Transcript MIKE 11
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 gAS0 C 2 2 Differential Equations MIKE 11 Q2 gQ Q Q ( A ) h 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 100sec g D V 9.8110 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’)