1-d dynamic modelling
DESCRIPTION
Mathematical Background. 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 - PowerPoint PPT PresentationTRANSCRIPT
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MIKE 11
1-D1-DDynamic Dynamic ModellingModelling
Mathematical Mathematical BackgroundBackground
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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
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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
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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.
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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)
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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
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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
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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
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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
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MIKE 11MIKE 11Kinematic Kinematic WaveWave
Includes: 1. Bed Friction Term 2. Gravity Term
Applications: + Steep Rivers - Backwater Effects NOT applicable - Tidal Flows NOT applicable
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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
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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
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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
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MIKE 11MIKE 11Solution Solution SchemeSchemeImplicit Abbot-Ionescu 6-point scheme
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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
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MIKE 11MIKE 11Solution Solution SchemeScheme
Solution method
Double Sweep algorithm
Nodal point solution
Grid point solution
Matrix bandwidth minimization
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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
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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
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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
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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.
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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
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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
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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)
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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’)