This progress report summarizes work on a sewer pipe model using a dynamic wave approach. The model uses the complete dynamic wave method to solve the Saint Venant equations for unsteady flow in open channels. A finite difference scheme with the Preissmann four-point implicit method is used to solve the governing equations numerically. Sensitivity analyses were conducted to analyze the effects of varying the spatial and temporal discretization steps (Δx and Δt) on the model results.

1. Progress Report (20180425 水曜日):
Fei Liu 劉非
Department of Socio-Cultural Environmental Studies
Graduate School of Frontier Sciences
the University of Tokyo, Chiba, Japan
• Sewer Pipe Model
(Complete Dynamic Wave Model)

2. Sewer pipe model
- Dynamic wave model
𝜕𝑄
𝜕𝑥
+
𝜕𝐴
𝜕𝑡
= 0
𝜕𝑄
𝜕𝑡
+
𝜕(𝑄2/𝐴)
𝜕𝑥
+ 𝑔𝐴(
𝜕 𝑦
𝜕𝑥
+ 𝑆𝑓 − 𝑆 ) = 0
Governing Equations:
Assumptions for St. Venant Equations:
1. Flow is one-dimensional
2. Hydrostatic pressure prevails and vertical accelerations are
negligible
3. Streamline curvature is small
4. Bottom slope of the channel is small
5. Manning’s equation is used to describe resistance effects
6. The fluid is incompressible
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow
1

3. Sewer pipe model
- Dynamic wave model
Finite Difference Formulation with Preissmann 4-Point Implicit Scheme:
Space derivatives:
𝜕𝐴
𝜕𝑡
= 0.5
𝐴𝑖+1
𝑗+1
− 𝐴𝑖+1
𝑗
∆𝑡
+ 0.5
𝐴𝑖
𝑗+1
− 𝐴𝑖
𝑗+1
∆𝑡
𝜕𝑄
𝜕𝑡
= 0.5
𝑄𝑖+1
𝑗+1
− 𝑄𝑖+1
𝑗
∆𝑡
+ 0.5
𝑄𝑖
𝑗+1
− 𝑄𝑖
𝑗+1
∆𝑡
Time derivatives:
𝜕𝑄
𝜕𝑥
= 𝜃
𝑄𝑖+1
𝑗+1
− 𝑄𝑖
𝑗+1
∆𝑥
+ (1 − 𝜃)
𝑄𝑖+1
𝑗
− 𝑄𝑖
𝑗
∆𝑥
𝜕𝑦
𝜕𝑥
= 𝜃
𝑦𝑖+1
𝑗+1
− 𝑦𝑖
𝑗+1
∆𝑥
+ (1 − 𝜃)
𝑦𝑖+1
𝑗
− 𝑦𝑖
𝑗
∆𝑥
𝜕(
𝑄2
𝐴)
𝜕𝑥
= 𝜃
(
𝑄2
𝐴)𝑖+1
𝑗+1
−(
𝑄2
𝐴)𝑖
𝑗+1
∆𝑥
+
(1 − 𝜃)
(
𝑄2
𝐴)𝑖+1
𝑗
−(
𝑄2
𝐴)𝑖
𝑗
∆𝑥
Constant terms:
𝑆𝑓 = 𝜃𝑆𝑓 𝑖+1/2
𝑗+1
+ (1 − 𝜃)𝑆𝑓 𝑖+1/2
𝑗
𝐴 = 𝜃
𝐴𝑖+1
𝑗+1
+ 𝐴𝑖
𝑗+1
2
+ 1 − 𝜃
𝐴𝑖+1
𝑗
+ 𝐴𝑖
𝑗
2
= θ𝐴𝑖
𝑗+1
+ (1 − 𝜃)𝐴𝑖
𝑗
(𝑆𝑓)
𝑖+
1
2
=
𝑛𝑖
2
𝑄𝑖 𝑄𝑖
𝐾2 𝐴𝑖
2
𝑅𝑖
4
3
𝐴𝑖 =
𝐴𝑖 + 𝐴𝑖+1
2
2
𝑄𝑖 =
𝑄𝑖 + 𝑄𝑖+1
2
𝑅𝑖 =
𝐴𝑖
𝐵𝑖

4. Sewer pipe model
- Dynamic wave model
Initial condition:
1. Specify as input
2. Uniform Flow 𝑆 = 𝑆𝑓
( Manning formula)
3. Steady Gradually Varied Flow
Boundary condition
1. Critical flow
2. Weir-type flow
3. Overfall
4. Rating curves
…
Upstream
Downstream
1. Discharge hydrograph
2. Stage or depth hydrograph
Relation between Q and y
𝜕
𝑄2
𝐴
𝜕𝑥
+ 𝑔𝐴(
𝜕 𝑦
𝜕𝑥
+ 𝑆𝑓 − 𝑆 ) = 0
3

5. Sewer pipe model
- Dynamic wave model
Solving technique:
Newton Iterative Algorithm
𝐹𝑖 𝑥1, 𝑥2, 𝑥3, … , 𝑥2𝑁 = 0,
i = 1,2,3, … , 2N
𝐹𝑖 𝑥 + 𝛿 𝑥 = 𝐹𝑖 𝑥 +
𝑘=1
2𝑁
𝜕𝐹𝑖
𝜕x 𝑘
𝜕𝑥𝑗 + 𝑂 𝜕 𝑥2
,
i = 1,2,3, … , 2N
𝐽 =
𝜕𝐹𝑖
𝜕x 𝑘
𝐹 𝑥 + 𝛿 𝑥 = 𝐹 𝑥 + 𝐽 ∙ 𝜕 𝑥 + 𝑂 𝜕 𝑥2
𝐽 ∙ 𝜕 𝑥 = − 𝐹
𝑥 𝑛𝑒𝑤 = 𝑥 𝑜𝑙𝑑 + 𝜕 𝑥
(Gaussian Elimination to
Solve the Linear Equations)
4

6. Sewer pipe model
- Dynamic wave model
𝜕𝐶1
𝜕𝑄1
𝜕𝐶1
𝜕ℎ1
𝜕𝑀1
𝜕𝑄1
𝜕𝑀1
𝜕ℎ1
𝜕𝐶1
𝜕𝑄2
𝜕𝐶1
𝜕ℎ2
𝜕𝑀1
𝜕𝑄2
𝜕𝑀1
𝜕ℎ2
𝜕𝑈𝐵
𝜕𝑄1
𝜕𝑈𝐵
𝜕ℎ1
𝜕𝐶𝑖
𝜕𝑄𝑖
𝜕𝐶𝑖
𝜕ℎ𝑖
𝜕𝑀𝑖
𝜕𝑄𝑖
𝜕𝑀𝑖
𝜕ℎ𝑖
𝜕𝐶𝑖
𝜕𝑄𝑖+1
𝜕𝐶𝑖
𝜕ℎ𝑖+1
𝜕𝑀𝑖
𝜕𝑄𝑖+1
𝜕𝑀𝑖
𝜕ℎ𝑖+1
𝜕𝐶 𝑁−1
𝜕𝑄 𝑁−1
𝜕𝐶 𝑁−1
𝜕ℎ 𝑁−1
𝜕𝑀 𝑁−1
𝜕𝑄 𝑁−1
𝜕𝑀 𝑁−1
𝜕ℎ 𝑁−1
𝜕𝐶 𝑁−1
𝜕𝑄 𝑁
𝜕𝐶 𝑁−1
𝜕ℎ 𝑁
𝜕𝑀 𝑁−1
𝜕𝑄 𝑁
𝜕𝑀 𝑁−1
𝜕ℎ 𝑁
𝜕𝐷𝐵
𝜕𝑄 𝑁
𝜕𝐷𝐵
𝜕ℎ 𝑁
⋱
𝛿𝑄1
𝛿ℎ1
𝛿𝑄2
⋮
𝛿𝑄𝑖
𝛿ℎ𝑖
⋮
𝛿ℎ 𝑁−1
𝛿𝑄 𝑁
𝛿ℎ 𝑁
=
−𝑅𝑈𝐵
−𝑅𝐶1
−𝑅𝑀1
⋮
−𝑅𝐶𝑖
−𝑅𝑀𝑖
⋮
−𝑅𝐶 𝑁−1
−𝑅𝑀 𝑁−1
−𝑅𝐷𝐵
Partial derivatives:
Forward Finite Difference
Approximation
𝜕𝑓
𝜕𝑥
=
𝑓 𝑥 + 𝑒𝑝𝑠 − 𝑓(𝑥)
𝑒𝑝𝑠
Gaussian Elimination to
solve the linear equations
𝐽 ∙ 𝜕 𝑥 = − 𝐹
𝑥 𝑛𝑒𝑤 = 𝑥 𝑜𝑙𝑑 + 𝜕 𝑥
⋱
5

7. Sewer pipe model
- Dynamic wave model
Solution of finite difference equations:
𝑈𝐵 𝑄1, ℎ1 = 0
𝐶1 𝑄1, ℎ1, 𝑄2, ℎ2 = 0
𝑀1 𝑄1, ℎ1, 𝑄2, ℎ2 = 0
…
𝐶𝑖 𝑄𝑖, ℎ𝑖, 𝑄𝑖+1, ℎ𝑖+1 = 0
𝑀𝑖 𝑄𝑖, ℎ𝑖, 𝑄𝑖+1, ℎ𝑖+1 = 0
…
𝐶 𝑁−1 𝑄 𝑁−1, ℎ 𝑁−1, 𝑄 𝑁, ℎ 𝑁 = 0
𝑀 𝑁−1 𝑄 𝑁−1, ℎ 𝑁−1, 𝑄 𝑁, ℎ 𝑁 = 0
𝐷𝐵 𝑄 𝑁, ℎ 𝑁 = 0
j=0
i=1 i=4i=3i=2
𝑄1
0
, ℎ1
0
𝑄2
0
, ℎ2
0
𝑄3
0
, ℎ3
0 𝑄4
0
, ℎ4
0
j=1
i=1 i=4i=3i=2
𝑄1
1
, ℎ1
1
𝑄2
1
, ℎ2
1
𝑄3
1
, ℎ3
1 𝑄4
1
, ℎ4
1
𝐶1 𝑄1, ℎ1, 𝑄2, ℎ2 = 0
𝑀1 𝑄1, ℎ1, 𝑄2, ℎ2 = 0
Unknows:
𝐶2 𝑄2, ℎ2, 𝑄3, ℎ3 = 0
𝑀2 𝑄2, ℎ2, 𝑄3, ℎ3 = 0
𝐶3 𝑄3, ℎ3, 𝑄4, ℎ4 = 0
𝑀3 𝑄3, ℎ3, 𝑄4, ℎ4 = 0
𝑈𝐵 𝑄1, ℎ1 = 0 D𝐵 𝑄4, ℎ4 = 0
6

8. main.f90
mod_input1.f90
mod_input2.f90
mod_input3.f90
mod_funcv.f90
mod_par.f90
mod_newton.f90
Sewer pipe model
- Program structure
mod_steady.f90
mod_intrpl.f90
7

9. Sewer pipe model
- Flow Chart
Call mod_input1
Call mod_input2
Call mod_input3
Compute initial
conditions
(steady GVF)
last time
step ?
Call mod_funcv
Call mod_newton
(solve matrix)
Check
convergence
Yes
No No
END
Yes
Choose DBC
8

10. Sewer pipe model
- Sensitivity analysis on ∆𝑥 𝑎𝑛𝑑 ∆𝑡
Fixed time step ? Or variable time step
On ∆𝑡
𝐿 = 100 𝑚
𝑏 = 1 𝑚
19 reaches (N =20)
∆𝑥 = 100/19 𝑚
𝑆0 = 0.012
Manning Roughness coefficient (n) = 0.013
Initial Condition:
Steady Gradually Varied Flow
𝑏 = 1 𝑚
upstream downstream
𝑄1, ℎ1 𝑄 𝑁, ℎ 𝑁
i=1 i=N
Downstream Boundary Condition:
Critical Flow
𝐹𝑟 =
𝑣
𝑔
𝐴
𝐵
= 1, 𝑄 = 𝑣𝐴
Upstream Boundary Condition:
Discharge Hydrograph
0
0.5
1
1.5
2
2.5
3
0:00 3:00 6:00 9:00 12:0015:0018:0021:00 0:00
FLow(m^3/s)
Time (hh:mm)
Input Discharge Hydrograph
A Rectangular Section
Sewer Pipe:
On ∆x
𝐿 = 100 𝑚
𝑏 = 1 𝑚
∆𝑡 = 20𝑠
𝑆0 = 0.012
Manning Roughness coefficient (n) = 0.013
(
𝑄2
𝐴
)𝑖+1−
𝑄2
𝐴 𝑖
+ 𝑔𝐴𝑖(ℎ𝑖+1 − ℎ𝑖 + 𝑆 𝑓 𝑖
∆𝑥) = 0
9

11. Sewer pipe model
- Sensitivity analysis on ∆𝑡
∆𝑡 = 1𝑠 ∆𝑡 = 5𝑠 ∆𝑡 = 10𝑠 ∆𝑡 = 20𝑠 ∆𝑡 = 60𝑠
∆𝑡 = 120𝑠 ∆𝑡 = 300𝑠 ∆𝑡 = 900𝑠 ∆𝑡 = 1800𝑠 ∆𝑡 = 3600𝑠
depth
10

12. Sewer pipe model
- Sensitivity analysis on ∆𝑡
∆𝑡 = 1𝑠 ∆𝑡 = 5𝑠 ∆𝑡 = 10𝑠 ∆𝑡 = 20𝑠 ∆𝑡 = 60𝑠
∆𝑡 = 120𝑠 ∆𝑡 = 300𝑠 ∆𝑡 = 900𝑠 ∆𝑡 = 1800𝑠 ∆𝑡 = 3600𝑠
flow
11

13. Sewer pipe model
- Sensitivity analysis on ∆𝑥
∆𝑡 = 20𝑠
12

14. Sewer pipe model
- Comparison between Models
0
0.5
1
1.5
2
2.5
3
0:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00 0:00
FLow(m^3/s)
Time (hh:mm)
Input Discharge Hydrograph
𝐿 = 100 𝑚
𝑏 = 1 𝑚
𝑆0 = 0.012
n = 0.013
Initial Condition: steady flow equation
DBC: Critical Flow
∆𝑡 = 20𝑠
N = 20
𝐿 = 100 𝑚
𝑏 = 1 𝑚
𝑆0 = 0.012
n = 0.013
Initial Condition: cannot choose
DBC: free outfall
Dynamic Wave Model
Routing ∆𝑡 = 60𝑠
Reporting time step = 15 min
N = cannot choose
In this Model In EPA Model (SWMM)
A Rectangular section sewer pipe:
13

15. Sewer pipe model
- Comparison between Models
rect
∆𝑡 = 20𝑠
N = 20
14

16. Sewer pipe model
- Comparison between Models
0
0.5
1
1.5
2
2.5
3
0:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00 0:00
FLow(m^3/s)
Time (hh:mm)
Input Discharge Hydrograph
𝐿 = 100 𝑚
𝑏 = 1.5 𝑚
𝑆0 = 0.012
n = 0.013
Initial Condition: steady flow equation
DBC: Critical Flow
∆𝑡 = 20𝑠
N = 20
𝐿 = 100 𝑚
𝑏 = 1.5 𝑚
𝑆0 = 0.012
n = 0.013
Initial Condition: cannot choose
DBC: free outfall
Dynamic Wave Model
Routing ∆𝑡 = 60𝑠
Reporting time step = 15 min
N = cannot choose
In this Model In EPA Model (SWMM)
A Circular section sewer pipe:
15

17. Sewer pipe model
- Comparison between Models
Circular
∆𝑡 = 20𝑠
N = 20
16

18. 𝜕𝑄
𝜕𝑥
+
𝜕𝐴
𝜕𝑡
= 0
𝜕𝑄
𝜕𝑡
+
𝜕(𝑄2/𝐴)
𝜕𝑥
+ 𝑔𝐴(
𝜕 𝑦
𝜕𝑥
+ 𝑆𝑓 − 𝑆 ) = 0
Governing Equations:
Dampening of Inertial Terms:
A weighting factor SIGMA between 0 and 1 is computed that
damps out the contribution of the inertial terms as the Froude
number (Fr) gets closer to 1.0 and ignores them completely when
Fr>1.0 (i.e., supercritical flow). The equation for SIGMA is:
𝑆𝐼𝐺𝑀𝐴 = 1.0, 𝐹𝑂𝑅 𝐹𝑟 < 0.5
𝑆𝐼𝐺𝑀𝐴 = 2 1 − 𝐹𝑟 , 𝐹𝑂𝑅 0.5 ≤ 𝐹𝑟 ≤ 1.0
𝑆𝐼𝐺𝑀𝐴 = 0, 𝐹𝑂𝑅 𝐹𝑟 > 1.0
Fr is computed based on the midpoint depth in the conduit.
𝜕𝑄
𝜕𝑥
+
𝜕𝐴
𝜕𝑡
= 0
𝑆𝐼𝐺𝑀𝐴 ∗ (
𝜕𝑄
𝜕𝑡
+
𝜕
𝑄2
𝐴
𝜕𝑥
) + 𝑔𝐴(
𝜕 𝑦
𝜕𝑥
+ 𝑆𝑓 − 𝑆 ) = 0
17

19. Thank you!
5564443124@edu.k.u-tokyo.ac.jp
fei.liu@whu.edu.cn