Mathematical Background
MIKE 11
1-D
Dynamic
Modelling
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
Fundamental Basis
MIKE 11
PHYSICAL SYSTEM
PHYSICAL LAWS
River Network
Flood Plains
Structures
Conservation of Mass
Conservation of
Momentum
SCHEMATIZE
DISCRETIZE
Represent by a simple
Equivalent System
Express as a Finite
Difference Relation
BOUNDARIES
NUMERICAL MODEL
OUTPUTS
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)
Conservation of Mass
MIKE 11
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)
h(t+dt)
h(t)
at time t+dt
at time t
Q
Q
Q
 dx
x
dx
  Q  d t    (Q 
I.e.:
Q
x
A

t
Q
A
dx )dt    dA  dx   
dx  dt
x
t
And:
Q
x
 B 
h
 0
t
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
h(t)
H
G
P
F
x
P+ P
z(t)
Conservation of
Momentum
Momentum
Momentum Flux
Pressure Force
Friction Force
Gravity Force
MIKE 11
= Mass per unit length * Velocity
= Momentum * velocity
= Hydrostatic Pressure P
= Force due to Bed Resistance
= Contribution in X-direction
M
( M U )
P
Ff
Fg




t
x
x
x
x
Momentum =
Momentum Flux
+ Pressure - Friction + Gravity
Conservation of
Momentum
MIKE 11
Momentum:
M    H  b U
Momentum Flux
Mf    H  b U U
Pressure Term:
P 
1
gbH
2
2
gU
Friction Term:
F  x b 
Gravity Term:
P  gAS 0
C
2
2
Differential
Equations
MIKE 11
Q2
 (
)
gQ Q
Q
h
A

 g  A

0
2
t
x
x
C A R
Wave Approximations:
Kinematic Wave
Diffusive Wave
Fully Dynamic Wave
Q
A

 q
x
t
Kinematic Wave
Includes:
1. Bed Friction Term
2. Gravity Term
Applications:
+ Steep Rivers
- Backwater Effects NOT applicable
- Tidal Flows NOT applicable
MIKE 11
Diffusive Wave
Includes:
1. Hydrostatic Gradient Term
2. Bed Friction Term
3. Gravity Term
Applications:
+ Relatively Steady Backwater Effects
+ Slowly Propagating Flood Waves
- Tidal Flows NOT applicable
MIKE 11
Fully Dynamic Wave
Includes:
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
MIKE 11
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
Solution Scheme
Implicit Abbot-Ionescu 6-point scheme
MIKE 11
Solution
Scheme
MIKE 11
Implicit Abbot-Ionescu 6-point scheme
t
unknown
n+1
dt
n
known
Q/h
dx
0
0
j-1
Q/h
h/ Q
dx
j
j+1
X
Solution Scheme
Solution method
Double Sweep algorithm
Nodal point solution
Grid point solution
Matrix bandwidth minimization
MIKE 11
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
MIKE 11
Stability
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:
Implicit methods:
Courant Number:
Conditionally stable (Cr < 1)
Unconditionally stable
Cr  ( g  D  v ) 
Example: D=10;V=1; dX=1000
t 
t
x
X
1000m

 100 sec
g  D V
9.81 10
Boundary Conditions
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
MIKE 11
Q
Q
Q
h or Q/h
In general, Boundaries should be located where key investigation area is not
directly affected by boundary condition!
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.
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
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
Accuracy:
• No quantitative criterion can be given
(very much dependent on data quality)
• Each case is unique
Main features :
• Timing of Peak
• Value of Peak
• Shape of Hydrograph
Calibration
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)
Verification
MIKE 11
Verify Model’s Performance - VERY IMPORTANT !
Do not use data from Calibration period!
Actions to perform before application of Model:
1)
2)
3)
4)
Setup of River Model
Calibration (preferably data from several periods)
Verification (do not use data from Calibration period)
Application (‘production runs’)