1-D Dynamic Modelling

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)

QQ

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

QQ

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’)