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MIKE 11
1-D1-DDynamic Dynamic ModellingModelling
Mathematical Mathematical BackgroundBackground
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
MIKE 11Fundamental BasisFundamental Basis
MIKE 11MIKE 11
PHYSICAL SYSTEM
River NetworkFlood PlainsStructures
PHYSICAL LAWS
Conservation of MassConservation of
Momentum
SCHEMATIZE
Represent by a simple Equivalent System
DISCRETIZE
Express as a Finite Difference Relation
NUMERICAL MODELBOUNDARIES OUTPUTS
Fundamental BasisFundamental Basis
MIKE 11MIKE 11Saint-Venant Saint-Venant EquationsEquations
Continuity Equation (Conservation of Mass)
Momentum Equation (Conservation of Momentum) (Newton’s 2’nd Law)
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.
MIKE 11
dx
Q
at time t
at time t+dth(t)
h(t+dt)
xdx
MIKE 11Conservation of Conservation of MassMass
Q dt QQ
xdx dt dA dx
A
tdx dt( )
Q
x
A
t
Q
xB
h
t 0
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)
MIKE 11
x
h(t)
F
PP+ P
z(t)
H
MIKE 11Conservation of Conservation of MomentumMomentum
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
MIKE 11
Momentum = Mass per unit length * VelocityMomentum Flux = Momentum * velocityPressure Force = Hydrostatic Pressure P Friction Force = Force due to Bed ResistanceGravity Force = Contribution in X-direction
MIKE 11Conservation of Conservation of MomentumMomentum
x
F
x
F
x
P
x
UM
t
M gf
)(
Momentum = Momentum Flux + Pressure - Friction + Gravity
MIKE 11MIKE 11Conservation of Conservation of MomentumMomentum
UbHM
P gbH1
22
F x bgU
C
2
2
Momentum:
Momentum Flux
Pressure Term:
Friction Term:
Gravity Term:
UUbHMf
0gASP
MIKE 11
Wave Approximations: Kinematic Wave
Diffusive Wave
Fully Dynamic Wave
0)(
2
2
RAC
QgQ
x
hAg
xAQ
t
Q
MIKE 11Differential Differential EquationsEquations
Q
x
A
tq
MIKE 11MIKE 11Kinematic Kinematic WaveWave
Includes: 1. Bed Friction Term 2. Gravity Term
Applications: + Steep Rivers - Backwater Effects NOT applicable - Tidal Flows NOT applicable
MIKE 11MIKE 11Diffusive Diffusive WaveWave
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
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 11Fully Dynamic WaveFully Dynamic Wave
MIKE 11MIKE 11High Order Fully Dynamic High Order Fully Dynamic WaveWave
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
MIKE 11MIKE 11Solution Solution SchemeSchemeImplicit Abbot-Ionescu 6-point scheme
MIKE 11
X
t
unknown
knownQ / h h/ Q
jj-1 j+1
n
n+1
MIKE 11Solution Solution SchemeScheme
dxdx
dt
00
Implicit Abbot-Ionescu 6-point scheme
Q / h
MIKE 11MIKE 11Solution Solution SchemeScheme
Solution method
Double Sweep algorithm
Nodal point solution
Grid point solution
Matrix bandwidth minimization
MIKE 11Model Data Model Data RequirementsRequirements
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
MIKE 11StabilityStability
Given: Initial Conditions and Finite DifferenceApproximation which is consistent
Then: Stability is the necessary and sufficientcondition for convergence
Stability analysis can only be done for linear differential eq.
Explicit methods: Conditionally stable (Cr < 1)Implicit methods: Unconditionally stable
MIKE 11
Cr g D vt
x ( )
Courant Number:
Example: D=10;V=1; dX=1000 sec1001081.9
1000
m
VDg
Xt
MIKE 11Boundary Boundary ConditionsConditions
MIKE 11
Q
h or Q/h
In general, Boundaries should be located where key investigation area is not directly affected by boundary condition!
Discharge, Q : Upstream of RiverLateral InflowClosed End (Q=0)Discharge ControlPump
Water Level, h : Downstream River boundaryOutlet in Sea (tide, wind)Water level control
Q/h Boundary : Downstream Boundary (Never upstr.)Critical Outflow from Model
MIKE 11Initial ConditionsInitial 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.
MIKE 11Data NeedsData Needs MIKE 11
Reliable Data required: ‘GARBAGE IN = GARBAGE OUT’
Topography Data: Width, Area, Volume of inundated plainsSchematization of ModelAerial/Satellite/Radar images of flood extentsReservoir data (control strategy, spillway etc.)Cross section dataDATUM - Same reference level for all data!
Hydraulic Data: Stage & Discharge hydrographsRating CurvesPeak Water level during significant eventsUsed for Boundary conditions and Calibration
MIKE 11CalibrationCalibration 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
MIKE 11CalibrationCalibration 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)
MIKE 11VerificationVerification 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 Model2) Calibration (preferably data from several periods)
3) Verification (do not use data from Calibration period)
4) Application (‘production runs’)