressures change with the relative movement between the block and debris flow (Fig. 8b). As the velocity of the block is lower than the velocity of debris flow, active earth pressure acts on the front of block, passive earth pressure acts on the rear end, and the direction of drag force is the same as the flow direction (State 1 in Fig. 8b). As the velocity of block is greater than the velocity of debris flow, the acting directions of the earth pressures and drag force are opposite (State 2 in Fig. 8b). Here, a function Sgn() is introduced in Eq. (7) to define the directions of the drag force and earth pressures. The movement of block can be divided into two states: 7Sgnvdn-vb=-1,vdn<vbState 10,vdn=vb1,vdn>vbState 2

Substituting the debris-flow movement governing Eq. (4) into the block's movement Eq. (6), a model of block movement can be obtained: 8-dΔvdt=CdρdBρbΔv2SgnΔv+12kp-ka1-λgρdBcos⁡θρb2H-hSgnΔv+Gd2n-1-ρdρbGb

The right-hand side of Eq. (8) comprises only three elements (Eq. 7 has four elements), resulting from the combination of gravity and basal friction (the third row of Eq. 8), because the two forces jointly affect block sliding on the slope. Then, Gb = gsin⁡θ-μbcos⁡θ1-λ is introduced into Eq. (8), which represents the equivalent acceleration of block sliding on the bed. The term vdn-vb is further replaced by Δv.

Established evidences indicate that the Froude number governs key dynamic characteristics of debris flows – including impact, superelevation, and overflow behavior. Therefore, the theoretical framework incorporates a simplification scheme predicated on gravitational and inertial dominance. By using the following dimensionless form of each parameter (^* denoting dimensionless): 9Δv*=Δvv0,t*=gtv0

A dimensionless form of Eq. (8) can be obtained: 10-dΔv*dt*=D*Δv*2SgnΔv*+K*SgnΔv*+G* Equation (10) expresses the dimensionless time derivative of the relative velocity between the block and debris flow on its left-hand side. Its solution determines the dimensionless velocity difference vd*n-vb*. Hence, with knowledge of debris-flow velocity and block-bed characteristics, the block velocity can be calculated. There are three dimensionless numbers on the right-hand side of Eq. (10): 11D*=Cdv02ρ*gB12K*=12kp-ka1-λhρ*Bcos⁡θ2Hh-113G*=Gd*2n-1-1ρ*Gb*

These three dimensionless numbers have distinct physical meanings. D*(Δv*)2 represents the magnitude of drag force relative to weight of block. K* represents the magnitude of earth pressure relative to weight of block. G* is the dimensionless deceleration difference between the debris flow Gd*/2n and block Gb*, with correction of the relative density ρ*. 14ρ*=ρbρd,Gd*=Gdg,Gb*=Gbg

G* > 0 means that the debris flow has a greater equivalent acceleration than that of block and G* < 0 indicates that the debris flow has a lower equivalent acceleration, and the present experimental investigation is confined to scenarios where G* < 0.

By comparing the magnitudes of the dimensionless dynamic drag force and the earth pressure at the initial time: 15D*Δv*2K*t*=0=Cdn2kp-ka1-λ1-h2Hv02gHcos⁡θ=αFr2 A relationship between D* and K* can be expressed in terms of the Froude number (Fr) of incoming flow. The coefficient α comprises the drag coefficient Cd, earth pressure coefficients ka and kp, degree of liquefaction λ, and ratio between block height h and debris-flow height H. A higher Froude number indicates a debris flow with high mobility where the dynamic drag force governs the block movement, while earth pressure dominates when Fr is lower (Table 2). When the dimensionless parameter αFr2 attains elevated magnitudes, the dynamic drag force D* is far larger than the earth pressure K*, and K* can be ignored. Equation (10) can be simplified: 16-dΔv*dt*=D*Δv*2SgnΔv*+G*

For configurations where αFr2 falls below critical thresholds, the effects of earth pressure K* are greater than those of dynamic drag force D*, and D* can be ignored. Equation (10) can be simplified as another form: 17-dΔv*dt*=K*SgnΔv*+G*

The general governing Eq. (10) is suitable for situations in which the contributions of dynamic drag force and earth pressure to block displacement cannot be ignored. We refer to the general form of Eq. (10) as Model I. The model with high value of αFr2 (Eq. 16) is named Model II, which is suitable for fast flow considering only the dynamic drag force. The model with low value of αFr2 (Eq. 17) is referred to as Model III and is suitable for slow-moving flow where earth pressure dominates.

The selection of parameters in the theoretical model is based on the physical characteristics of the experimental flume, the materials, and the measurements of the sensors (Table 4). In all experiments, the internal friction coefficient μd of debris flow solid particles is taken as 0.51, and the friction coefficient between debris flow solid particles and the bed is also 0.51. The friction coefficient μb between the block and bed is taken as 0.70. The drag coefficient Cd is assigned a typical value of 0.50, and the magnitude of the difference between the active and the passive earth pressure coefficient (kp-ka) amounts to 0.10.

Figure 10 compares the theoretical predictions with the experimental results. The horizontal axis represents the magnitude of the force acting on the block. Since the drag force and earth pressure exhibit a linear relationship (αFr2) (Eq. 15), the horizontal axis can equivalently represent either drag force or earth pressure. Consequently, comparisons of theoretical predictions from three models are performed for experiments with three distinct solid-phase concentrations.

In the model classification, both Model I and Model III incorporate earth pressure considerations. During the model-solving process, the solutions can be categorized into two distinct classes based on the relative magnitudes of K* and |G*|. In Fig. 10, the left hand side of the dashed line represents scenarios where K* < |G*|, while the right hand side corresponds to situations where K* > |G*|. The critical distinction between these two cases lies in whether the block velocity in State 2 achieves synchronization with the debris flow velocity.

For Models I and III, when K* < |G*|, the normalized block position L*/S* within the debris flow initially decreases and subsequently increases as the combined drag force and earth pressure intensify (Fig. 10). This phenomenon arises from the theoretical velocity prediction curve of the block (Fig. 9). In State 1, where the block's initial velocity is higher than that of the basal velocity of debris flow, both the drag force and earth pressure act as resistive forces. Larger magnitudes of these forces induce faster deceleration of the block. In State 2, the block's velocity falls below the basal flow velocity and asymptotically approaches a limited value. The difference between this asymptote and basal velocity curve is governed by the drag force and earth pressure: greater forces result in a smaller difference. Consequently, as the drag force and earth pressure increase, the block's velocity in State 2 gradually converges toward the debris flow velocity. Therefore, during State 1, the block's displacement exceeds the displacement of the debris flow front, whereas in State 2, the block's displacement becomes smaller than that of the flow front. The total block displacement is the sum of these two phases. The dynamic interplay between displacements in these two states governs the evolution of the relative position, ultimately leading to the non-monotonic trend (initial decline followed by an increase) in the L*/S* curve when K* < |G*|.

In contrast, when K* > |G*|, the earth pressure ensures synchronization between the block's motion in State 2 and debris-flow basal velocity. Consequently, the block position depends on the duration of the deceleration phase in State 1. Higher drag force and earth pressure enhance deceleration, shortening the duration of State 1 and reducing the displacement difference between the block and debris flow front. Thus, the block position decreases monotonically with increasing drag force and earth pressure under K* > |G*|.

For Model II, no such critical boundary exists (Fig. 10), and the block velocity consistently remains higher than that of the basal flow velocity. The drag force modulates both (1) the rate at which the block asymptotically approaches its terminal velocity and (2) the magnitude of the difference between this asymptote and the basal flow velocity. The dynamic equilibrium between these two effects still induces a non-monotonic trend in the normalized accumulation position L*/S*, characterized by an initial decrease followed by an increase.

The αFr2 of the 45 %-concentration experiment is 266.52 (Table 2), indicating that the dynamic drag force dominates the block movement. For the prediction of the 45 %-concentration experiment, the prediction of Model II is the closest (Fig. 10a). However, for the results of the 53 %-concentration experiment, the predictions of Model I and III almost overlap due to the negligible contribution of the dynamic drag force. The value of αFr2 is 0.04, indicating that earth pressure plays a dominant role (Fig. 10c). For the 50 %-concentration experiment with αFr2 = 71.45, the three theoretical predictions exhibit minor discrepancies (Fig. 10b). Neither of the two forces can be neglected, and Model I would provide appropriate prediction for the 50 %-concentration experiment. The relative deposition positions (L*/S*) for block across three solid concentrations yield: 0.53 (Test 45-40), 0.53 (Test 50-40), and 0.72 (Test 53-40), while the corresponding model predictions demonstrate close agreement: 0.49 (Test 45-40, Model II), 0.50 (Test 50-40, Model I), 0.75 (Test 53-40, Model III).

Experimental measurements establish the basal sliding velocity at 0.6 times the frontal velocity. The relative deposition positions (L*/S* ∼ 0.49–0.75) from both experiments and theoretical prediction exhibit significant correlation to basal velocity (0.6).

Conventional understanding posits that debris flows – as free-surface flows governed by gravitational and inertial forces – exhibit distinct regimes dictated by the dominance hierarchy between these forces, quantified through the Froude number (Fr). Consequently, the transport and deposition of blocks are presumed to demonstrate Fr-regime-dependent variability in relative deposition distance (L*/S*). Contrary to this paradigm, our theoretically derived curves for Models I–III exhibit remarkable congruence, showing negligible divergence across Fr regimes (Fig. 10).

Analysis of velocity time histories under varying solid concentrations reveals a universal characteristic: irrespective of whether transport is dominated by earth pressures or dynamic drag forces, block velocities invariably converge toward basal flow velocities at the block's equilibrium position (Fig. 9). This kinematic convergence results in deposition distances that remain invariant to gravitational-inertial dominance transitions. The basal sliding velocity exerts a dominant control on block position, fundamentally governing the prediction envelope of L*/S*. Specifically, adopting a basal sliding velocity of 0.6vd (where vd is frontal velocity) yields theoretical and experimental L*/S* values consistently converging at 0.6, demonstrating the control of basal sliding velocity on the position of building block. This practical finding enables emergency responders to predict final block position (and the trapped victims) solely from debris-flow velocity profiles (basal sliding velocity).