Debris flows are geological occurrences, made of high-concentration flows, that run down slopes and form thick deposits at the end of the flow. Debris flows can have high-bulk densities and can also occur in subaerial (air) and subaqueous (water) environments. Given their massive momentum, muddy debris flows have a great tendency to cause serious destruction and danger to both lives and properties on their way [1
], which could include communication cables, subsea pipelines, and offshore drilling rigs as well as increase maintenance cost of marine facilities. Therefore, there is a serious need for the predictions of debris flows, ranging from their velocity to the runout distance to be covered by such flows and erosional and depositional impacts, to avoid geohazards. Numerical modeling is an important strategy in the study of muddy debris flows, which will, therefore, be used in this current study.
To acquire a full resolution of the debris flow, full 3D modeling is suitable, even though it consumes a lot of computational efforts [1
]. A modeling strategy is the use of a depth-averaged solution to the governing equation with a thin-layer approximation. The thin layer can further be divided into one-dimensional (1D) e.g., [2
] and a two-dimensional (2D) e.g., [1
] models. Huang and Garcia [8
] proposed an analytical solution for a laminar Bingham flow from a set of continuity and momentum equations. It was reported that using a Newtonian flow may over-predict the propagation velocity due to zero yield stress, compared to a Bingham fluid with high yield stress. Instead of deriving the analytical solution, Imran et al. [3
] numerically solved the continuity and momentum equations for debris flow following the Herschel–Bulkley model, with a special focus on how the initial shape affects the final runout distance, shape, and frontal velocity of debris flow. The study showed that the initial shape did not affect the runout distance, but that a triangular shape has a higher velocity. De Blasio et al. [2
] investigated the hydroplaning property of a debris flow, which is suggested as a condition responsible for this long-runout distance. Their study revealed that the lubricating effect of ambient flow tends to decrease the local yield stress due to increased water pore pressures.
Further study on the importance of porosity in debris flow necessitated a study by Pastor et al. [9
], which showed that the failure of a landslide depends on the initial mass and initial pore pressure, whereby the porosity and presence of ambient fluid could aid the failure of the debris. Pudasaini [10
] developed a two-phase debris flow, which uses the Mohr–Coulomb plasticity to describe the solid stress, while the fluid stress is modeled like a solid-volume, fraction gradient-enhanced, non-Newtonian viscous stress. The study demonstrated that the viscous stress, generalized drag, and virtual mass make up the two-phase model and can be used to reproduce debris flow results and avalanches. Meng and Wang [11
] also enhanced the study of two-phase, gravity-driven flows using depth-integrated theory. The mixture theory was used to describe the mass and momentum of each phase, and the Mohr–Coulomb plasticity described the solid rheology, while the fluid phase was assumed to be a Newtonian fluid. Their study showed that the granular part moves faster in the presence of fluid and the deposition occurs at the downslope of an inclined plane. Additionally, during the flow, the fluid part goes faster than the granular phase, which creates more fluid at the front of the mixture. These studies enhance the knowledge of debris flows, even though they ignored the interaction between the flow and the bed.
Wang et al. [12
], in their study of hazard analysis and mitigation design against debris flows, used the full Navier–Stokes equations, which were combined with a rheological model of non-Newtonian fluid. Even though their model produced good results, they still considered the inadequacy of their model by the absence of entrainment equations. Braun et al. [13
] used a smoothed particle hydrodynamics (SPH) modeling method, combined with other different geomorphological index approaches to predict the forward simulation of the propagation and deposition of landslides, formation of landslide dams, and their evolution. Their findings showed that landslides that are deposited in a river can form a stable impoundment of the river, form a partial landslide dam, or cause a possible outburst of floods downstream. Studies by Qian and Das [7
] and Qian et al. [1
] revealed that yield stress, bottom slope, and initial failure height have an important effect on the runout distances by debris flow. Further understanding showed that the bottom slope and initial failure height are directly proportional, while yield stress is inversely proportional to runout distance. In these studies, the debris flows were assumed to be of constant density [1
] through the flow [14
]. Zhou et al. [15
], in their effort to understand the run-up mechanism and predict maximum run-up height for engineering and hazard mitigation of granular debris flow hazards along slit dams, used an analytical model, which was based on momentum approach, following a numerical study using the discrete element method (DEM). Their study emphasized one of the dangers of debris flow especially against slit dams and developed an equation to help engineers avoid dangerous overtopping. They further related Froude conditions with run-up at slit dams, whereby the higher the Froude number of an incoming inflow, the higher the maximum run-up along the slit dam. None of the abovementioned studies considered sediment erosion/deposition during the debris flow propagation and evolution. An experimental investigation by Lu et al. [16
] indicated that a debris volume increases at the end of a debris flow as a result of erosion of the bed materials. The impact erosion of bed materials is seen as a major factor in the entrainment by debris flow, while the erosion depth increases with an increase in debris mass. Toniolo et al. [17
] stressed the reworking of sediments into a debris flow. Their experimental study showed that a subaerial reworking can be as soon as the flow starts, while that of a subaqueous debris flow might be significant almost at the middle to the end of the flow where the deposit thickness is high. In the study by Takebayashi et al. [18
], they claimed that the consideration for entrainment of sediments is quite important, especially when the unstable sediment on a base rock is thick and the slope along the flow channel is steep. They used a 2D debris and land flow model, which considered both laminar and turbulence flow. They found out that fine materials during a debris flow could be transported by water flow from rain. They also showed that the volume of debris could grow during flow when entrainment occurs. Studies have shown that fluvial instabilities can arise as a result of a direct relationship at the interface of a flowing fluid and an erodible boundary. This could lead to the formation of particular structures or influence the flow at different scales [19
]. This phenomenon explains the onset of streambed erosion, whereby maximum stability is provided to the grains, leading to their entrainment into the flow. This increases the entrainment process or dynamics, thereby indicating that the onset of streambed erosion may not be an abrupt phenomenon [21
]. Since the majority of studies have ignored the interaction between debris flow and the bed during propagation, it was, therefore, essential to carry out a numerical study on the impacts of sediment erosion/deposition on debris flow propagation since the flow path of a debris flow in real-life cases can hardly be non-erodible. Additionally, this study revealed the comprehensive difference between turbidity currents and debris flows.
Some models have been developed to study the movement of debris flow over an erodible bed. However, they are either one-dimensional e.g., [14
], which do not express the lateral spreading of debris flow or ignored the feedback of bed deformation e.g., [9
], or did not consider the spatial and temporal change in concentration e.g., [3
]. Further, all these studies used different rheological models that have been proposed in the past, including quadratic shear stress model [28
], cohesive yield stress [29
], Bagnold’s dilatant fluid hypothesis [30
], and Chezy-type equation with a constant value of friction coefficient [31
], which leaves out the yield stress, an important parameter that determines the acceleration, deceleration, and deposition or the stopping of the debris flow when the channel slope reduces [14
]. This study, therefore, bridges this gap in the study of erosion and deposition impacts during the propagation of debris flow, while the debris rheology is described using the Herschel–Bulkley model, which is incorporated through the bed shear stress estimation. We, therefore, present three models that are based on the layer-averaged equations, including two debris models (QDnew and TCQD) and a turbidity current model (TC). The equations were then cast into a conservative hyperbolic system, and the finite volume method (FVM) and second-order, slope limiter-centered (SLIC) scheme were used to solve the equations numerically. Further, the weighted surface-depth gradient method (WSDGM) was included in the SLIC scheme to make the model well balanced [33
]. The models were then applied to the experimental cases by Wright and Krone [34
] and Mohrig et al. [35
] for model calibration. The three models (QDnew, TC, and TCQD) were also applied to a field case to study the impact of erosion and deposition on a debris flow. The TC and TCQD models were used to study the impact of erosion and deposition on debris flow while the QDnew model was used to describe a debris flow over a non-erodible bed. While TCQD expresses the yield stress and dynamic viscosity effects, TC, which represents a turbidity current model, lacks these. Finally, sensitivity analyses were carried out to investigate the impact of bed porosity, sediment size, concentration, and bed slope on the bed thickness.