1. Mathematical Background
1-D Dynamic Modelling

2. Fundamental Basis MIKE 11
Modelling of unsteady flow is based on three fundamental elements:
A differential relationship expressing the physical laws
A finite difference scheme producing a system of algebraic equations
A mathematical algorithm to solve these equations

3. Fundamental Basis MIKE 11 PHYSICAL SYSTEM River Network Flood Plains
Structures
PHYSICAL LAWS
Conservation of Mass
Conservation of
Momentum
SCHEMATIZE
Represent by a simple
Equivalent System
DISCRETIZE
Express as a Finite
Difference Relation
BOUNDARIES
NUMERICAL MODEL
OUTPUTS

4. Saint-Venant Equations
MIKE 11
General Assumptions:
Incompressible and homogenous fluid
Flow is mainly one-dimensional, (i.e. uniform velocity & WL horizontal in cross-section)
Bottom slope is small
Small longitudinal variation of cross-sectional parameters
Hydrostatic pressure distribution.
Continuity Equation (Conservation of Mass)
Momentum Equation (Conservation of Momentum) (Newton’s 2’nd Law)

5. Conservation of Mass MIKE 11 I.e.: And:
Net increase of Mass from Time1 to Time2 =
Net Mass Flux into control volume (Time1 to Time2) +
Net Mass Flux out of control volume (Time1 to Time2)
I.e.:
And:

6. Conservation of Momentum
MIKE 11
Net increase of Momentum from Time1 to Time2 =
Net Momentum Flux into control volume (Time1 to Time2) +
Sum of external forces acting over the same time G

7. Conservation of Momentum
MIKE 11
Momentum = Mass per unit length * Velocity
Momentum Flux = Momentum * velocity
Pressure Force = Hydrostatic Pressure P
Friction Force = Force due to Bed Resistance
Gravity Force = Contribution in X-direction
Momentum = Momentum Flux Pressure - Friction + Gravity

8. Conservation of Momentum
MIKE 11
Momentum:
Momentum Flux
Pressure Term:
Friction Term:
Gravity Term:

9. Differential Equations
MIKE 11
Wave Approximations: Kinematic Wave
Diffusive Wave
Fully Dynamic Wave

10. Kinematic Wave MIKE 11 Includes: Applications: 1. Bed Friction Term
2. Gravity Term
Applications:
+ Steep Rivers
- Backwater Effects NOT applicable
- Tidal Flows NOT applicable

11. Diffusive Wave MIKE 11 Includes: Applications:
1. Hydrostatic Gradient Term
2. Bed Friction Term
3. Gravity Term
Applications:
+ Relatively Steady Backwater Effects
+ Slowly Propagating Flood Waves
- Tidal Flows NOT applicable

12. Fully Dynamic Wave MIKE 11 Includes: Applications:
1. Acceleration Term
2. Hydrostatic Gradient Term
3. Bed Friction Term
4. Gravity Term
Applications:
+ Fast Transients
+ Tidal Flows
+ Rapidly changing backwater effects
+ Flood waves

13. High Order Fully Dynamic Wave
MIKE 11
Includes:
1. Acceleration Term
2. Hydrostatic Gradient Term
3. Bed Friction Term (Modified compared to Fully Dynamic Wave)
4. Gravity Term
Applications:
+ Fast Transients
+ Tidal Flows
+ Rapidly changing backwater effects
+ Flood waves
+ Steep Channels

14. Implicit Abbot-Ionescu 6-point scheme
Solution Scheme
MIKE 11
Implicit Abbot-Ionescu 6-point scheme

15. Solution Scheme MIKE 11 h Q h Q h Q h Momentum equation
Finite difference scheme: Abbott-Ionescu 6-point h Q h Q h Q h
Momentum equation
Continuity equation

16. Implicit Abbot-Ionescu 6-point scheme
Solution Scheme
MIKE 11
Implicit Abbot-Ionescu 6-point scheme t
n+1
unknown dt n
known
Q / h
h/ Q
Q / h dx dx X
j-1 j
j+1

17. Solution Scheme MIKE 11 Solution method Double Sweep algorithm
Nodal point solution
Grid point solution
Matrix bandwidth minimization

18. Model Data Requirements
MIKE 11
Solution of governing flow equations requires detailed descriptions of:
Catchment Delineation
River and Floodplain Topography
Hydrometric Data for Boundary Conditions
Hydrometric Data for Calibration / Validation
Man-made Interventions

19. Stability MIKE 11 Courant Number: Example: D=10;V=1; dX=1000
Given: Initial Conditions and Finite Difference Approximation which is consistent
Then: Stability is the necessary and sufficient condition for convergence
Stability analysis can only be done for linear differential eq.
Explicit methods: Conditionally stable (Cr < 1) Implicit methods: Unconditionally stable
Courant Number:
Example: D=10;V=1; dX=1000

20. Boundary Conditions MIKE 11
Discharge, Q : Upstream of River Lateral Inflow Closed End (Q=0) Discharge Control Pump
Water Level, h : Downstream River boundary Outlet in Sea (tide, wind) Water level control
Q/h Boundary : Downstream Boundary (Never upstr.) Critical Outflow from Model Q Q Q
h or Q/h
In general, Boundaries should be located where key investigation area is not directly affected by boundary condition!

21. Initial Conditions MIKE 11 Always specify h and Q for simulation:
Possibilities:
Specify manually (in HD Parameter Editor)
Select from HOTSTART file
Automatically calculated (Steady state approach)
Safest to Start with Lower Levels.
Never initialize a Flood problem with floodwaters in the flood plains.

22. Data Needs MIKE 11 Reliable Data required: ‘GARBAGE IN = GARBAGE OUT’
Topography Data: Width, Area, Volume of inundated plains Schematization of Model Aerial/Satellite/Radar images of flood extents Reservoir data (control strategy, spillway etc.) Cross section data
DATUM - Same reference level for all data!
Hydraulic Data: Stage & Discharge hydrographs Rating Curves Peak Water level during significant events Used for Boundary conditions and Calibration

23. Calibration
MIKE 11
Adjustment of Model parameters to obtain agreement between simulated and measures values.
Items:
Reservoirs/storage area - storage volume must be correct
Unsteady flow - agreement (simulated & measured) - usually adjust roughness parameters
Equivalent longitudinal conveyance - longitudinal profile shows obvious errors
Main features :
Timing of Peak
Value of Peak
Shape of Hydrograph
Accuracy:
No quantitative criterion can be given (very much dependent on data quality)
Each case is unique

24. Main parameter to Modify during Calibration process:
MIKE 11
Main parameter to Modify during Calibration process:
River Bed Roughness.
Modification of River Bed Roughness in MIKE 11:
Relative resistance (variation with cross section Width)
Resistance factor (variation with Water level)
Resistance number (longitudinal variation)
Time Series (seasonal variation)

25. Verification MIKE 11 Verify Model’s Performance - VERY IMPORTANT !
Do not use data from Calibration period!
Actions to perform before application of Model:
1) Setup of River Model 2) Calibration (preferably data from several periods) 3) Verification (do not use data from Calibration period) 4) Application (‘production runs’)