In this study, we modified Dan3D, a semi-empirical, depth-averaged, Lagrangian model that simulates landslide motion over 3D terrain (McDougall and Hungr, 2004). Dan3D treats the moving landslide mass as an equivalent fluid whose behaviour is governed by its internal and basal flow resistance (Hungr and McDougall, 2009). The momentum conservation equations are 1ρhDvxDt=ρhgx-kxσz∂h∂x+τzx-ρvxE,2ρhDvyDt=ρhgy-kyσz∂h∂y, where ρ is the bulk density; h is the bed-normal thickness; vx and vy are the x and y components of the velocity, where x is in the local direction of motion; gx and gy are the x and y components of gravity; kx and ky are the stress ratios (ratios of horizontal to vertical stresses in the x and y directions); σz is the bed-normal stress; τzx is the basal shear resistance; and E is the entrainment rate (Hungr and McDougall, 2009).

The coordinate system is bed-normal and aligned with the local direction of motion, so the basal resisting stress and entrainment terms only appear in Eq. (1).

Note that throughout this paper, we use the term thickness to refer to the distance in the local bed-normal direction and depth for the vertical distance. The internal rheology is represented by an internal friction angle, which is used to calculate the stress ratios (kx,y) within the moving mass as a function of longitudinal strains (McDougall and Hungr, 2004). The model allows for several possible basal shear resistance relationships to be selected, allowing for changes in material behaviour along the flow path (Hungr and McDougall, 2009). In this study, we use the Voellmy flow-resistance model, which is commonly used by researchers and practitioners to simulate debris-flow motion (see Dash et al., 2021, for a summary of debris-flow case histories calibrated with a Voellmy model).

The Voellmy flow-resistance model is defined as 3τzx=-σzf+ρgvx2ξ, where f is the Voellmy Coulomb friction parameter, and ξ is the Voellmy velocity-dependent resistance parameter, commonly referred to as the turbulence parameter.

As can be seen in Eq. (3), higher values of ξ lead to lower values of basal shear resistance.

Dan3D uses the smoothed particle hydrodynamics (SPH) numerical technique to discretize the moving mass and allow for behaviours such as flow splitting. SPH is a mesh-free continuum method, which discretizes the moving mass into a set of particles: forces are calculated at the particles, resulting in their displacement, while a free surface is interpolated between the particles to define the stress conditions that give rise to the forces at the particles. Dan3D calculates flow thickness using a Gaussian kernel at each particle, and the free surface at any location is the summation of each kernel's contribution at that location, as shown schematically in Fig. 1 (McDougall and Hungr, 2004).

The Dan3D numerical model was originally developed using a block start initial condition, where the debris-flow mass is fully fluidized at the starting time in the model (Hungr and McDougall, 2009). Here, we developed a modified version of Dan3D that allows for fluid particles to be added to the model throughout the simulation so that a wide variety of input hydrographs can be used. The smoothing length calculation that determines the size of the Gaussian kernel (Fig. 1), thus the contribution of each particle to the free surface calculation at a given point, is updated using the dynamic formula outlined by McDougall and Hungr (2004): 4l=B∑i=1NhiViN, where l is the smoothing length, B is the input smoothing length constant, N is the total number of particles, hi is the bed-normal thickness at particle i, and Vi is the volume of particle i.

The modified model calculates an initial value of the smoothing length at time zero based on the initial particle(s) in the model. The number of initial particles depends on the input hydrograph (discussed in Sect. 2.5). The smoothing length calculation can become unstable early in the simulation, and if the value becomes more than 10 times the initial value, the model resets the smoothing length to the initial value. We chose a limit of 10 based on initial testing that showed this value would prevent the smoothing length from approaching infinity early in the simulation while not interfering with the normal fluctuations in Eq. (4). Initial testing shows that the smoothing length calculation generally stabilizes within 20 s (model time) of the simulation start.

The original version of Dan3D utilized multiple flow-resistance models, which can be assigned to areas within the model domain (e.g., allowing for different flow-resistance behaviours in the source area and the deposition area). For this study, we have modified the model to allow for the particles to have different flow-resistance behaviours (i.e., the flow resistance is associated with the particle, not the location).

We developed an idealized model terrain to conduct numerical experiments on the effects of varying inflow conditions and flow-resistance parameters on the discharge, flow depth, and flow velocity downstream. This numerical flume has a longitudinal profile with a constant 40 % slope (22∘) for the first 780 m and then gradually transitions to a 17 % slope (10∘) over the remaining 1220 m (Fig. 2). The cross section of the model geometry used a smooth curve to define a 10 m deep channel that is 40 m across at the crest with a grid spacing of 3 m. The slopes and channel dimensions for the curved portion of the numerical flume are within the range observed in the upper fans of large debris-flow catchments in southwestern BC (Zubrycky et al., 2021a).

We used two approaches to generate the inflow hydrographs: an idealized triangular input and scaled hydrographs observed in the field. A triangular hydrograph is described by the peak discharge (Qp), the total inflow duration (tin), and the time to peak (tp), with the total volume (V) defined by the area of that triangle. Several empirical equations exist for estimating the peak discharge of a debris flow (see Rickenmann, 1999, for a summary), and in this study we utilize the equation based on Froude similarity from Rickenmann (1999): 5Qp=cV5/6, where c is a constant that ranges between 0.001 and 1, with values of 0.01 typical of muddy flows and 0.1 typical of granular flows (Ikeda et al., 2019). This equation generally agrees with other empirical relationships fit through purely statistical methods (e.g., Mizuyama et al., 1992; Bovis and Jakob, 1999).

With the volume and peak discharge, one can calculate the total inflow duration, but the time to peak must be selected. A recent study of debris flows in the Moscardo catchment in Italy from 2002 to 2019 showed a typical surge duration to be approximately 6 times the time from debris-flow initiation to the peak discharge (Marchi et al., 2021). In this study, the time to peak for the triangular hydrographs is taken as 20 % of the total inflow duration, similar to the general shape found by Marchi et al. (2021) and Hübl and Kaitna (2021).

For this study, we used a triangular hydrograph with a total volume of 100 000 m3 and a peak discharge of 1000 m3 s-1. This volume is within the range of relatively large natural debris flows in the southwestern BC area (Zubrycky et al., 2021a) and well within the range of events used to calibrate the empirical relationships from Rickenmann (1999). The peak discharge corresponds to a value of c=0.07 in Eq. (5), which is near the typical value for a stony debris flow of 0.1 reported by Ikeda et al. (2019). We conducted a parametric study by systematically varying the Voellmy parameters, f between 0.1 and 0.4 and ξ between 25 and 500 m s-2; extracting flow depths and velocities; and calculating the discharge at 1 s increments along cross sections distributed down the model slope. We input all particles at x=300 m, and tested the sensitivity to the inflow location by varying the input distance between x=300 and x=750 m. We also tested the effects of changing the inflow hydrograph by generating a triangular hydrograph with a peak discharge of 2940 m3 s-1 corresponding to a c=0.2 in Eq. (5) for the same volume of 100 000 m3. The c value we selected is an upper envelope from the real-event hydrographs compiled in this study (Sect. 3). We calculated Froude numbers and for both peak discharge cases and compared them to reported values for debris flows.

The effects of modelling a surge consisting of particles with basal resistance defined by two Voellmy materials were also examined. A triangular inflow hydrograph (Qp=1000 m3 s-1) was used as the basis for the two material simulations, with a higher f parameter on the rising limb than the falling limb of the hydrograph. This is meant as a simplified test of the idea of contrasting flow-resistance behaviours resulting in debris-flow surges proposed by Hungr (2000).

We assembled a dataset of real debris-flow events with high temporal resolution of velocity measurements coupled with flow depth measurements, allowing us to estimate discharge over time. We used these events as prototypes for the modelled inflow into the numerical flume by scaling the discharge. The three sites are Lattenbach, Austria; Dorfbach, Switzerland; and Spreitgraben, Switzerland. They are described in detail in Sect. 3. We kept the inflow duration constant and multiplied the instantaneous discharge by a scaling factor at each time step to obtain a target total volume. We used Eq. (5) to validate that the scaled peak discharge was within a reasonable range for the event volume with an assumed value of the constant c. If the scaled Qp value was unreasonable, we increased the inflow duration to maintain the Qp value within the target range.

As with the triangular hydrographs, we calculated Froude numbers and compared them to literature values for debris flows after applying the scaling. We calculated the intensity index, IDF=dv2, where d is the flow depth, and v is the depth-averaged velocity (Jakob et al., 2012), at various locations along the model channel. The intensity index is commonly used in hazard assessments to calculate building damage from debris flows.

Finally, we tested the sensitivity of simulated impact areas and flow intensities to varying inflow conditions for real, complex topography. The two examples modelled, Mount Currie and Neff Creek, are from southwestern BC, where mapped deposits and field observations provide estimates of flow depths and velocities that were compared to the simulation results. Both sites have at least partial airborne lidar coverage before and after the events for estimates of the deposited volume. We assumed a bulk density of 2000 kg m-3 for the deposited material and a solids density of 2600 kg m-3. We assumed the material in the deposit had a greater density than the flowing material due to drainage and consolidation of the deposited material over time. For the simulations, we assumed the flowing material was fully saturated with a solids content of 50 % by volume, resulting in a bulk density with flowing of 1800 kg m-3, or a total volume considering bulking and water content approximately 1.5 times greater than the deposit volume. For all cases, the model topography consisted of a 3 m DEM of pre-event conditions. We modelled each event using the 24 real input hydrographs scaled to the observed event volume and selected Voellmy parameters based on calibrations not considering variable inflows. To better represent the inferred deposition between stages of the second event (Neff Creek), we implemented a method to represent deposition during flow. We used a Monte Carlo approach that randomly divided the total event volume into four stages and randomly selected an input hydrograph for each of those stages. After each stage, we reduced the final deposit grid by a factor of 0.65 to account for drainage and consolidation (consistent with the assumptions for bulking for the simulation volume estimates) and merged it with the topography grid.

We numerically integrated the instantaneous discharge hydrographs for both the triangular and real hydrographs to create a time series of cumulative volume versus time. We then divided the time series by the total event volume to create unit hydrographs and scaled the unit hydrographs to achieve the desired inflow volume. We completed the scaling by multiplying the target volume by the unit hydrograph (assuming the inflow duration is fixed) or by adjusting the inflow time to result in a peak discharge corresponding to a target c value in Eq. (5).

We generated a cumulative inflow curve from the scaled hydrograph and discretized it into the SPH particles for the simulation. We used a total of 4000 particles in all simulations, with the total volume divided equally between the particles. The particles entered the model domain at a defined inflow line with initial particle positions sampled randomly along that line. We estimated the starting velocities, v, using the following equation (from Rickenmann, 1999): 6v=2.1Q0.33S0.33, where S is the channel slope.

We have summarized the runout modelling process as a flow chart in Fig. 3. The “Define Hydrograph” workflow is implemented in R, and the “Modified Dan3D” workflow is implemented in C++. We present a visualization of the hydrograph discretization process in Fig. A1 in the Appendix.

Hypothetical triangular inflow hydrographs were developed following the procedure outlined in Sect. 2.3. Discharge and peak discharge were calculated for all combinations of flow-resistance parameters (Sect. 2.3). Selected discharge results and peak discharge for all cases are shown in Fig. 5. Note that we smoothed the results with a loess function using a span of 15 s to remove small variations in the data due to numerical noise. We provide an example of the difference between the smoothed and raw model results in the Appendix (Fig. A2). In general, the results presented in Fig. 4 show that increasing f reduces the peak discharge, and increasing ξ increases the peak discharge. The modelled peak discharge had little sensitivity to the f or ξ parameters when the f value was significantly less than the model slope, shown by the relatively low variation in peak discharge in Fig. 5a, b, and d versus Fig. 5e.

We also tested the sensitivity of the extracted discharge to the inflow location, as shown in the Appendix (Fig. A3). The inflow location has little effect on the discharge at x=1000 m, except when the friction parameter f is greater than or approximately equal to the local slope at the measurement location. We tested the sensitivity of the extracted discharge to the initial velocity to see if the choice of Eq. (6) to estimate initial velocities had a significant effect on the results (also included Fig. A4 in the Appendix). We found low sensitivity to the input velocity, implying that the Voellmy parameters quickly regulate the simulated discharge. The remaining analyses shown for the numerical flume all used the x=300 m inflow location and Eq. (6) for the initial velocity for consistency.

The sensitivity of the discharge to the Voellmy parameters for the higher-inflow Qp was consistent with the results for the lower-inflow Qp discussed in the previous paragraph. However, the sensitivity to ξ was more prominent, with lower values leading to a more attenuated peak discharge (Fig. 6).

We calculated Froude numbers at the time of peak discharge at x=1000 m to compare with literature values. For the inflow Qp=1000 m3 s-1 cases, the calculated Froude values ranged from 0.32 to 4.06. Froude values ranged from 0.32 to 4.87 for the Qp=2940 m3 s-1 case (Fig. A5), with the higher values corresponding to lower flow-resistance parameters for both peak discharge scenarios. These Froude numbers are within the range of values reported for natural debris flows of 0.45 to 7.6 (Zhou et al., 2019).

By modelling a surge consisting of particles with basal resistance defined by two Voellmy materials, we found the results were most sensitive to the contrast between the ξ1 and ξ2 parameters (Fig. 7). The peak discharge was amplified when the resistance parameters were higher on the rising leg (higher f1, lower ξ1) relative to the inflow hydrograph and relative to the single flow-resistance cases (Fig. 5). This was a result of the lower resistance material on the falling leg pushing against the higher-resistance front. This amplification was most pronounced when the channel slope is steepest, and the peak rapidly attenuates as the channel flattens. The peak discharge is comparable to the peak discharge for the single material simulations when ξ1 is equal to ξ2.

We ran the models using the 24 debris-flow hydrographs summarized in Table 1, scaled to have a total volume of 100 000 m3 and a maximum Qp of 2940 m3 s-1. Figure 8 compares the extracted hydrographs at x=1000 m with f=0.2 and ξ=200 m s-2 for the Lattenbach inflow hydrographs. Figure 9 compares the extracted hydrographs with the Dorfbach and Spreitgraben inflow hydrographs. We selected these flow-resistance parameters because they did not result in attenuation of the triangular inflow hydrograph between Qp=1000 and 2940 m s-2 at the x=1000 m location (Figs. 5 and 6). There was some attenuation of the largest peak discharges, with the attenuation suspected to be related to the limit of precision in discretizing sharp peaks in the inflow hydrographs into particles and the tendency for the model to attenuate sharper peak discharges (e.g., the greater attenuation in the peak discharges shown in Fig. 6 versus Fig. 5). The Froude number for each case at the peak discharge ranged between 1.64 and 2.37 for the scaled, real hydrographs, which is within the expected range for a debris flow and what was found for the triangular hydrograph inputs.

Flow intensity indicators (depth, velocity, and IDF) were sensitive to the inflow conditions at four locations along the channel (Fig. 10). The sensitivity and variability in the maximum flow depth and IDF decreased with increasing distance down the channel. We attribute this to the material decelerating on the lower channel, implying the maximum depths are related to the final thickness of material. Since IDF is calculated with the square of velocity, as the velocities decreased, the variability in IDF related to variability in velocity also decreased. The cases with higher Qp and lower inflow duration tended to have greater maximum depth, velocity, and IDF; however, both relationships have significant scatter, as shown in the Appendix (Figs. A6 and A7).

There are four debris-flow fans on the north slope of Mount Currie, referred to as Currie A through D. The site is located approximately 4 km southeast of Pemberton, BC, Canada. The simulations shown here are for the eastern-most fan, Currie D, which has a watershed area of 1.7 km2 and extensive talus slopes composed of granitic rocks in the source area. A debris flow occurred on Currie D sometime between 3 and 12 July 2019, with a deposit volume of 100 000 m3 ± 5000 m3 (Zubrycky et al., 2021a). Zubrycky et al. (2021a) calculated the volume using lidar change detection with pre-event topography from 2017 and post-event topography collected in October 2019 utilizing a UAV-lidar system. Deposit mapping and UAV-lidar data for this event are available online (Zubrycky et al., 2021b). One velocity estimate from a superelevation calculation, two estimates of maximum flow depth, and two estimates of final deposit depth are from a field survey of the site in October 2019 (Fig. 11).

The inflow location was set near the fan apex, where local channel slope was 44 % (24∘). We used a friction parameter of f=0.13 based on a preliminary calibration, without variable-inflow conditions, to approximately match the event runout and three values of the ξ parameter. By varying the hydrograph inputs and Voellmy parameters, we observed variability in the simulated flow depths, deposit depths, and velocities at the points of the field observations (Fig. 12). The results for the area impacted are aggregated in the impact proportion plots in Fig. 12a–c. The proportion of cases impacting an area is represented as filled contours, with a value of 1 indicating all 24 simulations impacted that grid cell and a value of 0 indicating no impacts in any simulation. In the impact proportion plots, the area along the main channel was consistent between all input hydrographs and flow-resistance parameters; however, the three flow-resistance cases presented show slightly different avulsion patterns. The simulation results at the observation points (consistent with the field observations shown in Fig. 11) are shown in Fig. 12d–o.

Our results show some sensitivity to the velocity-dependent resistance, ξ; however, the results are generally much more sensitive to the variability in the inflow conditions, particularly higher in the channel. As with the results for the numerical flume, the variability decreases downslope as the material slows and deposits. Numerical results for each observation are provided in the Appendix, separated by input hydrograph (Tables A2 through A4). Similar to the results for the numerical flume, the highest depths and velocities tend to correspond to the largest peak discharges; however, there is significant scatter in that relationship.

Neff Creek is located approximately 25 km northeast of Pemberton, BC, Canada. A large debris flow occurred on the Neff Creek fan on 20 September 2015, with an estimated deposit volume of 220 000 m3 ± 30 000 m3 (Zubrycky et al., 2021a). The debris flow was triggered during a large-storm event following a dry summer (Lau, 2017). The event was characterized by significant erosion on the fan, with an estimated 40 % of the event volume entrained from the upper and medial portion of the fan and erosion depths of up to 14 m (Lau, 2017). The watershed area is 3.3 km2, and the source material is composed of sedimentary rocks.

Maximum flow depths and deposit depths on the lower fan were based on field estimates and change detection analysis for a portion of the fan (pre-event data from 2011 and post-event data from 2015) to check the deposit estimates. The impact area was mapped using satellite imagery (Fig. 13) and a field survey and is available online (Zubrycky et al., 2021b). Field observations of overlapping deposit lobes suggested that some material deposited during the event led to multiple avulsions and widening of the deposits. For the simulations, we selected the inflow location on the mid-fan, approximately where the event changed from being primarily erosional to primarily depositional, to avoid the added complexity of considering entrainment within the simulation. The local channel slope at the inflow location was 29 % (16∘).

In this study, we initially ran a set of simulations using Voellmy parameters f=0.10 and ξ=365 m s-2, calibrated assuming a single surge condition by Zubrycky et al. (2019), with 24 input hydrographs scaled with a c value of 0.2. We refer to these simulations as the variable-inflow case (Case 1). We then completed 10 iterations using the Monte Carlo sampling and deposition between runs outlined in Sect. 2.4 for the deposition-during-flow simulations (Case 2). Finally, we considered deposition with two materials (Case 3). For this case, we followed the same procedure as the deposition-during-flow case to modify the topography between flow stages but randomly assigned f=0.18 and ξ=365 m s-2 to 25 % of the particles to simulate the effect of heterogeneous flow resistance within an event. The results of these analyses are presented in Fig. 14.

Case 1 systematically underpredicts lateral runout extents, as well as flow and deposit thickness, at the field observation locations (Fig. 14). Cases considering deposition during an event resulted in greater lateral spreading, shorter runout distances, and thicker flow depths and deposits at the observation points, all of which are more consistent with the field observations than the single-hydrograph-input cases. The deposition cases have fewer runs than the variable-inflow case, but each run for the topography modification cases consisted of four randomly sampled hydrographs and stage volumes. This variation within each simulation could contribute to the increased variability at the observation points, despite having fewer runs total.

Our modelling results highlight the topographical and flow-resistance sensitivities of peak discharge estimates. We evaluated the interplay between Voellmy parameters and channel slope in a controlled manner using the numerical flume model topography. Figures 5 and 6 illustrate that the peak discharge is insensitive to the Coulomb friction parameter (f) at values lower than the channel slope but that peak discharge rapidly decreases as the channel slope becomes equal to or greater than the f value. This is an intuitive result as the best-fit f value in a Voellmy model can generally be estimated by the local slope where material deposition begins. The results also show that peak discharge increases as the velocity-dependent resistance parameter (ξ) increases, and the sensitivity to ξ increases with increasing peak discharge, all other factors equal. The positive correlation with peak discharge is expected as ξ generally controls the flow velocity, with higher ξvalues generating higher velocities.

The simulated peak discharges tend to decrease as the channel slope decreases (Figs. 5 and 6). The implication of this finding is that, when defining inflow conditions to calibrate the model to an observed debris-flow event where there is a downstream peak discharge estimate, the inflow peak discharge will have to be either equal to or greater than the downstream estimate if a single flow resistance is used. If the channel slope is steeper than the f value along the channel leading to the point where a discharge estimate is obtained, the downstream estimate peak discharge could be directly used as the inflow peak discharge upstream. However, if the local channel slope is near to or lower than the f value, the hydrograph will attenuate, and a higher inflow peak discharge will be necessary to match the downstream discharge observation. The exact value of the f parameter where significant attenuation will occur also depends on the ξ value used, with lower values of ξ leading to higher attenuation in the hydrograph. For example, the modelled peak discharge is approximately equal to the inflow peak discharge for all cases shown in Fig. 5a and b where the local channel slope is greater than the f value. Further in the simulation, the peak discharge attenuates significantly where the local channel slope is less than the f value (Fig. 5e) relative to the inflow hydrograph and the case where the local channel slope is still greater than the f value (Fig. 5d). Taken together, this suggests that, when calibrating a numerical runout model to a peak discharge estimate or using a design peak discharge estimate for forward analysis, the topography between the inflow location and the location of interest has an effect on the simulation. Furthermore, the effects of the flow-resistance parameters and the topography are interconnected.

We selected 24 inflow hydrographs from discharge-versus-time records for 21 debris flows observed in 3 natural channels. The observed discharge over time for each of these events arose from a unique combination of watershed conditions, triggering conditions, channel geometry, and measuring locations. Watershed characteristics and channel geometries are more or less constant for the records of multiple events from the Lattenbach site, and the same applies to the Dorfbach site. However, even with these two factors controlled, there was significant variability within the measured hydrographs at these sites (Table 1), which highlights the challenge of attempting to estimate a realistic inflow hydrograph for a runout analysis. The approach taken in this study is to use these real-event records as an indication of the potential variability in the inflow conditions that could occur while recognizing that they will not be exact analogs for events in other locations. Field monitoring of debris-flow discharge is becoming more common (Hürlimann et al., 2019), and collecting more of these data at more locations could provide valuable information for future modelling, allowing for a more refined selection of input hydrographs based on site-specific information.

We demonstrated the effect of changing inflow conditions based on the 24 inflow hydrographs by running each hydrograph through the idealized model geometry and extracting flow intensity metrics, consistent with those commonly used in hazard assessment or mitigation structure design. Flow depths, velocities, and impact intensities (as defined by Jakob et al., 2012) varied significantly (Fig. 10), and variability decreased as distance from the input location increased. The disaggregated results provided in the Appendix (Figs. A6 and A7) show that simulations with higher Qp and lower inflow duration tend to have higher impact intensities; however, there is significant scatter, suggesting that the variability is affected by other characteristics of the inflow hydrograph. When compared to the results using a simple, triangular hydrograph input, the maximum simulated flow depth, velocity, or intensity index for any case generally corresponded to the triangular hydrograph with a peak discharge set at the maximum allowable for the real hydrographs. This suggests that, when modelling flow intensities in channelized conditions, a triangular input hydrograph provides an adequate estimate of maximum intensity.

We examined two case histories, Mount Currie D from 2019 and Neff Creek from 2015, both in southwestern BC, Canada, using variable inflows based on the real hydrographs described in this study. The objective of the modelling was not to find the specific hydrograph input that resulted in the best match to the observed deposits but rather to examine the variability in impacts arising from the natural variability in discharge from the inflow hydrographs. The simulated flow intensities showed considerable variation, even with consistent Voellmy parameters (right panels in Figs. 12 and 14); however, the variation decreased towards the distal end of the deposit, which is consistent with the results from the idealized topography (Fig. 10).

The impact proportion plots for Currie D (left panel in Fig. 12) show distinct patterns of avulsions for the simulations with different ξ values. We attribute this behaviour to the channel being overwhelmed in the cases with lower ξ, which are slower, versus material leaving the channel after superelevating around bends with higher ξ. Along with the difference in avulsion locations, there were fewer avulsions with higher ξ. The simulations of the Neff Creek event showed the limitations of the variable-inflow conditions, with a consistent underprediction of the lateral extents and overprediction of the runout length of the event (Fig. 14), consistent with the single-surge results presented by Zubrycky et al. (2019). To overcome this limitation, we modelled a multiple-phase event, where the previous phases would modify the topography. This approach is reasonable as the observations from the real debris flows summarized in Table 1 indicate several instances of multiple debris flows within 1 d, modifying the topography for subsequent surges. We achieved a better match to the observed event by modelling the topographic modification. There are practical challenges to applying this topography modification method to a forward analysis as the number of phases where deposition will occur must be selected, increasing the number of model runs and model complexity. In the examples shown, considering 10 runs with random combinations of volume and inflow conditions resulted in a reasonable trade-off between capturing variability and runtime (the runtime for 10 runs with 4 phases each was approximately 32 h).

With the idealized topography, we found that a higher-resistance flow front amplified the peak discharge (Fig. 7). This approach demonstrates the idea of a coarse flow front leading to surging behaviour. The effect of mixing particles within a flow was also tested in the Neff Creek case study. Adding in a higher-resistance material with the f value approximately equal to the channel slope upslope of the roadway resulted in greater variation in deposit area and depths and enhanced the runout distance relative to the variable-inflow-with-deposition case. This increase in mobility is somewhat counter-intuitive, with 25 % of the particles having a higher frictional resistance. This result may be related to the higher-resistance particles maintaining higher flow depths, resulting in larger driving stresses within the flow. Future research could be conducted to examine how multiple flow-resistance particles mixed within a simulation can be used to represent flow behaviours more realistically.

We examined the sensitivity of the model results to assumptions regarding the location and velocity of the material entering the model. We found that there may be some numerical instability related to the smoothing length calculation early in the model, when only a small fraction of the particles have entered the model domain, or from initial accelerations or decelerations of the particles as they enter the model. We recommend placing the inflow location far enough upstream of any observations of interest to allow the simulated flow to “spin up”. We found that a distance of approximately 250 m upslope of the areas of interest had stable model results with little sensitivity to inflow location (e.g., Figs. A3 and A4); however, this is likely not a fixed value and will depend on the topography and flow characteristics in a given case.

We implemented a scaling method that used the shape of the observed hydrographs but allowed for a user-defined volume and peak allowable discharge. The empirical relationship to estimate peak allowable discharge (Eq. 5) is commonly used, and there is existing guidance for selecting the c parameter based on the debris-flow source material (Rickenmann, 1999; Ikeda et al., 2019). The data used to fit the regression between event volume and peak discharge are very scattered, and a wide range of c values fit a region of those data (Rickenmann, 1999; Ikeda et al., 2019), which limits the confidence in any specific peak discharge estimate. Despite this limitation, we used Eq. (5) because it provides a simple method to define an allowable peak discharge for the scaled hydrographs.

There are sources of uncertainty and simplifications relating to the modelling approach and field observations that should be considered when interpreting the results presented here. The single-phase, equivalent-fluid model used in this study considers the heterogeneous debris flow to be a fluid governed by simple flow-resistance models. Other models that consider solid and fluid motion independently still do not consider all the materials present in a flow, such as large woody debris or individual large boulders that can have an important influence on flows, for example, by creating channel blockages.

We applied hydrographs from specific locations to sites in different hydroclimatic and geomorphic settings. There is uncertainty associated with the field measurements, and the level of uncertainty is different for each field measurement. For example, the estimates of maximum flow depths are dependent on the observation of mud lines above the final deposit and the depth of the final deposit. The uncertainty in the final deposit depth is dependent on the quality of pre- and post-event topography data. Due to the complexity of variability in the natural systems leading to debris-flow events, our approach provides a practical way to explore the potential variability in debris-flow outcomes. When applying this approach to forward analysis, the significant uncertainty in the volume and mobility of future events must be recognized as these factors can vary widely, even at a single channel.

A further limitation of this work is that entrainment is not considered. Entrainment not only affects the volume of an event but also influences the dynamics through momentum transfer between the erodible bed and the flowing material as well as modification of effective basal resistance (Iverson and Ouyang, 2015). Future work could consider how entrainment interacts with these processes. Similar to the Monte-Carlo-type random sampling employed for the deposition within event cases for Neff Creek, a similar approach could be taken to also consider different entrainment rates. While this approach is computationally intensive, computational techniques such as GPU processing could significantly reduce runtimes and make large Monte Carlo simulations feasible for geohazard practitioners.