# SewerSedFoam: A Model for Free Surface Flow, Sediment Transport, and Deposited Bed Morphology in Sewers

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Hydrodynamics

#### 2.2. Computational Mesh and Finite Area Method (FAM)

#### 2.3. Sediment Fractions

_{A}, P

_{C}, and P

_{E}, can be made constant to suit different types of sewer systems or set to be variable based on the quantity of each sediment fraction that is eroded or deposited.

#### 2.4. Suspended Sediment Transport

_{i}is the concentration of type C or E sediment,

**u**is the flow velocity vector (all vectors in equations are represented with bold text), $\nabla $ is the gradient operator (i.e., ∂/∂x, ∂/∂y, ∂/∂z), ν is the kinematic viscosity of water, ν

_{t}is the turbulent kinematic viscosity from the turbulence model, and

**u**is the sediment settling velocity vector calculated for a sediment fraction i through Stokes’ law [30]:

_{si}_{si}is the density of a suspended sediment fraction i, ρ is the density of water, d

_{i}is the particle diameter for a sediment fraction i, and

**g**is the acceleration due to gravity in vector form.

#### 2.5. Bedload Transport

**q**is the bedload transport rate vector, d

_{b}_{A}is the median diameter of type A sediment, p

_{ef}is the probability of moving particles near the bed Equation (6), and

**u**is the mean velocity vector of a bedload particle Equation (8).

_{b}_{d}is the dynamic friction coefficient (set to a constant value of 0.6), θ

_{A}is the Shields parameter or dimensionless shear stress applied on bedload particles Equation (7), and θ

_{C,A}is the bed critical Shields parameter to initiate motion. It is important to note that the bedload transport rate calculations are only initiated when θ

_{A}values are greater than θ

_{C,A}.

_{B}is the bed sediment density,

**u**is the friction velocity vector (calculated through the turbulence model in OpenFOAM), s is the relative sediment density to the fluid (= ρ

_{f}_{s}/ρ), and ||

**X**|| represents the magnitude of a vector

**X**.

**u**can be approximated by [32]:

_{b}**u**is the near bed tangential velocity.

_{τ}**u**is more complicated and involves solving a system of nonlinear equations based on the forces acting on a bedload particle. This has been implemented in SSF based on the method detailed in Roulund et al. [31], but is not detailed in this manuscript due to the subsequent use of 2-D flow models.

_{b}#### 2.6. Critical Shields Parameter

_{C0,A}and d

_{*,A}are the initial critical Shields parameter and dimensionless diameter of a Type A sediment. The latter is defined for a sediment fraction i as

_{*i}, d

_{i}, and s

_{i}are the dimensionless diameter, diameter, and relative sediment density of a sediment fraction i, respectively.

_{0,E/C}and d

_{*E/C}is the initial critical Shields parameter and dimensionless diameter, respectively, of sediment Types C or E.

_{D}), an erosion factor (ϕ

_{E}), and the morphology of the deposited bed (ϕ

_{M}). Hence, the final critical Shields parameter was calculated through:

_{C,i}and θ

_{Co,i}are the final and initial critical Shields parameter, respectively, for a sediment fraction i.

_{D}is increased from a default value of one based on increasing applied shear stress from flow velocity until a user set maximum bed height is achieved:

_{i}, θ

_{io}, and θ

_{i,max}are the shear stresses applied on a sediment fraction i, expressed as a dimensionless Shields parameter, at the current time step, at the beginning of the simulation, and when deposited bed height is at h

_{max}respectively, h is the total height of the deposited bed, h

_{max}is a user-set maximum deposited bed height at which deposition scaling stops, t

_{dep}is the time for which deposition has been occurring, and t

_{dsp}is a user-set time after which the deposition scaling initiates. This equation results in an increase in the critical Shields parameter of a sediment after deposition occurs for t

_{dsp}seconds until a maximum bed height is reached, after which it remains constant.

_{E}is used to simulate an important behavior in the erosion of sewer sediments: the bed can be eroded relatively easily until a certain depth, where it has consolidated and/or settled, and needs a higher shear stress to erode further [18]. Hence, during erosion, the critical Shields parameter of the sediment fractions increase linearly from their initial values to a user-defined maximum value until a third of the bed is eroded, after which it remains constant at the maximum value. To achieve this change, ϕ

_{E}is calculated through:

_{er}is the current depth of erosion, h

_{esp}is the setpoint of erosion depth at which erosion scaling starts to occur, θ

_{C0,i}is the initial critical Shields parameter of a sediment fraction i calculated through Equations (9), (11), and (12), and θ

_{Cmax,i}is the maximum critical Shields parameter of a sediment fraction i set by the user.

_{M}is calculated through [31]:

_{s}is the static friction coefficient.

#### 2.7. Bed Morphology

**q**is the divergence of the bedload transport rate, SS

_{b}_{E}is the erosion rate of suspended sediment from the bed, and SS

_{D}is the deposition rate of suspended sediment onto the bed.

_{A}is the proportion of type A sediment in the bed, SS

_{E,C}and SS

_{E,E}are the erosion rate of suspended sediment types C and E, respectively, and SS

_{D,C}and SS

_{D,E}are the deposition rates of both suspended sediment types.

**q**is calculated from

_{b}**q**

_{b}values obtained through Equation (5). Some filtering is in place to prevent instabilities in bed height from the $\nabla $·

**q**term around the boundaries of the bed and to conserve mass by preventing deposition changes beyond the existing volume of bedload sediment.

_{b}_{E,C}and SS

_{E,E}are calculated empirically using C

_{E}, the equilibrium concentration of suspended sediment at a reference height [36]:

_{E,i}is the erosion rate of suspended sediment type i, P

_{i}is the proportion of suspended sediment i in the bed, and C

_{E,i}is defined as:

_{ref}is the reference height i.e., the height of the first cell from the bed in the computational mesh [33] and T

_{*,i}is the transport stage parameter of a sediment i defined as:

_{i}is the Shields parameter applied on a suspended sediment type i in the bed, calculated using Equation (7), and θ

_{Ci}is the critical Shields parameter of the same sediment type.

_{B,i}[16]:

_{B}values through [33]:

_{B,it}

_{+1}is the bed concentration in the next time step and C

_{B,it}is the bed concentration in the current time step for a sediment type i.

#### 2.8. Model Implementation

## 3. Model Validation

#### 3.1. Sediment Deposition

#### 3.1.1. Mesh Setup and Flow Initialization

**u**), turbulence boundary conditions (turbulent energy, k, and specific dissipation rate, ω), and volume fraction (α) are introduced through the lower section of the inlet to simulate the flow of wastewater through the channel. The remaining boundary conditions were set according to general two-phase pipe flow requirements for interFoam detailed in Shuard et al. [37]. The bed was modelled as a rough wall with a roughness height, k

_{s}, of 0.5 mm as this produced the closest results to the experimental data; this was achieved by modelling the turbulent viscosity, ν

_{t}, of the bed with OpenFOAM’s rough wall function, nutRoughWallFunction. The mesh was refined sufficiently to enable the use of wall functions for all turbulence parameters at the walls. Lastly, the timestep of the simulation and the mesh refinement were balanced to achieve a maximum Courant number of 0.8.

#### 3.1.2. Sediment and Deposited Bed Module Setup

^{3}. The effect of the mixing on the characteristics of the overall influent sediment appears to have been more complicated with a reduction in diameter of volatile solids and increase in diameter of non-volatile solids [35].

_{E}and SS

_{D}, were used to determine the inlet concentration of the Type C sediment fraction for Scenarios 6 and 7. The concentration values obtained and used in the simulations (Table 2) were significantly lower than the measured total suspended solids (TSS) concentrations in the pilot plant experiment. As discussed previously, this is not a cause for concern as SSF is not aiming to model all the sediment in the influent wastewater but only the fraction that is likely to settle. The bedload transport module has no inlet concentration or boundary conditions, instead the bedload transport rate is calculated at each timestep and it has a continuous impact on the deposited bed height through the divergence of the transport rate and the static proportion of sediments in the bed.

^{−9}m, it was found that a large scaling factor could be applied to the change in bed height during each time step to speed up the run times with only minor impacts on the final results. The main impact of scaling was a reduced overall stable deposited bed height with little to no change in the deposition pattern. Hence, to speed up the simulation, the change in bed height was scaled to 100 times its value each time step alongside a small increase in the inlet concentrations to counteract the reduced bed height.

_{max}, was changed to the stable deposited bed heights from the end of Day 2. The change in h

_{max}increased the critical Shields parameter for the deposited sediment based on Equation (14). Further, any deposition beyond a maximum bed height, 10 mm for Scenario 6 and 15 mm for Scenario 7, was suppressed to keep the bed height within reasonable limits. Again, SSF was run with the new values of h

_{max}until a stable deposited bed height was maintained for around 5 h, which occurred close to the end of Day 3 and was also used as the deposited bed height value for Day 4.

#### 3.1.3. Results and Discussion

#### 3.2. Sediment Erosion

_{+5}in Shahsavari et al. [38], which has little to no pipe slope. Further, the area of interest in this case study is the lower rectangular channel of the sewer where the deposition and erosion of sediment occur. Hence, in the SSF model, only this area is meshed to reduce the complexity of the simulation.

#### 3.2.1. Mesh Setup and Flow Initialization

_{s}, was increased to 5 mm based on the increased sediment diameters in this case and wall functions were used for all turbulence parameters at the walls. Besides these changes, the other boundary conditions are similar to the previous deposition case. A constant average flow velocity was calculated for the inlet boundary condition based on the available parameters, flow height and volumetric flow rate, and the cross-sectional area of the sewer to represent the peak flow velocity experienced during the flush [38]. However, this value does not consider the impacts of the sewer geometry which are likely to be significant. The calculated value was increased slightly to incorporate the likely higher flow velocities in the narrower lower channel to obtain a final inlet value of 1.8 m/s, representing the peak flow during the flushing. The lack of detailed flow velocity measurements within the channel is a major reason for the inability to use this case study as a complete validation for SSF’s erosion.

#### 3.2.2. Sediment and Deposited Bed Setup

_{Cmax}. The inlet suspended sediment concentration of Type E sediment, C

_{E}, was varied to change the quantity of suspended sediment in the sewer. A baseline C

_{E}value, 40 mg/L, was determined with initial model runs to result in a suspended sediment deposition rate that was less than half the magnitude of the erosion rate. Lastly, as the entire height of the flow is not modelled, the suspended sediment does not settle in a realistic way across the channel. To avoid this issue, SSF was modified so that the inlet suspended sediment does not settle through the impact of Equation (4) and remains constant across the channel. However, any eroded suspended sediment from the deposited bed settles normally. All five erosion cases were run until a stable bed height was achieved.

#### 3.2.3. Results and Discussion

_{Cmax}/θ

_{Co,i}, was not as significant, with less than a 2% decrease when it was changed from two to four times the initial critical Shields parameter (Figure 9b). The lack of impact from the bed consolidation multiplier is likely due to the applied Shields parameter from the flow being greater than several times the suspended sediment’s critical Shields parameter combined with the erosion of suspended sediment being the driver for most of the change in bed height. Overall, the quantity of the bed eroded in all the simulated cases, which range between 31% and 44.6% of the bed, is higher than the recorded value in the field, which was around 21% on average for 850 m of sewer after the flushing gate [38]. This indicates that the equivalent inlet sediment concentration and/or consolidation multiplier values used in the simulations probably need to be higher; however, the data limitations for this case study, particularly the velocity patterns in the lower channel, makes it difficult to establish this with any certainty. Overall, although SSF is not validated for a large magnitude erosion event, the initial results do appear to follow the expected patterns of erosion seen in the field and reasonable results can be obtained through the correct selection of simulation parameters.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of sewer sediment settling, erosion, and transport processes in gravity sewers along with their deposited bed interactions. Descriptions for suspended sediment transport, bedload transport, and bed stabilization processes based on Butler and Davies [13], Butler et al. [10], and Ashley et al. [2], respectively.

**Figure 2.**A flowchart representing the overall framework of the SSF solver and the interactions between the major components, i.e., flow, Suspended Sediment (SS) transport, Bedload (BL) transport, and bed morphology. The green shading indicates initial user inputs, the blue shading indicates processes that occur within the Finite Volume mesh, and the orange shading represents processes in the Finite Area deposited bed mesh.

**Figure 3.**Conceptual diagram of the A Coruña pilot sewer mesh illustrating the boundary definitions and water level. Please note that this diagram is not to scale.

**Figure 4.**Velocity magnitude results at the lower flow area after 1 day of deposition from the A Coruña pilot plant experiments and 6 m from the inlet of the model prior to any deposition occurring.

**Figure 5.**Measured average deposited bed heights from the A Coruña pilot sewer compared to the modelled SSF average deposited bed height for (

**a**) Scenario 6 and (

**b**) Scenario 7. Please note that the modelled deposited heights have been averaged for the entire length of the sewer channel besides the last 0.2 m in which the change in bed height was suppressed.

**Figure 6.**Modelled velocity magnitude 6 m from the inlet of the pilot sewer for Scenario 7 in the initial case, after 2 days of bed deposition, and after 4 days of bed deposition.

**Figure 7.**Conceptual outline of the boundary conditions and mesh representing a section of the Trunk Sewer from Paris.

**Figure 9.**Quantity of bed eroded at the end of the (

**a**) EC1, EC2, and EC3 simulations and the inlet Type E sediment concentrations used in the simulations and (

**b**) EC2, EC4, and EC5 simulations and the bed consolidation multiplier, the magnitude of increase between the reference, and the calculated critical Shields stress, i.e. θ

_{Cmax}/θ

_{Co,i}, used in the simulations.

**Table 1.**Description, likely characteristics, and transport method of the three sediment fractions implemented in SewerSedFoam (SSF).

Sediment Type | Description | Diameter Range (mm) | Density Range (kg/m^{3}) | Transport Mode |
---|---|---|---|---|

A | Coarse mineral sand and gravel that tends to dominate sewer deposited beds. | 1–6 ^{1} | 1600–2000 ^{1} | Transported exclusively as bedload. |

C | Fine-grained organic sediment in the silt or sand diameter range that tends to form easily eroded deposits on top of Type A sediment. May increase bed cohesion by introducing additional smaller diameter particles in the deposited bed and due to its high organic content. | 0.04–1 ^{1,2} | 1000–1200 ^{1,2} | Transported exclusively as suspended sediment. |

E | Fine-grained mineral sand. May increase bed cohesion by introducing additional smaller diameter particles in the deposited bed. | 0.063–2 ^{1,2} | 1600–2000 ^{1,2} | Transported exclusively as suspended sediment. |

**Table 2.**Sediment parameters used in SSF for the A Coruña sediment deposition validation Scenarios 6 and 7.

Sediment Module | Parameter | Value |
---|---|---|

Suspended Sediment Transport | Type C sediment diameter (d_{C}) | 75 µm |

Type C sediment density (ρ_{C}) | 1125 kg/m^{3} | |

Type C sediment inlet concentration (C) | Scenario 6: 19.7 mg/L Scenario 7: 23.5 mg/L | |

Type C sediment mass proportion in bed (P_{C}) | 0.5 | |

Bedload Transport | Type A sediment diameter (d_{A}) | Scenario 6: 280 µm Scenario 7: 405 µm |

Type A sediment density (ρ_{A}) | Both Scenarios: 1795 kg/m^{3} | |

Type A sediment mass proportion in bed (P_{A}) | 0.5 | |

Deposited bed morphology | Bed porosity (n) | 0.45 |

Initial maximum bed height for critical Shields parameter scaling (h_{max}) | 2 mm |

Sediment Module | Parameter | Value |
---|---|---|

Suspended Sediment Transport | Type E sediment diameter (d_{E}) | 0.2 mm |

Type E sediment density (ρ_{E}) | 2000 kg/m^{3} | |

Initial Type C sediment mass proportion in bed (P_{E}) | 0.5 | |

Bedload Transport | Type A sediment diameter (d_{A}) | 4.15 mm |

Type A sediment density (ρ_{A}) | 2317 kg/m^{3} | |

Initial Type A sediment mass proportion in bed (P_{A}) | 0.5 | |

Deposited bed morphology | Bed porosity (n) | 0.41 |

**Table 4.**Summary of difference in sediment consolidation magnitude, θ

_{Cmax,}and inlet suspended sediment concentration, C

_{E}, for the different erosion cases. θ

_{Ci}refers to the final critical Shields parameter calculated using Equation (13) for a sediment fraction i.

Erosion Case | θ_{Cmax} | C_{E} |
---|---|---|

EC1 | 2 × θ_{Ci} | 20 mg/L |

EC2 | 2 × θ_{Ci} | 40 mg/L |

EC3 | 2 × θ_{Ci} | 60 mg/L |

EC4 | 3 × θ_{Ci} | 40 mg/L |

EC5 | 4 × θ_{Ci} | 40 mg/L |

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**MDPI and ACS Style**

Murali, M.K.; Hipsey, M.R.; Ghadouani, A.; Yuan, Z. SewerSedFoam: A Model for Free Surface Flow, Sediment Transport, and Deposited Bed Morphology in Sewers. *Water* **2020**, *12*, 270.
https://doi.org/10.3390/w12010270

**AMA Style**

Murali MK, Hipsey MR, Ghadouani A, Yuan Z. SewerSedFoam: A Model for Free Surface Flow, Sediment Transport, and Deposited Bed Morphology in Sewers. *Water*. 2020; 12(1):270.
https://doi.org/10.3390/w12010270

**Chicago/Turabian Style**

Murali, Madhu K, Matthew R Hipsey, Anas Ghadouani, and Zhiguo Yuan. 2020. "SewerSedFoam: A Model for Free Surface Flow, Sediment Transport, and Deposited Bed Morphology in Sewers" *Water* 12, no. 1: 270.
https://doi.org/10.3390/w12010270