Intro to: Mathematical Modeling Basic Hydrologic/ Hydraulic Concepts HEC software systems Week 1 639.047 Loading HMS and DSSVue Touring HMS Running and

Intro to: Mathematical Modeling Basic Hydrologic/ Hydraulic Concepts HEC software systems

Week 1 639.047

Loading HMS and DSSVueTouring HMSRunning and viewing a simulation

Hands-on:

HMS: Linear reservoirs & Unit Hydrographs Precip Model options Basin Model Options

Week 2 639.047

Basin Model ExamplesBuild our first model from scratch

Hands-on:

Abstraction

Fidelity Behavior

Mechanism

What is a model?

A useful simplification of a complex reality

What’s the goal?

Abstraction

Fidelity Behavior

Mechanism

Art InsightBeauty

Mathematical Modeling

Abstraction

Fidelity

Table 2-2. What is a mathematical model?…simplified systems that are used to represent real-life systems and may be substitutesof the real systems for certain purposes. The models express [mathematically] formalized concepts of thereal systems (Diskin, 1970)

…a symbolic… mathematical representation of an idealized situation that has theimportant structural properties of the real system. (Woolhiser and Brakensiek, 1982)

…idealized representations…They consist of mathematical relationships that state atheory or hypothesis (Meta Systems, 1971)

Behavior

Mechanism

-HMS Technical Reference Model

Empirical (system theoretic) or Conceptual (mechanistic/theoretical)

This distinction focuses on the knowledge base upon which themathematical models are built. A conceptual model is built upona base of knowledge of the pertinent physical, chemical, andbiological processes that act on the input to produce the output.An empirical model, on the other hand, is built upon observationof input and output, without seeking to represent explicitly theprocess of conversion.

HEC-HMS includes both empirical andconceptual models. For example, Snyder’s unit hydrograph(UH) model is empirical: the model is fitted with observedprecipitation and runoff. The kinematic-wave runoff model isconceptual: it is based upon fundamental principles of shallowfree-surface flow.

Common distinctions made among mathematical models…

Lumped or Distributed

A distributed model is one in which the spatial (geographic)variations of characteristics and processes are consideredexplicitly, while in a lumped model, these spatial variations are averaged or ignored.

HEC-HMS includes primarily lumpedmodels. The ModClark model is an exception.


Event or Continuous

This distinction applies primarily to models of watershed-runoffprocesses. An event model simulates a single storm. Theduration of the storm may range from a few hours to a few days.A continuous model simulates a longer period, predictingwatershed response both during and between precipitationevents.

Most of the models included in HEC-HMS are event models. But the system is very flexible and can support continuous simulation via SMA approaches or external linkages.


Types of River Models

Hydrologic Hydraulic Load Biological (Channel & Floodplain)

Conservation of Mass{continuity}

predicts: Water discharge rateover time

Rational methodHEC-1HEC-HMSTR-20TR-55

Conservationof MassConservationof Momentum (energy)

predicts: Depth, Velocity distributions over time

WSP HEC-2HEC-RASHEC-4SWMM

Conservationof Momentumand Massfor solvent and solutes

predicts: Conc.& transportOver time

HEC-6SWMMAGNIPSSWATHEC-RASBASINS

HSIIFIMRIVPAKS{SEM}{MLR}

Various

predicts: habitat quality or Population sizeOr composition

The

ory

base

Basic Theoretical Concepts:

Conservation of Mass

Water Balance (Continuity Equation) Input rate – Output rate = dStorage/dt

Conservation of Momentum (Energy)Newton’s 2nd Law of Motion

external forces = Mass * acceleration

Q

P = precipitationE = evaporationT = transpirationR = runoffF = infiltrationG= groundwater flowQ = streamflow

Constructing a water balance equationfor a simple landscape...

Mass balance applied to a hydrologic system:

Q

I-O = dS (P +Gin) - ( T + E + R + Gout+ Q) = dSlake + dSG + dSR

if Gin, Gout, and Rout ~ 0

P-T-E-Q = dST

P-ET-Q = dST

P E T

QGin Gout


ST

Q

at equilibriumP-ET-Q = dS = 0P-ET-Q = 0

P = ET +QandQ = P - ET


But what if dS <>0?Dynamic simulation…

Basic concepts in storage

d/dt Storage = input - output

output [Q] = input - d/dt Storage

Storagetotal = (input - output) dt

time

Qin

Qout

Storagevolume

cfs

For any mass balance including a water balance

i.e. hydrologic storage is caused by time delay

All hydrographs can be thought of as being shaped by stormflow passing througha sequential series of simple storage compartments:e.g. catchments, channels, reservoirs,floodplains...

Approaches to accouting for storage effects generally fall into 2 groups: hydrologic and hydraulic routing methods

HEC Software Systems

Hydrologic Modeling System: HMS

Hydrologic Database Manager: DSSVue

Floodplain and Channel Hydraulics: RAS

HEC has 3 main Integrated “nextGEN”Modeling Products { and several more recent}

HMS Project ComponentsBasin ModelPrecipitation ModelControl specificationData Inputs

Each element has one or more alternate Methods (modeling methods)Basin model > Elements > Methods > Parameters

DssVue Main Database Window

Reads and writes **.dss files

Simulations are run from the MainWindow

Control Specification Window

Basin Model Window

639.047 Week 2639.047 Week 2

Synopsis of Models Included in HEC-HMS ProgramHEC-HMS uses a separate model to represent each component of the runoff process that is illustrated in Figure 3-2, including: ‧ Models that compute runoff volume; ‧ Models of direct runoff (overland flow and interflow); ‧ Models of baseflow; ‧ Models of channel flow.

Basic concepts in storage and routing




time

Qin

Qout

Storagevolume

cfs

Mass balance requires

Basic concepts in storage and routing:Mass balance constraints suggest a simple linear reservoir model

All hydrographs can be thought of as being shaped by excess precipitation passing through a sequential series of simple storage compartments where storage can be represented by

e.g. catchments >>channels>>reservoirs>>floodplains...


The number of compartments is arbitrary, and if a compartment's output is proportional to the the water is has in storage, then the resulting model is referred to as a Linear Reservoir Model

Mass balance constraints suggest water flow through theLandscape can be represented as a simple linear reservoir model

for a chain of n compartments

d/dt Storagen = inputn – outputn

d/dt Storagen = outputn-1 – outputn mass balance assumption

outputn = kn* Storagen linear rate assumption

n= 1…………….2…………….3…………….4……….

Implies

where

A system of i linear differential equations

DRO Hydrograph

Obs. Hydrograph

Unit HydrographTheory

[UH]

Assumptions:

DRO hydrographs are linear(i.e. proportional)and time invariant

D

Adjust Q togive 1 unit DROby dividing Q valuesby 1/DRO total as depth

Unit Hydrograph

DRO Hydrograph

Because of their assumed linearity...Unit hydrographs (UH) of short durationcan be used to generate longer duration UH

S-curve Method

S-curve Method

Hydrograph Convolution

UH’s can also be used to estimate DRO hydrographs from complexprecip events...

Qn = PiU n-i+1

n

i

Hydrograph Convolution

Qn = PiU n-i+1

n

i

Synthetic unit hydrographs

Empirical relationships for key parameters

Methods:SnyderSCSEpsey

Qp = Peak Q; tp = time to peak Q; Tr = rise timeD = precip duration; Tr + B = time base

tp(hrs)= Ct(L Lc )0.3

Qpeak(cfs) = 640 Cp AREA(mi2) tp

Cp= storage coeff. from .4 to .8Ct= coeff. ususally 1.8-2.2 [0.4-8.0]

Tbase(days) = 3 + tp/8

Lc=length along channel to watershed centroid

L= length of main stem to divide (ft)

Snyder’s Synthetic Unit Hydrograph method

SCS Synthetic Unit Hydrograph Method

time

Qpeak

lag time

Rise time Fall time

Trise B

tp

tp(hrs)= L .8 ( S+1) .7

1900 y .5

Qpeak(cfs) = 484 AREA Trise

S potentialabstraction= (1000/CN)-10

Trise(hrs) = D + tp

2

y = average watershed slope

L = length to divide (ft)

D

VOL.Q peak T rise

2

..Q peak 1.67T rise

2

abstraction1000

curve number10

SCS_runoff#

= 30 Units

45

57

70

82

94

100 sq mile catchment-2 hrs at 1 in/hr, uniformly distributed-same but SCS standard storm

ab

c

Can you get it to work? Try first without channels.How does lumping parameters change output?How does adding river routing change output?Use DSSView to compare results

No-where River

Sub-basin Channel length(miles)

Length toDivide(miles)

SCS-CN Averagewatershedslope

Catchmentarea

(sq miles)

tlag

a 10 13 50 0.50% 50 10.11

b 8 11 60 0.50% 35 8.55

c 4 6 70 0.10% 15 2.75

all 14 18 56.5 0.43% 100 19.11

639.047 Week 3639.047 Week 3Adding Reality…

Event versus Continuous simulations

Lumped versus distributed structure

Linear Reservoir Model

Continuous versus EventContinuous versus EventWhat’s the difference?What’s the difference?

• Events: short and wet… ignore ET and Events: short and wet… ignore ET and antecedant moisure (local water storage) antecedant moisure (local water storage) variationsvariations

• i.e. a place always has a characteristic unit i.e. a place always has a characteristic unit hydrographhydrograph

• Continuous: really hydrologic response Continuous: really hydrologic response depends on existing level of storage in the depends on existing level of storage in the landscape, and that varies between storms landscape, and that varies between storms with ET rate.with ET rate.

In SMA Methods Instead of a single I/O reservoir for each Sub-basin;

we add reality by including a series of linear reservoir with explicit water-balances for each

In HMS requires addition ofSMA units (and parameterizations)To the LOSS and BaseFlow Models

Lumped vs. Distributed

Resolution issuesAveraging ErrorTranslation error

Solution : Increase spatial resolution by disaggregating to more homogeneous units

Muskegon River Model41 sub-basins for 2400 sq miles

(82 external SMA units)41 Groundwater inputs41 Junctions42 20 explicit river reaches1 sink

MDEQ Cedar Creek Model13 sub-basins for 100 sq. miles

7 explicit river segments8 junctions4 reservoirs (pools)

Muskegon River Model41 sub-basins for 2400 sq miles

(82 external SMA units)41 Groundwater inputs41 Junctions42 20 explicit river reaches1 sink

HMS handles:Gridded Precipitation

external Hec Programs

Gridded LossesSCSSMA

Gridded TransformsModClark SUH

All Use a “Grid File” to define gridded structure by sub-basinCan be built manually or from Arview using GeoHMS extension.

Fully distributed Modeling requires a gridded Approach

DSS file

Ext Fortran Code for SMA

Ext MODFLOW

Groundwater Gage rec

Excess Precip Gage rec

Climate Data & Basin info

HEC-HMS

639.047 Week 4639.047 Week 4Adding Reality…

Channel routing issues

Adding Hydraulic constraints

Basic concepts in storage




time

Qin

Qout

Storagevolume

cfs

For any mass balance including a water balance

i.e. hydrologic storage is caused by time delay

Hydrologic routing based on mass balance constraint

output [Q] = f(input - d/dt Storage)

Simple hydrologic routing: as in our linear reservoirs model

Q= k * Storage

output [Q] = f( storage depth) rating curve, hydraulic geometry


Hydrologic routing through channels is more complex

based on mass balance constraintComplex hydrologic routing: channels

d/dt Storage = input - outputoutput [Q] = f(input - d/dt Storage)

output [Q] = f(input,output )


Q,V rising

falling

storage

wedge storage

Prism storage

DRO

Base

Complex Hydrologic routing based on mass balance constraint

Complex hydrologic routing: Muskingum RoutingMcCarthy (1938) proposed a method which uses the continuity constraint and a simple empirically fit storage function that depends on both input and output rate:

S = K [x I + (1-x)O] where x=weighting factor, ranges from 0 to 0.5;averages about 0.2K=travel time of flood wave through segment (days), =S/xI+(1-x)O


x In

put+

(1-x

)Out

put

storage

Q,V rising

falling

storage

Mass balance


Sum of external forces = momentum

d(Ff + Fgrav+ Fvp)/dt = d(mass*velocity)/dt

Type of Model/flow Momentum eq Kinematic wave Friction,gravity (fr# <2)

Diffusion (noinertial) model Friction,gravity,pressure Steady, nonuniform Unsteady, nonuniform

Friction,gravity,pressure,inertia (approx and exact)

Hydraulic routing based on mass balance constraint, and momentum eq {St. Venant eqs}

Theprectical basic for RAS modelling:

Kinematic Wave

Diffusion [Muskingum-Cunge]

Dynamic wave approximation [RAS]

Full Dynamic Wave [DWOPER, FLDWAV]

Upper Pere Marquette example:


A

BC

Upper Pere Marquette example: Landuse

Upper Pere Marquette example: SCS curve#

Area(sq miles)

SCS# %impervious

SCSTime lag(min)

Initial loss

combined 30.6 53 .01 2040 .1

Sub-basin A 17.25 61 2.00 2040 .1

Sub-Basin B 8.59 45 .04 1980 .1

Sub-Basin C 4.73 70 3.00 1020 0


Basin data

Upper endElev (m)

Lower endElev (m)

ChannelLength(m)

Type,width(m)sideslope

AverageCatchment% slope

combined 368 288 24140 1.1

Sub-basin A-upper 368 328 10800 Trap,2,3 1.22

Sub-Basin B-middle 328 304 6300 Trap,3,3 1.42

Sub-Basin C-lower 304 288 7040 Trap,4,3 .96


Channel data

Introduction to river modeling

Hydrologic Flood plain & Load Habitat Channel Hydraulics

Conservation of Mass

Q=P-ET +/- StorageQ=W D V

Runoff & Routing

predicts: DRO Q

Precip-runoffRational methodHEC-1HEC-HMS*TR-20TR-55

Conservationof Mass, Energy

Etotal = Z + D + V2/2gQ ~ Area SR/nQ=W D V

Routing & Energy

predicts: Depth, Vel

Flow conditionsWSP HEC-2HEC-RAS*HEC-4

Conservationof Momentumand Mass

predicts: Conc.& transport

Erosion,WQHEC-6SWMMAGNIPSSWAT*

BiologyHSIIFIMRIVPAKS{SEM}{MLR}

Empirical

predicts: utilization

HEC-HMS Set up and Application

To analyze a hydrologic system with HEC-HMS, the program user mustcomplete the following steps:

1. Start a new project;2. Create gage data;3. Enter basin model data;4. Enter precipitation model data;5. Enter control specifications;6. Configure a run (and name it)7. Compute!

Documents

Intro to: Mathematical Modeling Basic Hydrologic/ Hydraulic Concepts HEC software systems Week 1 639.047 Loading HMS and DSSVue Touring HMS Running and