A modular model for simulating continuous or event
Contents
1. Finite elements c Modules Fig 1 Alternative grids THEORETICAL BASIS Hydrological models range from statistical to conceptual embracing probabilistic curve fitting black boxes analogous e g cell type Diskin et al 1984 through to the more hydrodynamically correct Even the latter range from simplistic e g time area Watson 1981 though first approximation kinematic type diffusion equations and hydrodynamic equations SWMM Huber et al 1982 The latter are only necessary for surface runoff simulation and even then are not always warranted For runoff determination accelerations and backwater effects are not significant On the other hand the time area approach which derived from the rational method does not accommodate the effect of water depth on concentration time Mono time axes and linear rainfall runoff relationships have been taken to their extreme in unit hydrograph theory and as a result the hydraulic basis is often overlooked in sophisticated models e g OTTHYMO Wisner 1980 85 A modular model for simulating continuous or event runoff It is into the more hydraulically based models that the majority of research is now directed By suitable selection of module arrangement one dimensional flow can be assumed i e the module axis is taken in the general flow direction Lateral flow time is neglected which could introduce error in flood plane type modules Transverse i e lateral and vertical for horizo2. MANAGEMENT CAPABILITY A drawcard of a model prepared on the above lines WITSKM WITS STORM KINEMATIC MODULAR MANAGEMENT MODEL is its versatility when it comes to redirecting flows and attenuating hydrographs The facility of readily being able to redirect flows along different routes means channel storage or open versus closed conduit conveyance can be explored The re routing of flows along circuitous routes may increase channel storage This in turn increases concentration time and could reduce design peak flows New townships layouts could be varied until a suitable stormwater drainage pattern emerged The overflow facility also enables dual drainage to be used to maximum advantage Excess flow could be led to shallow channeis or roadways which will provide retardation or lead to channels which are only used in emergencies the overflow level can readily be varied to permit difference risk storms to be accommodated in the minor underground conduit system The aquifer opticn is also of use in urban catchment management studies Aquifers can be recharged by direct infiltration or with water led to them from less pervious areas In either case the absorption of the aquifer is only limited by the depth discharge characteristics and initial moisture conditions A useful module for hydrograph attenuation is the storage module Reservoir surface area dead storage and overspill crest level can be varied to achieve an optimum balance between ma
3. The kinematic equations are entirely adequate for this type of flow as dynamic effects are absent Data include aquifer reference number downstream module number initial flow depth length width conveyance defined as kS where k is permeability and S is gradient porosity aquifer depth and underneath aquifer number D Stephenson 90 OTHER FACILITIES A frequent source of error in stormwater programs arise when downstream catchment number is changed or forgotten A facility exists for displaying graphically on a PC colour screen the entire network once it is entered on the computer Each module is drawn according to the type e g pipe catchment channel and is connected upstream and downstream as indicated in the data Overflow routes are also indicated In general the model is designed for easy understanding and input and cross checking It is especially useful for stormwater management studies The groundwater modules enable continuous simulation to be performed which is useful for establishing antecedant moisture conditions for storms and dry weather flows Fig 4 GOOD PRACTICE BAD PRACTICE BP LE DISCONNECTED IMPERVIOUS AREA ROAD LAYOUT_ FLOOD PLANES ea ae DETENTION STORAGE 177 rrr TT DUAL DRAINAGE Fig 4 Management methods investigated with model 91 A modular model for simulating continuous or event runoff REFERENCES Brakensiek D L 1967 Kinematic flood routing Trans Am Soc Agr Engr
4. in accuracy Holden and Stephenson 1988 proposed a method for minimizing the numerical error and getting the best approximation to hydrodynamic diffusion The wave diffusion can be accounted for using the slightly more accurate equations namely the diffusion equations or the full dynamic equations However in some cases wave diffusion can be reproduced numerically From the mathematical point of view numerical diffusion can be controlled or minimised Explicit solution of the kinematic equations is often employed in preference to implicit solution as the friction equation is non linear and explicit schemes such as the backward centred or semi explicit such as the 4 point scheme of Brakensiek 1967 are reasonably accurate and fast Explicit schemes can be subject to numerical instability unless the time increment is small enough i e At lt Ax dx dt the Courant criterion where dx dt amy On the other hand the smaller At the greater the numerical diffusion as the numerical effect travels at a speed Ax At The optimum compromise is for Ax At dx dt This is not always possible in an equispaced grid as dx dt varies Ponce 1986 attempted to reproduce actual diffusion in kinematic equations by writing the finite difference equations for flow in a way similar to the Muskingum Cunge routing equation Adopting a more practical approach the kinematic diffusion process can be explained as follows The routing process which occurs with kin
5. New Directions for Surface Water Modeling Proceedings of the Baltimore Symposium May 1989 IAHS Publ no 181 1989 A modular model for simulating continuous or event runoff D Stephenson Water Systems Research Group University of the Witwatersrand Johannesburg South Africa ABSTRACT A review of stormwater model arrangements is made Computer programs able to assemble loosely connected elements are simplest to use and understand Arrangement of elements and assembly in parallel or series enables all possible types of models to be accommodated Hydraulic elements are used for surface runoff visualization and groundwater aquifers are simulated in parallei for continuous or long term simulations INTRODUCTION The accuracy of runoff models can be improved at the expense of more and more data There are many models availabie with differing levels of sophistication for such studies The author contends however the biggest cost of modelling is often in learning the ins and outs of a model and its principles and limitations On the lower levels of the learning curve many users may wish to dabble with a model and at the upper end there are many technical aspects to remember Time away from the model causes users to forget aspects and it is the ease of initial or re access which can inspire confidence in the user and enable the model to be used to its fullest Various types of models are discussed bearing these points in mind Sub catchment a
6. co ordinates across a section The stream between any two neighbouring points is treated as an independent section so that velocity varies depending on flow depth and roughness Flood planes are thus accommodated with slow moving storage on the banks and a more rapid stream between banks Data are module reference number downstream module initial depth length slope points co ordinates and roughness of each section This facility can be used to calculate normal depth in compound channels An impermeable catchment upstream with an area of 3600m x 100m is fed with Rmm of catchment rain where R is normal flow in m s and after a period of time the depth in the downstream channel stabilizes at normal depth Storage basins Where detention of retention is required on or off channel storage may be of use Data required are module reference number downstream module initial water depth length width conveyance a discharge depth coefficient m Q Way side slope of basin dead storage before discharge and crest level of dam wall By experimenting with the outlet e g crest or orifice spillway a best design may be achieved Aquifers Water may infiltrate to aquifers from catchments or be discharged directly into them from any conduit or overflow The aquifer acts as a conduit albeit with a much slower flow rate The aquifer will also have a maximum depth and may leak to a lower aquifer Stacking or cascading of aquifers is possible
7. ectly store inflow and separate storage modules are necessary CONTINUOUS SIMULATION Groundwater flow capability with aquifer modules makes possible long term simulation of catchment yield Groundwater contributions lag surface runoff by hours or even months Recession limbs of stormwater hydrographs can be due to contributions from perched water tables or interflow Longer term yields are from deeper aquifers Recharge of surface layers is however important from the point of view of antecedent moisture and permeability for forthcoming storms The continuous simulation capability therefore improves estimation of surface storm runoff Surface layer moisture is also important for estimating evaporation and losses The time scale of flow from deeper aquifers may be much longer than from the higher water tables and a greater time step could be used once surface runoff is reduced The problem then arises as to future storm problems and their infiltration However from the total yield point of view it is not critical if storm patterns are assumed ROUTING PROCESS Kinematic waves are theoretically not subject to diffusion i e spreading and attenuation as no dynamic effects are included in the equation There may be changes in wave shape since dx dt is a function of depth but there can be no change in peak flow unless there is an inflow The advantage of taking large distance increases with the kinematic method therefore results in a sacrifice
8. ematic modelling is similar to reservoir routing where discharge depends only on the stage at the outlet A unique stage 87 A modular model for simulating continuous or event runoff discharge relation is assumed i e no allowance is made for accelerations or water surface gradient A compromise could be made by setting discharge a function of stage at more than one point e g average of upstream and downstream stages The resulting effect is similar to that employed in the Muskingum method and in addition allows for non linearity in the stage discharge relationship It also has the advantage that the parameters in the equations are physically measurable and not empirical To overcome the non linear relationships the kinematic equations can be solved in two steps namely the continuity equation to determine change in water depth and discharge is obtained from stage using the selected discharge equation The discharge equation is not limited to a channel type equation such as that of Manning Thus using a general discharge equation of the form Q Kh if h is stage at discharge point and m 5 3 one has the Manning equation if m 5 2 one has a triangular weir m 3 2 is a rectangular weir m 1 2 is an orifice and m 1 is a deep rectangular channel If h is the difference between upstream and downstream stages then if m 1 2 one has turbulent pipe flow and if m 1 one has laminar flow in a closed conduit or contained aquifer
9. ntal flow direction accelerations are also neglected but this is quite satisfactory for all runoff modelling see Fig 2 Thus the hydrodynamic equations are narrowed down to the St Venant equations and their derivatives Accelerations are not of importance in overland or long river studies so these terms are omitted and backwater effects are only of importance in some channel situations so most modules are limited to kinematic type equations Flow is assumed uniform down the reach That is the water depth is constant down the reach There may be local backup which does affect system storage however Since inflow is thus spread over the full length of a reach each time step the routing effect can be unrealistic unless the time step is sufficiently large Methods of minimizing numerical routing or using it to approximate time routing have been investigated Holden and Stephenson 1988 LATERAL FLOW MAIN CHANNEL FLOOD PLANE LATERAL FLOW o VERTICAL ACCELERATION SURGE aes LONGITUDINAL ACCELERATION EEE a LIAT BACKWATER ay WITH Sy TERM cept pores ANOMALY 1 WITH KINEMATIC FLOW ACTUAL HYDROGRAPH oT TT XR ASSUMED NEW DEPTH ANOMALY 2 WITH KINEMATIC FLOW Fig 2 Problems with specific numerical methods D Stephenson 86 It should be noted the kinematic assumption of energy gradient parallel to conduit bed can cause complications at changes in slope Depressions do not corr
10. rrangement The interconnection of one sub catchment or element with another can be done in various ways Fig 1 i Finite difference grids In the case of a homogeneous type catchment a rectangular grid can be superimposed Thus flows and water depths are computed at grid point Either one or two directional flow can be assumed In general two flow vectors must be assumed An exception occurs if the flow is in one direction parallel to one of the axes For most undular topography two dimensional analysis is necessary ii Finite element The computations can be reduced and size and shape of element varied to suite the topography if a finite element approach is used In general a two direction flow pattern must be assumed although if the boundaries of elements are perpendicular to flow one directional flow can be assumed iii Modular The simplest and most versatile model is one made up of modules which can be linked up at the ends Generally the flow is one directional along the axis but two dimensional catchments can be made up of modules in parallel and series i e the orientation of the module is ignored because the directional momentum of the water is not considered It is the latter configuration on which the model describes here is based 83 D Stephenson 84 LONG FLOW DIRECTION ALTERNATIVE 2 d FLOW NEW SPACE GRID POSITION BACK BOUNDARY LATERAL DISTANCE x a Finite difference grid ALTERNATIVE 2 d OUT FLOW b
11. s 10 3 pp 340 343 Diskin M H Wyeseure G and Fayen J 1984 Application of a cell model to the Bellebeek watershed Nordic Hydrol 15 Green W H and Ampt G A 1911 Studies of soil physics 1 the flow of air and water through soils J Agric Science 4 1 p 1 24 Holden A P and Stephenson D 1988 Improved 4 point solution of the kinematic equations J IAHR 26 4 1 11 Huber W C Heaney J P Nix S J Dickenson P E and Pilman D J 1982 Stormwater Management Models User Manual Dept Environ Eng Sciences Univ of Florida Ponce V M 1986 Diffusion wave modelling of catchment dynamics Journal of Hydraulic Engineering ASCE 112 8 pp 716 727 Watson M D 1981 Application of Illudas to stormwater drainage design in S A Hydrol Res Unit University of the Witwatersrand Wisner P E 1980 OTTHYMO A planning model for master drainage plans in urban areas Toronto FI Fe
12. wetting front initial moisture content and aquifer module number are required An infiltration process based on the soil physics model of Green and Ampt 1911 is assumed Catchments can be linked in cascades in series for example changing slopes or disconnected impervious surface or in parallel for instance if portion has directly connected impermeable cover Circular conduits pipes Urbanized catchments are normally sewered with underground pipes which run part full for most of the time When they surcharge the 89 A modular model for simulating continuous or event runoff excess flow continues down roads and may be directed to channels Such a system fmajor minor system is common at high flows whether intentional or not and provides roads free of ponding for all but exceptional storms The capability of modelling such systems is therefore important Data required for this type of module are module reference number downstream module initial depth length diameter conveyance S n and overflow module number Trapezoidal channels Open channels are the most common conduits be they roadways gutters ditches or canals Where the channel is a simple trapezoid the data requirements are limited to module reference number downstream module initial flow depth length base width conveyance size slopes maximum depth and overflow module Compound channels Natural channels may be defined using an arbitrary number of
13. ximum water depth and dam cost The ability to vary the outlet discharge characteristic is however the most versatile facility of the storage module By means of a general discharge equation of the form m Q WA y any form of outlet control can be used For example an orifice is represented if m 1 2 and WA Ca y 2g where C is a discharge coefficient a is the orifice cross sectional area and g is gravity For detention attenuation which has a decreasing effect with D Stephenson 88 inflow m should be high and for high detention at all depths m should be small Again by trial an optimum compromise between dam cost and cost of conveying away the discharge can be achieved MODULES The versatility of the computer programme is enhanced by the possibility of fitting in various types of hydrological units or modules into a system Catchments aquifers conduits and storage basins can be linked in any order The various modules which can be built in are as follows Fig 3 y i E 5 Fig 3 Graphs output for connectivity check Catchments A basic catchment is a rectangular shape sloping in one direction The module reference number its downstream module initial water depth length width and discharge coefficient ratio of discharge to depth to the power of 5 3 e g V S n where S is gradient and n is Manning roughness are required as input data In addition the surface permeability suction at the ground
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