&EPA
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Athens QA 3O613
EPA/600/3-87/007
May 1987
Research and Development
The Enhanced Stream
Water Quality Models
QUAL2E and
QUAL2E-UNCAS:
Documentation and
User Model
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EPA/600/3-87/007
May 1987
THE ENHANCED STREAM WATER QUALITY MODELS QUAL2E AND QUAL2E-UNCAS:
DOCUMENTATION AND USER MANUAL
by
Linfield C. Brown* and Thomas 0. Barnwell, Jr.**
*Department of Civil Engineering
Tufts University
Medford, MA 02155
**Environmental Research Laboratory
U.S. Environmental Protection Agency
Athens, GA 30613
Cooperative Agreement No. 811883
Printed on Recycled Paper
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ATHENS, GEORGIA
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DISCLAIMER
The information in this document has been funded wholly or in part by
the United States Environmental Protection Agency. It has been subject to
the Agency's peer and administrative review, and it has been approved for
publication as an EPA document. Mention of trade names or commercial pro-
ducts does not constitute endorsement or recommendation for use by the U.S.
Environmental Protection Agency.
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FOREWORD
As environmental controls become more costly to implement and the penal-
ties of judgment errors become more severe, environmental quality management
requires more efficient management tools based on greater knowledge of the
environmental phenomena to be managed. .As part of this Laboratory's research
on the occurrence, movement, transformation, impact and control of environ-
mental contaminants, the Assessment Branch develops management or engineering
tools to help pollution control officials achieve water quality goals.
widely used for waste load
and other conventional pollu-
the introduction of QUAL-II in
have evolved. This manual
form of enhanced state-of-the-
The stream water quality model OUAL2E is
allocations, discharge permit determinations,
tant evaluations in the United States. Since
1970, several different versions of the model
presents the most recent modifications in the
art models called OUAL2E and QUAL2E-UNCAS. Both models have been developed
over the past three years through cooperative agreements between the National
Council for Air and Stream Improvement (NCASI), the Department of Civil
Engineering at Tufts University, and EPA. Distribution and maintenance of
the OUAL2E and QUAL2E-UNCAS computer programs, and training and assistance to
model users, will be provided by EPA's Center for Water Quality Modeling at
this Laboratory.
Rosemarie C. Russo, Ph.D.
Director
Environmental Research Laboratory
Athens, Georgia
111
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ABSTRACT
This manual is a major revision to the original QUAL2E Program Documen-
tation (EPA/6(K)/3-85/065) released in 1985. It includes a description of the
recent modifications and improvements to the widely used water quality models
QUAL-II and QUAL2E. The enhancements to QUAL-II that led to OUAL2E incorpo-
rated improvements in eight areas: (1) algal, nitrogen, phosphorus, and dis-
solved oxygen interactions; (2) algal growth rate; (3) temperature; (4) dis-
solved oxygen; (5) arbitrary non-conservative constituents; (6) hydraulics;
(7) downstream boundary concentrations; and (8) input/output modifications.
These are fully documented in this manual. The enhancements to QUAL2E, de-
scribed for the first time in this report, include (1) an extensive capabi-
lity for uncertainty analysis with the model QUAL2E-UNCAS, (2) an option for
reach-variable climatology input for steady state temperature simulation, and
(3) an option for plotting observed dissolved oxygen data on the line printer
plots of predicted dissolved oxygen concentrations.
OIIAL2E, which can be operated either as a steady-state or as a dynamic
model, is intended for use as a water quality planning tool. The model can
.
be used, for example, to study the impact of waste loads on instream water
quality or to identify the magnitude and quality characteristics of nonpoint
waste loads as part of a field sampling program. The user also can model
effects of diurnal variations in meteorological data on water quality (pri-
marily dissolved oxygen and temperature) or examine diurnal dissolved oxygen
variations caused by algal growth and respiration.
QUAL2E-UNCAS is an enhancement to QUAL2E that allows the user to perform
uncertainty analysis. Three uncertainty options are available: sensitivity
analysis, first order error analysis, and monte carlo simulation. With this
capability, the user can assess the effect of model sensitivities and of
uncertain input data on model forecasts.
This report was submitted in partial fulfillment of Cooperative Agree-
ment No. 811883 by Tufts University under the partial sponsorship of the
U.S. Environmental Protection Agency. This report covers a period from
June 1985 to January 1987, and work was completed as of January 1987.
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CONTENTS
FOREWORD • • • • m
ABSTRACT . . . iv
1. INTRODUCTION 1
1.1 QUAL2E Development 2
1.1.1 Current Release • 2
1.1.? History • 2
1.1.3 Enhancements to QUAL2E . . . .-,•,,• ••••••« 4
1.1.4 Information Sources ..... ..''.''. ....... 5
1.1.5 Organization of this Report 6
1.2 OUAL2E Computer Model • • 6
1.2.1 Prototype Representation • • 6
1.2.2 Model Limitations . . . , . .7
1.2.3 Model Structure and Subroutines 7
1.2.4 Program Language and Operating Requirements .... 9
2. GENERAL MODEL FORMULATION 10
2.1 Introduction • 10
2.2 Conceptual Representation 11
2.3 Functional Representation 11
2.3.1 Mass Transport Equation . . 11
2.4 Hydraulic Characteristics 15
2.4.1 Discharge Coefficients 15
2.4.2 Trapezoidal Cross Sections 16
2.4.3 Longitudinal Dispersion 16
2.5 Flow Augmentation 19
3. CONSTITUENT REACTIONS AND INTERRELATIONSHIPS 22
3.1 General Considerations 22
3.2 Chlorophyll a^ (Phytoplanktonic Algae) .......... 22
3.2.1 Algal Respiration Rate 24
3.2.2 Algal Specific Growth Rate ..'..' 24
3.2.3 Algal-Light Relationships 26
3.2.4 Algal-Nutrient Relationships 34
3.2.5 Temperature Dependence in Algae Simulation .... 35
3.3 Nitrogen Cycle . . . 35
3.3.1 Organic Nitrogen 35
3.3.2 Ammonia Nitrogen 36
3.3.3 Nitrite Nitrogen 36
3.3.4 Nitrate Nitrogen 37
3.3.5 Inhibition of Nitrification at Low DO 37
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CONTENTS (Cont'd)
Page
3.4 Phosphorus Cycle 33
3.4.1 Organic Phosphorus ".".".! 1 ".".! 38
3.4.2 Dissolved Phosphorus .*.'.'.' 39
3.5 Carbonaceous BOD '.'.'.'.' 39
3.6 Dissolved Oxygen .*.".*.*.*.'.*.*** * 40
3.6.1 Dissolved Oxygen Saturation Coefficient 1 '. '. '. *. *. 41
3.6.2 Atmospheric Reaeration Coefficient Estimation ... 42
3.6.3 Ice Cover 43
3.6.4 K2 Default Values .*.'.'.'.'.'.* 4R
3.6.5 Dam Reaeration *,".".! 49
3.7 Coliforms ,*.'.'.*.* 49
3.R Arbitrary Nonconservative Constituent ...*.".'. *. *. *. ". 50
3.9 Temperature 5n
3.10 Temperature Dependence of Rate Coefficients* ! *. 1 '. '. ". *. 51
3.11 Reaction Rates and Physical Constants ..... 52
4. FUNCTIONAL REPRESENTATION OF TEMPERATURE 57
4.1 Basic Temperature Equation * 57
4.2 Definition of % „*.*.*.". 58
4.3 Net Short-Wave Solar Radiation .'."!! ° * !!" 60
4.3.1 Extraterrestrial Radiation .' 61
4.3.2 Radiation Scattering and Absorption ..*.". 63
4.3.3 Cloudiness 55
4.3.4 Reflectivity .!....'.*. 65
4.4 Long-Wave Atmospheric Radiation ....'."'" 66
4.5 Water Surface Back Radiation .....*". 66
4.6 Evaporation ......*.* 67
4.7 Conduction * gg
4.8 QUAL2E Modifications for Reach Variable LocaVcfimatofogy'
and Temperature 59
5. COMPUTATIONAL REPRESENTATION 71
5.1 Prototype Representation .'.*.*.".*.". 71
5.2 Forcing Functions .*.*.*.".' 72
5.3 Model Limitations *.*.*.*. 73
5.4 Numeric Solution Technique .'.'.*.*.'.* 74
5.4.1 Formulation of the Finite Difference Scheme *. '. *. ". 74
5.4.2 Method of Solution 77
5.4.3 Boundary Conditions *.*.".*. 79
VI
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CONTENTS (Cont'd)
6. UNCERTAINTY ANALYSIS WITH QUAL2E 81
6.1 Introduction • 81
6.2 QUAL2E-UNCAS • • • 81
6.2.1 Sensitivity Analysis 82
6.2.2 First Order Error Analysis 83
6.2.3 Monte Carlo Simulation . 84
6.3 Input Variable Variances 85
6.4 Programming Strategy in QUAL2E-UNCAS 87
6.4.1 UNCAS Subroutines . . . 8.7
6.4.2 Internal UNCAS Data Files ..... 91
6.4.3 User Supplied UNCAS Data Files 91
6.5 Limitations and Constraints for QUAL2E-UNCAS 91
APPENDIX A. QUAL2E USER MANUAL ..... 93
APPENDIX B. QUAL2E-UNCAS USER MANUAL . 159
APPENDIX C. QUAL2E-UNCAS EXAMPLE APPLICATION 172
REFERENCES , 184
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LIST OF FIGURES
— Page
1-1 General Structure of QUAL2E 8
II-l Discretized Stream System 12
I1-2 Stream Network of Computational Elements and Reaches 13
III-l Major Constituent Interactions in QUAL2E , 23
III-2 OUAL2E Light Functions 29
IV-1 Heat Transfer Terms Associated with Interfacial Heat Transfer . 59
V-l Classical Implicit Nodal Scheme 74
VI-1 UNCAS Flow Diagram and Program Structure 88
LIST OF TABLES
•^ Page
II-l Values of Manning's "n" Roughness Coefficient .... 18
II-2 Experimental Measurements of Longitudinal Dispersion
in Open Channels 20
III-l Comparison of Dissolved Oxygen Saturation Concentrations ... 43
III-2 Default Temperature Correction Values for QUAL2E 53
II1-3 Typical Ranges for QUAL2E Reaction Coefficients 54
IV-1 Definition of Heat Transfer Terms Illustrated in Figure IV-1 . 60
IV-2 Empirical Coefficients for Determining Rs 65
VI-1 Summary of QUAL2E Input Variable Uncertainties 86
viii
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ACKNOWLEDGMENT
Over the years, many investigators have" contributed to the development
of what has become OUAL2E. The foundation upon which the model has been
built was laid by the Texas Water Development Board in the late 1960s in the
QUAL-I model. Many versions of the model emerged in the 1970s. The lineage
of QUAL2E can be traced to work done for the Southeast Michigan Council of
Governments (SEMCOG) by Water Resources Engineers, Inc. (now Camp, Dresser,
McKee Inc.). QUAL-II/SEMCOG was chosen for distribution by the Center for
Water Quality Modeling (CWQM) in the late 1970s and began to receive wide use
in water quality modeling and wasteload allocation programs.
QUAL-II/SEMCOG was throughly reviewed, tested, and documented by the
National Council of the Paper Industry for Air and Stream Improvement, Inc.
(NCASI), as discussed in NCASI Technical Bulletin No. 391. Changes arising
from this review were incorporated in a model called OUAL-II/NCASI, which
was adopted for distribution by the Center for Water Quality Modeling.
Because of a mutual interest in the program, CWQM partially sponsored an
NCASI review of other versions of the QUAL-II computer program and incor-
porated useful features of these versions in the program called QUAL2E.
Appendix A of this documentation report, the OUAL2E users manual, is
modeled after NCASI Technical Bulletin No. 457, "Modifications to the QUAL-2
Water Quality Model and User Manual for OUAL2E Version 2.2." We express our
appreciation to NCASI for permission to use and modify this material in this
report.
The OUAL2E program also has been made available for IBM PC-compatible
microcomputer. The microcomputer installation of this program was performed
by Mr. Bruce Bartell and Mr. David Disney of Computer Sciences Corporation,
Inc. and was made possible through the support of Mr. King Boynton of the
U.S. EPA's Office of Water and through an agreement with the US-Spain Joint
Committee for Scientific and Technical Cooperation.
The current release of the program incorporates modifications to the
1985 release to accommodate large elevation differences along a river funded
through an agreement with the US-Spain Joint Committee for Scientific and
Technical Cooperation. The major extension to the program documented herein,
the uncertainty analysis capability, was begun by the first author while on a
sabbatical year (1984) from Tufts University at the Athens Environmental
Research Laboratory and completed on his return to academic work.
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1. INTRODUCTION
QUAL2E is a comprehensive and versatile stream water quality model. It
can simulate up to 15 water quality constituents in any combination desired
by the user. Constituents which can be simulated are:
1. Dissolved Oxygen
2. Biochemical Oxygen Demand
3. Temperature
4. Algae as Chlorophyll _a
5. Organic Nitrogen as N
6. Ammonia as N
7. Nitrite as N
8. Nitrate as N
9. Organic Phosphorus as P
10. Dissolved Phosphorus as P
11. Coliforms
12. Arbitrary Nonconservative Constituent
13. Three Conservative Constituents
The model is applicable to dendritic streams that are well mixed. It assumes
that the major transport mechanisms, advection and dispersion, are signifi-
cant only along the main direction of flow (longitudinal axis of the stream
or canal). It allows for multiple waste discharges, withdrawals, tributary
flows, and incremental inflow and outflow. It also has the capability to
compute required dilution flows for flow augmentation to meet any prespeci-
fied dissolved oxygen level.
Hydraulically, QUAL2E is limited to the simulation of time periods
during which both the stream flow in river basins and input waste loads are
essentially constant. OUAL2E can operate either as a steady-state or as a
dynamic model, making it a very helpful water quality planning ;tool. When
operated as a steady-state model, it can be used to study the impact of
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waste loads (magnitude, quality and location) on instream water quality and
also can be used in conjunction with a field sampling program to identify the
magnitude and quality chracteri sties of nonpoint source waste loads. By
operating the model dynamically, the user can study the effects of diurnal
variations in meteorological data on water quality (primarily dissolved
oxygen and temperature) and also can study diurnal dissolved oxygen varia-
tions due to algal growth and respiration. However, the effects of dynamic
Tnrm.5.9ocUnCtl0nSj SUGh as headwater fl°ws or point loads, cannot be modeled
I n UUAL.£C •
QUAL2E-UNCAS is a recent enhancement to QUAL2E which allows the modeler
to perform uncertainty analysis on the steady state water quality simula-
tions. Three uncertainty options are available: sensitivity analysis first
order error analysis, and monte carlo simulations. With this capability, the
user can assess the effect of model sensitivities and of uncertain input data
on model forecasts. Quantifications of the uncertainty in model forecasts
will allow assessment of the risk (probability) of a water quality variable
being above or below an acceptable level. The uncertainty methodologies
provide the means whereby variance estimates and uncertainty prediction can
become as much a part of water quality modeling as estimating expected values
is today. An evaluation of the input factors that contribute most to the
level of uncertainty will lead modelers in the direction of most efficient
data gathering and research. In this manner the modeler can assess the risk
of imprecise forecasts, and recommend measures for reducing the magnitude of
that imprecision.
1.1 QUAL2E DEVELOPMENT
1.1.1 Current Release
The current release of OUAL2E (Version 3.0) was developed under a coop-
erative agreement between Tufts University, Department of Civil Engineering
and the EPA Center for Water Quality Modeling (CWQM), Environmental Research
n?,.?^?,1?* Athens» GA- It includes modifications to prior releases of
QUAL2E (Version 2.2, Brown and Barnwell, 1985) as well as an extensive capa-
bility for uncertainty analysis (UNCAS) of its steady state simulation output.
I,,«,Sor e,1,e,ase of OUAL2E and its companion program for uncertainty analysis,
mm. T r ' 1S 1ntended to supercede all prior releases of QUAL2E and
L"* 1 1 *
1.1.2 History
^ oMn, nuAL-n model was an extension of the stream water quality
model QUAL-I developed by F. D. Masch and Associates and the Texas Water
Development Board (1971) and the Texas Water Development Board (1970). In
1972, Water Resources Engineers, Inc. (WRE) under contract to the U.S. •
Environmental Protection Agency, modified and extended QUAL-I to produce the
first version of QUAL-I I. Over the next 3 years, several different versions
of the model evolved in response to specific user needs. In March 1976 the
Southeast Michigan Council of Governments (SEMC06) contracted with WRE to
make further modifications and to combine the best features of the existing
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versions of QUAL-II into a single model. The significant modifications made
in the SEMCOG version by WRE (Roesner et a_]_., 1981a and b) were:
• Option of English or metric units on input data
• Option for English or metric output—choice is independent of input
units
• Option to specify channel hydraulic properties in terms of trapezoidal
channels or stage-discharge and velocity-discharge curves
• Option to use Tsivoglou's computational method for stream reaeration
• Improvement in output display routines
• Improvement in steady-state temperature computation routines
The SEMCOG version of QUAL-II was later reviewed, documented, and revised
(NCASI, 1982). The revised SEMCOG version has since been maintained and
supported by the EPA Center for Water Quality Modeling (CWQM). In 1983, EPA,
through the CWQM, contracted with NCASI to continue the process of modifying
QUAL-II to reflect state-of-the-art water quality modeling. Extensive use of
QUAL-II/SEMCOG had uncovered difficulties that required corrections in the
algal-nutrient-light interactions. In addition, a number of modifications to
the program input and output had been suggested by users. The enhanced
QUAL-II model was renamed Q1IAL2E (Brown and Barnwell, 1985) and incorporated
improvements in eight areas. These enhancements are fully documented in this
report and summarized as follows:
1. Algal, nitrogen, phosphorus, dissolved oxygen interactions
• Organic nitrogen state variable
• Organic phosphorus state variable
• Nitrification inhibition at low DO
• Algal preference factor for NH3
2. Algal growth rate
Growth rate dependent upon both NHs and N03 concentrations
Algal self-shading
Three light functions for growth rate attenuation
Three growth rate attenuation options
Four diurnal averaging options for light
3. Temperature
• Link to algal growth via solar radiation
• Default temperature correction factors
4. Dissolved Oxygen (DO)
• 16th Edition Standard Methods DO saturation function
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• Traditional SOD units (g/m2-day or g/ft2-day)
t Dam reaeration option
5. Arbitrary non-conservative constituent
• First order decay
• Removal (settling) term
• Benthal source term
6. Hydraulics
• Input factor for longitudinal dispersion
t Test for negative flow (i.e., withdrawal greater than flow)
• Capability for incremental outflow along reach
7. Downstream boundary
• Option for specifying downstream boundary water quality
constituent concentrations
8. Input/output modifications
• Detailed summary of hydraulic calculations
t New coding forms
t Local climatological data echo printed
• Enhanced steady-state convergence
• Five part final summary including components of DO deficit and
plot of DO and BOD
1.1.3 Enhancements to OUAL2E
Since the first release of QUAL2E in 1985, enhancements to the model
have continued. The modifications, listed below, are designed to improve
the computational efficiency of the code, as well as to assist the user in
model calibration and verification. The reach variable climatology modifi-
cations were added in response to applications of QUAL2E to the river network
1"™dr}d> Spain. In that system, large changes in elevation presented
difficulties in calibrating QUAL2E for temperature and dissolved oxygen. The
major addition to the current release of QUAL2E is the uncertainty analysis
capability. Inclusion of this feature resulted from a project which investi-
gated various methodologies for incorporating uncertainty analysis as an
integral part of the water quality modeling process. The QUAL2E model was
chosen for this application because it is a general purpose computer code
widely used by consultants and state regulatory agencies in waste load alloca-
tion and other planning activities.
Enhancements to QUAL2E in the current release include:
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1. Option for reach variable climatology input .for steady state
temperature simulation.
2. Option for including observed dissolved oxygen data on the line
printer plots of predicted dissolved oxygen concentrations.
3. Changing the steady state convergence criterion for algal, nitrifi-
cation, and dissolved oxygen simulations from an absolute error to a
relative error.
4. Updating the formulation for estimating reaeration effects of water
flowing over a dam.
Capabilities of the uncertainty analysis model, OUAL2E-UNCAS, include
the following:
1. Sensitivity analysis—with an option for factorially designed
combinations of input variable perturbations.
2. First order error analysis—with output consisting of a normalized
sensitivity coefficient matrix, and a components of variance matrix.
3. Monte carlo simulation—with summary statistics and frequency
distributions of the output variables.
1.1.4 Information Sources
Major sources of information for this revised documentation are:
1. Roesner, L. A., Giguere, P. R. and Evenson, D. E. Computer
Program Documentation for Stream Quality Modeling (QUAL-II).
_ 1?.,—_ .-. • _ •• -•_,_' _••'•. - nj_i — .». r* n c~nn cnn fC\ Q1 __m XI
81-014,
program um.uinefiiat.iuii • ui on can iiuui-i^j <^^^,,,,^, x.;~.._
U.S. Environmental Protection Athens, GA. EPA-600/9-S1-
February 1981. •
2. JRB Associates. Users Manual for Vermont QUAL-II Model.
Prepared for U.S. Environmental Protection .Agency, Washington,
DC. June 1983.
3. National Council for Air and Stream Improvement. A Review of
the Mathematical Water Quality Model QUAL-II and Guidance for
its Use, NCASI, New York, NY, Technical Bulletin No. 391,
December 1982.
4. Brown, L. C. and T. 0. Barnwell, Jr., Computer Program Docu-
mentation for the Enhanced Stream Water Quality Model QUAL2E,
U.S. Environmental Protection Agency, Environmental Research
Laboratory, Athens, GA, EPA/600-3-85/065, August 1985.
This documentation of QUAL2E updates the report distributed with the
prior version of the model (Brown and Barnwell, 1985) and consolidates
material from these and other sources into a single volume. The basic
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theory and mechanics behind the development of QUAL2E are described in this
volume. The two appendices contain user manuals for QUAL2E and QUAL2E-UNCAS
descrit1on of 1nPut data requirements, as well as
«* ,
UNrR rnSESo. 39 T* Thls. reP°rt, a copy of the QUAL2E and QUAL2E-
UNCAS computer code, and sample input/output data files are available from
Fnvirnn r/?rpWat6r S"f ^ P* Model1"9' u-s- Environmental Protection Agency,
Environmental Research Laboratory, Athens, GA 30613. «a*»uos
1.1.5 Organization of this Report
are dlSriiffpJT?! ffogram future specifications, and limitations of OUAL2E
are discussed in the remainder of this chapter. Chapter 2 describes the con-
fSEr? tand fSn;ht10na] ^Presentation of ^UAL2E as w"" a* the hydraul c char-
J2?n *£ i*^ m0d^' The mathem^Tcal basis of the water quality con-
stituent formulations is presented in Chapter 3. Chapter 4 presents the framp
work for modeling temperature. With the exception of Sect on TI. It is ex-
tracted essentially verbatim from Roesner et al . , 1981. Chapter 5 describes
rePresentat1on of the model-^nd the numerical sol
algoriS rePresentat1on of the model-^nd the numerical solution
analysis caPab11it1es of QUAL2E-UNCAS are documented in
fnv.mcA?Pen*i!X A contains a user manual complete with revised input coding
forms for the current release Version 3.0) of QUAL2E. Appendix B is the
0" QUAL2E-UNCAS' APPend1x C describes an eSple application of
«^tf!Tethe Conven1ence of the majority of users, all of the units speci-
fications are given in the English system of measurement. QUAL2E, however
will recognize either English or metric units.
1.2 QUAL2E COMPUTER MODEL
1-2.1 Prototype Representation
tPm ^^ScT1"^ s?mula*11°!? of an^ branching, one-dimensional stream sys-
in?A The.first s^P m modeling a system is to subdivide the stream system
into reaches, which are stretches of stream that have uniform hydraulic ch a r-
-
Ea?i reaCS 1s then d1vided 1nto computational elements of equal
elements reaches must consist of an integer number of computational
There are seven different types of computational elements:
1. Headwater element
2. Standard element
3. Element just upstream from a junction
6
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4. Junction element
5. Last element in system
6. Input element
7. Withdrawal element
Headwater elements begin every tributary as well as the main river system,
and as such, they must always be the first element in a headwater reach. A
standard element is one that does not qualify as one of the remaining six
element types. Because incremental flow is permitted in all element types,
the only input permitted in a standard element is incremental flow. A,type 3
element is used to designate an element on the main stem just upstream of a
junction. A junction element (type 4) has a simulated tributary entering it.
Element type 5 identifies the last computational element in the river system;
there should be only one type 5 element. Element types 6 and 7 represent
inputs (waste loads and unsimulated tributaries) and water withdrawals, re-
spectively. River reaches, which are aggregates of computational elements,
are the basis of most data input. Hydraulic data, reaction rate coefficients,
initial conditions, and incremental flows data are constant for all computa-
tional elements within a reach.
1.2.2 Model Limitations
QUAL2E has been designed to be a relatively general program; however,
certain dimensional limitations have been imposed during program develop-
ment. These limitations are:
• Reaches: a maximum of 25
• Computational elements: no more than 20 per reach or a total
of 250
• Headwater elements: a maximum of 7
• Junction elements: a maximum of 6
• Input and withdrawal elements: a maximum of 25
QUAL2E incorporates features of ANSI FORTRAN 77 that allow these limita-
tions to be easily changed.
1.2.3 Model Structure and Subroutines
QUAL2E is structured as one main program supported by 51 different
subroutines. Figure 1-1 illustrates the functional relationships between
the main program and the subroutines. New state variables can be added
or modifications to existing relationships can be made with a minimum of
model restructuring through the simple addition of appropriate subroutines.
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Q
U
A
L
2
E
Ver.
(3.0)
low Augmentation
Program Return Loop for F
Q2EZ
nulation
Program Return Loop for Dynamic Sir
Steady State Convergence 1
— »
WRPT3A
WRPT3B
-* PRPLO
UNCAS
H INDATA 1 . h
H HYDRAU 1 »! CHANL J »
H TRIMAT |
>[ R'FAFnfPH ••
Steady State i '
Convergence Loop
QCALC1 '
H nr-Ai r^ I
>) SSCONV I r *l enow 1
LIGHT
— *t DOS
' — >t SOVMATJ
M WRPTI 1 J \A/r?pT** 1
H FLOAUG|
^ (Sbb Fly. VI- 1 f or pr ogram str
»
»
*l
"- * HEATEX
•
» HEATER
* TEMPSS
1 » CONSVTI
* BODS
»1 COLIS
»l ANCS
^
* GROW
LIGHT
*\ ALGAES
H PORGS
M PO4S
I1
H NH2S
»| NH3S
H (M62S |
w NO3S 1
V
»l DOS 1
1 INDOO 1
IND01
INDIA
IND02
IND03
IND04
IND05
IND06
IND07
IND08
IND09
iMn i n
IND11
IND12
IND13
ow
R
V
P
M
T
A
2
T
ucture)
Figure I-1 General Structure of QUAL2E
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The structural framework of QUAL2E has been modified from prior ver-
sions of QUAL-II. The large MAIN program and subroutine INDATA have been
divided into smaller groups of subroutines, each with a more narrowly
defined task. The new subroutines in QUAL2E include the algal light func-
tions (GROW/LIGHT), the steady state algal output summary (WRPT1), the or-
ganic nitrogen and phosphorus state variables (NH2S, PORG), and the line
printer plot routine (PRPLOT). This reorganization of QUAL2E into smaller-
programmatic units is the first step in adapting the model to micro and
minicomputers that have limited memory.
QUAL2E Version 3.0 retains this modular program structure. QUAL2E may ,
be obtained with or without the UNCAS capability. The program structure and
subroutine descriptions for UNCAS are described in Chapter 6 of this report.
1.2.4 Program Language and Operating Requirements
QUAL2E is written in ANSK FORTRAN 11 and is compatible with mainframe
and personal computer systems that support this language. QUAL2E typically
requires 256K bytes of memory and uses a single system input device (cards or
disk file) and the system's line printer (or disk file) as the output device.
If the system's normal FORTRAN input device unit is not unit 1 or the
output unit is not unit 7, then the variables "NI" and "NJ" in the main
program (files Q2E3P0 or Q2U3P0) should be changed to reflect the system s
I/O unit identifiers.
-------
2. GENERAL MODEL FORMULATION
2.1 INTRODUCTION
tn n., any Stream water quality model development is
h!*En?n^ 5° +that ha* the caPabimy for simulating the behavior of the
hydro] ogic and water quality components of a stream system. The development
nn I 15 *?i S1"!ulate Prototype behavior by applying a mathematical model
&g1n I comPu^ Proceeds through three general phases (Water Resources
1. Conceptual representation
2. Functional representation
3. Computational representation
tun* Conceptual representation involves a graphic idealization of the proto-
type by description of the geometric properties that are to be modeled and by
identification of boundary conditions and interrelationships between various
Pntn H?L t6 P^t0tyPf-,, U|Ually' th1s process entails dividing the prototype
S5 i c*ete ele|"e"*s of a S12e compatible with the objectives that the
model must serve, defining these elements according to some simple geometric
rules and designating the mode by which they are connected, either physically
or functionally as integral parts of the whole. A part of this conceptual
structuring is the designation of those boundary conditions to be considered
in ine simulation.
Functional representation entails formulation of the physical features
processes, and boundary conditions into sets of algebraic equations. It
involves precise definition of each variable and its relationship to all
shis parameters that characterize the model or its input-output relation-
Computational representation is the process whereby the functional model
is ^ranslated into the mathematical forms and computational procedures re-
quired for solution of the problem over the desired time and space continuum.
^mSe7Kd'^th devel°Pment of a specific solution technique that can be
accommodated by the computer and with codification of the technique in compu-
ter language. ^
urfn hJ Sc r?j"ajnfler <£ th1s section the Conceptual Representation of QUAL2E
will be described together with its general functional representation for
mass transport, hydraulic characteristics, and longitudinal dispersion
10
-------
Chapter 3 will discuss specific constituent reactions and interactions.
Chapter 4 will develop the functional representation of stream temperature as
simulated in QUAL2E.
2.2 CONCEPTUAL REPRESENTATION
Figure II-l shows a stream reach (n) that has been divided into a
number of subreaches or computational elements, each of length AX. For each
of these computational elements, the hydrologic balance can be written in
terms of flows into the upstream face of the element (Q-j_i), external sources
or withdrawals (Qx-j), and the outflow (Qi) through the downstream face of the
element. Similarly, a materials balance for any constituent C can be written
for the element. In the materials balance, we consider both transport (Q-C)
and dispersion (A DL j)C) as the movers of mass along the stream axis. Mass
Ax 3x
can be added to or removed from the system via external sources and with-
drawals (QxCx)-f and added or removed via internal sources or sinks (Si) such
as benthic sources and biological transformation. Each computational element
is considered to be completely mixed.
Thus, the stream can be conceptualized as a string of completely mixed
reactors—computational elements--that are linked sequentially to one another
via the mechanisms of transport and dispersion. Sequential groups of these
reactors can be defined as reaches in which the computational elements have
the same hydrogeometric properties—stream slope, channel cross section,
roughness, etc.--and biological rate constants—BOD decay rate, benthic
source rates, algae settling rates, etc.--so that the stream shown at the
left of Figure 11-2 can be conceptually represented by the grouping of reaches
and computational elements shown on the .right of Figure II-2.
2.3 FUNCTIONAL REPRESENTATION
2.3.1 Mass Transport Equation . . .
The basic equation solved by QUAL2E is the one dimensional advection-
dispers'ion mass transport equation, which is numerically integrated over
space and time for each water quality constituent. This equation includes
the effects of advection, dispersion, dilution, constituent reactions and
interactions, and sources and sinks. For any constituent, C, this equation
can be written as:
8M
8t
_9C
3x)
3x
3(AX u C) dC
dx + (Ax dx) — + s II-l
9x dt
11
-------
Computational
Element i
FLOW
BALANCE
Ox
i-t
MASS
BALANCE
(QC).
Figure II-l. Discretized Stream System
12
-------
Most UMtrtam
Point
Rtoeh
Numbtr
Computation's!
(Itmtnt Nurnbir"
Figure 11-2. Stream Network of Computational Elements and Reaches
13
-------
where
M
x
t
C
u
s
mass (M)
distance (L)
time (T)
concentration (M L~3)
cross-sectional area (L2)
dispersion coefficient (L2 T"1)
mean velocity (L T-l)
external source or sinks (M T-l
Because M = VC, we can write
aM
a(vc) ac av
at at at at
II-2a
where
V = AX dx = incremental volume (L3)
If we assume that the flow in the stream is steady, i.e., an/at
the term aV/at = 0 and equation ll-2a becomes
8M ac
_ s V —•
at at
= P, then
II-2b
Combining equations II-l and Il-2b and rearranging,
ac
at
JC
a(AxDL ax) a(Ax u c) dc s
- — + _
Ax 8x Ax 8x dt V
II-3
J °n the/19ht-hand side of the equation represent, respec-
n !,,sPers^on' advection, constituent changes, external sources/sinks,
and dilution. The dC term refers only to constituent changes such as
dt gp
growth and decay, and should not be confused with the term --, the local
at
concentration gradient. The latter term includes the effect'of constituent
changes as well as dispersion, advection, sources/sinks, and dilutions.
14
-------
Under steady-state conditions, the local derivative becomes equal to
zero; in other words:
9C
_ = o
3t
n-4
Changes that occur to individual constituents or particles independent of
advection, dispersion, and waste inputs are defined by the term
dC
— = individual constituents changes
dt
II-5
These changes include the physical, chemical, and biological reactions and
interactions that occur in the stream. Examples of these changes are
reaeration, algal respiration and photosynthesis, and col i form die-off.
2.4 HYDRAULIC CHARACTERISTICS
QUAL2E assumes that the stream hydraulic regime is steady-state; i.e.,
30 /at = 0, therefore, the hydrologic balance for a computational element
can be written simply as (see Figure II-l):
3Q
(-) = (Ox).
3X • 1
n-6
where (Qx) is the sum of the external inflows and/or withdrawals to that
element, i
2.4.1 Discharge Coefficients
Once equation 1 1-6 has been solved for 0, the other hydraulic
characteristics of the stream segments can be determined by equations of
the form:
1 1-7
Ax = Q/u
and
d =
n-9
where a, b, o- and p are empirical constants, and d is the stream depth.
These constants usually can be determined from stage-discharge rating
curves.
15
-------
2.4.2 Trapezoidal Cross Sections
Alternatively, if the cross-sectional properties of the stream segment
are available as a function of the depth d, u can be obtained as a function
of discharge by the trial and error solution of Mannings equation-
where
1.486
Q = - Aw Rj/3
n
x x
1/2
11-10
Ax = cross-section area of the channel or canal, ft2
Rx s mean effective hydraulic radius, ft
n = Manning roughness factor (usual range 0.010 to 0,10)
Se = slope of the energy grade line (dimensionless)
0 = discharge, ft3/sec
The value for U is then determined from equation II-8.
2.4.3 Longitudinal Dispersion
Dispersion is basically a convective transport mechanism. The term
dispersion is generally used for transport associated with spatially
averaged velocity variation, as opposed to "diffusion," which is reserved
for transport that is associated primarily with time-averaged velocity
fluctuations.
Tayi°r (1956) den'ved a predictive equation for the longitudinal disper-
sion coefficient, DL, in long straight pipes, as
DL = 10 r0 u*, ft2/sec n_n
where r0 is the pipe radius and u* is the average shear velocity defined as
u* = /T0/p, ft/sec 11-12
where
TO = boundary shear stress, lb/ft2, and
p = mass fluid density, Ib-sec2/ft4
Some investigators have attempted to apply Taylor's expression to stream-
Tlow. Such applications are only approximate, however, because of the
difference Between the geometry or velocity distributions in streamflow
and those in a pipe.
16
-------
Elder (1959) assumed that only the vertical velocity gradient was
important in streamflow and developed an expression analogous to Taylor's
expression:
nL = Kdu*
where d is the mean depth in feet of the stream.
5.93 for K in this equation.
11-13
Elder used a value of
Other investigators have derived similar expressions for D|_ and found
it to be extremely sensitive to lateral velocity profiles. Elder's
expression, however, seems adequate in one-dimensional situations where
the channel is not too wide. For very wide channels, Fisher (1964) has
shown that half-width rather than depth is the dominant scale and there-
fore is important to the definition of the longitudinal dispersion coeffi-
cient. Equations 11-11 and 11-13 can be written in terms of the Manning
Equation and other variables characteristic of stream channels.
As an
example, for steady-state open-channel flow.
u* = C / RSP
11-14
where
C = Chezy's coefficient
R = the hydraulic radius
Se = the slope of the energy grade line
Chezy's coefficient is given by:
Rl/6
n
C =
11-15
where n is the Manning roughness coefficient tabulated for different types
of channels in Table II-l.
Se, the slope of the energy gradient, is given by
s = (, _ )2
6 1.486 R2/3
11-16
where u is the mean velocity. Substituting equations 11-14, 11-15 and II
16 into equation 11-13 and letting R = d for a wide channel yields the
expression
PL = 3.82 K n u d5/6
11-17
17
-------
TABLE II-l
VALUES OF MANNING'S "n" ROUGHNESS COEFFICIENT
After Henderson (1966)
Artificial Channels
n
Glass, plastic, machined metal
Dressed timber, joints flush
Sawn timber, joints uneven
Cement plaster
Concrete, steel troweled
Concrete, timber forms, unfinished
Untreated gunite
Brickwork or dressed masonry
Rubble set in cement
Earth, smooth, no weeds
Earth, some stones, and weeds
0.010
0.011
0.014
0.011
0.012
0.014
0.015-0.017
0.014
0.017
0.020
0.025
Natural River Channels
n
Clean and straight
Winding with pools and shoals
Very weedy, winding and overgrown
Clean straight alluvial channels
(d »
0.025-0.030
0.033-0.040
0.075-0.150
0.031 dl/6
D-75 size in ft.
diameter that 75
percent of parti-
cles are smaller
than)
18
-------
where
DL = longitudinal dispersion coefficient, ft2/sec
K = dispersion constant (dimensionless)
n = Manning's roughness coefficient (dimensionless)
u" = mean velocity, ft/sec
d = mean depth, ft ,
Typical values for dispersion coefficients, DL, and values of the
dispersion constant, K, cited by Fisher et al. (1979), are given in Table
11-2. Note that the dispersion constant, K, shown in this table is one to
three orders of magnitude greater than that used by Elder.
2.5 Flow Augmentation
When the DO concentration in a stream drops below some required target
level, such as the state water quality standard for DO, it may be desirable
to raise this DO concentration by augmenting the flow of the stream.
According to the originators .of the flow augmentation routine in QUAL2E,
Frank D. Masch and Associates and the Texas Water Development Board (1971),
the amount of flow necessary to bring the DO concentrations up to required
standards cannot be calculated by an exact functional relationship. A good
approximation of the relationship is used in QUAL2E and has the following
quadratic form:
and
where,
DOR = DOT - D0min
DOR DOR
Q = Q [__ + Q.15 (—)2]
DOy -n°T
11-18
11-19
= dissolved xoygen concentration required to meet target
conditions, mg/L
DOy = required target level of DO, mg/L
D0ml-n= minimum DO concentration (critical level) in the oxygen sag
curve, mg/L
QR = amount of flow augmentation required, ft3/sec
o
°
- flow at the critical point in the oxygen sag curve, ft°/sec
19
-------
TABLE II-2
EXPERIMENTAL MEASUREMENTS OF LONGITUDINAL DISPERSION IN OPEN CHANNELS
(After Table 5.3, Fisher et al., 1979)
Channel
Chicago Ship
Channel
Sacramento
River
River Derwent
South Platte
River
Yuma Mesa
A Canal
Trapezoidal
Laboratory
Channel with
roughened
sides
Green-Duwanish
River
Missouri River
Copper Creek
(below gage)
Clinch River
Copper Creek
(above gage)
Powell River
Clinch River
Coachella Canal
Bayon Anacoco
Nooksack River
Wind/Bighorn
Rivers
John Day River
Depth
d
(ft)
26.5
13.1
0.82
1.5
11.3
0.115
0.154
0.115
0.115
0.069
0.069
3.61
8.86
1.61
2.79
1.61
2.79
6.89
6.89
1.31
2.79
1.90
5.12
3.08
2.98
2.49
3.61
7.09
1.90
8.10
Width
W
(ft)
160
_-
__
--
_ _
1.31
1.41
1.31
1.12
1.08
0.62
66
660
52
59
52
154
197
174
62
112
118
79
85
121
210
194
226
82
112
Mean
Velocity
u
(ft/sec)
0.89
1.74
1.25
2.17
2.23
0.82
1.48
1.48
1.44
1.48
1.51
--
5.09
0.89
1.97
0.85
1.05
3.08
2.62
0.52
0.49
0.69
2.33
1.12
1.31
2.20
2.89
5.09
3.31
2.69
Shear
Velocity
u*
(ft/sec)
0.063
0.17
0.46
0.23
1.13
0.066
0.118
0.115
0.114
0.108
0.127
0.16
0.24
0.26
0.33
0.26
022
034
0.35
0.38
0.18
0.16
0.14
0.22
0.22
0.89
0.39
0.56
0.46
0.59
Dispersion
Coefficient
DL
(ft2/sec)
32
161
50
174
8.2
1.3
2.7
4.5
0.8
4.3
2.4
70-92
16,000
215
226
102
151
581
506
97
102
87
103
355
420
377
452
1722
151
700
Dispersion
Constant
K
20
74
131
510
8.6
*J • W
174
150
338
205
392
270
120-160
7500
500
250
245
235
245
210
220
200
280
140
524
640
170
318
436
172
146
20
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TABLE 11-2
EXPERIMENTAL MEASUREMENTS OF LONGITUDINAL DISPERSION IN OPEN CHANNELS
(After Table 5.3, Fisher et al., 1979) (Continued)
Channel
Comite River
Sabine River
Yadkin River
Depth
d
(ft)
1.41
6.69
15.6
7.71
12.6
Width
W
(ft)
52
341
417
230
236
Mean
Velocity
u
(ft/sec)
1.21
1.90
2.10
1.41
2.49
Shear
Velocity
u*
(ft/sec)
0.16
0.16
0.26
0.33
0.43
Dispersion
Coefficient
Di
(ft2/sec)
151
3390
7200
1200
2800
Dispersion
Constant
K
650
3100
1800 '
470 .
520
The model augments the stream flow by first comparing, after steady-
state conditions have been reached, the simulated DO concentration with
the prespecified target level of DO in each reach. If the calculated DO
is below the target level, the program finds those upstream sources that
the user has specified for dilution purposes, and adds water equally from
all these sources. The DO calculations are then repeated. This process
continues until the DO target level is satisfied. (NOTE: The flow-
augmentation subroutine can be used for DO only.)
21
-------
3. CONSTITUENT REACTIONS AND INTERRELATIONSHIPS
3.1 GENERAL CONSIDERATIONS
One of the most important considerations in determining the waste-
assimilative capacity of a stream is its ability to maintain an adequate
dissolved oxygen concentration. Dissolved oxygen concentrations in streams
are controlled by atmospheric reaeration, photosynthesis, plant and animal
respiration, benthal demand, biochemical oxygen demand, nitrification
salinity, and temperature, among other factors.
The most accurate oxygen balance would consider all significant factors.
The QUAL2E model includes the major interactions of the nutrient cycles, algae
production, benthic oxygen demand, carbonaceous oxygen uptake, atmospheric
aeration and their effect on the behavior of dissolved oxygen. Figure III-l
illustrates the conceptualization of these interactions. The arrows on the
figure indicate the direction of normal system progression in a moderately
polluted environment; the directions may be reversed in some circumstances
for some constituents. For example, under conditions of oxygen supersatura-
tion, which might occur as a result of algal photosynthesis, oxygen might be
driven from solution, opposite to the indicated direction of the flow path.
Coliforms and the arbitrary nonconservative constituent are modeled as
nonconservative decaying constituents and do not interact with other consti-
tuents. The conservative constituents, of course, neither decay nor interact
in any way with other constituents.
and
^ mathematical relationships that describe the individual reactions
interactions are presented in the following paragraphs.
3.2 CHLOROPHYLL at (PHYTOPLANKTONIC ALGAE)
Chlorophyll a^ is considered to be directly proportional to the concen-
tration of phytoplanktonic algal biomass. For the purposes of this model
algal biomass is converted to chlorophyll a^ by the simple relationship:
Chi a =
A
III-l
22
-------
where
Chlj[ = chlorophyll ^concentration, ug-ChljJ/L
A = algal biomass concentration, mg-A/L
a0 = a conversion factor, ug Chlji/mg A
The differential equation that governs the growth and production of algae
(chlorophyll a) is formulated according to the following relationship.
ORG-N
N H
N O
N O
a.p
Atmospheric
Reaeration
,rIV2
D
I
S
S
O
L
V
E
D
0
X
Y
G
E
N
ORG-P
DIS-P
Chla
ALGAE
77'/'/'/
Figure III-l. Major Constituent Interactions in QUAL2E
23
-------
dA
—
dt
- PA -- A
d
in_2
where
A = algal biomass concentration, mg-A/L
t = time, day
p = the local specific growth rate of algae as defined below,
which is temperature dependent, day-l
p = the local respiration rate of algae, which is temperature
dependent, day-l
GI = the local settling rate for algae, which is temperature
dependent, ft/day
d = average depth, ft
3.2.1 Algal Respiration Rate
In QUAL2E, the single respiration rate parameter, p, is used to approxi-
mate three processes: (a) the endogenous respiration of algae, (b) the
conversion of algal phosphorus to organic phosphorus, and (C) the conversion
of algal nitrogen to organic nitrogen. No attempt is made to use separate
rate coefficients for these three processes, as is done in the State of
Vermont, revised Meta Systems version of QUAL-II (JRB Associates, 1983: and
Walker, 1981).
3.2.2 Algal Specific Growth Rate
The local specific growth rate of algae, y, is known to be coupled to
the availability of required nutrients (nitrogen and phosphorus) and light.
A variety of mathematical expressions for expressing multiple nutrient-light
limitations on algal growth rate have been reported (De Groot, 1983; Scavia
and Park, 1976; and Swartzman and Bentley, 1979). QUAL2E has the capability
of modeling the interaction among these limiting factors in three different
ways.
Growth Rate Option 1. Multiplicative. The kinetic expressions used to
represent the effects of nitrogen, phosphorus, and light are multiplied
together to determine their net effect on the local algal growth rate. This
option has as its biological basis the multiplicative effects of enzymatic
processes involved in photosynthesis:
(FL) (FM) (FP)
24
-------
where
FL
FN
FP
= maximum specific algal growth rate,
= algal growth limitation factor for light
= algal growth limitation factor for nitrogen
= algal growth limitation factor for phosphorus
This formulation is used in the SEMCOG version of QUAL-II.
Growth Rate Option 2. Limiting Nutrient. This option represents the
local algal growth rate as limited by light and either nitrogen or phosphorus,
but not both. Thus, the nutrient/light effects are multiplicative, but the
nutrient/nutrient effects are alternate. This formulation mimics Liebig's
law of the minimum:
(FL) Min (FN,FP)
Thus, the algal growth rate is controlled by the nutrient (N or P) with the
smaller growth limitation factor. This option is used in the State of
Vermont version of QUAL-II.
Growth Rate Option 3. Harmonic Mean. This option, a compromise
between options 1 and 2, is a modification of an intuitive form suggested by
Scavia and Park (1976) and is mathematically analogous to the total resistance
of two resistors in parallel. In this option, an effective nutrient limita-
tion factor is computed as the average of the inverse reciprocals of the
individual nitrogen and phosphorus growth limitation factors, i.e.,
(FL) [-
1/FN + 1/FP
Thus, the algal growth rate is controlled by a multiplicative relation
between light and nutrients, but the nutrient/nutrient interactions are
represented by a harmonic mean. This option has been used by Water
Resources Engineers in the application of a OUAL-II-like model, WREDUN, to
Lake Dunlap (Brandes and Stein, no date; see also Bowie fft al_.» 1985).
Walker (1983) has cautioned against using the harmonic mean option in
systems where one nutrient is in excess (say nitrogen, so that FN^-1.0) and
the other is extremely limiting (say phosphorus, so that FP^-0.0). In this
case the value of the nutrient attenuation factor approaches 2 FP, rather
than FP, as expected.
25
-------
3.2.3 AT gal-Light Relationships
3.2.3.1 Light Functions
A variety of mathematical relationships between photosynthesis and light
have been reported in the literature (Jassby and Platt, 1976; Field and
Effler, 1982). Although they differ in mathematical form, the relationships
exhibit similar characteristics. All show an increasing rate of photosynthe-
sis with increasing light intensity up to a maximum or saturation value. At
high light intensities, some of the expressions exhibit photoinhibition,
whereas others show photosynthetic activity remaining at the maximum rate.
QUAL2E recognizes three options for computing the algal growth limi-
tation factor for light, FL in Equations III-3a,b,c. Light attenuation
effects on the algal growth rate may be simulated using a Monod half-
saturation method, Smith's function (Smith, 1936), or Steele's equation
(Steele, 1962).
Light Function Option 1. Half Saturation. In this option, the algal
growth limitation factor for light is defined by a Monod expression:
FL,
KL
where
FLZ = algal growth attenuation factor for light at intensity I2
I2 = light intensity at a given depth (z), Btu/ft2-hr
KL = half saturation coefficient for light, Btu/ft2-hr
z = depth variable, ft
Light Function Option 2. Smith's Function. In this option, the algal
growth limitation factor for light is formulated to include second order
effects of light intensity:
FL,
i2)l/2
where
K|_ = light intensity corresponding to 71% of the maximum growth
rate, Btu/ft2-hr
with the other terms as defined in Equation III-4a.
26
-------
Light Function Option 3. Steel's Equation. This option incorporates
an exponential function to model the effect of photoinhibition on the algal
growth rate:
FLZ = (—) exp (1 - -)
KL KL
where
KL = saturation light intensity at which the algal growth rate is
a maximum, Btu/ft2-hr
with the other terms as defined in Equation III-4a.
Note: The parameter KL, which appears in all three light function equations
is defined differently in each.
All of the light functions in Equations III-4a,b,c express the value
of FL for an optically thin layer. In QUAL2E photosynthesis occurs throughout
the depth of the water column. Light intensity varies with depth according
to Beer's law:
III-5
where
Iz = I exp (-X z)
Iz = light intensity at a given depth (z), Btu/ft2-hr
I = surface light intensity, Btu/ft2-hr
X = light extinction coefficient, ft-1
z = depth variable, ft
When Equation III-5 is substituted into Equations III-4a,b,c and
integrated over the depth of flow, the depth-averaged light attenuation
factor is obtained. The resulting expressions for the three options are:
Option 1: Half Saturation
FL = (1/Xd) In [
KL + I
KL + Ie-x
KL = light intensity at which growth rate is 50%
of the maximum growth rate.
27
-------
Option 2: Smith's Function
FL = (1/xd) ln[-
I/K
(I/KL)2)V2
K]_ = light intensity at which growth rate is
of the maximum growth rate.
Option 3: Steel's Equation
_. 2'718 _ . -Ad(I/K.) -I/K.
FL = [e-(e L ) _ e L]
Xd
K[_ = light intensity at which growth rate is
equal to the maximum growth rate.
where
FL = depth-averaged algal growth attenuation factor for light
KL = light saturation coefficient, Btu/ft2-hr
X = light extinction coefficient, ffl
d = depth of flow, ft
I = surface light intensity, Btu/ft2-hr
The relative merits of these light functions are discussed by various
authors (Bannister, 1974; Platt et al_., 1981; Swartzmann and Bentley, 1979;
and Field and Effler, 1982). The half saturation method is the form used
in the SEMC06 version of QUAL-II. Evidence shows that the use of Smith's
function is preferrable over the half saturation method if photoinhibition
effects are unimportant (Jassby and Platt, 1976). The mathematical forms
of Equations III-4a,b,c are compared graphically in Figure III-2. All
three equations have a single parameter, KL; however, it is defined differ-
ently in each equation. In Figure III-2 the values of KL are selected so
that each curve passes through a common point, namely FL = 0.5 at I = 5
intensity units (i.e., a half saturation rate equal to 5 light intensity
units).
3.2.3.2 Light Averaging Options
Steady state algal simulations require computation of an average value
of FL, the growth attenuation factor for light, over the diurnal cycle.
28
-------
There are four options in QUAL2E for computing this average. The options
arise from combinations of situations regarding two factors:
• The source of the solar radiation data used in the computation,
i.e., whether it is supplied externally by the user or calculated
internally in the temperature heat balance.
• The nature of the averaging process, i.e., whether hourly values of
FL are averaged, or a single daylight average value of solar radia-
tion is used to estimate the mean value of FL.
The four daily light averaging options are defined below. In each case,
the half saturation light function is used as an example; in practice any of
the three light functions may be employed.
Option 1: FL is computed from one daylight average solar radiation
value calculated in the steady state temperature heat balance:
FL = AFACT * f *
= — In [-
Xd KL + Ial g(
Saturation
^ 0.8-
Half Saturation
1 = Half Saturation ; KL = 5.0
2 = Smith's Function ; KL = 8.66
3 = Steele's Equation ; KL = 21.55
Light Intensity, I (arbitrary units)
Figure 111-2. QUAL2E Light Functions
29
-------
Tal g = TFACT * Tt
emp
where
FL - algae growth attenuation factor for light, adjusted for
daylight hours and averaging method
A FACT = a light averaging factor, used to provide similarity
between calculations using a single average value of solar
radiation and computations using the average of hourly
values of FL
f = fraction of daylight hours
Fl_i = growth attenuation factor_for light, based on daylight
average light intensity (Iaig)
X = light extinction coefficient, ffl
d = mean depth of stream, ft
KL = half saturation coefficient for light, Btu/ft2-hr
Ta-|g = daylight average, photosynthetically active, light
intensity, Btu/ft2-hr
TFACT = fraction of solar radiation computed in the temperature
heat balance that is photosynthetically active
rtemp = daylight average light intensity as computed in the
temperature heat balance, Btu/ft2-hr
Option 2; FL is computed from one daylight average solar radiation
value supplied externally by the user. The calculations required to obtain
JFL in option 2 are the same as those for option 1, except that the value of
Ialg is computed directly from user input of photosynthetically active solar
"*al =
1 1 1-8
where
*tot = total daily photosynthetically active solar radiation,
Btu/ft2
N = number of daylight hours per day, hr
Both I-tot and N are supplied by the user as input information.
Equations III-8, III-7b, and III-7a are used to compute the value of FL.
Because the user input value of Itot is assumed to be the photosynthetically
active radiation, the factor TFACT is not used in option 2.
30
-------
Option 3: FL is obtained by averaging the hourly daylight values of FL
that are computed from the hourly daylight values of solar radiation calcu-
lated in the steady state temperature heat balance:
FL = f * FL2
1 N 1
FL2 = - E [•
N 1=1 Ad *
g,i
.i = TFACT * Hemp.i
where
FL2
Talg,i
Itemp,i
average of N hourly values of FL, based on
hourly values of light intensity (Ialg,i)
hourly value of photosynthetically active light
intensity, Btu/ft2-hr
hourly value of light intensity as computed in
the steady state temperature heat balance, Btu/
ft2-hr
with other terms are defined in Equations III-7a,b,c, and 111-8.
Because the average FL computed in option 3 (and 4) is an average of
diurnally varying values of FL, the factor A FACT is not used in the
calculations.
Option 4: FL is obtained by averaging the hourly daylight values of FL
that are computed from the hourly daylight values of solar radiation calcu-
lated from a stngle value of total daily, photosynthetically active, solar
radiation and an assumed cosine function. The calculations required to
obtain FL are the same as those for option 3, except that the values of
n -i are computed from an internally specified cosine function:
y, i
- Ttot/N (1
COS 2
N + 1
i = 1,N
111-10
As in the case of option 2, both Itot and N are supplied by the user.
Equations III-^vIII-9b, and HI-9a are then used to compute the value of FL.
Because the user specified value of Itpt is assumed to be photosynthetically
active, the factorTFACT is not used with option 4.
31
-------
Three empirical factors--diurnal cosine function AFACT and TFAPT
used in the formulations of the four light avenging options.' TFACT"
...Tw° Jiurnal cosine functions were evaluated for use in OUAL2E- m a
modified form of the one in the SEMCOG version of QUAL-II and (2) the fom
ccSulln QUAL'TX (Texas Water Development Board, 1984). The function in
davf?nhf hnn°d1f1ed-t0 P^0d,UCe ^^ Solar Cation vSfues f^ each
daylight hour, as given in Equation 111-10. The form used in QUAL-TX is-
Hot IT (1-1) iri
lalg.i = [COS( ) - COS(-)] ,
2N N N
1=1,N
III-ll
™r
AFACT).
nn nn?" HI-11 were evaluated by comparing simulated values
ng options 2 and 4 (i.e., in effect computing values of
ations were performed over a range of values of Ki , A, d I*,,*
- j « «« as for eacn of the three light functions,. The values of AFflrf
averaged 0.92 and 0.94 for the SEMCOG and Texas equations, respective?v
There was no compelling reason to include both functions (with the user soeci-
fying the one to be used). The diurnal cosine function used In QUAL2E there
fore, is the modified SEMCOG version given in Equation 111-10 '
^h^ 1S the adJ'ustmenJ factor accounting for the nonlinear averaging
inherent in computing a daily average value of FL. From the simulations
just described, a resonable value of AFACT is 0.92, with a range from 0 85
wSlJ^noSSV16-^ (1-85) rep°rt an 1mplied vafue of 1-0 °8
Walker (1983) suggests using a value of 0.85.
tinn hn Photosynthetically active fraction of total solar radia-
tion. When performing algae simulations, it is important that the value of
I±h J^e??lty ?"d Il9!!t Saturat1on coefficient, KL, be in units of phot £
synthetically active radiation, PAR (Bannister, 1974: Field and Effler 198V
and Stefan etal., 1983). Because the temperature heat balance computes '
total radiation over a wide spectrum, this value must be adjusted to PAR if
h*J %Dfl°D^US6d in *?e algae Slmulat1°n. The ratio of energy in the visible
n "£ l^n^0^"6^ 1n the complete (standard) spectrum is approximately
0.43 to 0.45 (Bannister, 1974 and Stefan et al_. , 1983). TFACT is Tuser
input variable; thus a value to meet site specific conditions may be used.
of Daily Averaging Options:
ar9.e'y.on t^e extent to which the user wishes to account for
"Intensity. Options 1 and 2 use a single
The selection of a light averaging
i ! °n an avera9e" da"y solar radiation value. Options
n calculate hourly values of FL from hourly values of solar radiation
and then average the hourly FL values to obtain the daily average value
rKfc1 a?i3 UKS€UthV°lar radfat1°n fr°m the temperatu% hSS balance
routines. (Thus both algae and temperature simulations draw on the sam?
nZ?HepJ°hvS,°Har rad1;t1on;> °Pt1ons 2 and 4 use the solar raSiatlon value
provided by the user for algae simulation. Thus, either option 2 or 4 must
be selected when algae are simulated and temperature is not. The light
32
-------
averaging factor (AFACT) is used to provide similarity in FL calculations
between options 1 and 2 versus options 3 and 4. The solar radiation factor
(TFACT) specifies the fraction of the solar radiation computed in the heat
balance, which is photosynthetically active. It is used only with options 1
or 3.
In dynamic algae simulations, photosynthetically active radiation is
computed hourly using Equation III-9c unless temperature is not simulated,
in which case photosynthetically active solar radiation data must be
supplied with the local climatology data.
3.2.3.3 Algal Self Shading
The light extinction coefficient, x, in Equations III-6a,b,c is coupled
to the algal density using the nonlinear equation
X =
(aQA)2/3
111-12
where
Xo =
A =
non-algal portion of the light extinction coefficient, ft"1
linear algal self shading coefficient, ft'1 (ug-Chla/L)-1
nonlinear algal self shading coefficient, ft"1 (ug-Chlja/L)"2'3
conversion factor, ug-Chlji /mg A
algal biomass concentration, mg-A/L
Appropriate selection of the values of \i and \2 allows modeling of a
variety of algal self-shading, light-extinction relationships:
• No algal self shading (QUAL-II SEMCOG)
\l = \2 = °
• Linear algal self shading (JRB Associates, 1983)
\l ? 0 , \2 = °
• Nonlinear algal self shading (Riley Eq., in Bowie et al_., 1985)
\! = 0.00268, ft'1 (ug-Chla/L)-1
X2 = 0.0165, ft"1 (ug-ChU/L)-2/3
or
33
-------
Xx = 0.0088, m"1 (ug-ChU/L)'1
X2 s 0.054, nT1 (ug-Chl_a/L)-2/3
3.2.4 Algal Nutrient Relationships
The algal growth limitation factors for nitrogen (FN) and for phos-
phorus (FP) are defined by the Monod expressions:
FN =
111-13
and
where
FP
P2 + Kp
111-14
Ne = the effective local concentration of available inorganic
nitrogen, mg-N/L
KN = the Michaelis-Menton half-saturation constant for nitrogen,
mg-N/L
P? = the local concentration of dissolved phosphorus, mg-P/L
Kp = the Michaelis-Menton half-saturation constant for
phosphorus, mg-P/L
Algae are assumed to use ammonia and/or nitrate as a source of in-
organic nitrogen. The effective concentration of available nitrogen is
given by:
where
HI
N3
N3
concentration of ammonia nitrogen, mg-N/L
concentration of nitrate nitrogen, mg-N/L
111-15
The empirical half-saturation constants for nitrogen, KM, and phos-
phorus, Kp, are used to adjust the algal growth rate to account for those
34
-------
factors that can potentially limit algal growth. Each constant is actually
the level at which that particular factor limits algal growth to half the
maximal or "saturated" rate (Bowie et al_., 1985). Table III-3 at the end of
this chapter lists typical values of the half-saturation constants for nitro-
gen and phosphorus. If algal concentrations are simulated and either nitro-
gen, phorphorus, or both are not simulated, the program assumes that the
parameter not simulated is not limiting.
3.2.5 Temperature Dependence in Algae Simulation
The algal growth rate and death rates are temperature dependent. They
are corrected within the model, as are all other temperature dependent
systems variables, according to the procedure explained in Section 3.10.
3.3 NITROGEN CYCLE
In natural aerobic waters, there is a stepwise transformation from
organic nitrogen to ammonia, to nitrite, and finally to nitrate. The nitro-
gen cycle in QUAL2E contains all four of these components, as shown in Figure
III-l. The incorporation of organic nitrogen as a state variable, an organic
nitrogen settling term, and an algal nitrogen uptake preference factor are
the primary enhancements to the nitrogen cycle in OUAL2E compared to the
SEMCOG version of QUAL-II. The differential equations governing transforma-
tions of nitrogen from one form to another are shown below.
3.3.1 Organic Nitrogen
where
- = 01 P A -
dt
N4 - 04 N4
111-16
N4 = concentration of organic nitrogen, mg-N/L
33 = rate constant for hydrolysis of organic nitrogen to
ammonia nitrogen, temperature dependent, day-1
«i = fraction of algal biomass that is nitrogen, mg-N/mg-A
Algal respiration rate, day-1
algal biomass concentration, mg-A/L
p
A
04
rate coefficient for organic nitrogen settling, temperature
dependent, day-1
35
-------
3.3.?? Ammonia Nitrogen
- = P3N4 -
dt
+ a3/d - FI cqpA
111-17
where
+ (1 - PN)N3) m_18
NI = the concentration of ammonia nitrogen, mg-N/L
NS = the concentration of nitrate nitrogen, mg-N/L
N4 = the concentration of organic nitrogen, mg-N/L
Pl = rate constant for the biological oxidation of ammonia nitrogen
temperature dependent, day-I '
P3 = organic nitrogen hydrolysis rate, day"1
01 = fraction of algal biomass which is nitrogen, mg-N/mg-A
o3 = the benthos source rate for ammonia nitrogen, mg-N/ft2-day
d = mean depth of flow, ft
FI = fraction of algal nitrogen uptake from ammonia pool
y = the local specific growth rate of algae, day-1
A = algal biomass concentration, mg-A/L
PN = preference factor for ammonia nitrogen (0 to 1.0)
_The OUAL2E model includes an algal preference factor for ammonia, PM
(Bowie etal 1985; JRB Associates, 1983). The ammonia preference factor
f 2V fu ° the fractlon of algal nitrogen uptake from the ammonia
pool when the concentrations of ammonia and nitrate nitrogen are equal.
3.3.3 Nitrite Nitrogen
dt
111-19
36
-------
where
HI = the concentration of ammonia nitrogen, mg-N/L
N;? = the concentration of nitrite nitrogen, mg-N/L
31 = rate constant for the oxidation of ammonia nitrogen,
temperature dependent, dayl
32 = rate constant for the oxidation of nitrite nitrogen,
temperature dependent, dayl
3.3.4 Nitrate Nitrogen
where
dN3
dt
= 32N2 - (1 - F)amA
111-20
F =
fraction of algal nitrogen taken from ammonia pool, as
defined in Section 3.3.2
fraction of algal biomass that is nitrogen, mg-N/mg-A
local specific growth rate of algae, dayl
3.3.5 Inhibition of Nitrification at Low Dissolved Oxygen
QUAL2E has the capability of inhibiting (retarding) the rate of
nitrification at low values of dissolved oxygen. This inhibition effect
has been reported by others (Department of Scientific and Industrial
Research, 1964; Texas Water Development Board, 1984).
Nitrification rates are modified in QUAL2E by computing an inhibition
correction factor (having a value between zero and one) and then applying
this factor to the values of the nitrification rate coefficients, 3^, and
32. The nitrification rate correction factor is computed according to
a first order equation:
where
CORDO = l.n - exp(-KNITRF * DO)
CORDO = nitrification rate correction factor
exp = exponential function
111-21
37
-------
KNITRF = first order nitrification inhibition coefficient, mg/L'1
DO = dissolved oxygen concentration, mg/L
The correction factor is applied to the ammonia and nitrite oxida-
tion rates by:
Ammonia: (3i)1nhib. = CORDO * (ei)1nput
Nitrite: (32)inhib. = CORDO * (02)input
111-22
111-23
A value of 0.6 for KNITRF closely matches the inhibition formulation in
QUAL-TX, the Texas Water Development Board version of QUAL-II, whereas, a
value of 0.7 closely simulates the data for the Thames Estuary (DSIR, 1964).
3.4 PHOSPHORUS CYCLE
The phosphorus cycle operates like the nitrogen cycle in many respects.
Organic forms of phosphorus are generated by the death of algae, which then
convert to the dissolved inorganic state, where it is available to algae for
primary production. Phosphorus discharged from sewage treatment plants is
generally in the dissolved inorganic form and is readily taken up by algae
(Bowie et rt_., 1985). QUAL2E revises the SEMCOG version of QUAL-II, which
included only dissolved phosphorus, to simulate the interactions between
organic and dissolved phosphorus. Below are the differential equations
governing transformations of phosphorus from one form to another.
3.4.1 Organic Phosphorus
dPj
— = a2 p A -
dt
where
- 05?!
111-24
PI = the concentration of organic phosphorus, mg-P/L
a2 = phosphorus content of algae, mg P/mg-A
p = algal respiration rate, day-1
A = algal biomass concentration, mg-A/L
P4 » organic phosphorus decay rate, temperature dependent, day'1
05 = organic phorphorus settling rate, temperature dependent,
day-1
38
-------
3.4.2 Dissolved Phosphorus
dP2
—
dt
111-25
where
?2 = concentration of inorganic or dissolved phosphorus, mg-P/L
02 = benthos source rate for dissolved phosphorus, temperature
dependent, mg-P/ft2-day
d = mean stream depth, ft
y = algal growth rate, day-1
A = algal biomass concentration, mg-A/L
3.5 CARBONACEOUS BOD
The QUAL2E model assumes a first order reaction to describe deoxygen-
ation of ultimate carbonaceous BOD in the stream. The BOD function as
expressed in the model also takes into account additional BOD removal due
to sedimentation, scour and flocculation, which do not exert an oxygen
demand (Thomas, 1948):
dL
_
dt
- K3L
111-26
where
L = the concentration of ultimate carbonaceous BOD, mg/L
KIL = deoxygenation rate coefficient, temperature dependent, day~l
KS = the rate of loss of carbonaceous BOD due to settling,
temperature dependent, day-1
QUAL2E simulates ultimate BOD in the general case; however, the user may
choose to use 5-day BOD values for input and output. In this case, the model
will make the necessary coversions from 5-day to ultimate BOD. The conversion
equation is:
BOD5 = BODU (1.0 - exp(5 * KBOD))
111-27
39
-------
where
ROD5 = 5-day BOD, mg/L
BODU = ultimate BOD, mg/L
KBOD = BOD conversion rate coefficient, day-l-
SEMC06 version of QUAL-II uses a value of 0.23 day! for KBOD. With
QUAL2E, the user may specify the appropriate value for this conversion. Note:
when modeling 5-day BOD, the same conversion coefficient is applied to all
input BODs forcing functions (headwaters, incremental flows, point loads, and
the downstream boundary condition).
3.6 DISSOLVED OXYGEN
The oxygen balance in a stream system depends on the capacity of the
stream to reaerate itself. This capacity is a function of the advection and
diffusion processes occurring within the system and the internal sources and
sinks of oxygen. The major sources of oxygen, in addition to atmospheric
reaeration, are the oxygen produced by photosynthesis and the oxygen contained
in the incoming flow. The sinks of dissolved oxygen include biochemical
oxidation of carbonaceous and nitrogenous organic matter, benthic oxygen
demand and the oxygen utilized by algae respiration (Bowie ejt al_., 1985).
The differential equation used in QUAL2E to describe the rate of change
of oxygen is shown below. Each term represents a major source or sink of
oxygen.
dO
dt
= K2(0*-0) + (03 y - a4P) A - KI L - K4/d - a5
- a(5 32 N2 1 1 1-28
where
n
0*
the concentration of dissolved oxygen, mg/L
the saturation concentration of dissolved oxygen at the
local temperature and pressure, mg/L
the rate of oxygen production per unit of algal photo-
synthesis, mg-0/mg-A
the rate of oxygen uptake per unit of algae respired, mg-0/mg-A
the rate of oxygen uptake per unit of ammonia nitrogen
oxidation, mg-0/mg-N
40
-------
06
w
p
A
L
d
K2
K4
Pi
P2
Ml
N2
the rate of oxygen uptake per unit of nitrite nitrogen
oxidation, mg-0/mg-N
algal growth rate, temperature dependent, dayl
algal respiration rate, temperature dependent, dayl
algal biomass concentration, mg-A/L
concentration of ultimate carbonaceous BOD, mg/L
mean stream depth, ft
carbonaceous BOD deoxygenation rate, temperature dependent,
dayl
the reaeration rate in accordance with the Fickian diffusion
analogy, temperature dependent, dayl
sediment oxygen demand rate, temperature dependent, g/ft^-day
ammonia oxidation rate coefficient, temperature dependent,
day-1
nitrite oxidation rate coefficient, temperature dependent,
day-1
ammonia nitrogen concentration, mg-N/L
nitrite nitrogen concentration, mg-N/L
3.6.1 Dissolved Oxygen Saturation Concentration
The solubility of dissolved oxygen in water decreases with increasing
temperature, increasing dissolved solids concentration, and decreasing
atmospheric pressure (Bowie et afU, 1985). QUAL2E uses a predictive
equation for the saturation "(Equilibrium) concentration of dissolved oxygen
(APHA, 1985).
lnn* = -139.34410 + (1.575701 x 1Q5/T) - (6.642308 x 107/T2)
+ (1.243800 x 1Q10/T3) - (8.621949 x IQH/T^) 111-29
where:
0* = equilibrium oxygen concentration at 1.000 atm, mg/L
T = temperature (°K) = (°C+273.150) and °C is within the
range 0.0 to 40.0°C
41
-------
For non-standard conditions of pressure, the equilibrium concentration
of dissolved oxygen is corrected by the equation 111-30:
Op = 0*P [
U-PWV/P)
(1-Pwv)
111-30
where
and
where
Op = equilibrium oxygen concentration at non-standard pressure,
mg/L
0* = equilibrium oxygen concentration at 1.000 atm, mg/L
P = pressure (atm) and is within 0.000 to 2.000 atm
Pw = partial pressure of water vapor (atm), which may be
computed from:
InP = 11.8571 - (3840.70/T) - 216961/T2) 111-31
= 0.000975 - (1.426 x 10-5t) + (6.436 x 10-8t2) 111-32
t = temperature, °C
The equations in Standard Methods (1985) for computing dissolved oxygen
saturation concentrations also include corrections for salinity and chloride.
Because neither salinity nor chloride is explicitly modeled, OUAL2E does not
correct 0* for chloride or salinity. Furthermore, the pressure correction to
0* (Equation 111-30) is made only when temperature is modeled, because baro-
metric pressure data are a primary requirement of the heat balance equations.
The dissolved oxygen saturation concentrations computed from the Texas
and SEMCOG versions of QUAL-II are compared to those from the Standard Methods
formulations of QUAL2E in Table III-l.
3.6.2 Atmospheric Reaeration Coefficient Estimation
The reaeration coefficient (K2> is most often expressed as a function of
stream depth and velocity. QUAL2E provides eight options for estimating or
reading in i<2 values, which are discussed in the sections below. A compara-
tive study of reaeration prediction equation performance has been reported by
St. John et aj_. (1984).
42
-------
TABLE III-l
COMPARISON OF DISSOLVED OXYGEN SATURATION CONCENTRATIONS
(Barometric Pressure = 1 atm, Chloride = O.Omg/L,
Equilibrium with Air Saturated with Water Vapor)
Temperature,
°C
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
QUAL-II
SEMCOG
14.631
14.227
13.837
13.461
13.100
12.752
12.418
12.096
11.787
11.489
11.203
10.927
10.661
10.406
10.159
9.922
9.692
9.471
9.257
9.050
8.849
8.655
8.465
8.281
8.101
7.925
7.753
7.584
7.417
7.252
7.089
6.927
6.765
6.604
6.442
6.280
6.116
5.950
5.782
5.612
5.438
OUAL-TX
Texas
14.584
14.187
13.806
13.441
13.091
12.755
12.433
12.124
11.828
11.544
11.271
11.009
10.758
10.517
10.285
10.062
9.848
9.642
9.444
9.253
9.069
8.891
8.720
8.555
8.396
8.241
8.092
7.948
7.807
7.672
7.540
7.412
7.288
7.167
7.049
6.935
6.823
6.715
6.609
6.506
6.406
OUAL2E
Std. Meth.
14.621
14.217
13.830
13.461
13.108
12.771
12.448
12.139
11.843
11.560
11.288
11.027
10.777
10.537
10.306
10.084
9.870
9.665
9.467
9.276
9.093
8.915
8.744
8.578
8.418
8.264
8.114
7.969
7.828
7.691
7.559
7.430
7.305
7.183
7.065
6.949
6.837
6.727
6.620
6.515
6.413
43
-------
Kp Option 1
Option 1 allows the user to read in l<2 values that have been previously
selected by the modeler. This option is useful in modeling unusual situations
such as ice cover (see Section 3.6.3).
K? Option 2
Using data collected in field measurements of stream reaeration,
Churchill, Elmore, and Buckingham (1962) developed the following expression
for K2 at 20°C.
where
K220 = 5.026 u °-969 d-1-673 x 2.31 IH-33
u = average velocity in the stream, ft/sec.
d = average depth of the stream, ft
l<2 = reaeration coefficient, day"-'-
K? Option 3
O'Connor and Dobbins (1958) proposed equations based on the turbulence
characteristics of a stream. For streams displaying low velocities and
isotropic conditions, Equation 111-34 was developed:
I/ 20
K2
m
dl.50
111-34
For streams with high velocities and nonisotropic conditions, the rela-
tionship is:
where
Ort
20 =
480D
05
dl-25
111-35
S0 = slope of the streambed, ft/ft
d = mean stream depth, ft
u ~ mean velocity, ft/day
Kp » reaeration coefficient, dayl
44
-------
and Dm is the molecular diffusion coefficient (ft2/day), which is given by:
Dm = 1.91 x W6 (1.037)
T-20
111-36
Equation 111-34 has been found to be generally applicable for most cases and
is the equation used in QUAL2E for Option 3. Equation 111-35 can be used to
calculate l<2 outside the model and input it directly under Option 1.
K? Option 4
Based on the monitoring of six streams in England, Owens et al. (1964)
obtained reaeration estimates for shallow, fast moving streams. Combining
their data with that of Churchill et al., they developed an euation for
streams exhibiting depths of 0.4 to 11.0 feet and velocities of 0.1 to 5.0
ft/sec:
K220 = 9.4 u
x 2.31
111-37
where
u" = mean velocity, ft/sec
d = mean depth, ft
Kp Option 5
Thackston and Krenkel (1966) proposed the following equation based on
their investigation of several rivers in the Tennessee Valley Authority
system.
u*
K2,0 = 10.8 (1 + F0'5) — x 2.31
d
where F is the Froude number, which is given by:
111-38
F =
111-39
and u* is the shear velocity, ft/sec.:
u* = V d Seg =
u n /~g
1.49
111-40
45
-------
where
d = mean depth, ft
g = acceleration of gravity, ft/sec^
Se = slope of the energy gradient
IT = mean velocity, ft/sec
n = Manning's coefficient
Kp Option 6
Langbien and Durum (1967) developed a formula for K2 at 20°C:
K220 = 3.3 u/d1-33 x 2.31 111-41
H a mean velocity, ft/sec
d = mean depth, ft
where
K? Option 7
This option computes the reaeration coefficient from a power function
of flow. This empirical relationship is similar to the velocity and depth
correlations with flow used in the hydraulics section of QUAL2E, i.e.,
where
K2 = aQb 111-42
a = coefficient of flow for K2
Q = flow, ft3/sec
b = exponent on flow for K2
Kp Option R
The method of Tsivoglou and Wallace (1972) assumes that the reaeration
coefficient for a reach is proportional to the change in elevation of the
water surface in the reach and inversely proportional to the flow time through
the reach. The equation is:
46
-------
Ah
K220 = c -
tf
where
c = escape coefficient, ft~l
Ah = change in water surface elevation in reach, ft
tf = flow time within reach, days
Assuming uniform flow, the change in water surface elevation is
Ah = Se Ax
where
Se = slope of the energy gradient, ft/ft
Ax = reach length, ft
and the time of passage through a reach is
I I 1-43
II 1-44
AX
=•
u
where
u = mean velocity in reach, ft/sec
Substituting the above in equation II1-43 gives
K-20 = (3600 x 24) cSp IT
111-45
111-46
Equation 111-46 is the form of Option 8 used in QUAL2E. The constants
3600 and 24 convert velocity to units of feet per day. The slope may be
input directly for computing K£ with this option, or it can be calculated
from Manning's equation as follows
u n2
Se =
(1.49)2 d4/3
111-47-.
47
-------
where
d = mean depth, ft
n = Manning's coefficient
The escape coefficient is usually treated as a variable and determined
empirically. TenEch (1978) recommends the following guideline in determining
c values, analogous to that recommended for uncalibrated stream segments by
Tsivoglou and Neal (1976):
c = 0.054 ft-1 (at 20°C) for 15 < = Q < = 3000 «3/sec
c = 0.110 ft-1 (at 20°C) for 1 < = 0 < = 15 ft3/sec
3.6.3 Ice Cover
Ice cover on streams during winter low flow conditions may significantly
affect reaeration. Reaeration rates are decreased because ice cover reduces
the surface area of the air-water interface through which reaeration occurs
(TenEch, 1978). Approaches recommended by TenEch (1978) for estimating the
extent of ice cover include:
•
t
Statistical analyses of past records
Steady state heat budget analysis (including the U.S. Army Corps of
Engineers differential equations)
Extensive field observations
To adjust the reaeration rate for winter ice cover conditions in the
QUAL2E model, the calculated reaeration rate must be multiplied by an "ice
cover factor" and input under Option 1. TenEch recommends factors ranging
from 0.05 for complete ice cover to 1.0 for no ice cover. Depending on the
extent of cover, reaeration values can be greatly reduced.
3.6.4 K? Default Values
There are no default Kg values in QUAL2E. In some versions of OUAL-II,
a default value of Kg is computed, accounting for the influences of wind-
induced turbulence and diffusion under low-velocity conditions. In those
models, when the calculated values of Kg are less than two divided by the
depth of the reach (2/d), Kg is set equal to 2/d. This feature has not
always proved useful, particularly when simulating the very low reaeration
rates; thus it is not included in QUAL2E.
48
-------
3.6.5 Dam Reaeratlon
QUAL2E has the capability of modeling oxygen input to the system from
reaeration over dams. The following equation described by Butts and Evans
(1983) and attributable to Gameson is used to estimate oxygen input from
dam reaeration.
D Db=[l , — : ] Oa 111-48
1 + 0.116abH(l,- 0.034H)(1 + 046T)
where
Da = oxygen deficit above dam, mg/L
Ob = oxygen deficit below dam, mg/L
T = temperature, °C
H = height through which water falls, ft
a = empirical water quality factor
= 1.80 in clean water
= 1.60 in slightly polluted water
=1.0 in moderately polluted water
= 0.65 in grossly polluted water
b = empirical dam aeration coefficients
= 0.70 to 0.90 for flat broad crested weir
= 1.05 for sharp crested weir with straight slope face
= 0.80 for sharp crested weir with vertical face
= 0.05 for sluice gates with submerged discharge
The factors H, a and b are input for each dam. The model includes a
provision for bypassing some or all of the flow around the dams (e.g.,
through generators). The fraction of the total flow that spills over the
dam is supplied as an input variable.
3.7 COLIFORMS :
Coliforms are used as an indicator of pathogen contamination in sur-
face waters. Expressions for estimating coliform concentrations are
49
-------
T1rst order decay functions, which only take into account coliform
die-off (Bowie et al_., 1985). The OUAL2E model uses such an expression:
dE
"*"""
dt
111-49
where
E = concentration of coliforms, colonies/100 ml
K5 = coliform die-off rate, temperature dependent, day"1
3.8 ARBITRARY NONCONSERVATIVE CONSTITUENT
QUAL2E has the provision for modeling an arbitrary nonconservative
constituent (ANC). In addition to a first order decay mechanism, there are
source and sink terms in the mass balance. The differential equation
describing the interactions for an arbitrary nonconservative constituent is-
dR
— = -K6 R -
dt
cr7/d
HI-BO
where
R =
concentration of the nonconservative constituent, mg-ANC/L
Kg = decay rate for the constituent, temperature dependent, day'1
<*6 = rate coefficient for constituent settling, temperature
dependent, day-1
cfy » benthal source for constituent, temperature dependent,
mg-ANC/ft2-day
d = mean stream depth, ft
3.9 TEMPERATURE
Temperature is modeled by performing a heat balance on each computa-
tional element in the system. The heat balance accounts for temperature
inputs and losses from the forcing functions as well as the heat exchanged
between the water surface and the atmosphere. The air-water heat balance
terms include long and short wave radiation, convection, and evaporation
using:
50
-------
where
Hc
He
- H - H
111-51
Hn = net heat flux passing the air water surface, Btu/ft2-day
Hsn = net short wave solar radiation after losses from absorption and
scattering in the atmosphere and by reflection at the interface,
Btu/ft2-day
net long wave atmosphere radiation after reflection, Btu/ft2-day
outgoing long wave back radiation, Btu/ft2-day
convective heat flux, Btu/ft2-day
an
heat loss by evaporation, excluding sensible heat loss,
Btu/ft2-day
In order for QUAL2E to perform the heat balance computations, the user
must supply a variety of data, including the longitude and latitude of the
basin, the time of year, evaporation coefficients, and a dust attenuation
coefficient. Local climatological information in the form of time of day,
wet and dry bulb air temperatures, atmospheric pressure, cloud cover and wind
velocity also must be provided.
In the dynamic mode, local climatological data must be supplied at
regular (typically 3 hour) intervals. In this manner the source/sink term
for the heat balance is updated in time to simulate the diurnal response of
the steady hydraulic system to changing temperature conditions.
In the steady state mode, average local climatological data must be
supplied by the user. The program uses linear approximations for the long-
wave back radiation and evaporation terms for solution of the steady state
heat balance. The reader is referred to Chapter 4 of this report for a
detailed treatment of the temperature simulation.
In the dynamic mode, local climatology data are applied uniformly over
the entire river basin (i.e., there is no spatial variation). In the steady
state mode, local climatology data may vary spatially by reach.
3.10 TEMPERATURE DEPENDENCE OF RATE COEFFICIENTS
The temperature values computed in QUAL2E are used to correct the rate
coefficients in the source/sink terms for the other water quality variables.
These coefficients are input at 20°C and are then corrected to temperature
using a Streeter-Phelps type formulation:
51
-------
XT . X20 0
111-52
where
XT = the value of the coefficient at the local temperature (T)
X20 - the value of the coefficient at the standard temperature (20°C)
9 = an empirical constant for each reaction coefficient
The values of the temperature correction factors, 9, may be specified by
the user. In the absence of user specified values, the default values
shown in Table III-2 are employed. For comparison purposes, the 9 values
used in the SEMCOG version of QUAL-II are also listed in Table 111-2.
flf temperature is not simulated, the temperature value specified for
the initial condition is assumed to be the temperature for the simulation.
3.11 REACTION RATES AND PHYSICAL CONSTANTS
The chemical and biological reations that are simulated by QUAL2E are
represented by a complex set of equations that contain many system parameters;
some are constant, some are spatially variable, and some are temperature
dependent. Table 111-3 lists these system parameters and gives the usual
range of values, units, and types of variation. Kramer (1970), Chen and
Orlob (1972), and Bowie et al_. (1985) give detailed discussions of the basic
sources of data, ranges and reliabilities of each of these parameters. Final
selection of the values for many of these system parameters or measurement of
sensitive ones should be made during model calibration and verification.
52
-------
TABLE 111-2
DEFAULT TEMPERATURE CORRECTION, 9, VALUES FOR QUAL2E
Rate Coefficient
BOD Decay
BOD Settling
Reaeration
SOD Uptake
Organic N Decay
Organic N Settling
Ammonia Decay
Ammonia Source
Nitrite Decay
Organic P Decay
Organic P Settling
Dissolved P Source
Algal Growth
Algal Respiration
Algal Settling
Col i form Decay
ANC
AMC
ANC
Symbol
Kl
KS
K2
K4
33
CT4
31
°3
32
34
°5
°2
V
P
°1
KB
K6
°6
°7
Default Values
SEMCOG qUALZt
1.047 1.047
1.024
1.0159 1.024
1.060
1.047
1.024
1.047 1.083
- 1.074
1.0471 1.047
1.047
1.024
1.074
1.047 1.047
1.047 1.047
1.024
1.047 1.047
1.047 1.000
1.024
1.000
Note: - = not temperature dependent in QUAL-II SEMCOG.
ANC = Arbitrary Nonconservative Constituent
53
-------
TABLE III-3
TYPICAL RANGES FOR QUAL2E REACTION COEFFICIENTS
Variable Description
«0
«1
W
p
KL
KN
Kp
AQ
Xx
Ratio of chlorophyll -a
to algal biomass
Fraction of algal biomass
that is Nitrogen
Fraction of algal biomass
that is Phosphorus
Og production per unit of
algal growth
0;? uptake per unit of
algae respired
02 uptake per unit of
NHs oxidation
Oj> uptake per unit of
N02 oxidation
Maximum algal growth rate
Algal respiration rate
Michaelis-Menton half-
saturation constant
for light (Option 1)
Michaelis-Mention half-
saturation constant
for nitrogen
Michaelis-Menton half-
saturation constant
for phosphorus
Non-algal light extinc-
tion coefficient
Linear algal self-shading
coefficient
Units
ug-Chla
mg A
mg-N
mg A
mg-P
mg A
mg-0
mg A
mg-0
mg A
mg-0
mg N
mg-0
mg N
day-1
day-1
Btu/ft2-
min
mg-N/L
mg-P/L
ft-1
I/ft
ug ChU/L
Range of Variable
Values by Reach
10-100
0.07-0.09
0.01-0.02
1.4-1.8
1.6-2.3
3.0-4.0
1.0-1.14
1.0-3.0
0.05-0.5
0.02-0.10
0.01-0.30
.001-0.05
Variable
0.002-0.02
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Temperature
Dependent
No
No
No
No
No
No
No
No
No
No
No
No
No
No
54
-------
TABLE III-3 (cont'd)
TYPICAL RANGES FOR QUAL2E REACTION COEFFICIENTS
Vari-
able
PN
0^1
0O
(Jo
*4
'5
°6
°7
Kl
K2
K3
K4
K5
Description
Nonlinear algal self-
shading coefficient (
Algal preference factor
for ammonia
Algal settling rate
Benthos source rate for
dissolved phosphorus
Benthos source rate for
ammonia nitrogen
Organic nitrogen
settling rate
Organic phosphorus
settling rate
Arbitrary non-conserva-
tive settling rate
Benthal source rate for
arbitrary non-conserva-
tive settling rate
Carbonaceous deoxygenera-
tion rate constant
Reaeration rate constant
Rate of loss of BOD due
to settling
Benthic oxygen uptake
Col i form die-off rate
Arbitrary non-conserva-
Units
I/ft
xg Chla/L)^
ft/day
mg-P
ft^-day
mg-0
ftf-day
day-1
day-1
day'1
rnq-ANC
ft^-day
day-1
day-1
day-1
mg-0
.day.-1
day'1
Range
of Variable Temperature
Values by Reach Dependent
0.0165
'3 (Riley)
0.0-1.0
0.5-6.0
Variable
Variable
0.001-0.1
0.001-0.1
Variable
, Variable
0.02-3.4
0.0-100
-0.36-0.36
Variable
0.05-4.0
Variable
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
tive decay coeffici
55
-------
TABLE III-3 (cont'd)
TYPICAL RANGES FOR QUAL2E REACTION COEFFICIENTS
Variable Description
Range of Variable Temperature
Units Values by Reach Dependent
Rate constant for the
biological oxidation
of NH3 to N02
Rate constant for the
biological oxidation
of N02 to N03
Rate constant for the
hydrolysis of organic-
N to ammonia
Rate constant for the
decay of organic-P
to dissolved-P
day -1 0.10-1.00 Yes
day-1 0.20-2.0 Yes
day-1 0.02-0.4 Yes
day-1 0.01-0.7 Yes
Yes
Yes
Yes
Yes
56
-------
4. FUNCTIONAL REPRESENTATION OF TEMPERATURE
4.1 BASIC TEMPERATURE EQUATION
The basic mass transport equation for QUAL2E was given in Section II as
(see equation II-3):
9C
at
3C
3(AXDL 3x)
3(AX u C)
dC
—
dt
s
-
V
IV-1
In temperature modeling, C is taken as the concentration of heat (HL-3) and
can be equated to temperature through the relationship
C = P c (T - T0)
IV-2
where
p = the density of water (M L-3)
c = the heat capacity of water (HM-1
T = the water temperature
T0 = an arbitrary base temperature
M = mass
H = heat energy flux
D = degrees
The parameters p and c can be considered constant for practical purposes,
Also, the internal heat generation dC, which results from viscous dis'si-
"Ht
pation of energy and boundary friction, is generally small enough to be
57
-------
considered negligible. Thus setting dC = 0 in equation IV-1 and substituting
dt 9
equation IV-2 for C gives us (after some simplification):
3T 3(AXDL 3x)
3t Ay 3x
3(AX u T) Is
.. + _ _
Ax 3x pc V
IV-3
The source term s/V (with units of HL-Sy-l) accounts for all heat trans-
ferred Across the system boundaries, i.e., heat transferred across the air-
water interface and heat conducted across mud-water interface. Heat transfer
across the mud-water interface is generally insignificant; hence, s/V takes
on the identity of the net rate of heat input per unit volume of stream
through the air-water interface.
*, !t/MSxm5st Conven1ent to represent the interfacial heat transfer rate as
a flux 0%) having units of HL^T'1. For a stream element of length dx and
mean surface width W, HN is related to s/V as follows.
The total rate of heat input across the air-water interface is HN dx W.
This heat is distributed uniformly throughout the underlying volume of Av dx
Where Ax is the mean cross-sectional area of the element. Thus the rate of
heat gain per unit volume of water, s/V, is computed as-
HN (Wdx)
Ax dx
HN
d
IV-4
Where d = AX/W is the hydraulic depth of the stream. Substituting equation
IV-4 into equation IV-3 gives the generalized form of the temperature
equation: K
3T
at
_3T
3(AXDL 3x)
Ax 3x
3(AX u T) HN
Ax 3x pcd
IV-5
4.2 DEFINITION OF HN
Heat is transferred across the air-water interface of a surface water
^ b¥-three_difference processes: radiation exchange, evaporation, and
conduction. The individual heat terms associated with these processes are
shown in Figure IV-1 and are defined in Table IV-1 with the typical ranqes of
their magnitudes in northern latitudes also listed.
58
-------
The expression that results from the summation of these various energy
fluxes is:
where
u
Hc
H
= Hsn + Han - (Hb ± Hc + He)
IV-6
net energy flux passing the air-water interface,
Btu/ft2-day
net short-wave solar radiation flux passing through the
interface after losses due to absorption and scattering
in the atmosphere and by reflection at the interface,
Btu/ft2-day
net long-wave atmospheric radiation flux passing through
the interface after reflection, Btu/ft2-day
outgoing long-wave back radiation flux, Btu/ft2-day
conductive energy flux passing back and forth between the
interface and the atmosphere, Btu/ft2-day
energy loss by evaporation, Btu/ft2-day
These mechanisms by which heat is exchanged between the water surface and the
atmosphere are fairly well understood and are adequately documented in the
literature by Edinger and Reyer (1965). The functional representation of
these terms has been defined by Water Resources Engineers, Inc. (1967).
H
H
sn
an
H
Hsr
\
'1
"or
A
I
r '
A He
'
" t
1 AIR-WATER
INTERFACE
r
Figure IV-1.
Heat Transfer Terms Associated with
Interfacial Heat Transfer
59
-------
TABLE IV-1
DEFINITION OF HEAT TRANSFER TERMS
ILLUSTRATED IN FIGURE 1
Heat Term
Units
Magnitude
(BTU/ft2-dayl)
total incoming solar or
short-wave radiation
HL-2-r-i
400-2800
H
sr
H
reflected short-wave radiation
total incoming atmospheric
ratiation
HL-2T-1
40-200
2400-3200
H
'ar
reflected atmospheric radiation HL~2T-1
back radiation from the water
surface
70-120
2400-3600
H,
heat loss by evaporation
150-3000
heat loss by conduction to
atmosphere
HL-2T-1
-320 to +400
The formulations reported here were extracted from that more detailed work
by Frank D. Masch and Associates and the Texas Water Development Board
(1971).
4.3 NET SHORT-WAVE SOLAR RADIATION
The net incoming solar radiation is short-wave radiation which passes
directly from the sun to the earth's surface. Its magnitude depends on:
the altitude of the sun, which varies daily as well as seasonally for a
fixed location on the earth; the dampening effect of scattering and
absorption in the atmosphere due to cloud cover, and the reflection from
the water surface.
60
-------
The net amount of solar radiation which reaches the surface of the earth
may be represented functionally on an hourly basis by:
Hsn ' Ho
where
Hsn
HO
at
RS
CL
(1) (11) (11D
net short-wave solar radiation flux, Btu/ft2-hr
amount of radiation flux reaching the earth's
atmosphere, Btu/ft2-hr
atmospheric transmission term
Albedo or reflection coefficient
cloudiness as a fraction of sky covered
It is appropriate for purposes of this discussion to identify and treat
separately the four components in equation IV-7 as (1) extraterrestrial solar
radiation, (ii) radiation scattering and absorption, (iii) reflectivity, and
(iv) cloudiness.
4.3.1 Extraterrestrial Radiation
The short-wave solar radiation flux that strikes the earth's outer
atmosphere over a given period of time is given by Water Resources Engineers,
Inc. (1967) as:
HSC irtp
H0 = — { sin sin 6 (te -
r2 180
12 irCp iftg
+ _ cos — cos 6 [sin (—) - sin (—)]} r
TT .180 12 '' 12
IV-8
where
sc
solar constant = 438.0 Btu/ft2-hr
normalized radius of the earth's orbit
latitude of the site, degrees
61
-------
6 = declination of the sun, degrees
tb,te = hour angles corresponding to the beginning and end,
respectively, of any time interval between sunrise
and sunset
r = a correction factor for diurnal exposure to radiation
flux
Listed below are several parameters in equation IV-8 requiring further
definition as described by Water Resources Engineers, Inc. (1967).
a. Relative Earth-Sun Distance--
2TT
r = 1.0 + 0.017 cos [— (186-Dy)]
365
where Dy is the number of the day of the year (beginning January 1)
IV-9
b. Declination—
23.45 ZTT
TT cos [— (173-Dy)]
180 365
IV-10
c, Hour Angles--
STb - Ats + ET - 12
IV-11
and
STP - Ate + ET - 12
IV-12
where STb, STe are the standard times at the beginning and end of the time
interval selected
ET
an expression for time from a solar ephemeris that
represents the difference in hours between "true solar
time" and that computed on the basis of a yearly average.
It is given for each day of the year, Dy, by
ET
2TT
0.000121 - 0.12319 sin [--- (Dy-1) - 0.0714]
365
62
-------
where
At<
Lsm
Llm
4TT
0.16549 sin [- — (Dy-1) + 0.3088]
365
difference between standard and local civil time
in hours as determined from:
IV-13
Ate
e
15
-1 for west longitude
+1 for east longitude
longitude of standard meridian, degrees
longitude of local meridian, degrees
IV-14
d. Diurnal Exposure--
r = 1 when STr _< Sit, or STe <^ STS
r = 0 when STS £ STb or STe <_ STr
IV-15
IV-16
where STr and STS are the standard times of sunrise and sunset, respectively,
as determined from:
12
TT(j)
STr = 12 -•— arc cos [tan (—) tan <$] + Ats
TT 180
IV-17
and
ST. = 24 - STr + 2At<
IV-18
4.3.2 Radiation Scattering and Absorption
The atmospheric transmission term, at, is given by Water Resources
Engineers, Inc. (1967) as:
a" + 0.5 (1 - a' - d)
1 ,-,0.5 Rs (1 - a' + d)
IV-19
63
-------
in which a" is the mean atmospheric transmission coefficient after scatterina
and absorption, given by:
a" = exp { - [0.465 + 0.0408 Pwc]
[0.179 + 0.421 exp (-0.721 Qam)] Qam}
where Oam is the optical air mass given by the expression:
exp (-Z/2531)
in which
sin a + 0.15 (180a + 3.885)-l«253
IT
elevation of the site in feet
sun's altitude in radians, given by:
IV-20
IV-21
a = arc sin [sin — — sin 6 + cos —
180 180
irt
COS 6 COS — ]
12
IV-22
in which t is the hour angle, described by an equation similar to equation
IV-11 and Iv-12.
pwc 1n equation IV-20 is the mean daily precipitable water content in
the atmosphere, given >by the expression:
we
0.00614 exp (0.0489Td)
IV-23
where ?d is the dewpoint in °F, which can be obtained from the expression:
Td = In [(ea + 0.0837)/0.1001]/0.03 IV-24
where ea is the water vapor pressure of the air.
The mean atmospheric coefficient, a1, can also be represented by an
equation of the form of equation IV-20 as:
64
-------
exp { - [0.465 + 0.0408 Pwc.)
[0,129 +0.171 exp (-0.880 9am)] 9am} IV-25
Dust attenuation of the solar radiation flux, which is represented in
equation IV-19 by the quantity d, varies with optical air mass, season of the
year, and geographic location. Water Resources Engineers, Inc. (1967) gives
a range of 0-0.13 for several locations.
4.3.3 Cloudiness
The dampening effect on the solar radiation flux is given by Water
Resources Engineers, Inc. (1967) as
1.0 - 0.65
IV-26
where C|_ is the decimal fraction of the sky covered. Water Resources
Engineers, Inc. (1967) reports that equation IV-26 gives satisfactory results
except for heavy overcast conditions, i.e., when C[_ approaches 1.0.
4.3.4 Reflectivity
The reflection coefficient, Rs, can be approximately computed as a
function of the solar altitude, a, by Anderson's (1954) empirical formula:
Aa
B
IV-27
where a is in degrees, and A and B are functions of cloudiness, C|_. Values
for A and B given by Anderson (1954) are shown in Table IV-2.
TABLE IV-2
EMPIRICAL COEFFICIENTS FOR DETERMINING Rs
After Anderson (1954)
Cloudiness
cL
Coefficients
0 0.1 - 0.5
Clear Scattered
A B A B .
1.18 :-0.77 2.20 -0.97
0.6 - 0.9
• Broken
A -• B
s 0.95; --0.75-
1.0
Overcast
•:-. A B
, 0.35 -0.45
65
-------
4.4 LONG-WAVE ATMOSPHERIC RADIATION
The long-wave radiation emitted by the atmosphere varies directly with
the moisture content of the atmosphere. Although it is primarily dependent
on air temperature and humidity, it can also be affected by ozone, carbon
dioxide, and possibly other materials in the atmosphere. Anderson (1954)
indicated that the amount of atmospheric radiation is also significantly
affected by cloud height. The amount of long-wave atmospheric radiation that
is reflected is approximately a constant fraction of the incoming radiation,
found by Anderson (1954) to be approximately 0.03.
The net atmospheric radiation flux can be expressed as:
where
Han = [2.89 x ID'6] a (Tg + 460)6 (1.0 + 0.17C^) (1-RL) IV-2R
Han
a
Ta
R[_
net long-wave atmospheric radiation flux, Btu/ft2-hr
Stefan-Boltzman constant, 1.73 x 10-9 Btu/ft2/hr/°Rankine4
air temperature at a level 6 feet above the water surface, °F
reflectivity of the water surface for atmospheric radiation =
0.03
cloudiness, fraction of cloud cover
4.5 WATER SURFACE BACK RADIATION
The third source of radiation transfer through the air-water interface
is long-wave back radiation from the water surface, %, which represents a
loss of heat from the water. It can be seen from Table IV-1 that back
radiation accounts for a substantial portion of the heat loss from a body
of water. This loss is expressed by the Stefan-Boltzman Fourth Power
Radiation Law for a blackbody as:
0.97 CT (T,. + 460)4
IV-29
where
water surface back radiation flux, Btu/ft^-hr
water surface temperature, °F
66
-------
Equation IV-29 can be linearized over a given temperature range as
Hb
32 T
IV-30
where
32
constants defined over the range 35 to 135 °F
In the steady-state temperature solution, this linearized version of the
back radiation equation is used to allow the temperature dependent terms to
be separated out of the equation. Sets of «2, 32 are specified for 21 5°F
temperature intervals between 35°F and 135°F. For dynamic simulations the
heat flux term calculations are based on the temperature at the beginning of
the time step.
4.6 EVAPORATION
A water body also loses heat to the atmosphere by evaporation. Each
pound of water that leaves as water vapor carries its latent heat of vapori-
zation (approximately 1050 BTU at 60°F) plus its sensible heat. This signif-
icant heat loss due to evaporation can be expressed as:
Y HLE + Hv
IV-31
where
Y
HL
HL
E
H..
specific weight of the water being evaporated, Ib/ft3
latent heat of vaporization, Btu/lb, given by
1084 - 0.5 Ts
evaporation rate, ft/hr
sensible heat loss Btu/ft2-hr
The evaporation rate, E, is most often expressed as
(a + bW) (es - ea)
IV-32
where
a,b =
constants
67
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w
and
wind speed, in mph, measured 6 feet above the water
surface
saturation vapor pressure of the air, in. of Hg, at the
temperature of the water surface, as given by
0.1001 exp (0.03 Ts) - 0.0837
water vapor pressure, in. of Hg, at a height of 6 feet
above the water surface, given as
ewb - 0.000367 Pa (Ta - Twb)
where
ewb =
ewb =
Twb - 32
(1.0 + )
1571
IV-34
saturation vapor pressure, in. of Hg, at the wet bulb
temperature from the expression
0.1001 exp (0.03 T^) ~ 0.0837
local barometric pressure, in. of Hg
wet'bulb air temperature, °F
dry bulb air temperature, °F
IV-35
The literature contains a wide range of values for the evaporation constants
a and b. Roesner (1969) reports that a good average value of a would be 6.8
x 10-4 ft/hr-in. of Hg, while b would best be represented by 2.7 x 10-4 ft/
hr-in. of Hg.-mph.
To linearize the variation of evaporation rate with surface water
temperature Ts, equation IV-34 is approximated over 5°F intervals as:
31 T
IV-36
Sets of ai, 3i are specified for twenty-one 5°F intervals between 35°F and
135°F. The linearized evaporation expression is used in the steady-state
temperature solution.
68
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The sensible evaporative heat loss can be expressed simply as:
c y E (Ts - TO)
IV-37
where
Tf
heat capacity of water = 1 Btu/lb-°F
reference temperature, °F
Sensible heat loss is very small compared to the other heat loss components
in the energy budget and,thus is not included in the QUAL2E temperature
computation.
4.7 CONDUCTION
Heat that is transferred between the water and the atmosphere due to
a temperature difference between the two phases and not related to water
vapor exchange is normally called conduction. Using the fact that transfer
by conduction is a function of the same variables as evaporation, it is
possible to arrive at a proportionality between heat conduction and heat loss
by evaporation. This proportionality, known as Bowen's ratio, is expressed
as:
Hc Ts - Ta Pa
- = CB [ ]
He es - ea 29.92
IV-38
where Cg is a coefficient == 0.01.
By using Bowen's ratio, the rate of heat loss to the atmosphere by
heat conduction, Hc, can be defined as:
Pa
Hc = T HL (a+bW) (0.01 ) (Ts - T-a) IV-39
29.92
For practical purposes, the ratio (Pa/29.92) can be taken as unity.
4.8 QUAL2E MODIFICATIONS FOR REACH VARIABLE LOCAL CLIMATOLOGY AND TEMPERATURE
Prior versions of QUAL-II and QUAL2E have assumed that the input variables
for temperature simulation were uniform over the entire river basin (global
inputs). These input variables consist of climatological, geographical, and
heat balance information as follows: basin elevation, dust attenuation
69
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coefficient, evaporation coefficients, dry and wet bulb air temperatures,
atmospheric pressure, cloud cover, and wind speed. In the current version
of QUAL2E most of these inputs, with the exception of the evaporation coeffi-
cients are reach variable. Thus, for systems in which variable ambient
temperature and climatology may be important, for example in modeling rivers
with large changes in elevation, different values for these factors may be
supplied for each reach in the river. The overall heat balance computations
are performed as described in Sections 4.1-4.7 of this chapter, using the
reach specific values of each input variable. When reach variable tempera-
ture simulation inputs are used, a detailed temperature and heat balance
summary is provided with the QUAL2E final output.
The user has a number of options in specifying the input variables for
temperature simulation. Global values may be used (all reaches having the
same values for each of the temperature simulation inputs), or different
input values may be explicitly specified for each reach in the system. In
the case where reach specific values of atmospheric pressure are not known
or available, OUAL2E has the capability of estimating the value of atmo-
spheric pressure for each reach from its elevation and air temperature.
These estimates are computed from the ideal gas law integrated over the
change in elevation relative to a datum (Plate, 1982).
eC-(g/RT)(z - z0)]
IV-40
Where:
P = atmospheric pressure at elevation z (in Hg),
g = gravitational constant (32.2 ft/sec2),
R = gas law constant (1715 ft2/sec2-QR),
T = dry bulb air temperature (°R),
z = elevation of reach (ft),
ZQ> PO = datum elevation and pressure, respectively,
The principal assumptions used in deriving Eq. IV-40 are that air temperature
and specific humidity are constant. Thus, the value of the gas constant, R,
is that for dry air and the value of dry bulb air temperature, T, is the
average of the dry bulb temperatures at elevations z and z0. Although re-
finements to this methodology are possible, they were deemed premature until
more experience with this option is obtained. If the reach variable values
of atmospheric pressure are computed from Eq. IV-40, they are echo-printed
with the QUAL2E output.
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5. COMPUTATIONAL REPRESENTATION
5.1 PROTOTYPE REPRESENTATION
To expand upon the basic conceptual representation presented in Sections
1 and 2, QUAL2E permits any branching, one-dimensional stream system to be
simulated. The first step involved in approximating the prototype is to
subdivide the stream system into reaches, which are stretches of stream that
have uniform hydraulic characteristics. Each reach is then divided into
computational elements of equal length so that all computational elements in
all reaches are the same length. Thus, all reaches must consist of an integer
number of computational elements.
There are seven different types of computational elements:
1. Headwater element
2. Standard element
3. Element just upstream from a junction
4. Junction element
5. Last element in system
6. Input element
7. Withdrawal element
Headwater elements begin every tributary as well as the main river system,
and as such, they must always be the first element in a headwater reach. A
standard element is one that does not qualify as one of the remaining six
element types. Because incremental flow is permitted in all element types,
the only input permitted in a standard element is incremental flow. A type 3
element is used to designate an element on the mainstem that is just upstream
of a junction. A junction element (type 4), has a simulated tributary en-
tering it. Element type 5 identifies the last computational element in the
river system (downstream boundary); there should be only one element type 5.
Element types 6 and 7 represent elements which have inputs (waste loads and
unsimulated tributaries) and water withdrawals, respectively.
71
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River reaches, which are aggregates of computational elements, are the
basis of most data input. Hydraulic data, reaction rate coefficients, initial
conditions, and incremental flow data are constant for all computational
elements within a reach.
5.2 FORCING FUNCTIONS
Forcing functions are the user specified inputs that drive the system
being modeled. These inputs are specified in terms of flow, water quality
characteristics, and local climatology. OUAL2E accommodates four types of
hydraulic and mass load forcing functions in addition to local climatological
factors—headwater inputs, point sources or withdrawals, incremental inflow/
outflow along a reach, and the (optional) downstream boundary concentration.
1. Headwater Inputs - Headwater inputs are typically the upstream
boundary conditions at the beginning of the system. They are the conditions
required to generate the solution of the mass balance equations for the first
computational element in each headwater reach. Headwaters are also the
source of water for flow augmentation.
2. Point Sources and/or Withdrawals - These loads are used to represent
point source discharges into the system (i.e., sewage and industrial waste,
or storm water runoff) and losses from the system resulting from diversions.
In QUAL2E point source discharges may represent either raw or treated waste
loads. If raw waste loads are used, the effect of treatment can be simulated
by applying a specific fract-ional removal for carbonaceous BOD to each point
source load.
3. Incremental Inflow - OUAL2E has the capability to handle flow
uniformly added or removed along a reach. The total flow increment along
a reach is apportioned equally to all computational elements in the reach.
This feature can be used to simulate the effects of non-point source inputs '
to the system, or the effect of loss of stream flow to the groundwater.
4. Downstream Boundary Concentration (optional) - OUAL2E has the
capability of incorporating known downstream boundary concentrations of the
water quality constituents into the solution algorithm. This feature is
useful in modeling systems with large dispersion in the lower reaches (e.g.,
estuaries). When downstream boundary concentrations are supplied, the solu-
tion generated by QUAL2E will be constrained by this boundary condition. If
the concentrations are not provided, the constituent concentrations in the
most downstream element will be computed in the normal fashion using the zero
gradient assumption (see Section 5.4.3).
Local climatological data are required for the simulation of algae and
temperature. The temperature simulation uses a heat balance across the
air-water interface and thus requires values of wet and dry bulb air tempera-
tures, atmospheric pressure, wind velocity, and cloud cover. The algal
simulation requires values of net solar radiation. For dynamic simulations,
these climatological data must be input at regular time intervals over the
72
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course of the simulation and are applied uniformly over the entire river
basin. For modeling steady-state temperature and algae, average daily local
climatological data are required and may vary spatially over the basin by
reach.
5.3 MODEL LIMITATIONS
QUAL2E has been developed to be a relatively general program; however,
certain dimensional limitations have been imposed upon it during program
development. These limitations are as follows:
Reaches: a maximum of 25
Computational elements: no more than 20 per reach or 250 in total
Headwater elements: a maximum of 7
Junction elements: a maximum of 6
Input and withdrawal elements: a maximum of 25 in total
(Note: These limitations may be modified, if necessary, by the user by
altering the PARAMETER statement specifications in file MAIN.VAR of the
program and recompiling.
QUAL2E can be used to simulate any combination of the following
parameters or groups of parameters.
1. Conservative minerals (up to three at a time)
2. Temperature
3. BOD
4. Chlorophyll ji
5. Phosphorus cycle (organic and dissolved)
6. Nitrogen cycle (organic, ammonia, nitrite, and nitrate)
7. Dissolved oxygen
8. Col iforms
9. An arbitrary nonconservative constituent
All parameters can be simulated under either steady-state or dynamic
conditions. If either the phosphorus cycle or the nitrogen cycle are
not being simulated, the model presumes they will not limit algal growth.
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5.4 Numerical Solution Technique
At each time step and for each constituent, Equation 11-3 can be written
I times, once for each of the I computational elements in the network.
Because it is not possible to obtain analytical solutions to these equations
under most prototype situations, a finite difference method is used—more
specifically, the classical implicit backward difference method (Arden and
Astill, 1970; Smith, 1966; and Stone and Brian, 1963).
The general basis of a finite difference scheme is to find the value of
a variable (e.g., constituent concentration) as a function of space at a time
step n+1 when its spatial distribution at the nth time step is known. Time
step zero corresponds to the initial condition. Backward difference or im-
plicit schemes are characterized by the fact that all spatial derivatives
(8/3x) are approximated in difference form at time step n+1.
5.4.1 Formulation of the Finite Difference Scheme
The finite difference scheme is formulated by considering the consti-
tuent concentration, C, at four points in the mnemonic scheme as shown in
Figure V-l.
Three points are required at time n+1 to approximate the spatial
derivatives. The temporal derivative is approximated at distance step i.
DOWNSTREAM 4-
UPSTREAM
element i + 1
element
i
N
: t
At
o
i
N : t -
At
Figure V-l. Classical Implicit Nodal Scheme
74
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Equation 11-3 can be written in finite difference form in two steps.
First, the advection and diffusion terms are differentiated once with
respect to x, giving:
3C-j
3t
3C 3C
(ADL --) - (ADL --)
9X 9X _
(A u C)i - (A u C)i_i
where
dCi s-j
f t _|_ , r
dt V-
V-j = A.J A x-j
V-l
Secondly, expressing the spatial derivative of the diffusion terms in finite
difference and thence the time derivative of C in finite difference, there
results:
At
.] Cfft - C(ADL)i] Cf 1
L)!.!] cfi -'
A X
, Cn+1
11
Cn+1 + Pi
~ V-2
YI_
In equation V-2, the term dC/dt is expressed as:
dC-j
dt
where
rj = first order rate constant
Pi = internal constituent sources and sinks (e.g., nutrient
loss from algal growth, benthos sources, etc.)
75
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Note that the dC/dt for every constituent modeled by OUAL2E can be expressed
in this form.
If equation V-2 is rearranged in terms of the coefficients of Cn+L
Cf+1, and Cf{J, we obtain the equation: 1~1
i Cf 1
V-3
where
b1 » 1.0 + [(ADL)i
At Qi_i At
+ — — ]
At
AX-
At
Qi ri At
At
At
Pi At
The values of a1, b-, c-, and Z- are all known at time n, and the Cn+1
terms are the unknowns at time step n+1. 1
In the case of a junction element with a tributary upstream element,
the basic equation becomes:
hi
= Z,-
V-4
where
dj = - [(AD)j
At
At
j = the element upstream of junction element i
C!j+1 = concentration of constituent in element j at time n+1
It can be seen that the dj term is analogous to the a-,- term. Both
terms account for mass inputs from upstream due to dispersion and advec-
tion.
76
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Under steady-state conditions, — = 0 in equation V-l. Working
9t
through the finite difference approximations, and rearranging terms as
before, the steady-state version of equation V-3 is derived:
where
., rn+l + K. rn+l + r. rn+l = z-
ai H-l Di H i i+1 i
V-5
(ADL)i-l
= -C +
(ADL)i
C - +
V-jAx-j
+ -
V-
(ADL)i
C ]
Z1 > - + Pi
Note that equation V-5 is the same as equation V-3, with three
changes:
o At = 1.0
o the constant 1.0 in b-j = 0.0
o the initial concentration Clj1 in Z^ = 0.0
5.4.2 Method of Solution
Equations V-3 and V-5 each represent a set of simultaneous linear
equations whose solution provides the values of Cn+1 for all i's.
Expressed in matrix form, this set of equations appears as:
77
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bl cl
do bp Cp
a3 b3 c3
ai bi ci
al bl
zi
V-6
The left matrix is a tri-diagonal matrix. An efficient method that readily
lends itself to a computer solution of such a set of equations is:
Divide through the first equation in V-6 by bj to obtain:
Cn+1 + Wx
V-7
where
and R =
Combine the expression for b-j (see V-3) and the second equation in V-6
to eliminate 32 and the result is:
Cf 1 + W2 Cf 1 = G2
V-8
where
W2 =
C2
and
Z2 - 92
b2 - 32
Combine equation V-8 and the third equation in V-6 to eliminate 33 and
the result is:
78
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cg+1 + w3 cf i = GS
V-9
where
b3 - a3
and
Z3 - a3 G2
t>3 - a3 W2
Proceed through the equations, eliminating a-j and storing the values
of W-j and G-j given by:
W1 =
ci
and
b-j. - a.-j Wi_
zi - ai Ri-
'-, i = 2, 3, . . . ,
V-10
V-ll
The last equation is solved for
by
C{
+1 =
V-12
Solve for Cn+}-, Cn^, • • • .
by back substitution.
Cn+1 = G1 - W1 Cn^, i = 1-1, 1-2, . . . , 1
V-13
5.4,3 Boundary Conditions
In most situations of interest, transport is unidirectional in nature,
i.e., there is no significant transport upstream. Therefore, the concen-
tration at some point just upstream from the beginning or end of the stream
reach of interest can be used as the boundary condition.
5.4.3.1 Upstream Boundary (Headwater Elements)
For headwater elements there is no upstream, i-1, element. Thus, the
headwater driving force is substituted in Equation V-3 for the upstream
concentration G-J_I. Because the headwater concentrations are fixed, they
are incorporated on the right hand side of Equation V-3 in the known term
for headwater elements as follows.
At - a1 C(
V-14
79
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where C0 is the upstream boundary condition (headwater concentration).
5.4.3.2 Downstream Boundary (Last Element in the System)
QUAL2E has two options for modeling the downstream boundary. One uses a
zero gradient assumption; the other incorporates fixed downstream constituent
concentrations into the solution algorithm.
Zero Gradient Assumption (Arden and Astill, 1970)-~For the last computa-
tional element in the system, there is no downstream, i+1, element. At this
boundary, a zero gradient assumption is made that replaces C1+1 with Cj_i.
In this manner, the downstream boundary acts as a mirror to produce a zero
gradient for the concentration of the constituent variable. The coefficient
a-j, therefore, is modified to include the dispersion effect normally found in
the coefficient c-j for the last element in the system. Thus, the equation
for a-j in V-3 becomes:
= -C((ADL)i_i + (AnL)j)
Qi-lAt
V-15
and
= 0
V-16
where I = index of the downstream boundary element
Fixed Downstream Constituent Concentrations—For this boundary option,
the user supplies known downstream boundary concentrations C[_B for each water
quality constituent. Thus, the value of C-j+i in Equation V-3 becomes
= CLB
V-17
Because the boundary concentrations are known in this option, they are
incorporated on the right hand side of Equation V-3 in the known term Z-j for
the downstream boundary element then results as
z = c
SjAt
LB
V-18
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6. UNCERTAINTY ANALYSIS WITH QUAL2E
6.1 INTRODUCTION '
Uncertainty analysis for model simulations is assuming a growing
importance in the field of water quality management. The impetus for this
concern is provided by recent public awareness over health risks from
improper disposal of toxic wastes as well as by the continuing emphasis
within EPA on risk assessment. One of the first steps in the chain of risk
assessment is the quantification of the error in predicting water quality.
Unfortunately, uncertainty analysis of water quality model forecasts has
not received as much attention in practice as has the prediction of expected
(average) values.
Uncertainty analysis has been the subject of much discussion in the
ecosystem modeling literature (Rose and Swartzman, 1981 and O'Neill and
Gardner, 1979). In the water resources literature, lake eutrophication
models have been used to compare various methods of uncertainty analysis
(Reckhow, 1979; Scavia et al_., 1981; and Malone et _a]_., 1983). The method-
ologies described in this chapter represent a systematic approach to uncer-
tainty analysis for the general purpose stream water quality model OUAL2E.
The objective is to provide some of the tools for incorporating uncertainty
analysis as an integral part of the water quality modeling process. The
QUAL2E model was chosen for this application because it is a general purpose
computer code, widely used by consultants and state regulatory agencies in
waste load allocation and other planning activities. The resulting uncer-
tainty model is named QUAL2E-UNCAS.
6.2 QUAL2E-UNCAS
Three uncertainty analysis techniques can be employed in QUAL2E-UNCAS—
sensitivity analysis, first order error analysis, or monte carlo simulation.
The user is provided this array of options for flexibility, because the
methods differ in their assumptions and will not always agree with each
other. Discrepancies may be explained by errors in the first order approxi-
mation or by errors due to biased variance calculations. Monte carlo simula-
tion has the advantage of output frequency distributions, but it carries a
high computational burden. First order error propagation provides a direct
estimate of model sensitivity, but that variability is usually more indica-
tive of the variance of model components than of the dynamics of the model
structure.
81
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The methodology provided in QUAL2E-UNCAS allows the model user to per-
form uncertainty analysis with relative ease and efficiently manages the
output from the analysis. Although the application is specific to the QUAL2E
model, the methodology is general. The preprocessing and postprocessing
algorithms used are, in principle, applicable to many water quality models.
The preprocessor allows the user to select the variables and/or parameters
to^be altered, without having to manually restructure the input data set.
This task is performed automatically by the preprocessor for as many uncer-
tainty conditions as the user wishes to simulate. The postprocessor stores
and manipulates only the output of interest, thus reducing potential volumi-
nous output. The user must select the important variables and locations in
the stream network where uncertainty effects are desired for analysis.
6.2.1 Sensitivity Analysis
In normal usage sensitivity analysis is accomplished using a one-
variable-at-a-time approach (Duke, 1976). Sensitizing more than one input
variable at a time is an attractive method for assessing their interaction
effects on the output variable. When many input parameters and variables are
altered, however, the number of combinations to be investigated becomes
large, thus complicating interpretation of the results. Experimental design
strategies can be efficiently applied in this situation to elicit main and
interaction effects of input variables.
With the sensitivity analysis option in QUAL2E-UNCAS, the user may vary
the inputs singly, in groups, or using factorial design strategies. The
input requirements for sensitivity analysis consist of identifying the input
variables to be perturbed and specifying the magnitude of the perturbation.
The output for each sensitivity simulation consists of the changes (i.e., the
sensitivities) in the value(s) of each output variable (AY) resulting from
the changes in the value(s) of the input variables (AX). This output is
provided in tabular format, similar to the QUAL2E final summary, except that
the table entries are sensitivities rather than the values of the output
variables.
QUAL2E-UNCAS also has the capability of assessing the main and interac-
tion effects of input variables on various output variables by sensitizing
the inputs according to 2-1 eve! factorial design strategies. Currently
QUAL2E-UNCAS accommodates only 2-variable (i.e., 22) and 3-variable (i.e.,
23) factorial designs. As in normal sensitivity analysis, the user specifies
the names of the input variables to be perturbed and the magnitude of the
perturbation. The factorial design computations for main and interaction
effects are performed using standard statistical procedures (Box et al.,
1978; and Davies, 1967).
Because QUAL2E computes values of each output variable for every
computational element in the system, the factorial design output would be
voluminous if performed for each element. Thus, the user must specify
particular locations (maximum of 5) in the basin where this analysis is to
be performed. The critical locations, such as the dissolved oxygen sag
point, or the location below the mixing zone of a tributary junction or
t
82
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point discharge, are usually included among those chosen for analysis.
6.2.2 First Order Error Analysis
First order error analysis utilizes the first order approximation to the
relationship for computing variances in multivariate situations. The input
variables are assumed to act independently (covariances are ignored) and the
model to be linear (the higher order terms of the Taylor expansion are omit-
ted). The first order approximations to the components of output variance is
often,good (Walker, 1982). ,
The QUAL2E-UNCAS output for first order error analysis consists of two
parts--(a) a tabulation of normalized sensitivity coefficients and (b) a
listing of the components of variance. The normalized sensitivity coeffi-
cients represent the percentage change in the output variable resulting from
a 1 percent change in each input variable, and are computed as follows.
^ = (AYj/Yj)/(AXi/Xi)
VI-1
where:
= normalized sensitivity coefficient for output Yj to input X-j,
= base value of input variable,
= magnitude of input perturbation,
= base value of output variable,
= sensitivity of output variable.
The components of variance for each output variable Y are the percent
ages of output variance attributable to each input variable X, computed in
the following manner.
where:
Var(Yj) = I Var(Xi)(AY.j/AXi)2
Var(Yj) = variance of output variable Yj,
Var(X-j) = variance of input variable X-j,
Yj and,X-| are as defined in Eq. VI-1.
VI-2
83
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As can be seen from Eq. VI-2, each term in the summation is a component
of the variance of the output variable, Yj. contributed by the input Sable
tie inlVr?anne^S, °u "rv ?"*£"' Var1an"' Var(Yj). represent a Seighti'ng o?'
to inobt fAY'/A? ?' Vrr(Xl)s by ^e squai:e of the sensitivity of model output
cWloiW^
variance or a large (small) sensitivity coefficient, or b09thi Performing
first order error analyses with differing values of X,- will provide
---....ate of the strength of model nonlinearities. Outputs that are linear
XT will have unchanging sensitivity coefficients, (AYj/AXi), as AX? changes.
In normal applications of first order error analysis, all of the irmut
™b* if .are Pert«tfd- In th1S manner> the contributions to output variance
from all input vanab es are computed. QUAL2E-UNCAS has the capability!
order SSl5salnin?hi!;e1^eJ.of ^V^ables to be included ft a
ana'ysis- This limitation is achieved by allowing the user to
n»n- fouP °£ ^P.^s (i.e., "hydraulic variables," "reaction
1n the analysis.^1 °rCln9 funct1ons'" etc-) that are to be perturbed
The input requirements for first order error analysis consist of (a)
vari1^0^^6.1"^ P?rturi"»«°n. AX1t and (b) the value of the
variance of the input vanab e, Var X,) The value of ^ (default value
nS«r aii'fnn i ""* 7 Zu '°5) 1S sP6C7fied by the user and applied uniformly
£SJ f • Pf ?r the PUrp°Se Of GOmPutl'n9 sensitivities Default values
£^h8 i"??* }[anances are Provided with the QUAL2E-UNCAS model (see
bectlpn 6.3); however, users are cautioned to use values appropriate to thpir
'rnuft cnh?0aSPePltnC.ai10n;- F1nally^ 3S 1n ^e factor1al design^ fthe user "
must choose the locations (maximum of 5 in the basin at which the first
order error analysis for the output variables is to be performed.
6.2.3 Monte Carlo Simulation
Monte carlo simulation is a method for numerically operating a comolex
has.ra"dom components. Input variables are sampled at randSm
distributions (with or without correlation)
°t output values from repeated simulations is analyzed
' 1s
th
statilttf a"nH SJTl? Slmujat?"°!1 computations in QUAL2E-UNCAS provide summary
loStion? in th! 1? fnCy distributions for the state variables at specific
limnl^P^ M« y" -m' h6 -Ummary Stat1st1cs Include: mean (base and
of variatkn 9nH ™1 niraum. pximum, range, standard deviation, coefficient
in .* skew coefficient. Frequency and cumulative frequency
ions are tabulated in increments of one-half a standard deviation
f th|.standa;d deviation estimates from monte carlo slS at ons
pn ^ 1* orde!T.error analysis provide an indication of the
extent of model nonlinearities. Cumulative frequency distributions are
XJ«el«in S^S^'^u0^11 d1sPersl0" in the model predictions and in
assessing the likelihood of violating a water quality standard.
84
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The input requirements for the monte carlo simulation option in QUAL2E-
UNCAS consist of (a) the variance of the input variable, Var(X-j), (b) the
probability density function of the input variable, and (c) the number of
simulations to be performed. Specification of input variances is done in
the same manner as that for first order error analysis. Currently there
are two options for the input probability density functions: normal and
log-normal. The distribution for each input variable can be specified from
either of these options. The default option is the normal distribution.
The number of monte carlo simulations must be large enough to avoid large
errors in the estimated values of output variance, yet small enough to avoid
unduly long computation times. Preliminary experience with UNCAS indicates
that about 2000 simulations are required to achieve estimates of output
standard deviations with 95% confidence intervals of 5$.
QUAL2E-UNCAS assumes that all inputs act independently. Thus, each
input is randomized independently from the others. In normal usage, all
input variables are randomized in monte carlo simulation. As in the case of
first order error analysis, however, the user may constrain the number of
inputs to be varied by specifying that only certain generic groups of inputs
be randomized. Lastly, the user must specify the locations (maximum of five)
in the basin at which monte carlo simulation results are to be tabulated.
6.3 Input Variable Variances
One of the fundamental requirements for performing uncertainty analyses
in water quality modeling is a knowledge of the uncertainty characteristics
of the model inputs. Information on model input uncertainty is not widely
available in the literature, although recent articles show an increasing
tendency to publish such information (Kennedy and Bell, 1986). Three reports
(Koenig, 1986; NCASI, 1982; and McCutcheon, 1985) have been examined to
compile an uncertainty data base for use with QUAL2E-UNCAS. A summary of
this information is shown in Table VI-1. These values represent ranges in
the uncertainty of model inputs caused by such factors as spatial variation,
temporal variation, sampling error, analytical error, and bias in measurement
or estimation technique.
In QUAL2E-UNCAS, uncertainty information is provided in two forms: (a)
the value of the variance of the input variables and (b) the specification of
a probability density function for each input. The model reads this informa-
tion, as required, from a data file named "INVAR.DAT." An example of this
file, containing a set of default values for all QUAL2E inputs, is provided
with the QUAL2E-UNCAS model. These data are consistent with the typical
ranges of uncertainty shown in Table VI-1 and are provided only as a guide
for beginning the process of estimating the uncertainty associated with
QUAL2E input variables. All users are CAUTIONED not to assume that these
values are appropriate to all modeling situations. The burden of verifying
and confirming input variance estimates for a particular application lies
with the user. Efforts to develop a better understanding of input variable
uncertainties are continuing. ;
85
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TABLE VI-1
SUMMARY OF QUAL2E INPUT VARIABLE UNCERTAINTIES
Input Variable QUAL2E
or Parameter Data Type
Algae, Nutrient, Light
Coefficients 1A
Temperature Coefficients IB
Hydraulic Data 5
Temperature/LCD 5A
Reaction Coefficients 6
Constituent Concentrations 8,10,11
Temperature
DO
CBOD
N Forms
P Forms
Algae
Col i form
Conservative Minerals
Relative Standard Deviation, %
Low Typical High
5
1
1
1
5
1
2
5
10
10
5
20
1
10-20
2-5
5-15
2-10
10-25
2-3
5-10
10-20
15-30
15-40
10-25
25-50
5-10
50
10
50
20
100
5
15
40
75
75
50
100
15
Summary of data compiled from APHA, 1985; Koenig, 1986; McCutcheon, 1985;
and NCASI, 1982a.
86
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In the general case, QUAL2E-UNCAS accepts input variability information
in relative rather than absolute units. Thus, the input perturbations for
first order error analysis and input variances for first order analysis and
monte carlo simulation are supplied as percent perturbation and coefficient
of variation, respectively. The transformation equations between relative
and absolute units are: .
RP * Xi
.) = (CVi *
VI-3
VI-4
where:
RP = relative perturbation for input variable X-j
CV = coefficient of variation for input variable X-j
X-j = value of input variable used in base case simulation
The specific manner in which the input data requirements are supplied
to QUAL2E-UNCAS, including the data file "INVAR.DAT," are described in
Appendix B-User Manual for QUAL2E-UNCAS.
6.4 PROGRAMMING STRATEGY IN QUAL2E-UNCAS
QUAL2E-UNCAS has been structured in a manner to minimize the tedious
requirements for user adjustments to the QUAL2E input data file used in the
base case simulation. The UNCAS portion of QUAL2E-UNCAS consists of two
parts: (a) a package of 16 subroutines that perform the necessary book-
keeping and computations as well as printing the uncertainty results and (b)
one data file to decode and link UNCAS requests with QUAL2E. The user must
supply two input data files—the first provides the general specifications
for the uncertainty analysis to be performed, and the second contains the
input variance information. In addition, during execution, UNCAS creates
two disk files for storing and retrieving the simulation information used
in computing the uncertainty analysis results. The flow chart for UNCAS in
Figure VI-1 shows the relationships among the subroutines and data files.
Each component of the UNCAS package and its function is described in the
following sections.
6.4.1 UNCAS Subroutines
a. Subroutine UNCAS.
Subroutine UNCAS manages
sTmulations, computations, and
the execution of the
output reports for
uncertainty analysis , , .
QUAL2E-UNCAS. It calls the appropriate subroutines, for reading the
uncertainty data files, for screening the input and output variables for
consistency and compatibility with the OUAL2E model options selected in
87
-------
u
N
0.
o
c
_
£
_
C+-«
UNDATA
or
INSENS
SETUP
UECHO
Q2EZ
DATSAV
or
WRPT3A/B
URPT3
RSTOR
SENS
FDES
or
or
FOEA
MCSIM
OMATCH
ZEROP
NRGEN
SWAP
SUBROUTINE / DATA FILE INTERACTIONS
1. UCODE.DAT
OMATCH, INSENS, IFOAMC,
SETUP
2. BASE.DAT
OMATCH, SETUP, RSTOR, SENS
3. STORE.DAT
OMATCH, INSENS, IFOAMC,
DATSAV, FDES, FOEA, MCSIM
4. INVAR.DAT
IFOAMC
UNDATA, INSENS
Figure VI-1 UNCAS Flow Diagram and Program Structure
88
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the base case simulation, for performing the uncertainty simulations, and for
computing and printing the appropriate uncertainty results.
b. Subroutine UNDATA. This subroutine reads the user-supplied input
data file, *****.DAT, which contains the general specifications required
for uncertainty analysis. It sets the appropriate flags and conditions for
the type of uncertainty analysis to be performed.
c. Subroutine OMATCH. Subroutine OMATCH retrieves, purges, and stores
(on disk file) the values of the appropriate output variables from the base
case simulation. For sensitivity analysis, it saves the complete output
from the base case simulation in the file BASE.DAT. For first order error
analysis and monte carlo simulation it stores only the values of the output
variables at the locations (maximum of five) in the basin where uncertainty
results are desired and only for those that were modeled in the base case
simulation (STORE.DAT). These data are subsequently used by subroutines
FDES, FOEA, and MCSIM for their respective uncertainty analysis computations.
d. Subroutine INSENS. This subroutine controls the input specifications
for sensitivity analysis. It reads the user-supplied input data file,
*****.DAT, f0r the input variables that are to be perturbed for sensitivity
analysis. It determines the total number of sensitivity simulations to be
performed as well as the levels of all variables to be perturbed in each
simulation.,
e. Subroutine IFOAMC. Subroutine IFOAMC controls the input specifications
for first order error analysis and for monte carlo simulation. It searches
through a list of all input variables and purges (a) those variables that
are not requested to be perturbed and (b) those input or model options that
were not used in the base case simulation.
f. Subroutine ZEROP. This subroutine examines the numerical value of
each input variable. If the value is such that the variable is not used in
the base simulation (i.e., zero, or 1.0 for a temperature coefficient), the
input variable is purged from the uncertainty analysis simulations.
g. Subroutine SETUP. Subroutine SETUP sets up the input condition for
the current uncertainty simulation. Using the list of relevant inputs
developed in either INSENS or IFOAMC, each input variable is perturbed or
randomized as specified. It then calls subroutine SWAP to replace the base
•case value with the new value of the input variable.
h. Subroutine SWAP. This subroutine swaps the newly perturbed or
randomized value of the input variable(s) for the base value(s). Swapping
is done in memory by input data type and EQUIVALENCED arrays. Base case
values are either saved in memory (sensitivity or first order options) or
stored in the disk file BASE.DAT (monte carlo).
i. Suijroutine NRGEN. This subroutine generates either normally or log-
no rma11 y"!Tfstrl¥uted~random numbers for each input variable to be randomized
in a monte carlo simulation. It uses a machine-specific random number
generator.
89
-------
., J- Su^utineJJECHO. Subroutine UECHO prints, as intermediate output,
the input conditions of the current uncertainty simulation. This output
includes the name of the input variable being altered and its base and
perturbed value. This output is optional.
k. Subroutine 02EZ. This subroutine is not new to UNCAS. It is that
portion of the QUAL2E model that performs the simulation computations (see
Figure 1-1). v
. ]• Subroutine DATSAV. This subroutine stores the appropriate output
variables from each uncertainty simulation on the disk file STORE.DAT for
later processing by FDES, FOEA, or MCSIM.
nnni "?' Subroutines WRPT3A and WRPT3B. These subroutines are from the
QUAL2E model, ana write the final output summary for an UNCAS simulation.
This output is optional and is not available in the monte carlo option.
n. Subroutine URPT3. Subroutine URPT3 writes a limited intermediate
out put ^ summary of each uncertainty simulation. The summary consists of a
comparison of (a) the steady state convergence characteristics for tempera-
ture and algae and (b) the base and new values of the output variables at the
locations specified. This output is optional and is available only for the
sensitivity analysis using factorial design and first order error analysis.
o. Subroutine RSTOR. This subroutine restores the value of the per-
turbed input to its base case value after completion of an uncertainty simu-
lation. Thus, it prepares the input data for the next UNCAS simulation.
p. Subroutine SENS. Subroutine SENS writes the UNCAS final report for the
sensitivity analysis option. It is similar in format to the OUAL2E output pro-
duced^ subroutines WRPT3A/B, but consists of the change in output variable
(sensitivity) resulting from the input perturbations of the sensitivity analysis.
. q. Subroutine FDES. This subroutine performs the analysis of a facto-
rial ly designed set of sensitivity analysis simulations. It writes the UNCAS
final report for the factorial design, including the main and interaction
effects of the sensitized input variables on each output variable at the user
specified locations in the basin.
.. 㣥 |ybroutine_FOEA. Subroutine FOEA performs the computations and writes
the UNCAS final report for the first order error analysis option. The output
consists of the normalized sensitivity coefficient matrix and the components
of variance analysis for all inputs affecting each output variable at the
user-specified locations in the basin.
.. s* Subroutine MCSIM. This subroutine performs the computations and
writes the UNCAS final report for the monte carlo simulation option. The
output consists of summary statistics, including base and simulated mean
bias, minimum, maximum, range, standard deviation, coefficient of variation
and skew coefficient as well as the frequency distribution (in one-half
standard deviation steps) for each output variable at the user-specified
locations in the basin.
90
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6.4.2 Internal UNCAS DATA Files
a. File UCODE.DAT. This internal data file is supplied with the UNCAS
package. It is a master file that contains information for identifying,
matching, and screening the inputs to be modified in an UNCAS simulation. It
also serves as the primary information source for linking UNCAS requests to
the QUAL2E input data file.
b. File BASE.DAT. This internal data file stores information for the
base case simulation. In the sensitivity analysis option, it stores the
values of all the output variables for the QUAL2E base simulation. In the
monte carlo simulation option, it stores the base values of the input
variables that have been randomized. This data file is not used with the
first order error analysis option.
c. File STORE.DAT. This internal data file stores the values of output
variables at the user-specified locations for the base simulation and for each
uncertainty simulation. When all uncertainty simulations are completed, these
data are then used for the appropriate uncertainty output computations, i.e.,
factorial design for the sensitivity analysis option, or normalized sensiti-
vity coefficients and components of variance for the first order error analy-
sis option, or summary statistics and frequency distributions for the monte
carlo option.
6.4.3 User-Supplied UNCAS Data Files
a. File INVAR.DAT. This data file contains the uncertainty information
for each input varable in QUAL2E. These data consist of the variable name,
its coefficient of variation, and its probability density function. An
example of this file, containing a set of default data, is provided with the
UNCAS package. Instructions for adjusting the uncertainty inputs to user
specifications are provided in Appendix B--User Manual for QUAL2E-UNCAS.
b. File*****.DAT. This data file, named and prepared by the user,
contains the general requirements for performing a QUAL2E-UNCAS simulation.
This information consists, in part, of specifying the uncertainty analysis
option, the type of intermediate output, any constraints on input variables
to be modified, the output variables and locations for computing and printing
uncertainty results, the number of monte carlo simulations, and the magnitude
of the input variable perturbation. Instructions for assembling this data
file are provided in Appendix B--User Manual for QUAL2E-UNCAS.
6.5 LIMITATIONS AND CONSTRAINTS FOR QUAL2E-UNCAS
Because of the general purpose nature of the QUAL2E and UNCAS computer
codes, there are a few constraints in using the models that arise from the
program structure and bookkeeping strategies used. These limitations are
related to the level of detail the modeler may use in perturbing specific
input variables.
91
-------
!• Reach or Source Variable Inputs and Forcing Functions. In QUAL2E-
UNCAS, input variables are treated in the general case rather than
individually. For example, if the user wishes to perform uncertainty
analysis on the CBOD rate coefficient, or the point load flows, then all
input values (over the entire basin) of the rate coefficient and flows are
perturbed. UNCAS does not have the capability of perturbing only one (or a
few) of these inputs; i.e., the value of the CBOD rate coefficient in reach
3 or the flows for the second and fourth point loads. In short, the user
specifies the name of the variable to be perturbed and the magnitude of the
perturbation, then all values of that input variable are modified by the
amount specified.
2. First Order Error Analysis. In first order error analysis, the
user specifies the magnitude of the input perturbation, AX, for computing
sensitivity coefficients. UNCAS applies this value of AX uniformly to all
input variables. The modeler is not allowed to use one value of AX for one
group of inputs and another value for a different group of inputs. (Note:
The variance of each input variable can be specified uniquely, but as stated
in subsection 1, that variance applies equally to all values of the variable
in the basin.)
3. Input Variables Having a Numerical Value of Zero. Input variables
whose values are determined by QUAL2E-UNCAS to be zero (either blanks in
the input data file or an actual input value of zero) are assumed to be
non-modeled inputs. Those variables will not be perturbed in any UNCAS
simulation, and thus will not contribute to the uncertainty of the modeled
output.
92
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APPENDIX A
•
QUAL2E User Manual '
The following sections illustrate the coding of input data forms for the
QUAL2E model.
A. Title Data
All 16 cards are required in the order shown. The first two are title
cards, and columns 22 through 80 may be used to describe the basin, date of
simulation, etc. Title cards 3 through 15 require either a "YES" or "NO" in
columns 10 through 12 and are right justified. Note that each of the
nitrogen and phosphorus series must be simulated as a group.
For each conservative substance (up to three) and the arbitrary non-
conservative, the constituent name must be entered in columns 49 through 52.
Corresponding input data units are entered in columns 57 through 60 (e.g.,
mg/L).
QUAL2E simulates ultimate BOD in the general case. If the user wishes to
use 5-day BOD for input and output, the program will internally make the
conversions to ultimate BOD. This conversion is based upon first order
kinetics and a decay rate that can be specified by the user (Type 1 Data,
line 8). If no value is specified, the program uses a default value of
0.23 per day, base e. It is recommended that users work only with ultimate
BOD unless they have detailed knowledge of the river water and point source
BOD kinetics. To use the 5-day BOD input/output option, write "5-DAY
BIOCHEMICAL OXYGEN DEMAND" on the title 7 card beginning in column 22.
Card 16 must read ENDTITLE beginning in column 1.
*From: Modifications to the QUAL-2 Water Quality Model and User Manual for
QUAL-2E Versio.n 2.2. National Council of the Paper Industry for Air and
Stream Improvement, Inc., New York, NY. NCASI Tech. Bulletin No. 457. April
1985. Used by permission.
"frk
Further modified to include enhancements to QUAL2E resulting in Version
3.0 of the model, January 1987.
93
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B. Data Type 1 - Program Control
Type 1 Data define the program control options and the characteristics of
the stream system configuration, as well as some of the geographical/meteoro-
logical conditions for modeling temperature. There are a maximum of 17 Data
1 cards. The first 13 are required; the last four are necessary only
if temperature is being simulated.
The QUAL2E program recognizes Type 1 Data by comparing the first four
characters (columns 1-4) of each data card wijth a set of internally fixed
codes. If a match between the code and characters occurs, then the data are
accepted as supplied on the card by the user. If a match does not occur,
then the program control options will revert to default values and the
system variables for the unmatched codes will be assigned a value pf zero
(0.0).
The first seven cards control program options. If any characteristics
other than those shown below are inserted in the columns 1 through 4, the
actions described will not occur.
LIST - Card 1, list the input data.
WRIT - Card 2, write the intermediate output report, WRPT2 (see SUBROUTINE
WRPT2 in the QUAL-II documentation report (Roesner et al., 1981), or
NCASI Technical Bulletin No. 391).
FLOW - Card 3, use the flow augmentation option.
STEA - Card 4 shows this is a steady-state simulation. If it is not to be a
steady-state, write DYNAMIC SIMULATION or NO STEADY STATE, and it is
automatically a dynamic simulation.
TRAP - Card 5, cross-sectional data will be specified for each reach. If
discharge coefficients are to be used for velocity and depth computa-
tions, write DISCHARGE COEFFICIENTS, or NO TRAPEZOIDAL CHANNELS,
beginning in column 1.
PRIN - Card 6, local climatological data specified for the basin simulation
will appear in the final output listing.
PLOT - Card 7, dissolved oxygen and BOD will be plotted in final output
listing.
The next two cards provide further program flags and coefficients. This
information is supplied in two data fields per card; columns 26-35, and 71-
80. Note that the character codes in columns 1-4 must occur as shown in
order for the data to be accepted by the program.
94
-------
FIXE - Card 8, specifies: (a) whether the downstream boundary water quality
constituent concentrations are fixed (user specified), and (b) the
value of the rate coefficient for converting input 5-day BOD to ulti-
mate BOD. A value of 1.0 (or larger) in columns 26-35 specifies that
the downstream boundary water quality constituent concentrations will
be supplied in Data Types 13 and 13A. A valtie less than 1.0 (usually
0.0 or blank) in these columns means that the downstream boundary
concentrations are not user specified. In this case, the concentra-
tions in the most downstream element (Type 5) will be computed in the
normal fashion using the zero gradient assumption (Section 5.4.3.2).
The second value on this card, columns 71-80, is the rate coefficient
for converting 5-day to ultimate BOD. It is used only when 5-day BOD
is being modeled (Title Card 7). If the columns are left blank, the
model uses a default value of 0.23 per day, base e. Note that this
conversion factor is applied to all input BODc forcing functions
(headwaters, incremental flows, point loads, and the downstream boun-
dary condition).
INPU - Card 9, specifies whether the input and/or output will be in metric or
English units. The value of 1.0 (or larger) in card columns 26-35
specifies metric input. The value of 1.0 (or larger) in card column
71-80 specifies metric units for output. Any value less than 1.0
(usually 0.0 or blank) will specify English units.
The next four cards describe the stream system. There are two data
fields per card, columns 26-35 and 71-80. The program restrictions on the
maximum number of headwaters, junctions, point loads, and reaches are defined
by PARAMETER statements in the Fortran code. These statements may be modi-
fied by the user to accommodate a particular computer system or QUAL2E simu-
lation application. The values of the constraints in the code as distributed
by EPA are:
Maximum number of headwaters 7
Maximum number of junctions 6
Maximum number of point loads 25
Maximum number of reaches 25
Maximum number of computational elements 250
NUMB - Card 10, defines the number of reaches into which the stream is
segmented and the number of stream junctions (confluences) within the
system.
NUM_ - Card 11 shows the number of headwater sources and the number of
inputs or withdrawals within the system. The inputs can be small
streams, wasteloads, etc. Withdrawals can be municipal water
supplies, canals, etc. NOTE: Withdrawals must have a minus sign
ahead of the flow in Data Type 11 and must be specified as withdrawals
in Data Type 4 by setting IFLAG = 7 for that element. Note, the code
for Card 11 is 'NUM_ ' (read: NUM space) to distinguish it from the
code for Card 10, NUMB.
95
-------
TIME - Card 12 contains the time step interval in hours and the length of
the computational element in miles (kilometers). The time step
interval is used only for a dynamic simulation, thus it may be omitted
if the simulation is steady-state.
MAXI - Card 13 provides information with different meanings depending on
whether a dynamic or a steady-state simulation is being performed. For
a dynamic simulation, the maximum route time is specified in columns
26-35. This value represents the approximate time in hours required
for a particle of water to travel from the most upstream point in the
system to the most downstream point. The time increment in hours for
intermediate summary reports of concentration profiles is specified in
columns 71-80. For a steady-state simulation, the maximum number of
iterations allowed for solution convergence is entered in columns 26-
35. The value in columns 71-80 may be left blank because it is not
required in the steady-state solution.
The next four cards provide geographical and meteorological information
and are required only if temperature is being simulated. There are two data
fields per card, columns 26-35 and 71-80. Note: the character codes in
columns 1-4 must occur as shown in order for the data to be accepted by the
program.
1ATI - Card 14 contains the basin latitude and longitude and represent mean
values in degrees for the basin.
STAN - Card 15 shows the standard meridian in degrees, and the day of the
year the (Julian date) simulation is to begin.
EVAP - Card 16, specifies the evaporation coefficients.
AE - 6.8 x
- _ . _r Typical values are
10"4 ft/hr-in Hg and BE = 2.7 x 10"4ft/hr-in Hg-mph of wind
for English units input, or AE = 6.2 xlO m/hr-mbar and BE = 5.5 x
10" m/hr-mbar-m/sec of wind for metric units input.
ELEV - Card 17 contains the mean basin elevation in feet (meters) above mean
sea level, and the dust attenuation coefficient (unitless) for solar
radiation. The dust attenuation coefficient generally ranges between
zero and 0.13. Users may want to consult with local meteorologists
for more appropriate values.
Note: If the reach variable climatology option (steady-state
simulations only) is used, the elevation data and dust attentuation
coefficient for each reach are supplied in Data Type 5A and the value
supplied in Data Type 1A are overridden.
Data Type 1 must end with an ENDATAl card.
96
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C. Data Type 1A - Global Algal, Nitrogen, Phosphorus, and Light Parameters
These parameters and constants apply to the entire simulation and
represent the kinetics of the algal, nutrient, and light interactions. It is
important to note that proper use of all options in QUAL2E requires detailed
knowledge of the algal growth kinetics appropriate for the water body being
simulated.
These data cards are required only if algae, the nitrogen series
(organic, ammonia, nitrite, and nitrate), or the phosphorus series (organic
and dissolved) are to be simulated. Otherwise they may be omitted, except
for the ENDATA1A card. Information is supplied in two data fields per card,
columns 33-39 and 74-80. As with Type 1 Data, QUAL2E recognizes Type 1A Data
by comparing the first characters (columns 1-4) of, each card with a set of
internally fixed codes. If a match between the codes and the characters
occurs, then data are accepted as supplied on the card by the user. If a
match does not occur, then the system variables for the unmatched codes will
be assigned the value zero (0.0). Note: the spaces (under bars) are an
integral (necessary) part of the four character code.
0_UP - Card 1 specifies the oxygen uptake per unit of ammonia oxidation, and
oxygen uptake per unit of nitrite oxidation.
0_PR - Card 2 contains data on oxygen production per unit of algae growth,
usually 1.6 mg 0/mg A, with a range of 1.4 to 1.8. It also contains
data on oxygen uptake per unit of algae respiration, usually 2.0mg
0/mg A respired, with a range of 1.6 to 2.3.
N_CO - Card 3 concerns the nitrogen content and phosphorus content of
algae in mg N or P per mg of algae. The fraction of algae biomass
that is nitrogen is about 0.08 to 0.09, and the fraction of algae
biomass that is phosphorus is about 0.012 to 0.015.
ALG_ - Card 4 specifies the growth and respiration rates of algae.
The maximum specific growth rate has a range of 1.0 to 3.0 per day.
The respiration value of 0.05 is for clean streams, while 0.2 is used
where the Ng and ?2 concentrations are greater than twice the half
saturation constants.
N_HA - Card 5 contains the nitrogen and phosphorus half saturation coeffi-
cients. The range of values for nitrogen is from 0.01 to 0.3 mg/L and
for phosphorus the values typically range from 0.001 to 0.05 mg/L.
LIN - Card 6 contains the linear and nonlinear algal selfshading light
extinction coefficients. The coefficients X-, and \2 are defined
below.
A-]_ = linear algae self-shading coefficient
(l/ft)/(ug chla/L), or (l/m)/(ug chla/L)
\2 — nonlinear algae self-shading coefficient
(l/ft)/ug chla/L)2/5, or,(l/m)/(ug chla/L)2/3
97
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These two self-shading coefficients are used with AQ, the non-algal
light extinction coefficient (Type 6B Data) in the general light extinc-
tion equation shown below:
A - A
0
+ A2(c*0A)2/3
where A is the total light extinction coefficient and A is the algae
biomass concentration in mg A/L and a is the chlorophyll a to algae
biomass ratio as ug chla/mg A. Appropriate selection of the values of
AQ, An, and \n allows a variety of light extinction relationships to be
simulated as follows.
* No self-shading (Roesner et al, SEMCOG)
Al = A2
0
* Linear algal self -shading (JRB Assoc. Vermont)
* Nonlinear self -shading (Riley Eq. , metric units)
A]L - 0.0088
A2 = 0.054
LIGH - Card 7 contains the solar light function option for computing the
effects of light attenuation on the algal growth rate, and the light
saturation coefficient. QUAL2E recognizes three different solar light
function options. The light saturation coefficient is coupled to the
selection of a light function, thus care must be exercised in
specifying a consistent pair of values.
The depth integrated form of the three light functions and the
corresponding definitions of the light saturation coefficient are
given in Section 3.2.3.1, Eq. III-6a,b,c and outlined in the following
table .
Light Function Option
(Columns 33-39)
1 (Half Saturation)
2 (Smith's Function)
3 (Steele's Function)
Light Saturation Coefficient*
. (Columns 74-80)
Half Saturation Coefficient
Light intensity corresponding
to 71% of maximum growth rate
Saturation Light Intensity
* Units of the Light Saturation Coefficient are as
follows:
English: BTU/ft -min and Metric: Langleys/min
98
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Light Function Option 1 uses a Michaelis-Menton half saturation
formulation for modeling the algal growth limiting effects of light
(FL). It is the method used in the SEMCOG version of QUAL-2. Option 2
is similar to Michaelis-Menton, but uses a second order rather than
first order light effect. Both options 1 and 2 are monotonically
increasing functions of light intensity. Option 3 includes a photo-
inhibition effect at high light intensities and has been reported in
Bowie et al. (1985).
BAIL - Card 8, contains the light averaging option (columns 33-39) and the
light averaging factor (columns 74-80). These values are used only in
a steady-state simulation. The light averaging option allows the user
to specify the manner in which the light attenuation factor is
computed, from the available values of solar radiation. (See Section
3.2.3.2). A summary of these options is given below.
Option
Description
1 FL is computed from one daily average solar
radiation value calculated in the steady-
state temperature subroutine (HEATER).
2 FL is computed from one daily average solar
radiation read from Data Type 1A.
3 FL is obtained by averaging the 24 hourly
values of FL, that are computed from the 24
hourly values of solar radiation calculated
in the steady-state temperature subroutine
(HEATER).
4 FL is obtained by averaging the 24 hourly
values of FL, that are computed from the 24
hourly values of solar radiation computed
from the total daily solar radiation (Data
Type 1A) and an assumed cosine function.
Note: that if options 1 or 3 are selected, temperature must be
simulated.
The light averaging factor (columns 74-80) is used to make a
single calculation using daylight average solar radiation (Option 1 or
2) agree with average of calculations using hourly solar radiation
values (Option 3 or 4). The factor has been reported to vary from
0.85 to 1.00.
The selection of a daily (diurnal) light averaging option depends
largely on the detail to which the user wishes to account for the
diurnal variation in light intensity. Options 1 and 2 utilize a
single calculation of FL based on an average daylight solar radiation
value. Options 3 and 4 calculate hourly values of FL from hourly
values of solar radiation and then average the hourly FL values to
99
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obtain the average daylight value. Options 1 and 3 use the solar
radiation from the temperature heat balance routines (thus both algae
and temperature simulations draw on the same source for solar
radiation). Options 2 and 4 use the solar radiation value in Data
Type 1A for the algae simulation. Thus either option 2 or 4 must be
selected when algae are 'simulated and temperature is not. The light
averaging factor is used to provide similarity in FL calculations
between options 1 and 2 versus options 3 and 4. The solar radiation
factor (Data Type 1A, card 11) specifies the fraction of the solar
radiation computed in the heat balance that is photosynthetically
active. It is used only with options 1 or 3.
In dynamic algae simulations, option 3 is used (default) unless
temperature is not simulated, in which case solar radiation data are
read in with the local climatology data.
Card 9 contains the number of daylight hours (columns 33-39), and the
total daily radiation (BTU/ft2, or Langleys) (columns 74-80). This
information is used if light averaging options 2 or 4 are specified
for the simulation.
ALGY - Card 10 contains the light-nutrient option for computing the algae
growth rate (columns 33-39), and the algal preference factor for
ammonia nitrogen (columns 74-80). The light-nutrient interactions for
computing algae growth rate are as follows (see also Section 3.2.2).
NUMB -
Option
1
2
3
Description
Multiplicative: (FL) * (FN) * (FP)
Limiting Nutrient: FL * [minimum (FN, FP)]
Harmonic Mean FL * 2
1/FN + 1/FP
Option 1 is the form used in QUAL-II SEMCOG, while option 2 is
used in the revised META Systems Version of QUAL-II (JRB Associates,
1983). Option 3 is described by Scavia and Park (1976).
The algal preference factor for ammonia (columns 74-80) defines
the relative preference of algae for ammonia and nitrate nitrogen (see
also Section 3.3.2). The user defines this preference by specifying a
decimal value between 0 and 1.0, for example:
Algal Preference
Factor for Ammonia
0.0
0.5
1.0
Interpretation
Algae will use only nitrate for growth.
Algae will have equal preference for ammonia
and nitrate.
Algae will use only ammonia for growth.
100
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ALG/ - Card 11 contains the factor for converting the solar radiation value
from the heat balance to the solar radiation value appropriate for the
algae simulation (columns 33-39) and the value of the first order
nitrification inhibition coefficient (columns 74-80).
The solar radiation factor specifies the fraction of the solar
radiation computed in the heat balance (subroutine HEATER) that is
photosynthetically active (i.e., used by algal cells for growth). It
is required only in steady-state simulations when light averaging
options 1 or 3 (Data Type 1A, card 8) are selected. A decimal value
between 0 and 1.0 specifies the value of this fraction. Typically the
value of this fraction is about 0.45 (Bannister, 1974).
The first order nitrification inhibition coefficient is the value
of KNITRF in the following equation (see Section 3.3.5).
CORDO =1.0 - exp (-KNITRF * DO)
where:
DO = dissolved oxygen concentration (mg/L), and
CORDO = correction factor applied to ammonia and nitrite
oxidation rate coefficients.
The following table contains values of CORDO as a function of DO
(row) and KNITRF (column).
DO
(mg/L)
0.1
0.2
0.3
0.4
0.5
0.7
1.0
1.5
2.0
3.0
4.0
5.0
7.0
10.0
0.5
.05
.10
.14
.18
.22
.30
.39
.53
.63
.78
.86
.92
.97
.99
0.7
.07
.13
.19
.24
.30
.39
.50
.65
.75
.88
.94
.97
.99
1.00
KNITRF
1.0
.10 .
.18
.26
.33
.39
.50
.63
.78
.86
.95
.98
.99
1.00
1.00
2.0
.18
.33
.45
.55
.63
.75
.86
.95
.98
1.00
1.00
1.00
1.00
1.00
5.0
.39
.63
.78
.86
.92
.97
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
10.0
.63
.86
.95
.98
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
A value of 0.6 for KNITRF closely matches the inhibition formula
tion in QUAL-TX (TWDB, 1984) while a value of 0.7 closely matches the
data for the Thames Estuary (DSIR, 1964). The default value of KNITRF
is 10.0, i.e., no inhibition of nitrification at low dissolved oxygen.
ENDA - The last card in Data Type 1A must be an ENDATA1A card, regardless of
whether algae, nitrogen, or phosphorus are simulated.
101
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D. Data Type IB - Temperature Correction Factors
Several of the processes represented in QUAL2E are affected
by temperature. The user may elect to input specific temperature correction
factors. In the absence of such information, default values are used as
noted in Table A-l. The user need supply only those values that are to be
changed.
Data Type IB information is supplied as follows:
Alphanumeric code for each temperature
coefficient as noted in Table A-l:
User specified temperature coefficient
Columns 10-17
Columns 19-26
The last card in Data Type IB must be an ENDATA1B card, regardless of
whether any of the default values are modified.
TABLE A-l DEFAULT THETA VALUES FOR QUAL2E
RATE COEFFICIENT
BOD Decay
BOD Settling
Reaeration
SOD Uptake
Organic N Decay
Organic N Settling
Ammonia Decay
Ammonia Source
Nitrite Decay
Organic P Decay
Organic P Settling
Dissolved P Source
Algae Growth
Algae Respiration
Algae Settling
Coliform Decay
Non-cons Decay
Non-cons Settling
Non-cons Source
DEFAULT
SEMCOG
1.047
-
1.0159
-
-
-
1.047
-
1.047
-
-
-
1.047
1.047
-
1.047
1.047
-
-
VALUES
QUAL-2E
1.047
1.024
1.024
1.060
1.047
1.024
1.083
1.074
1.047
1.047
1.024
1.074
1.047
1.047
1.024
1.047
1.000
1.024
1.000
CODE
BOD DECA
BOD SETT
OXY TRAN
SOD RATE
ORGN DEC
ORGN SET
NH3 DECA
NH3 SRCE
N02 DECA
PORG DEC
PORG SET
DISP SRC
ALG GROW
ALG RESP
ALG SETT
COLI DEC
ANC DECA
ANC SETT
ANC SRCE
102
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E. Data Type 2 - Reach Identification and River Mile/Kilometer Data
The cards of this group identify the stream reach system by name and
river mile/kilometer by listing the stream reaches from the most upstream
point in the system to the most downstream point. When a junction is
reached, the order is continued from the upstream point of the tributary.
There is one card per reach. The following information is on each card:
Reach Order or Number
Reach Identification or Name
River Mile/Kilometer at Head of Reach
River Mile/Kilometer at End of Reach
Columns 16-20
Columns 26-40
Columns 51-60
Columns 71-80
A very useful feature of QUAL2E pertaining to modifications of reach
identification once the system has been coded is that existing reaches may be
subdivided (or new reaches added) without renumbering the reaches for the
whole system. If, for example, it is desired to divide the river reach
originally designated as REACH 3 into two reaches, the division is
made by calling the upstream portion REACH 3 and the "new reach" downstream
REACH 3.1. Up to nine such divisions can be made per reach (3.1-3.9); thus
REACH 3 (or any other reach) can be divided into as many as 10 reaches
numbered 3, 3.1-3.9. This option of dividing a reach is useful particularly
when new field data indicate a previously unknown change in geomorphology, or
when the addition of a new or proposed load alters the biochemistry in the
downstream portion of the reach. If this option is invoked, the number of
reaches specified in Data Type 1 must be changed to the new total number of
reaches.
Note: It is important to realize that this option cannot be used to
subdivide a reach into more (and thus smaller) computational elements, in an
attempt to provide greater detail to the simulation. All computational
elements must have the same length (as specified in Type 1 Data).
This option also will allow the user to add a new reach to the system;
for example, taking a tributary that was initially modeled as a point source
and changing it to a modeled reach (or reaches) in the basin. This type of
modification adds a junction to the system and thus the junction information
in Data Types 1, 4, and 9 must be modified accordingly.
This group of cards must end with ENDATA2.
103
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*". Data Type 3 - Flow Augmentation Data
These cards, except ENDATA3, are required only if flow augmentation is
to be used. The cards in this group contain data associated with determining
flow augmentation requirements and available sources of flow augmentation
There must be as many cards in this group as in the reach identification
group. The following information is on each card.
Reach Order or Number
Augmentation Sources (the number of
headwater sources which are avail-
able for flow augmentation)
Target Level (minimum allowable
dissolved oxygen concentration (mg/L)
in this reach)
Order of Sources (order of avail-
able headwaters, starting at most
upstream points
Columns 26-30
Columns 36-40
Columns 41-50
Columns 51-80
This card group must end with ENDATA3, even if no flow
augmentation is desired.
104
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G. Data Type 4 - Computational Elements Flag Field Data
This group of cards identifies each type of computational element in
each reach. These data allow the proper form of the routing equations to be
used by the program. There are seven element types allowed, they are listed
below.
IFIAG
1
2
4
5
6
7
Type
Headwater source element.
Standard element, incremental inflow/
outflow only.
Element on mainstream immediately upstream
of a junction.
Junction element.
Most downstream element.
Input (point source) element.
Withdrawal element.
Each card in this group (one for each reach), contains the
following information:
Reach Order or Number
of Elements in the Reach
Element Type (these are the
numbers, (IFLAG above), which
identify each element by type) .
Columns 16-20
Columns 26-30
Columns 41-80
Remember that once a system has been coded, reaches can be divided
or new ones added without necessitating the renumbering of the entire system
(see Data Type 2 - Reach Identification and River Mile/Kilometer Data for
application and constraints). When this option is invoked, the element types
and number of elements per reach for the affected reaches must be adjusted in
Data Type 4 to reflect the changes.
This card group must end with ENDATA4.
105
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H. Data Type 5 - Hydraulics Data
Two options are available to describe the hydraulic characteristics of
the system. The first option utilizes a functional representation, whereas
the second option utilizes a geometric representation. The option desired is
specified in Data Type 1, card 5. The code "TRAPEZOIDAL" specifically
denotes the geometric representation. Any other code, such as "NO
TRAPEZOIDAL," or "DISCHARGE COEFFICIENTS," specifies the functional
representation.
Note: With either option, the effect is global (for the entire system).
This option is not reach variable.
If the first option is selected, velocity is calculated as V
_ • _£? . _ .1 * TX. «ft. - _
aQD and
depth is found by D = cQd. Each card represents one reach and contains the
^TQltt/lO /^ "P *^ 'W A ^._JJ __ J__. _ _ • 1 11 i
values of a, b, c, and d, as described below.
Reach Order or Number
Dispersion Constant
a, coefficient for velocity
b, exponent for velocity
c, coefficient for depth
d, exponent for depth
Mannings "n" for reach (if
not specified, the program
default value is 0.02)
Columns 16-20
Columns 23-30
Columns 31-40
Columns 41-50
Columns 51-60
Columns 61-70
Columns 71-80
The dispersion constant is the value of K in the general expression
relating the longitudinal dispersion coefficient to the depth of flow and
shear velocity (See Section 2.4.3).
where:
Kdu
" longitudinal dispersion coefficient,
(ftz/sec,
K — dispersion constant, dimensionless
d - mean depth of flow, (ft,m)
u - shear velocity, (ft/sec, m/sec) = (gdS)1/2
g - gravitational constant (ft/sec2, m/sec2)
S - slope of the energy grade line (ft/ft, m/m)
106
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Substitution of the Manning equation for S leads to the following expression
for the longitudinal dispersion coefficient, DL.
DL = 3.82 Knud5/'6'
where:
n = Mannings roughness coefficient, and
V = Mean stream velocity (ft/sec, m/sec).
Typical values of K .range from 6 to 6000. A value of 5.93 leads to the
Elder equation for longitudinal dispersion, which is the one used in the
SEMCOG version of QUAL-II.
The coefficients a, b, c, and d should be expressed to relate velocity
depth and discharge units as follows.
System
Metric
English
V D
m/sec m/sec m
ft3/sec ft/sec ft
If the second option is selected, each reach is represented as a
trapezoidal channel. These data are also used to specify the trapezoidal
cross-section (bottom width and side slope), the channel slope, and the
Manning's "n" corresponding to the reach. The program computes the velocity
and depth from these data using Manning's Equation and the Newton-Raphson
(iteration) method.
One card must be prepared for each reach:
Reach Order or Number
Dispersion Constant, K
Side Slope 1 (run/rise; ft/ft, m/m)
Side Slope 2 (run/rise; ft/ft, m/m)
Bottom Width of Channel,
(feet, meters)
Channel Slope (ft/ft, m/m)
Mannings "n" (Default - 0.020)
Columns 16-20
Columns 23-30
Columns 31-40
Columns 41-50
Columns 51-60
Columns 61-70
Columns 71-80
This group of data cards must end with an ENDATA5 card.
107
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HA.
Data Type 5A - Temperature and Local Climatology Data
This group of data supplies the reach variable air temperature and
climatological information for steady-state water temperature simulation.
If QUAL2E is to be used in the dynamic/diurnal mode, the air temperature and
climatological inputs must be global constants and are supplied in a separate
data file according to the format described in Section X.- Climatological
Data. The data in this group consist of geographical and meteorological data
required for performing the energy balance for heat transfer across the air-
water interface.
There are three options in QUAL2E for providing the input variables for
steady state temperature simulation.
Option 1: Reach Variable Temperature Inputs. In this option the user
specifies explicitly the values of the temperature simulation inputs for all
reaches in the system. One card (line of data) is necessary for each reach
and contains the following information.
Reach Order or Number
Reach Elevation (ft,m)
Dust Attenuation Coefficient
Cloudiness, fraction in tenths
of cloud cover
Dry Bulb Air Temperature (F, C)
Wet Bulb Temperature (F, C)
Barometric (atmospheric) Pressure
(inches Hg, millibars)
Wind Speed (ft/sec, m/sec)
Columns 16-20
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
Option 2a: Global Values - Current Version of QUAL2E With this
option the user may specify a single value for each of the temperature
simulation inputs and QUAL2E will assume that these values apply to all reaches
in the system being modeled. The required input data for this option is the
same as that for option 1, with the exception that only one line of data
is necessary.
Option 2b: Global Values - Prior QUAL2E Versions The current
version of QUAL2E will accept without modification input data files for
steady-state temperature simulations from prior versions of QUAL2E Because
prior versions treated the temperature simulation inputs as global constants
so also_will the current version. In this option the required temperature '
simulation inputs are supplied according to the specifications in Section X
- Climatological Data.
108
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Option 3: Reach Variable Temperature Inputs with Estimation of Pressure
Variation with Elevation. In the case where reach variable temperature
simulation inputs are desired, but atmospheric pressure values are either
unknown or unavailable, QUAL2E has the capability of estimating the value of
atmospheric pressure for each reach from its elevation and temperature. These
estimates are computed from the ideal gas law integrated, at constant
temperature and specific humidity, over the change in elevation relative to a
datum (see Section 4.8). The input requirements for this option are the same
as for option 1, with the exception that the value of atmospheric pressure is
supplied for only one reach. This value serves as the datum or reference
from which atmospheric pressures for the other reaches are estimated. If this
option is used, the computed values of reach atmospheric pressure will appear
in the QUAL2E echo-print of the input data.
Notes:
1. It is important to realize that the user does not explicitly specify
whether options 1,2, or 3 for steady-state reach variable temperature
simulation are to be used. Rather, QUAL2E examines the format in which the
temperature/climatology input information are provided in the input data file,
matches it with one of the options described above, and then proceeds with the
appropriate computational strategy.
2. This data group (Data Type 5A) must end with ENDATA5A. If option 2b
is to be used (input data files from prior versions of QUAL2E), this data type
is eliminated entirely. Data Type 5A is also not allowed for dynamic/diurnal
QUAL2E simulations.
3. Values for elevation and dust attenuation' coefficient appear in two
places, here in Data Type 5A and also in Data Type 1. The values in Data Type
5A are used with options 1, 2a, and 3 and always override those in Data Type 1.
The values in Data Type 1 are used only in option 2b - input data files from
prior versions of QUAL2E.
109
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I- Type 6 - BOD and DO Reaction Rate Constants Data
This group of cards includes reach information on the BOD decay rate
coefficient and settling rate, sediment oxygen demand, as well as the method
of computing the reaeration coefficient. Eight options for reaeratlon
coefficient calculation are available (see Section 3.6.2) and are listed
below.
K2 OPT Method
1 Read in values of K2.
2 Churchill.
3 O'Connor and Dobbins.
4 Owens, Edwards, and Gibbs
5 Thackston and Krenkel.
6 Langbien and Durum.
7 Use equation K2 - aQb
8 Tsivoglou-Wallace.
One card is necessary for each reach, and contains the following;
information:
Reach Order or Number
BOD Decay Rate Coefficient (I/day)
BOD Removal Rate by Settling (I/day)
Sediment Oxygen Demand
(g/ft^-day, g/m2-day)
Option for K2 (1-8, as above)
K2 (Option 1 only) Reaeration
Coefficient, per day, base e, 20C
a, Coefficient for K2 (Option 7)
or Coefficient for Tsivoglou
(Option 8)
b, Exponent for K2 (Option 7) or
Slope of the Energy Gradient, S
(Option 8) e
Columns 16-20
Columns 21-28
Columns 29-36
Columns 37-44
Columns 45-48
Columns 49-56
Columns 57-64
Columns 65-72
110
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The units of a and b vary depending on whether option 7 or 8 is used and
on whether the input data are in English or Metric units, as follows:
Units of a:
English
Metric
Option 7 (Coefficient)
Option 8 (Coefficient)
Units of b:
Option 7 (Exponent)
Option 8 (S )
Consistent with
flow in cfs
I/ft
English
Consistent with
flow in cfs
Dimens ionless
Consistent with
flow in cms
1/m
Metric
Consistent with
flow in cms
D imens i onl ess
For option 8 (Tsivoglou's option), the energy gradient, Se need not be
specified if a Manning "n" value was assigned under Hydraulic Data Type 5.
S will be calculated from Manning's Equation using the wide channel
approximation for hydraulic radius.
This group of cards must end with ENDATA6.
Ill
-------
J- Data Type 6A - N and P Coefficients
This group of cards is required if algae, the nitrogen series (organic
nitrogen, ammonia, nitrite, and nitrate), or the phosphorus series
(organic and dissolved) are to be simulated. Otherwise, they may be
omitted. Each card of this group, one for each reach, contains the following
information:
Reach Order or Number
Rate Coefficient for Organic-N
Hydrolysis (I/day)
Rate Coefficient for Organic-N
Settling (I/day)
Rate Coefficient for Ammonia
Oxidation (I/day)
Benthos Source Rate for Ammonia
(mg/ft -day, mg/m -day)
Rate Coefficient for Nitrite
Oxidation (I/day)
Rate Coefficient for Organic
Phosphorus Decay (I/day)
Rate Coefficient for Organic
Phosphorus Settling (I/day)
Benthos Source Rate for Dissolved
Phosphorus (as P, mg/ft2-day, mg/m2-day)
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
Columns 74-80
Note that the benthos source rates are expressed per unit of bottom
area. Other versions of QUAL-II use values per length of stream. To convert
to the areal rate, divide the length value by the appropriate stream width.
This group of cards must end with ENDATA6A, even if algae, nitrogen, or
phosphorus are not simulated.
112
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K. Data Type 6B - Algae/Other Coefficients
This group of cards is required if algae, the nitrogen series, the
phosphorus series, coliform, or the arbitrary non-conservative is to be
simulated. Otherwise, they may be omitted. Each card of the group, one per
reach, contains the following information:
Reach Order or Number
Chlorophyll a to Algae Ratio
(ug chla/mg algae)
Algal Settling Rate (ft/day, m/day)
Non-Algal Light Extinction
Coefficient (I/ft, 1/m)
Coliform Decay Coefficient (I/day)
Arbitrary Non-Conservative Decay
Coefficient (I/day)
Arbitrary Non-Conservative Settling
Coefficient (I/day)
Benthos Source Rate for Arbitrary
Non-Conservative (mg/ft -day, mg/m -day)
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
* If not specified, the QUAL2E default value is 50 ug Chl-a/mg algae.
** If not specified, the QUAL2E default value is 0.01 ft which corresponds
approximately to the extinction coefficient for distilled water.
This group of cards must end with ENDATA6B, even if algae, nitrogen,
phosphorus, coliform, or the arbitrary, non-conservative are not simulated.
113
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L. Data Type 7 - Initial Conditions - 1
This card group, one card per reach, establishes the initial conditions
of the system, with respect to temperature, dissolved oxygen concentration,
BOD concentration, and conservative minerals. Initial conditions for
temperature must always be specified whether it is simulated or not. The
reasons for this requirement are: (a) when temperature is not simulated, the
initial condition values are used to set the value of the temperature
dependent rate constants; (b) for dynamic simulations the initial condition
for temperature, and every other quality constituent to be simulated, defines
the state of the system at time zero; and (c) for steady state simulations of
temperature, an initial estimate of the temperature between 35 F and 135 F
is required to properly initiate the heat balance computations
Specifying 68F or 20C for all reaches is a sufficient initial condition for
the steady-state temperature simulation case. The information contained is
as follows.
Reach Order or Number
•&"4*
Temperature (F or C)
Dissolved Oxygen (mg/L)
BOD (mg/L)
Conservative Mineral I*
Conservative Mineral II*
Conservative Mineral III
Arbitrary Non-Conservative
Coliform (No./lOO ml)
*
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
Columns 74-80
* - Units are those specified on the Title Card.
** - If not specified, the QUAL2E default value is 68 F, 20 C.
This group of cards must end with ENDATA7.
114
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M. Data Type 7A - Initial Conditions - 2
This group of cards is required if algae, the nitrogen series, or the
phosphorus series are to be simulated. The information is coded as follows:
Reach Order or Number
Chlorophyll a (ug/L)
Organic Nitrogen as N (mg/L)
Ammonia as N (mg/L)
Nitrite as N (mg/L)
Nitrate as N (mg/L)
Organic Phosphorus as P (mg/L)
Dissolved Phosphorus as P (mg/L)
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
This group of cards must end with ENDATA7A, even if algae, nitrogen, or
phosphorus are not simulated.
115
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N. Data Type 8 - Incremental Inflow - 1
This group of cards, one per reach, accounts for the additional flows
into the system not represented by point source inflows or headwaters.
These inflows, which are assumed to be uniformly distributed over the reach,
are basically groundwater inflows and/or distributed surface runoff that can
be assumed to be approximately constant through time.
An important new feature to QUAL2E is that incremental outflow along a
reach may be modeled. This option is useful when field data show a
decreasing flow rate in the downstream direction indicating a surface flow
contribution to groundwater.
Each card, one for each reach, contains the following information:
Reach Order or Number
Incremental Inflow (cfs,
m /sec) outflows are indicated
with a minus "-" sign.
•-ar
Temperature (F, G)
Dissolved Oxygen (mg/L)
BOD (mg/L)
Conservative Mineral I
Conservative Mineral II
Conservative Mineral III
Arbitrary Non-Conservative
Coliform (No./lOO ml)
This group of cards must end with ENDATA8.
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-44
Columns 45-50
Columns 51-56
Columns 57-62
Columns 63-68
Columns 69-74
Columns 75-80
116
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0. Data Type 8A - Incremental Inflow - 2
This group of cards is a continuation of Data Type 8 and is required
only if algae, the nitrogen series or the phosphorus series are to be
simulated. Each card, one per reach, contains the following information.
Reach Order or Number
Chlorophyll a Concentration (ug/L)
Organic Nitrogen as N (mg/L)
Ammonia as N (mg/L)
Nitrite as N (mg/L)
Nitrate as N (mg/L)
Organic Phosphorus as P (mg/L)
Dissolved Phosphorus as P (mg/L)
Columns 20-24
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
This group of cards must end with ENDATA8A, even if algae, nitrogen, or
phosphorus are not simulated.
117
-------
P. Data Type 9 - Stream Junction Data
This group of cards is required if there are junctions or confluences in
the stream being simulated. Otherwise, they may be omitted. The junctions
are ordered starting with the most upstream junction. For systems
containing a junction(s) on a tributary, the-junctions must be ordered in
the manner indicated in Figure A-l; that is, the junctions must be ordered so
that the element numbers just downstream of the junction are specified in
ascending order. In Figure A-l. the downstream element numbers for Junction
1, 2 and 3 are 29, 56, and 64, respectively. There is one card per junction,
and the following information is on each card:
Junction Order or Number
Junction Names or Identification
Order Number of the Last Element
in the reach immediately
upstream of the junction (see
Figure A-l). In the example,
for Junction 1, the order number
of the last element immediately
upstream of the junction is
number 17. For Junction 2, it
is number 49. For Junction 3,
it is number 43.
Order Number of the First Element
in the reach immediately down-
stream from the junction. It is
these numbers that must be
arranged in ascending order.
Thus, for Figure A-l these order
numbers for Junctions 1, 2, and 3
are 29,56, and 64 respectively.
Columns 21-25
Columns 35-50
Columns 56-60
Columns 66-70
Order Number of the Last Element Columns'76-80
in the last reach of the tribu-
tary entering the junction.
For Figure A-l these order
numbers for Junctions 1, 2,
and 3 are 28, 55, and 63,
respectively.
This group of cards must end with ENDATA9, even if there are
no junctions in the system.
118
-------
Most Upstream
Point
Reach
1 Number
Computational
Element Number
FIGURE A-1 STREAM NETWORK EXAMPLE TO ILLUSTRATE DATA INPUT
119
-------
Q- Data Type 10 - Headwater Sources Data - 1
This group of cards, one per headwater, defines the flow, temperature,
dissolved oxygen, BOD, and conservative mineral, concentrations of the
headwater. The following information is on each card.
Headwater Order or Number
Starting at Most Upstream Point
Headwater Name or Identification
Flow (cfs, m /sec)
Temperature (F, C)
Dissolved Oxygen Concentration (mg/L)
BOD Concentration (mg/L)
Conservative Mineral I
Conservative Mineral II
Conservative Mineral III
This group of cards must end with ENDATA10.
Columns 15-19
Columns 20-35
Columns 36-44
Columns 45-50
Columns 51-56
Columns 57-62
Columns 63-68
Columns 69-74
Columns 75-80
120
-------
R. Data Type 1OA - Headwater Sources Data - 2
This group of cards supplements the information in Data Type 10 and
is required if algae, the nitrogen series, the phosphorus series, coliform,
or arbitrary non-conservative are to be simulated. Each card, one per
headwater, contains the following data.
Headwater Order or Number
Arbitrary Non-Conservative
Coliform, (No./lOO mi)
Chlorophyll a (ug/L)
Organic Nitrogen as N (mg/L)
Ammonia as N (mg/L)
Nitrite as N (mg/L)
Nitrate as N (mg/L)
Organic Phosphorus as P (mg/L)
Dissolved Phosphorus as P (mg/L)
Columns 16-20
Columns 21-26
Columns 27-32
Columns 33-38
Columns 39-44
Columns 45-50
Columns 51-56
Columns 57-62
Columns 63-68
Columns 69-74
This group of cards must end with ENDATA10A, even if algae,
nitrogen, phosphorus, coliform, or arbitrary non-conservative
are not simulated.
121
-------
S. Data Type 11 - Point Load - 1
This group of cards is used to define point source inputs
and point withdrawals from the stream system. Point sources include both
wasteloads and unsimulated tributary inflows. One card is required per
inflow or withdrawal. Each card describes the percent of treatment (for
Wastewater treatment), inflow or withdrawal, temperature, and dissolved
oxygen, BOD, and conservative mineral concentrations. They must be ordered
starting at the most upstream point. The following information is on each
card.
Point Load Order or Number
Point Load Identification or Name
Percent Treatment (applies only to
influent BOD values)
Point Load Inflow or Withdrawal
(cfs, m /sec) (a withdrawal must
have a minus ("-") sign
Temperature (F, C)
Dissolved Oxygen Concentration (mg/L)
BOD Concentration (mg/L)
Conservative Mineral I
Conservative Mineral II
Conservative Mineral III
This group of cards must end with ENDATA11,
Columns 15-19
Columns 20-31
Columns 32-36
Columns 37-44
Columns 45-50
Columns 51-56
Columns 57-62
Columns 63-68
Columns 69-74
Columns 75-80
122
-------
T. Data Type 11A - Point Load - 2
This group of cards supplements Data Type 11 and contains the algal,
nutrient, coliform, and arbitrary non-conservative concentrations of the point
source loads. This information is necessary only if algae, the nitrogen
series, the phosphorus series, coliform, or the arbitrary non-conservative
are to be simulated. Each card, one per waste load (withdrawal), contains the
following information.
Point Load Order or Number
Arbitrary Non-Conservative
Coliform (Nd./lOO ml)
Chlorophyll a (ug/L)
Organic Nitrate as N (mg/L)
Ammonia as N (mg/L)
Nitrite as N (mg/L)
Nitrate as N (mg/L)
Organic Phosphorus as P (mg/L)
Dissolved Phosphorus as P (mg/L)
Columns 16-20
Columns 21-26
Columns 27-32
Columns 33-38
Columns 39-44
Columns 45-50
Columns 51-56
Columns 57-62
Columns 63-68
Columns 69-74
This group of cards must end with ENDATA11A, even if algae,
nitrogen, phosphorus, coliform, or arbitrary non-conservative
are not simulated.
123
-------
U. Data Type 12 - Dam Reaeration
This group of cards is required if oxygen input from reaeration over dams
is to be modeled as a component of the dissolved oxygen simulation. Dam
reaeration effects are estimated from the empirical equation attributed to
Gameson as reported by Butts and Evans, 1983 (see Section 3.6.5).
The following inputs are required.
Dam Order or Number
Reach Number of Dam
Element Number Below Dam
Columns 20-24
Columns 25-30
Columns 31-36
ADAM Coefficient: Columns 37-42
ADAM - 1.80 for clean water
— 1.60 for slightly polluted water
— 1.00 for moderately polluted water
— 0.65 for grossly polluted water
BDAM Coefficient: Columns 43-48
BDAM - 0.70 to 0.90 for flat broad crested weir.
— 1.05 for sharp crested weir with straight slope face.
— 0.80 for sharp crested weir with vertical face.
— 0.05 for sluice gates with submerged dishcarge.
Percent of Flow Over Dam
(as a fraction 0.0-1.0)
Height of Dam (ft, m)
Columns 49-54
Columns 55-60
This group of cards must end with ENDATA12, even if oxygen input from
dam reaeration is not to be modeled.
124
-------
V. Data Type 13 - Downstream Boundary - 1
This data card supplies the constituent concentrations at the downstream
boundary of the system. It is required only if specified in Data Type 1,
card 8. This feature of QUAL2E is useful in modeling systems with large
dispersion in the lower reaches (e.g., estuaries). When downstream boundary
concentrations are supplied, the solution generated by QUAL2E will be
constrained by this boundary condition. If the concentrations are not
provided, the constituent concentrations in the most downstream element will
be computed in the normal fashion using the zero gradient assumption (see
Section 5.4.3.2).
Downstream boundary values for, temperature, dissolved oxygen, BOD,
conservative mineral, coliform, and arbitrary non-conservative aTe required
as follows.
Temperature (F, C)
Dissolved Oxygen (mg/L)
BOD Concentration (mg/L)
Conservative Mineral I
Conservative Mineral II
Conservative Mineral III
Arbitrary Non-Conservative
Coliform (No./lOO ml)
Columns
Columns
Columns
Columns
Columns
Columns
Columns
Columns
25-31
32-38
39-45
46-52
53-59
60-66
67-73
74-80
This data group must end with an ENDATA13 card, even if the
fixed downstream boundary concentration option is not used in
the simulation.
125
-------
W. Data Type 13A - Downstream Boundary - 2
This group of data (one card) Is a continuation of Data Type 13. It is
required only if the fixed downstream boundary condition is used and if
algae, the nitrogen series, and the phosphorus series are to be simulated.
This card contains the downstream boundary concentrations for algae,
nitrogen, and phosphorus as follows.
Chlorophyll a (ug/L)
Organic Nitrogen as N (mg/L)
Ammonia as N (mg/L)
Nitrite as N (mg/L)
Nitrate as N (mg/L)
Organic Phosphorus as P (mg/L)
Dissolved Phosphorus as P (mg/L)
Columns 25-31
Columns 32-38
Columns 39-45
Columns 46-52
Columns 53-59
Columns 60-66
Columns 67-73
This data group must end with an ENDATA13A card, even if the fixed
downstream boundary condition is not used, and if algae, nitrogen, or
phosphorus are not simulated.
126
-------
X. Climatological Data
Climatological data are required for:
1. Temperature simulations, both steady-state and dynamic,
2. Dynamic simulations where algae is being simulated, and
temperature is not.
If neither temperature nor dynamic algae are being simulated, these cards may
be omitted.
For steady-state temperature simulations, these data may be supplied here
(as in prior versions of QUAL2E) or in Data Type 5A, but not both. If the
data are provided at this point in the input file, QUAL2E assumes that the
climatological inputs are global constants. Only one card (line of data) is
required, which gives the basin average values of climatological data, as
follows.
Month
Day
Year (last two digits)
Hour of Day
Net Solar Radiation*
(BTU/ft2-hr, Langleys/hour)
Cloudiness**, fraction in
tenths of cloud cover
Dry Bulb Temperature** (F, C)
Wet Bulb Temperature** (F, C)
>$£•&
Barometric pressure
(inches Hg, millibars)
Wind speed** (ft/sec, m/sec)
Columns 18-19
Columns 21-22
Columns 24-25
Columns 26-30
Columns 31-40
Columns 41-48
Columns 49-56
Columns 57-64
Columns 65-72
Columns 73-80
* Required only if dynamic algae is simulated and
temperature is not.
** Required if temperature is simulated.
127
-------
For dynamic/diurnal simulations, the climatological input data must be
read from a separate input file (FORTRAN Unit Number 2). This input
procedure is different from that used with prior versions of QUAL-II and
QUAL2E and is designed to assist user interaction with QUAL2E by modularizing
the variety of input data QTJAL2E may require. The time variable climatology
input data file is structured in the following manner. The first line
consists of a descriptive title (80 alphanumeric characters) that identifies
the data contained in the file. Subsequent lines provide the time variable
basin average climatology data, chronologically ordered at 3-hour intervals.
There must be a sufficient number of lines of data to cover the time period
specified for the simulation (Data Type 1, card 13, MAXIMUM ROUTE TIME). The
format for these data is the same as that described above for steady state
temperature simulations.
There is no ENDATA line required for the climatological data.
128
-------
Y. Plot Reach Data
This data type is required if the plotting option for DO/BOD is selected
(Data Type 1, card 7, PLOT DO/BOD). The following information is required
for QUAL2E to produce a line printer plot.
1. Card 1 - BEGIN RCH
Reach number at which plot
is to begin
2. Card 2
PLOT RCH
Columns 11-15
a. Reach numbers in their
input order (1, 2, 3..NREACH)
b. If a reach is not to be
plotted, (i.e., a tributary)
replace the reach number
with a zero.
Columns 11-15
Columns 16-20
21-26
etc.
76-80
c. Use additional PLOT RCH cards
if there are more than 14
reaches in the system.
3. Additional plots can be obtained by repeating the
sequence of BEGIN RCH and PLOT RCH cards.
As an example of the plotting option, suppose that for the river system
shown in Figure A-l, one wishes to obtain two DO/BOD plots: one for the main
stream (Reaches 1, 2, 5, 6, 10, and 11) and one for the second tributary
(Reaches 7 and 9). The plot data would appear in the following order.
BEGIN RCH 1
PLOT RCH 120056000 10 11
BEGIN RCH 7
PLOT RCH 00000070900
No ENDATA card is required for the PLOT information.
129
-------
YA. Plot Observed Dissolved Oxygen Data. The current version of QUAL2E has
the capability to plot observed values of dissolved oxygen concentrations
on the line printer plots produced for the computed values from the model.
This feature is useful in assisting the user in model calibration. The
observed DO data are read from a separate input data file (FORTRAN unit
number 2) structured in a manner to be compatible with the Plot Reach Data
(Section Y).
The first line, "DO TITLE:", consists of a descriptive title (70
alphanumeric characters) that identifies the data contained in the file.
The second line, "NUM LOGS:", specifies the number of locations (n-i) for the
first plot for which observed DO data are available. The next n-^ lines, "DO
DATA", provide the observed DO data plotting information. One line is
required for each location and contains the following data.
River location (mi, km)
Minimum DO (mg/L)
Average DO (mg/L)
Maximum DO (mg/L)
Columns 11-20
Columns 21-30
Columns 31-40
Columns 41-50
If only a single value of DO is available at a given location, it may be
entered in either the minimum or average data position. Then by default,
QUAL2E will set the minimum, maximum, and average values all equal to the
value entered. When more than one line printer plot is specified in the Plot
Reach Data, the observed DO values for these plots are provided on the lines
following that for the first plot. The information is entered by repeating
the sequence of "NUM LOGS:" and "DO DATA" lines for the data in the current
plot.
130
-------
Z. Summary
Constructing a consistent and correct input data set for. a QUAL2E
simulation must be done with care. This user's guide is designed to assist
the user in this process. It has been NCASI's and EPA's experience that two
of the most frequently made errors in constructing a QUAL2E input data set
are:
(a) Usinga numerical value that is inconsistent with the
input units option selected, and
(b) Notadheringto the 4-character input codes forData
Types 1 and 1A.
As an aid to the units problem, Table A-2 is included in this report.
It provides a complete summary of all the input variables whose dimensions
are-dependent on whether English or metric units are selected. Finally, the
user is encouraged to check and recheck the input codes in Data Types 1 and
1A for accuracy, especially the codes for cards 10 and 11 of Data Type 1
(i.e., "NUMB" and "NUM_")-
131
-------
TABLE A-2. LIST OF QUAL2E INPUT VARIABLES THAT ARE ENGLISH/METRIC UNIT DEPENDENT
Data
Type
1
1
1
1
1A
1A
1A
2
Card or
Line
8
8
11
15
15
16
6
6
7
9
all
all
5 all
(Discharge
Coefficient)
5 all
(Trapezoidal)
5A
6
6
6
6A
6B
7
8
10
11
12
13
LCD
all
all
all
all
all
all
all
all
all
all
all
1
all
Variable Description
Input Units Specification
Output Units Specification
Length of Computational Element
Evaporation Coefficient
Evaporation Coefficient
Basin Elevation
Linear Algal Extinction Coeff
Non- linear Algal Extinction
Coefficient
Light Saturation Coefficient
Total Daily Solar Radiation
River Mile/km to Head of Reach
River Mile/km to End of Reach
FORTRAN
Code Name
METRIC
METOUT
DELX
AE
BE
ELEV
EXALG1
EXALG2
CKL
SONET
RMTHOR
RMTEOR
Coefficient on Flow for Velocity COEFQV
Exponent on Flow for Velocity EXPOQV
Coefficient on Flow for Depth COEFQH
Exponent on Flow for Depth EXPOQH
Bottom Width of Channel
Reach Elevation
Dry Bulb Temperature
Wet Bulb Temperature
Barometric Pressure
Wind Speed
SOD Rate
Option 7 for k,
Coefficient onflow for k,
Exponent on flow for k_ '
Option 8 for K_
Coefficient for Tsivoglou Eq.
Slope of Energy Gradient
Benthal Source Rate for
Ammonia-N
Benthal Source Rate for
Phosphorus
WIDTH
RCHELV
RCHTDB
RCHTUB
RCHATH
RCHWND
CK4
COEQK2
EXPQK2
COEQK2
EXPQK2
SNH3
SPHOS
Algal Settling Rate ALGSET
Non-algal Extinction Coefficient EXCOEF
Arbitrary Nonconservative
Benthal Source Rate
Initial Condition - Temperature
Incremental Inflow
Flow Rate
Temperature
Headwater Conditions
Flow Rate
Temperature
Point Source/Withdrawal
Flow Rate
Temperature
Height of Dam
Downstream Boundary- Temperature
Solar Radiation
Dry Bulb Temperature
Wet Bulb Temperature
Barometric Pressure
Wind Speed
SRCANC
TINIT
QI
TI
HWFLOW
HWTEMP
WSFLOU
WFTEMP
HDAM
LBTEMP
SOLHR
DRYBLB
WETBLB
ATHPR
WIND
132
Units
English
0
0
mile
ft/hr-in Hg
ft/hr-in Hg-mph
ft
1/ft-ug-Chla/L
1/ft-(ug-Chla/L)2/3
BTu/ft2-min
Btu/ft2
mile
mile
Consistent with
flow, velocity
and depth in
cfs, fps, ft
respectively
ft
ft
F
F
in Hg
ft/sec
gm/ft -day
Consistent with
flow in cfs
1/ft
ft/ft
mg/ft2-day
mg/ft2-day
ft/day
I/ft
mg/ft2-day
F
cfs
F
cfs
F
cfs
F
ft
F
BTu/ft2-hr
F
F
in Hg
ft/sec
Metric
1
1
kilometer
m/hr-mbar
m/hr-mbar-m/sec
meters
1/m-ug-Chla/L
1/m-(ug-Chla/L)2/3
langley/min
Lang leys
kilometer
kilometer
Consistent with
flow, velocity,
and depth in
cms, mps. m
respectively
meters
meters
C
C
mbar
m/sec
2
gm/m -day
Consistent with
flow in cms
1/meter
meter/meter
mg/m -day
mg/m -day
m/day
1/meter
mg/m -day
C
cms
c
cms
c
cms
c
meters
C
langleys/hr
C
mbar
m/sec
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ALITY ROUTING MODEL - QUAL2E INPUT DATA CODING FORMS
DATA TYPE 1 PROGRAM CONTROL DATA
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APPENDIX B
USER MANUAL FOR QUAL2E-UNCAS
I. Introduction
The following sections provide instructions for assembling the two
application-specific input data files for an UNCAS simulation. The first
provides the general specifications for the uncertainty analysis to be
performed, and the second contains the input uncertainty information for each
input variable.
II. General Specification File; ****.DAT
This data file, named and prepared by the user, contains the general
requirements for performing a QUAL2E-UNCAS simulation. This input data file
consists of nine data types, as follows.
UNCAS
Data Type
1
2
3
4
5
6
7
8
9
Description
Heading
System Title
Uncertainty Option
Input Condition
Intermediate Output
Output Variables
Output Locations
Input Variables
Ending
Data Types 1 through 7 are read by subroutine UNDATA, whereas Types 8 and 9
are read by subroutines INSENS or IFOAMC as necessary. In all UNCAS data
types, the first 30 columns contain default data type descriptive information
(see UNCAS Input Coding Form).
A. UNCAS Data Type 1 - Heading.
This data type is a default header line for the beginning of the UNCAS
general specification file. It consists of one line and is prepared in the
following format.
- Text
"UNCAS1 ^HEADING
*.,
"QUAL2E UNCERTAINTY ANALYSIS"
Note: The underscore, "__" indicates a space.
159
Position
Columns 1-30
Columns 31-57
-------
B. UNCAS Data Type 2 - System Title.
This data type contains a user-supplied descriptive title (50 alpha-
numeric characters) for the uncertainty simulations. It consists of one line
and is formatted as follows.
"UNCAS2 *SYSTEM_TITLE_
User Title
*.,
Position
Columns 1-30
Columns 31-80
C. UNCAS Data Type 3 - Uncertainty Option
Data type 3 is where the user specifies the particular type of
uncertainty analysis to be performed. The descriptive text for this data
type appears in the first 30 columns as follows.
"UNCAS3 *UNCERTAINTY_OPTION-*"
There are three uncertainty options--sensitivity analysis, first order error
analysis, and monte carlo simulation. Also, if first order or monte carlo
are selected, the user must supply the magnitude of the input pertubation, or
number of monte carlo simulations, respectively. Data type 3 consists of one
line prepared with the descriptive text described above, followed by one of
these three options.
Entry
"SENSITIVITY ANALYSIS"
Position
Columns 31-50
or
"FIRST ORDER ERROR ANALYSIS;"
Magnitude of input perturbation,
" % PERTURBATION"
Columns 31-57
Columns 59-64
Columns 66-79
or
"MONTE CARLO SIMULATION:"
Number of monte carlo simulations
"SIMULATIONS"
Columns 31-53
Columns 59-64
Columns 66-76
(* Enter as a percent.
of 5% is used.)
If not specified, a default value
Note: UNCAS tests the four alphanumeric characters in columns 31-34 (i.e.
"SENS", "FIRS", or "MONT") to determine the uncertainty analysis option
desired.
160
-------
D. UNCAS Data Type 4 - Input Condition.
This data type provides UNCAS with information concerning the
particulars of the inputs to be modified. The 30 column descriptive
text for this line of data is:
"UNCAS4
INPUT CONDITION
*.
If the sensitivity analysis option is being exercised, data type 4
conveys to UNCAS whether the inputs (specified in Data Type 8) are to be
perturbed (a) singly or in groups or (b) using a factorial design strategy.
For the factorial design option, the user must specify the number of input
variables in the design. Currently UNCAS accommodates only 2 or 3 variable
factorial designs. For sensitivity analysis, UNCAS data type 4 is completed
with one of the following two selections.
Entry
"SINGLE/MULTIPLE PERTURBATIONS'
Position
Columns 31-59
or
"2-LEVEL FACTORIAL DESIGN"
Number of input variables (2 or 3)
"VARIABLES"
Columns 31-54
Column 63
Columns 64-73
If the first order error analysis or the monte carlo simulation option is
selected, data type 4 is used to specify which of the generic groups of input
variables are to be varied. These groupings are defined according to the
QUAL2E input data types and are specified using the following alphanumeric
code.
QUAL2E Input QUAL2E
Variables Data Types
Global 1, 1A, IB
Hydraulic/Climatology 5, 5A
Reaction Coefficient 6, 6A, 6B
Incremental Flow 8, 8A
Headwater Conditions 10, 10A
Point Loads 11, 11A
Dams 12
UNCAS Alphanumeric
Code
GLBL
HYDR
RXNC
FFIF
FFHW
FFPL
FFDM
161
-------
For the first order and monte carlo options, data type 4 is completed
with one of the following two selections.
"ALL INPUTS"
or
"GENERIC GROUPS"
1st alphanumeric code
2nd alphanumeric code
3rd alphanumeric code
4th alphanumeric code
5th alphanumeric code
6th alphanumeric code
7th alphanumeric code
Position
Columns 31-40
Columns
Columns
Columns
Columns
Columns
Columns
Columns
Columns
31-44
47-50
52-55
57-60
62-65
67-70
72-75
77-80
Any number (from 1-7) of groups may be specified and only the QUAL2E inputs in
that (those) group(s) will be perturbed in the uncertainty analysis. Note:
UNCAS tests the four alphanumeric characters in columns 31-34 (i.e. "SING,"
"2-LE," "ALL_" or "GENE") to determine the input condition desired.
E. UNCAS Data Type 5 - Intermediate Output
With data type 5, the user can specify whether any intermediate output
is desired. Intermediate output is defined as line printer output for each
uncertainty simulation. The 30 column descriptive text for this line of data
is:
"UNCAS5
INTERMED OUTPUT
*.
UNCAS recognizes three options for intermediate output: none, a complete
QUAL2E final summary, and a limited output summary. The limited intermediate
output summary consists of an echo print of the inputs that have been
perturbed for the uncertainty simulation, a summary of the steady-state
temperature and algae convergence computations, and a tabulation of the base
and new values of the ouptut variables at the locations specified (UNCAS Data
Type 7). Entries for data type 5 are completed with one of the following 3
selections.
"NONE"
or
"COMPLETE QUAL2E FINAL SUMMARY"
or
"LIMITED"
Position
Columns 31-34
Columns 31-59
Columns 31-37
Note: because of the potential for voluminous output, the second and third
options are not available for monte carlo simulation. UNCAS tests the four
alphanumeric characters in columns 31-34 (i.e. "NONE", "COMP", or "LIMI") to
determine the intermediate output desired.
162
-------
F. UNCAS Data Type 6 - Output Variables.
Data type 6 is used to constrain the list of output variables for which
uncertainty results will be computed. These constraints are applied in a
manner analogous to the input variable constraints in data type 4. The user
simply specifies the generic groups of output variables for which uncertainty
results are desired. The 30 column descriptive text for this line of data
is:
"UNCAS5
OUTPUT VARIABLES
.<
The generic output groups are named "HYDRAULIC," "QUALITY," AND "INTERNAL."
The hydraulic group consists of 10 output variables (flow, depth, velocity,
dispersion,etc.) associated with the hydraulic output from QUAL2E. The
quality group consists of the values of the 17 state variables simulated by
QUAL2E. The internal group is made up of 9 diagnostic or internal variables
associated with the algal, nutrient, light interactions in QUAL2E (i.e. algal
growth rate p minus r and p/r ratio, light and nutrient factors in the growth
rate computation, nitrification inhibition factor, etc.). This data type is
completed by adding the names of the generic output variable groups to the
data type 6 line as follows.
Generic Output Group 1
Generic Output Group 2
Generic Output Group 3
Position
Columns 31-40
Columns 46-55
Columns 61-70
Note: UNCAS tests the four alphanumeric characters in columns 31-34, 46-49,
and 61-64 (i.e., "HYDR," "QUAL," or "INTE") to determine the generic group of
output variables to be analyzed. They may be placed in any order in the
appropriate positions.
163
-------
G. UNCAS Data Type 7 - Output Locations.
This data type is used to define the locations in the basin where the
output variables are to be examined for uncertainty analysis. The 30 column
descriptive text for UNCAS data type 7 is:
"UNCAS7
OUTPUT LOCATIONS
UNCAS will accept a maximum of 5 locations in the basin for output analysis.
They are supplied as a single line in the form of reach and element number as
follows.
Entry
Location 1 (Reach and Element Number)
Location 2 (Reach and Element Number)
Location 3 (Reach and Element Number)
Location 4 (Reach and Element Number)
Location 5 (Reach and Element Number)
Position
Columns 33-35, 36-38
Columns 41-43, 44-46
Columns 49-51, 52-54
Columns 57-59, 60-62
Columns 65-67, 68-70
Note: Reach and element numbers must be right-justified in their appropriate
column fields.
164
-------
H. UNCAS Data Type 8 - Input Variables
This data type is used to supply UNCAS with the input variable
specifications for performing sensitivity analysis. It is not required for
the first order error analysis and monte carlo simulation options. The 30-
column descriptive text for UNCAS data type 8 is:
"UNCAS8
*INPUT VARIABLES*"
This data type will consist of one or more lines, depending on how many
sensitivity simulations are desired and/or on how many variables are to be
sensitized in a given simulation.
The information in this data type is designed to handle any of three
different input conditions for sensitivity analysis: one variable at a time,
variables in groups, or factorially designed. The data on.each line consists
of specifying the input condition, the number of variables to be sensitized,
the name of the input variable, and the magnitude of the perturbation.
For a one variable at a time simulation, one line of input is required
as follows.
Entry
"SINGLE"
Number of inputs perturbed
Input variable code
Magnitude of perturbation, %
Position
Columns 31-36
Column 45
Columns 48-56
Columns 58-63
The number of inputs perturbed with this option is always 1. The input
variable codes are 8 alphanumeric characters as shown in Table B-l. This line
of data may be repeated for one variable at a time sensitivity simulations
with other variables or other levels of perturbation.
For sensitivity analyses where more than one variable is perturbed, one
line of input is required for each input variable to be altered, as follows.
"MULTIPLE"
Number of inputs perturbed
Input variable code
Magnitude of perturbation, %
Position
Columns 31-38
Column 45
Columns 49-56
Columns 58-63
165
-------
UNCAS limits the number of inputs perturbed for this option to be either 2 or
3, thus requiring 2 or 3 lines of UNGAS data type 8, respectively. The input
variable codes are shown in Table B-l. As with one variable at a time
simulations, groups of multiple variable sensitivity simulations may appear
one after the other in this data type.
For sensitivity analysis using variables in a factorically designed
configuration, one line of input is required for each input variable as
follows.
Position
"FACTORIAL"
Number of Inputs perturbed
Input variable code
Magnitude of perturbation, %
Columns 31-39
Column 45
Columns 49-56
Columns 58-63
UNCAS limits the number of inputs perturbed in the factorial design option
to be either 2 or 3, thus requiring 2 or 3 lines of UNCAS data type 8,
respectively. The input variable codes are shown in Table B-l. UNCAS
automatically sets up conditions for each of the 4 or 8 factorial design
simulations. As with the other sensitivity analysis options, groups of
factorial design conditions may appear one after the other in this data type.
Note: UNCAS tests the four alphanumeric characters in column 31-34 (i.e.
"SING", "MULT", and "FACT") to determine the sensitivity analysis option
desired. UNCAS also allows the user to mix the sensitivity analysis option
types in a single execution of the program; however, the maximum number of
sensitivity simulations is 120. This data type is not required for the first
order error analysis or monte carlo simulation options.
I. UNCAS Data Type 9 - Ending.
This data type is a default ending line that signifies the end of the
general specification file. It consists of one line and is prepared in the
following format.
- Text
"UNCAS 9 ENDING_
"ENDUNCERTAINTY"
*.,
Position
Columns 1-30
Columns 31-44
166
-------
III. Input Variance Data File; INVAR.DAT.
This data file contains the uncertainty information for each input
variable in QUA12E. An example of this file containing a set of default data
is provided with the UNCAS package. However, the user must adjust the default
data to values suitable for the particular case being modeled. The data
contained in INVAR.DAT consists of the variable code name, its QUA12E data
type, its coefficient of variation, and its probability density function. The
first two lines of the file are title and header lines. Subsequent lines
contain the variance information, formatted as follows.
Position
Input Variable Name
Input Variable Code
QUAL2E Data Type
Coefficient of Variation
Probability Density Function
Columns 3-30
Columns 36-43
Columns 49-50
Columns 56-60
Columns 68-69
The input variable codes are shown in Table B-l. The two character codes for
probability density function are "NM" for normal distribution and "LN" for
log-normal.
167
-------
TABLE B-l INPUT VARIABLE NAME CODES
Input variable Name
Evaporation coef - AE
Evaporation coef - BE
Oxygen uptake by NH3 oxdtn
Oxygen uptake by N02 oxdtn
Oxygen prod by algae grwth
Oxygen uptake by algy resp
Nitrogen content of algae
Phosphorus content of algy
Algy max spec growth rate
Algae respiration rate
Nitrogen half sat'n coef
Phosphorus half sat'n coef
Linear alg self shade coef
Non-lin alg self shade co
Light sat'n coefficient
Light averaging factor
Number of daylight hours
Total daily solar radt'n
Alg pref for ammonia-N
Alg to temp solar factor
Nitrification inhib fact
5-D to ult BOD conv r-cof
Temp coef BOD decay
Temp coef BOD settling
Temp coef 02 reaeration
Temp coef sed 02 demand
Temp coef organic-N decay
Temp coef organic-N set
Temp coef ammonia decay
Temp coef ammonia srce
Temp coef nitrite decay
Temp coef organic-P decay
Temp coef organic-P set
Temp coef diss-P source
Temp coef algy growth
Temp coef algy respr
Temp coef algy settling
Temp coef coli decay
Temp coef ANG decay
Temp coef ANC settling
Temp coef ANC source
Daily averaging option
Light function option
Algae growth calc option
Input Code
ECOEF-AE
ECOEF-BE
NH30XYUP
N020XYUP
AGYOXYPR
AGYOXYUP
AGYNCON
AGYPCON
AGYGROMX
AGYRESPR
NHALFSAT
PHALFSAT
AGYEXTLN
AGYEXTNL
LSATCOEF
IAVGFACT
NUMBDLH
TDYSOLAR
APREFNH3
A/TFACT
NHIBFACT
5TOUBODK
TC/BODDC
TC/BODST
TC/REAER
TC/SOD
TC/NH2DC
TC/NH2ST
TC/NH3DC
TC/NH3SC
TC/N02DC
TC/PRGDC
TC/PRGST
TC/P04SC
TC/ALGRO
TC/ALRES
TC/ALSET
TC/CLIDC
TC/ANCDC
TC/ANCST
TC/ANCSC
DIURNOPT
LFNOPTN
AGYGROPT
QUAL2E Data Type
1
1
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1A
1
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
IB
1A
1A
1A
168
-------
Input Variable Name
Dispersion corr constant
Coef on flow for velocity
Expo on flow for velocity
Coef on flow for depth
Expo on flow for depth
Manning's roughness n
Side slope 1
Side slope 2
Bottom width
Slope of channel
Mean elevation of reach
Dust attenuation coef
Fraction of cloudiness
Dry bulb air temperature
Wet bulb air temperature
Barometric pressure
Wind speed
CBOD oxidation rate
CBOD settling rate
SOD uptake rate
Reaeration rate option 1
Coef on flow for K2 opt-7
Expo on flow for K2 opt-7
Coef for K2 (TSIV) opt-8
Slope for K2(TSIV) opt-8
Organic-N hydrolysis rate
Organic-N settling rate
Ammonia-N decay rate
Ammonia-N bethai source
Nitrite-N decay rate
Organic-P hydrolysis rate
Organic-P settling rate
Dissolved-P Benthal srce
Chla to algae ratio
Algae settling rate
Light ext coefficient
Coliform decay rate
ANC decay rate
ANC settling rate
Initial temperature
Reaeration equation opt.
Incremental flow
Incr-temperature
Incr-dissolved oxygen
Table B-l (continued)
Input Code
DISPSN-K
COEFQV-A
EXPOQV-B
COEFQH-C
EXPOQH-D
MANNINGS
TRAP-SSI
TRAP-SS2
TRAP-WTH
TRAP-SLP
ELEVATIN
DUSTATTN
CLOUD
DRYBULB
WETBULB
ATMPRES
WINDVEL
BOD DECA
BOD SETT
SOD RATE
K2-OPT1
CQK2-OP7
EQK2-OP7
K2COEF-8
K2SLOP-8
NH2 DECA
NH2 SETT
NH3 DECA
NH3 SRCE
N02 DECA
PORG DEC
PORG SET
DISP SRC
CHLA/ART
ALG SETT
LTEXTNCO
COLI DEC
ANC DECA
ANC SETT
INITTEMP
K20PTION
INCRFLOW
INCRTEMP
INCRDO
QUAL2E Data Type
5
5
5
5
5
5
5
5
5
5
5A
5A
5A
5A
5A
5A
5A
6
6
6
6
6
6
6
6
6A
6A
6A
6A
6A
6A
6A
6A
6B
6B
6B
6B
6B
6B
7A
6
8
8
8
169
-------
Input Variable Name
Incr-BOD
Incr-consv min 1
Incr-consv min 2
Incr-consv min 3
Incr-arbitrary non-cons
Incr-coliform
Incr-algae
Incr-organic-N
Incr-ammonia-N
Incr-nitrite-N
Incr-nitrate-N
Incr-organic-phos
Incr-dissolved-phos
Headwater flow
Hwtr-temperature
Hwtr-dissolved oxygen
Hwtr-BOD
Hwtr-consv min 1
Hwtr-consv min 2
Hwtr-consv min 3
Hwtr-arbitrary non-cons
Hwtr-coliform
Hwtr-algae
Hwtr-organic-N
Hwtr-ammonia -N
Hwtr-nitrite-N
Hwtr-nitrate-N
Hwtr-organic-phos
Hwtr-dissolved-phos
Ptld-trtmnt factor
Point load flow
Ptld-temperature
Ptld-dissolved oxygen
Ptld-BOD
Ptld-consv min 1
Ptld-consv min 2
Ptld-consv min 3
Ptld-arbitrary non-cons
Ptld coliform
Ptld-algae
Ptld-organic-N
Ptld-ammonia-N
Ptld-nitrite-N
Ptld-nitrate-N
Ptld-organic phos
Ptld-dissolved-phos
Dam coefficient a
Dam coefficient b
Fraction of flow over dam
Table B-l (continued)
Input Code
INCRBOD
INCRCM1
INCRCM2
INCRCM3
INCRANC
INCRCOLI
INCRCHLA
INCRNH2N
INCRNH3N
INCRN02N
INCRN03N
INCRPORG
INCRDISP
HWTRFLOW
HWTRTEMP
HWTRDO
HWTRBOD
HWTRCM1
HWTRCM2
HWTRCM3
HWTRANC
HWTRCOLI
HWTRCHIA
HWTRNH2N
HWTRNH3N
HWTRN02N
HWTRN03N
HWTRPORG
HWTRDISP
PTLDTFCT
PTLDFLOW
PTLDTEMP
PTLDDO
PTLDBOD
PTLDCM1
PTLDCM2
PTLDCM3
PTLDANC
PTLDCOLI
PTLDCHLA
PTLDNH2N
PTLDNH3N
PTLDN02N
PTLDN03N
PTLDPORG
PTLDDISP
DAMSACOF
DAMSBCOF
DAMSFRAG
QUAL2E Data Type
8
8
8
8
8
8
8A
8A
8A
8A
8A
8A
8A
10
10
10
10
10
10
10
10A
10A
10A
10A
10A
10A
10A
10A
10A
11
11
11
11
11
11
11
11
11A
11A
11A
11A
11A
11A
11A
11A
11A
12
12
12
170
-------
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171
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APPENDIX C
QUAL2E-UNCAS Example Application
A. Introduction
The material in this appendix provides an example of how the uncertainty
methodologies in QUAL2E-UNCAS can be applied to a QUAL2E data set. The
sole purpose of this section is to demonstrate the utility of uncertainty
analysis rather than to provide a definitive analysis of the river system
from which the data were obtained. The example input data files and some
of the output data files that were used in this application are provided
with the model code distributed by the Center for Water Quality Modeling
B. Withlacoochee River Basin
The data used to demonstrate the capabilities of QUAL2E-UNCAS were obtained
from a USEPA survey of the Withlacoochee River during October 1984 (Koenig,
1986). In this study, water quality simulations were examined for portions
of the river subjected to both municipal and industrial waste loads. In
addition there is a significant accretion of flow from groundwater inputs.
The river has a uniform low slope, but is characterized by alternating
shoals and pools (often in excess of 25 feet deep). Average depths during
the survey periods were 5.2 to 14.8 feet, widths were 90 to 140 feet, and
flows varied from 150 cfs at the headwater to 660 cfs at the end of the
system. Water quality is affected by algal activity resulting from
municipal waste discharges above the section of stream studied. The
addition of industrial waste at RM 24, however, dramatically reduces light
penetration to the extent that the algal population diminishes in the
downstream direction.
A location map of the basin is shown in Figure C-l and a plot of observed
and modeled dissolved oxygen concentrations is presented in Figure C-2.
Ten state variables were simulated in this study, temperature, dissolved
oxygen, carbonaceous BOD, four nitrogen forms, (organic, ammonia, nitrite,
and nitrate), two phosphorus forms, (organic and dissolved), and algae as
chlorophyll a_. A summary of the calibrated inputs and their variance
estimates for the uncertainty analysis is shown in Table C-l. The
calibrated values in general were obtained by adjusting field or laboratory
measurements of the specific model inputs. The variance estimates were
computed from replicate data taken during the survey period and by
inference from other published data. (McCutcheon, 1985 and Bowie et al
1985) --
172
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Madi son
iee-^
Suwannee
River
Fig. C-1. Location map of the Withlacoochee River basin,
0)
o>
I 4
-o
O)
o
I/I
I/J
S 2
RM 26
RM 2
RM 20
Industrial
Waste
Spring
Dilution
.1
30
20 15 10
River Location (mile)
Fig. C-2. Observed and predicted dissolved oxygen concentrations.
173
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C. First Order Error Analysis (FOEA)
Table C-2 shows the first order error analysis (FOEA) results for the
output variables of CBOD and DO at three locations in the Withlacoochee
system: an upstream location (RM 26), a midpoint near the dissolved oxygen
sag (RM 20), and a downstream location (RM 2). For the CBOD sensitivity
coefficients in Table C-2a, it is clear that the input forcing functions
dominate model sensitivity. In general, point load and headwater flows and
CBOD have the largest sensitivity coefficients, however, their effects
change with location in the system. Headwater inputs dominate sensitivity
in the upper reaches of the river and decrease in importance as one
TABLE C-l Summary of Input Data for QUAL2E-UNCAS Simulations -
Withlacoochee River Survey 1984
Input Parameter or
Coefficient
Hydraulic Data (7)*
Flows (qfs)
Depths (ft)
Velocities (fps)
Others
Reaction Coefficients (8)
CBOD Decay (I/day)
Reaeration (I/day)
SOD (gm/ft2-day)
N, P, Algae
Algae, Nutrient, Light Coefficients (17)
Maximum Growth Rate (I/day)
Respiration Rate (I/day)
Others
Climatology, Temperature Inputs (23)
Wet, Dry Bulb Air Temps (°F)
Temperature Coefficients
Others
Headwater, Incremental, Point Loads (27)
DO, Temperature
CBOD, N, P, Algae
Base Case (Mean)
Values
150 - 660
5.2 - 14.8
.12 - .78
a,b
.04 - .10
.08 - .80
.04 - .13
a,b
1.3
.15
a,b
64.3, 74.5
1.00 - 1.083
a,b
a
a
Relative Standard
Deviations (%)
3%
8%
8%
10 - 20%
15%
13%
12%
15 - 25%
10%
10%
10%
2%
3%
1 - 15%
1 - 3%
8 - 25%
(a) Basin specific values from Koenig, 1986.
(b) Typical values from Table III-3 of this report.
* Value in parentheses is the number input variables of the type indicated.
174
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proceeds downstream. At the downstream location, the sensivity of CBOD to
point load and incremental flow inputs is strong. The sensitivity to the
biochemical reaction coefficient grows in magnitude in the direction of
flow, but is substantially smaller than the values associated with the
point load forcing functions.
Table C-2a also presents the components of variance for the modeled CBOD
output. These results show a similar, but somewhat modified pattern as the
sensitivity coefficients. The headwater CBOD is the dominant contributor
(99%) to CBOD variability in the upper reaches of the basin. The point
load CBOD values are the primary variance component elsewhere in the river
(84% at RM 20 and 79% at RM 2'). The variance contribution from the CBOD
rate coefficient grows in importance as one proceeds downstream, but is at
least an order of magnitude lower than that from the CBOD point loads. In
the downstream portion of the basin, the variance contributions from the
headwater inputs are small, as one would expect. It is interesting to note
that although the hydraulic inputs (incremental, point load, and headwater
flow) have sensitivity coefficients that rank high, their contribution to
CBOD variance is low because the relative standard deviation of these
inputs is low (3%) compared to the CBOD loads (15%). The sensitivity
coefficients and components of variance results at the sag point (RM 20)
clearly show the upstream to downstream transition of the dominant input
components. The total variability in simulated CBOD estimated by the first
order analysis, when expressed as a standard deviation, varies from 0.35
mg/L to 0.76 mg/L to 0.27 mg/L as one proceeds through the basin. This
prediction error is approximately 15% and is comparable to the magnitude of
the error in the CBOD input forcing functions.
The FOEA results for dissolved oxygen are presented in Table C-2b. As
contrasted with CBOD, the only forcing functions having large DO
sensitivity coefficients are the headwater inputs, not the point load
inputs. Furthermore, DO is much more sensitive to temperature inputs than
is CBOD. As with CBOD, practically all the DO sensitivity in the upper
reaches can be attributed to headwater DO; however as one proceeds
downstream, DO loses sensitivity to the headwater condition. Next in
importance in terms of DO sensitivity are the reaeration rate coefficient
and velocity, both characteristic of system hydraulics. The biochemical
factors of sediment oxygen demand and CBOD rate coefficient follow in rank.
Similar patterns of dissolved oxygen sensitivity are apparent from
examining the components of variance (Table C-2b). The importance of
reaeration and SOD is striking as is the relatively small impact of CBOD
decay. Jhe temperature inputs, while having large sensitivity
coefficients, provide a minimum contribution to DO variance. Although
algae dynamics were simulated in this application, their effect on DO
uncertainty was negligible both in terms of sensitivity coefficient and
components of variance. The total variability in simulated DO when
expressed as a standard deviation increases in the downstream direction
varying from 0.18 mg/L to 0.30 mg/L and averaging about 5% of the simulated
uu •
176
-------
D. Effect of Model Non-linearity
First order error analysis uses the linear approximation to compute an
estimate of output variance. The validity of that approximation can be
assessed by computing the sensitivity coefficients for both large and small
values of AX, the input perturbation (see Eq. VI-2). Small changes in the
normalized sensitivity coefficient indicate near linearity of the state
variable over the range of perturbed input values, whereas large changes in
sensitivity reflect important nonlinear effects. Table C-3 contains values
of the normalized sensitivity coefficients for the state variables DO and
chlorophyll a for input pertubations, AX, ranging from -20 to +20 percent.
The input variables selected for analysis are those having the largest
sensitivity coefficients.
For dissolved oxygen (Table C-3a), the reaeration and headwater temperature
inputs show the largest relative changes in sensitivity, indicating that
these variables have the largest nonlinear effects on DO. The relative
changes in sensitivity coefficient for the two inputs, however, are only 9
and 16%, respectively, suggesting that the nonlinear effects are not
TABLE C-3
Normalized Sensitivity Coefficients for Various Sizes
of Input Perturbations (Withlacoochee RM 20)
(a) Simulation Variable: Dissolved Oxygen (ug/L)
Input Variable
CBOD Decay
SOD
Reaeration
HW Temp
HW DO
Magnitude of Input Perturbation
-20% -V/o +1% +20%
.12
.23
.33
.66
.55
•.12
•.23
.31
•.69
.55
.12
.22
.31
.70
.55
•.12
.23
.30
.77
.55
Std. Dev. (mg/L) .28 .27 .27
(b) Simulation Variable: (Chlorophyll a_ (ug/L)
.26
Relative
Change (%)
0
0
-9
+16
0
-7
Max Growth Rate
Respiration
Chi a/Agy-B
HW Flow
HW Chi a
.40
-.37
-1.24
.28
.96
.41
-.36
-1.01
.24
.95
.42
-.35
-.98
.25
.96
.43
-.34
-.83
.21
.94
+7
-8
-33
-25
-2
Std. Dev. (ug/L) 3.72
3.12
3.06
2.64
-29
177
-------
strong. The other three variables, CBOD decay, SOD, and headwater DO have
normalized sensitivity coefficients that are essentially constant. Thus
their impacts are, for practical purposes, linear for the conditions of
this simulation. The net effect from all model input nonlinearities is
manifest in the FOEA estimate of dissolved oxygen standard deviation, which
decreases by 7% as the magnitude of the input perturbation changes fom -20
to +20 percent.
Similar, but more pronounced patterns are observed for the state variable,
chlorophyll a_ (Table C-3b). Two input variables, the ratio of chlorophyll a
to algal biomass (Chla/Agy-B) and headwater flow exhibit large nonlinear ~
effects on chlorophyll^. The maximum algal growth rate and the algal
respiration rate show modest nonlinearities in sensitivity, while
sensitivity to headwater chlorophyll a is essentially constant. The net
FOEA estimate of standard deviation o?~ chlorophyll a_ decreases by 29% over
the range of input perturbations. Thus the effects of model nonlinearities
appear to be stronger with chlorophyll a. than with dissolved oxygen.
Analysis of other state variables showed changes in FOEA estimates of
standard deviation of about 7% for algal growth rate, 5% for temperature
and less than 5% for all others, including CBOD, the nitrogen forms and the
phosphorus forms (see Table C-5). Note that, in all cases, the FOEA
estimate of standard deviation decreases as the magnitude of the input
perturbation increases over the range of -20 to +20%. It is curious that
the large effect of model nonlinearities to chlorophyll a are not reflected
in the dissolved oxygen sensitivites. This observation Ts perhaps
explained by the fact that the largest input contributor to nonlinearity
effects on chlorophyll a_ is a units conversion factor—the ratio of
chlorophyll a to algal biomass. This factor does not serve as a linkage
between the chlorophyll £ and dissolved oxygen kinetic expressions in
QUAL2E. The algal growth and respiration rates do provide that linkage,
however, and the extent of their nonlinearities are comparable with that of
dissolved oxygen, about 7%.
E. Monte Carlo Simulations
The monte carlo simulation output in QUAL2E-UNCAS provides summary
statistics and frequency distributions for the state variables at specific
locations in the basin. Table C-4 contains the mean, minimum, maximuim,
range, standard deviation, coefficient of variation, and skew coefficient
for simulated dissolved oxygen and chlorophyll a_ at the upstream, midpoint,
and downstream locations in the Withlacoochee basin. All summary
statistics are based on 2000 monte carlo simulations using the same input
variances that were employed in the first order error analysis. Input
probability distributions were assumed to be normal.
There is very good agreement between the calibrated mean and simulated mean
for dissolved oxygen. Differences are less than 0.5%. The differences
between calibrated and simulated means for chlorophyll a_ average about 3%
and may be attributed in part to the previously described nonlinearities in
chlorophyll a.. For dissolved oxygen, the standard deviation grows in the
178
-------
TABLE C-4 Summary Statistics from 2000 Monte Carlo
Simulations for Wlthlacoochee River
Dissolved Oxygen (mg/L)
Chlorophyll a (ug/.;
Statistic
Calibrated Mean
Simulated Mean
RM 26
5.83
5.82
RM 20
4.48
4.47
RM 2
5.06
5.05
RM 26
18.1
18.9
RM 20
14.4
15.0
RM 2
6.6
6.6
Minimum 5.26 3.47 3.69 10.2 2.8 3.0
Maximum 6.41 5.31 5.89 53.8 41.4 22.2
Range 1.15 1.84 2.20 45.6 33.6 19.2
Std. Deviation 0.18 .28 .31 4.25 3.48 1.87
Coef. Variation 3.0% 6.2% 6.2% 23.5% 24.2% 28.4%
Skew Coef. .01 -.15 -.20 1.73 1.60 1.46
Std. Deviation 0.18 0.27 0.30 3.54 2.94 1.62
from FOEA
downstream direction. This phenomenon is attributable to the fact that
dissolved oxygen never recovers to approach saturation (it lies in the 50
to 70% range) and to the cumulative effect of model input uncertainty as it
propagates through the system. For chlorophyll a., the standard deviation
decreases steadily in the downstream direction principally because the
algal biomass concentration is also decreasing. The decrease in algal
biomass concentration results from a lower algal growth rate attributable
to reduced light penetration caused by color in the industrial waste
discharge at RM 24 and to the dilution effects from groundwater inflow.
The coefficient of variation for chlorophyll a_ averages about 25%
throughout the basin, whereas that for dissolved oxygen is about 5%. The
dissolved oxygen data exhibit little skew, but the chlorophyll a. data show
marked positive skewness.
Estimates of output variance by monte carlo simulation are not affected by
model nonlinearities. Thus a comparison of monte carlo generated standard
deviations with those produced by first order error analysis should provide
information on the extent of any nonlinearities. As shown in Table C-4,
these two estimates differ by less than 5% for DO and by about 20% for
chlorophyll _a. This comparison indicates weak nonlinearities associated
with dissolved oxygen and more substantial ones with chlorophyll a, thus
supporting the previous sensitivity coefficient observations in the first
order error analysis. As shown in Table C-5, for the output variables of
temperature, CBOD, and algal growth rate, the monte carlo estimate of
standard deviation differs by less than 5% from the FOEA estimate. These
179
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differences are within the 95% confidence interval for the monte carlo
estimates, thus implying negligible nonlinear effects for the conditions of
this simulation. The frequency distributions for dissolved oxygen
generated by the monte carlo analysis are shown graphically in Figure C-3.
These distributions are useful in providing a visual representation
of the distribution of model output at different locations in the system.
In the case of dissolved oxygen shown in Figure C-3, the distributions
appear nearly symmetric and the dispersion in the upper reaches of the
basin is substantially smaller than that in the middle and lower reaches.
Similar plots (not shown) for chlorophyll a_ data in Table C-4 clearly show
the decreasing dispersion and pronounced positive skew in the simulated
data.
F. Number of Monte Carlo Simulations.
A number of experiments were performed with the Withlacoochee data set to
determine the number of monte carlo simulations required to achieve a given
precision in the computed standard deviation of each output state
variable. Twenty replicate sets of 25, 50, 100, 200, and 500 monte carlo
simulations were conducted. The approximte 95% confidence interval (based
on the assumption of normality) was computed for each replicate set and
then plotted versus the total number of simulations performed. The results
for dissolved oxygen and CBOD are presented in Figure C-4. The smooth
curve represents an envelope for the upper limit of the 95% CI for
simulated standard deviation from repeated monte carlo simulations. For
both DO and CBOD it can be seen that about 1000 simulations are required to
estimate the output standard deviation to within 5% of the mean. With this
criterion as a goal, 2000 monte carlo simulations were conservatively and
routinely performed for the preceding analyses.
TABLE C-5 Differences in Standard Deviation Estimates for
Output Variables - Withlacoochee River Survey - 1984
Output Variables
Between FOEA Input
Perturbations from -20
to +20%
Between FOEA (5%)
and Monte Carlo
Simulations (2000)
Temperature
Dissolved Oxygen
CBOD
Nitrogen Forms
Phosphorus Forms
Chlorophyll a
Algal Growth Rate
5.4
7.7
0.8
*
*
29
6.9
1.8
0.6
1 4
1 • ~
16
2
- 4 3
™ • \J
- 4 5
~ • *J
-26
C. • \J
*
*
- 21
C, I
- 4
*Expected values of standard deviations too small to compute meaningful
relative differences, although values are certainly less than 10% and
likely less than 5%.
180
-------
RM 26
6.5
7 DO (mg/L)
RM 20
tt>
CP
3.5
4.5
5.5 DO (mg/L)
RM 2
-------
£ 6
(A rtJ
Ul i-
O 10
TJ
W C
LO 03
cn +J
Q.<*-
0.0
100
Dissolved Oxygen
8
1000
Number of Simulations
10000
£= 6
m o
10 to
Ul •<-
.OC1
1-1 I.
o ra
XJ
M c
in ra
cn 4->
in
x. 2
100
BOO
1000 10000
Number of Simulations
Fig. C-4. Convergence characteristics of monte carlo simulations
with QUAL2E-UNCAS (Withlacoochee River).
182
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6. Summary
The following observations summarize experience to date with uncertainty
analysis using QUAL2E. QUAL2E-UNCAS has been shown to provide a useful
framework for performing uncertainty analysis in steady state water quality
modeling. Application of the first order error analysis and monte carlo
simulation methodologies to a data set from the Withlacoochee River Basin
has highlighted some of the useful features of uncertainty analysis. These
include the changing sensitivities and components of variance in different
portions of the river basin, the assessment of model nonlinearities, and
the convergence characteristics of monte carlo methods. Better
understanding of input variance and probability density functions, model
nonlinearities and input parameter correlations are needed for more
confident application of these techniques. An evaluation of the input
factors which contribute the most to the level of uncertainty in an output
variable will lead modelers in the direction of most efficient data
gathering or research. In this manner the modeler can assess the risk of
imprecise forecasts and recommend measures for reducing the magnitude of
that imprecision.
H. Acknowledgements
The material presented in this Appendix is taken from a paper entitled
"Uncertainty Analysis in Water Quality Modeling Using QUAL2E", written by
the first author, for presentation at the WATERMATEX 87 Symposium, London,
June 30-July 2, 1987. The author also wishes to acknowledge Barbara
Notini, graduate student, for her work in compiling the input variance data
base and in performing the many monte carlo simulations.
183
-------
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189
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