Steady-state Dissolved Oxygen Model 

of the Willamette River, Oregon 

By Stuart W. McKenzie, Walter G. Hines, 

David A. Rickert, and Frank A. Rinella 

RIVER-QUALITY ASSESSMENT OF THE WILLAMETIE RIVER BASIN, OREGON 

GEOLOGICAL SURVEY CIRCULAR 715-J 

1979 
United States Department of the Interior 

CECIL D. ANDRUS, Secretary 

Geological Survey 

H. William Menard, Director 

Steady-state dissolved oxygen model of the Willamette River, Oregon. 

(River-quality assessment of the Willamette River Basin, Oregon) (Geological Survey 

circular; 715-J) 

Bibliography: p. J25. 

1. Water-Dissolved oxygen-Mathematical models. 2. Stream self-purificationMathematical models. 3. Water-quality-Oregon--Willamette River watershed. I. 

McKenzie, Stuart W. II. Series. III. Series: United States. Geological Survey. 

Circular; 715-J. QE75.C5 no. 715-J [TD423] 551.4'83'09 

Free on application to Branch of Distribution, U.S. Geological Survey, 1200 South Eads Street, Arlington, VA 22202 
FOREWORD 

The American public has identified the enhancement and protection of river 

quality as an important national goal, and recent laws have given this commitment considerable force. As a consequence, a considerable investment has been 

made in the past few years to improve the quality of the Nation's rivers. Further 

improvements will require substantial expenditures and the consumption of 

large amounts of energy. For these reasons, it is important that alternative 

plans for river-quality management be scientifically assessed in terms of their 

relative ability to produce environmental benefits. To aid this endeavor, this 

circular series presents a case history of an intensive river-quality assessment 

in the Willamette River basin, Oregon. 

The series examines approaches to and results of critical aspects of riverquality assessment. The first several circulars describe approaches for providing 

technically sound, timely information for river-basin planning and management. Specific topics include practical approaches to mathematical modeling, 

analysis of river hydrology, analysis of earth resources-river quality relations, 

and development of data-collection programs for assessing specific problems. 

The later circulars describe the application of approaches to existing or potential 

river-quality problems in the Willamette River basin. Specific topics include 

maintenance of high-level dissolved oxygen in the river, effects of reservoir 

release patterns on downstream river quality, algal growth potential, distribution of toxic metals, and the significance of erosion potential to proposed future 

land and water uses. 

Each circular is the product of a study devoted to developing resource information for general use. The circulars are written to be informative and useful to 

informed laymen, resource planners, and resource scientists. This design stems 

from the recognition that the ultimate success of river-quality assessment 

depends on the clarity and utility of approaches and results as well as their basic 

scientific validity. 

Individual circulars will be published in an alphabetical sequence in the 

Geological Survey Circular 715 series entitled nRiver-Quality Assessment of 

the Willamette River Basin, Oregon." 

J. S. Cragwall, Jr. 

C hie{ Hydrologist 

III 
Cover: Willamette River as it winds through Portland, Oregon. Photograph taken by 

Hugh Ackroyd. 
CONTENTS 

Page 

Abstract ______________________________________________ J1 

Introduction _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 

Conceptual development of the steady-state dissolved-oxygen model _ _ _ _ _ _ _ _ _ _ 1 

Formulation of the WIRQAS DO Model _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 

Configuration _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 

Computational procedure ___________________________ -.- _ 5 

Hydrologic characteristics _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 

Streamflow_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 

Channel geometry__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 

Water temperature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 

Calibration _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 8 

Verification _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 11 

Sensitivity analysis _______________________________________ i1 

Discussion and summary _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 

References cited _________________________________________ 25 

Explanation of WIRQAS Model printout ___________________________ 26 

ILLUSTRATIONS 

Page 

FIGURE 1. Sketch showing idealized concepts of water and materials transport used in river DO models _ _ _ _ _ _ _ J3 

2. Map of Willamette River basin, Oregon, emphasizing principal tributaries, major reservoirs, and river mile 

locations on main stem Willamette River _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 7 

3-15. Graphs showing: 

3. Comparison ofWIRQAS Model predictions with percent DO saturation measured during calibration conditions, August 197 4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10 

4. Comparison of predicted BODult loads with loads measured during calibration conditions, August 197 4 _ 10 

5. Comparison of WIRQAS Model predictions with percent DO saturation measured during verification con- ditions, July-August, 1973 _ _ ___ _ __ __ _ _ _ _ ______ _ _ _ _ _ _ _ __ __ __ _ _ _ _ __ __ _ _ 13 

6. DO profiles of the Willamette River for selected levels of streamflow _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 14 

7. DO profiles of the Willamette River assuming different percent DO saturations at RM 86.5 _ _ _ _ _ _ _ 15 

8. DO profiles of the Willamette River for a temperature range from 3°C above to 3°C below standard conditions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 16 

9. DO profiles of the Willamette River between RM's 86.5 and 26.5 using four methods of calculating reaeration -------------------------------------------------------- U 

10. DO profiles of the Willamette River for a fourfold range of BODult loading _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 18 

11. DO pt'ofiles of the Willamette River for selected rates of carbonaceous deoxygenation _ _ _ _ _ _ _ _ _ _ _ 19 

12. DO profiles of the Willamette River for a fourfold range of ammonia-nitrogen loading _ _ _ _ _ _ _ _ _ _ _ 20 

13. DO profiles of the Willamette River for selected rates of nitrification _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 21 

14. DO profiles of the Willamette River between RM's 24.0 and 5.0 for various conditions of backwater _ _ _ 22 

15. DO profiles of the Willamette River with and without an assumed oxygen demand between RM's 12.8 and 

~2--------------------------------------------------------- ~ 

v 
TABLES 

Page 

TABLE 1. Summary and explanation of parameters, major input data, and information used in calibration of the WIRQAS DO Model _________________________________________________ - J9 

2. Reach-by-reach account of DO gains and losses for calibration conditions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 12 

3. Summary and explanation of parameters, major input data, and information used in verification of the WIRQAS DO Model _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 12 

4. Sensitivity analysis summary of the WIRQAS DO Model_____________________________ 24 

5. Example of the computer printout of WIRQAS DO Model ---------------------------- 26 

SYMBOLS 

(Definition of symbols and abbreviated terms used in this circular) 

BOD-biochemical oxygen demand 

BODu~t-ultimate BOD 

K1-carbonaceous deoxygenation rate (loge), as measured in BOD bottle at 20°C 

k1-carbonaceous deoxygenation rate (log10), as measured in BOD bottle at 20°C 

kr-river carbonaceous deoxygenation rate (log10), corrected for temperature 

Kn-nitrogenous deoxygenation rate (loge) in river, at 20°C 

kn-nitrogenous deoxygenation rate (log10) in river, at 20°C 

K2-reaeration coefficient (loge) 

k~reaeration coefficient (logw) 

Q-stream discharge 

RM-river mile 

CONVERSION FACTORS 

Factors for converting English units to the International System of Units (SI) are given below to 

four significant figures. However, in the text the metric equivalents are shown only to the 

number of significant figures consistent with the values for the English units. 

English 

ft (feet) 

ft2 (square feet) 

ft3/s (cubic feet per second) 

in2 (square inches) 

lb (pounds) 

mi (miles) 

mi2 (square miles) 

Multiply by 

3.048 X 10-1 

9.290 X 10-2 

2.832 X lQ-2 

6.452 

4.536 X lQ-1 

1.609 

2.590 

VI 

Metric (Sl) 

m (meters) 

m2 (square meters) 

m3/s (cubic meters 

per second) 

cm2 (square centimeters) 

kg (kilograms) 

km (kilometers) 

km2 (square kilometers) 
STEADY-STATE DISSOLVED-OXYGEN MODEL 

OF THE WILLAMETIE RIVER, OREGON. 

By Stuart W. McKenzie, Walter G. Hines, 

David A. Rickert and Frank A. Rinella 

ABSTRACT 

A steady-state dissolved-oxygen model, which is based on 

synoptic data collected during the summers of 1973 and 

1974, has been formulated and tested for the Willamette 

River, Oregon. The model employs the accounting scheme of 

C. J. Velz in conjunction with a conceptually simple Lagrangian reference system. The model was calibrated and verified with separate data sets from the synoptic studies. 

Verification resulted in excellent agreement of model predictions with measured dissolved-oxygen data. Sensitivity 

analysis was used to examine the relative importance of the 

individual self-purification processes and model parameters. 

The verified model provides a planning and management 

tool for the critical summer low flow, high temperature conditions of the river. 

INTRODUCTION 

The summertime DO (dissolved-oxygen) regimen of the Willamette River, Oregon is presently dominated by basinwide secondary treatment, 

low-flow augmentation, nitrification, and a 

benthic-oxygen demand in Portland Harbor. 

The entire regimen and all sources and sinks of 

DO were recently described in detail in a companion report (Hines and others, 1977), which is 

hereafter referred to as Circular 715-1. 

Using the information presented in 715-I as a 

basis, the present report describes a mathematical model that quantitatively simulates the lowflow DO regimen of the Willamette. Discussion 

focuses on the description, formulation, and testing of the model, referred to hereafter as the 

WIRQAS (Willamette Intensive River Quality 

Assessment Study) Model. The final goal of the 

model is the development of a practical management tool useful for assessing the impacts on 

river DO of planning and management alternatives. The impact assessment phase of the modeling effort will be subsequently described in 

Circular 715-K. 

J1 

As used here, formulation and testing deal 

with four major elements of model development: 

1. Configuration 

2. Calibration 

3. Verification (validation) 

4. Sensitivity analysis 

Before dealing with these elements, it is helpful 

to introduce the conceptual basis of the WIRQAS Model. In doing so it is convenient to consider first a general conceptual model of river 

self-purification processes. Then, as a means for 

specifying the configuration of the WIRQAS 

Model, the general model is modified for compatibility with the physical and biochemical 

characteristics of the Willamette River system. 

Because of the technical detail of the following 

section, some readers may wish to proceed directly to the section entitled "Formulation of the 

WIRQAS DO Model." 

CONCEPTUAL DEVELOPMENT 

OF THE STEADY -STATE 

DISSOLVED-OXYGEN MODEL 

The understanding and explanation of the DO 

regimen of rivers involves the consideration of 

several complex physical, chemical, and biological processes. One approach for describing such a 

system is the development of a conceptual 

model. In other words, one seeks to formulate an 

integrated, rational set of concepts that satisfactorily describe the "real" system. Mathematics 

are used to provide an internally consistent, rigorous definition of the concepts and to allow a 

quantitative simulation. 

Through more than 50 years of empirical observation and thought, a rich conceptual model 

of the river DO regimen has evolved. This general model incorporates a description of five selfpurification processes (Circ. 715-I, Supp. A): 
1. Carbonaceous deoxygenation 

2. Nitrification 

3. Benthic-oxygen demand 

4. Plant photosynthesis and respiration 

5. Atmospheric reaeration 

For purposes of modeling, these processes can be 

considered as DO sources (producers) and sinks 

(consumers); the sources being photosynthesis 

and reaeration, the others being DO sinks. 

Based on the need to quantitatively simulate 

the five self-purification processes, a number of 

mathematical DO models have been developed. 

One model, now in widespread use, was originally 

described by Bella and Dobbins (1968). Their 

model was based on the classic Streeter-Phelps 

(1925) equations for carbonaceous deoxygenation and atmospheric reaeration, plus terms 

for the other DO sources and sinks. The model 

also includes equations for simulating the transport characteristics of the river, including the inflow and outflow of water, DO, and oxygendemanding materials. The mathematical framework of the Bella-Dobbins model is shown in 

equation 1: 

a(AC) + a(UAC) = _j_ (AE ac) +K A(C -C) 

at ax ax ax 2 s 

where 

A = area of stream cross section perpendicular to flow, 

c = average DO concentration in cross section, 

u = average cross-sectional velocity in the 

longitudinal direction, 

E = longitudinal dispersion coefficient, 

K2 = atmospheric reaeration rate (loge), 

Cs = oxygen saturation in water at the prevailing temperature, 

Kl = carbonaceous deoxygenation rate (loge), 

L = BODult concentration (from carbonaceous material) in cross section, 

Kn = nitrogenous deoxygenation rate (loge), 

N = nitrogenous oxygen demand (a function 

of ammonium- and nitrite-ion concentration) in cross section, 

p = oxygen added by photosynthesis per unit 

area, 

R = oxygen consumed by aquatic plant respiration per unit area, 

B = oxygen demand by benthal deposits per 

unit area, 

Db = oxygen added along the stream by 

inflows, 

t = time, and 

x = distance in the longitudinal direction. 

In order to obtain values for L and N to use in 

equation 1 (L and N are constantly undergoing 

first-order decay along the course of the river in 

the x-direction), it is necessary to first solve 

equations 2 and 3: 

aL 

a(:tL) a(A!ax) + a(~~L) -Kl(AL) + ALa(2) 

aN 

a(AN) a(AEax) + a(AUN) -K (AN) AN 

at ax ax n + a 

(3) 

where 

La = rate of BODutt addition along the stream by 

inflows, 

Na =rate of addition of nitrogenous-oxygen demand along the stream by inflows, 

and all other terms are as defined in equation 1. 

The DO model described by equations 1-3 is 

based upon the idealized concept of a one-dimensional stream. That is, variations in velocity, 

concentrations, and process-rate coefficients are 

assumed to occur only in the longitudinal direction (x), and not in the horizontal or vertical directions. Equations 1-3 are derived using the 

laws of conservation of mass and momentum by 

performing a mass balance for a time interval, dt, 

on a stream segment of cross-sectional area, A, 

and length, dx (see fig. lA). As dt and dx approach infinitesimally small values, the three 

partial differential equations that describe the 

temporal and spatial distribution of C, L, N, P, 

R, B, and Db are obtained. 

J2 

In many applications, the assumption of a onedimensional stream is reasonable and scientifically sound. This is so because data-averaging 

techniques can be used to "smooth out" local and 

short-term nonhomogeneity in the vertical and 

horizontal dimensions. 

The application of equations 1-3 to an actual 

river situation usually involves a fixed reference 

frame known as the Eulerian system (fig. lA). 

With this system, the river is divided into short 

segments by establishing numerous cross-sectional planes normal to the direction of flow. 
(A) Eulerian system 

(B) Lagrangian system 

Water and materials move 

as time variant fluxes 

Water and materials move as a 

time invariant discrete unit 

FIGURE !.-Idealized concepts of water and materials transport used in river DO models. A, Eulerian system. B, 

Lagrangian system. 

J3 


where 

V = average velocity ~ average timeof-travel 

Q = streamflow 

A= cross-sectional area 

Consistent with the Velz-Lagrangian system, 

the WIRQAS Model was applied to the river between RM's 86.5 to 5.0 under conditions of summer low flow. As described elsewhere (Gleeson, 

1972; Circular 715-1), no significant DO problems have occurred upstream of RM 86.5, nor 

have problems been noted at times other than 

during the summer. RM 5.0 is the approximate 

location of the lowest DO concentration in the 

Willamette during low-flow conditions. Below 

this point, mixing with Columbia River water 

(Circular 715-1, figs. 21 & 22) causes a rapid increase in DO concentrations. 

HYDROLOGIC CHARACTERISTICS 

A key consideration in the configuration of any 

DO model is the method by which the quantitative definition of streamflow, channel morphology, and water temperature is accomplished. 

These river characteristics determine the "immediate environment" (Hines and others, 1976) 

of the river within which the self-purification 

processes occur and are largely controlled. 

STREAMFLOW 

For the WIRQAS Model, streamflow data 

were obtained from direct measurements made 

routinely at U.S. Geological Survey gaging stations on tributaries and the main-stem Willamette River. (See Circular 715-1, section on 

"Streamflow.") The Survey streamflow gage at 

Salem (fig. 2) was used as the reference gage. 

Flow measured at Salem was routed downstream 

on the basis of times-of-travel calculated by the 

volumetric-displacement technique. Measured 

inflows from tributaries and waste-water outfalls 

were simply added to the mainstem flow to obtain a cumulative flow at any downstream location. During summer low-flow periods no major 

diversions occur in the segment of interest, RM 

86.5-5.0. Water losses from evaporation, riverto-groundwater seepage, and irrigation are small, 

and they were considered to be balanced by the 

small volume of unmeasured surface and 

ground-water inflows. 

J6 

CHANNEL GEOMETRY 

Definition of channel geometry was accomplished as follows: 

1. In the Tidal Reach (RM's 0-26.5), recently 

compiled (1972) channel geometry (width and 

depth) maps were available from the Portland 

District, U.S. Army Corps of Engineers. Soundings on these maps are referenced to a specific 

river-discharge and tidal condition. This facilitated adjustments to obtain channel geometry 

data consistent with the discharge and tidal conditions encountered during model calibration 

and verification. 

2. In the Newberg Pool (RM's--26.5-52.0), new 

channel geometry data were obtained by making 

cross-sectional traverses in a boat equipped with 

a recording fathometer. The traverses were made 

at longitudinal intervals of approximately 0.2 mi 

during periods of low flow in the summer and 

early fall of 1973. Auxiliary staff gages were installed to develop stage-discharge ratings. 

3. As described in Circular 715-1, the Upstream Reach (RM's 52.0-187) is shallow, meandering, and characterized by year-to-year shifts 

in channel shape. Therefore, for the modeled 

portion, a detailed definition of channel geometry with the recording fathometer was considered of dubious value. Consequently, time-oftravel values for use in the model were calc~lated 

on the basis of dye-tracer data reported by Harris (1968). Generalized values for average crosssectional depth, cross-sectional area, and segment volume were obtained using the continuity 

equation (equation 7) in conjunction with width 

measurements made from high-resolution aerial 

photos taken under known conditions of streamflow. 

Once compiled, the channel geometry data for 

the three reaches were collated so as to define 

relatively homogeneous 0.2-1.0 mi segments for 

which representative values of depth, width, and 

channel volume could be assigned. The averaged 

values were those actually used in the model for 

computational purposes. Besides channel geometry data, the location of waste-water outfalls 

and tributaries was used as a criterion for establishing river segment boundaries. In all, 260 discrete segments were established between RM's 

86.5 and 5.0. 

WATER TEMPERATURE 

Water-temperature data were obtained from 

measurements made during the course of data 
w 

~ <( 

0 

u 

II 

INDEX MAP OF OREGON 

10 20 MILES 

~-..... , __J.I--"T'"i -~, I 

10 20 30 KILOMETERS 

FIGURE 2.-Willamette River basin, Oregon, emphasizing principal tributaries, major reservoirs, and river mile locations on main-stem Willamette River. 

J7 

w 

0 

<( 

u 

V'J 

<( 

u 
collection for model calibration and verification. 

Average daily temperatures were used in the 

model as calculated from individual values having a maximum range at any site of ± 3°C. 

CALIBRATION 

Calibration is the procedure whereby model 

parameters are quantified and adjusted so that 

model outputs (for a specified set of input data 

such as streamflow, water temperature, and 

waste-water loads) approximate a set of observed 

DO and BOD data. The quantification of model 

parameters and the range within which they can 

be realistically adjusted should be based primarily on field and laboratory data, an understanding of the particular river system, and sound 

professional judgment. For the calibration to be 

credible, it should not be based on arbitrary parameter optimization routines and computerized 

curve fitting (see Hines and others, 1975). This 

means that parameter values should be similar to 

and consistent with (though not necessarily 

identical to) thoae values calculated f1·om field 

and laboratory studies of the river system. Further, if adjustments of parameters are necessary 

to make model outputs approximate observed 

data, they should be readily explainable. 

Model parameters, input data, and other pertinent information related to the calibration of 

the WIRQAS Model are summarized in table 1. 

The model parameter values were developed 

from the results of the synoptic studies reported 

in Circular 715-1. Specifically, the model calibration involved data on flow, waste-water loading, and measured in-river parameters from the 

synoptic studies of August 6, 7 and 12-14, 197 4. 

As discussed in Circular 715-1, the 1974 data 

were used for calibration because the nitrogen 

and BOD results were of better quality than 

those obtained in 1973. 

Streamflow in the Willamette River was low 

(approximately 6,760 ft3/s at Salem) and steady 

during the early-August 197 4 calibration period. 

On the basis of precalibration sampling (see 

Rickert and others, 1976), BOD, nitrogen, and 

DO concentrations in the river and waste-water 

effluents were also stable on a daily average 

basis. 

Using the information in table 1, a model computer run was made to produce a predicted DO 

profile of the Willamette River between RM's 

86.5 and 5.0. The predicted profile is compared 

in figure 3 with the ranges and averages of the 

measured calibration data. In general, there is 

good agreement between the two profiles; the 

largest discrepancy occurs at RM 72 where the 

predicted DO concentration is 5 percent 

lower than the daily average. 

A further calibration check was made by comparing results of predicted and measured loads 

of ultimate BOD (BODuit). As shown in figure 4, 

the two profiles compare reasonably well, except 

in the segment between RM's 13 to 7 where 

measured loads are approximately 27,000 lb/d 

higher than predicted. As described elsewhere 

(Circular 715-1, in the section on "Anomalously 

High BOD in the Tidal Reach") oxygen-demanding benthic materials are suspected as one major 

cause of this discrepancy. 

A third, though somewhat limited, calibration 

check involved the comparison of predicted and 

measured nitrogen concentrations (Circular 715-

1, fig. 14) in the zone of active nitrification (RM's 

86.5-55.2). As noted in table 1 and explained in 

Circular 715-1 (in the section on "Nitrification"), 

the WIRQAS Model incorporates an effective nitrification rate that does not generate segmentto-segment predictions for nitrate-, nitrite-, and 

ammonia-N concentrations. However, a predicted ammonia-N concentration can be obtained at 

RM 55 by flow routing and decaying the cumulative loadings from above this point. On this basis, 

the WIRQAS Model predicted an ammonia-N 

concentration of 0.36 mg/L at RM 55, whereas 

the measured value was 0.22 mg/L. Considering 

the likelihood of ammonia-N losses to algal assimilation, this is considered a reasonable check. 

Table 2 presents a reach-by-reach accounting 

of changes in the concentration (mg/L) and mass 

(lb-02/d) of DO in the Willamette during calibration conditions. In terms of oxygen loss, the 

percentages over the 81.5 mi are nitrification38 percent, carbonaceous deoxygenation-4 7 

percent, and benthic demand-15 percent. All 

the nitrogenous demand occurs in the Upstream 

Reach, and all the benthic demand occurs in lowJ8 
TABLE 1.-Summary and explanation of parameters, major input data, and information used in calibration of the 

WIRQAS DO Model 

[Data collected August 1-15, 1974. Unless otherwise noted, see Circular 715-1 for details on the cited parameters and information] 

1. Carbonaceous deoxygenation. Simulated with first-order decay kinetics with the following rate coefficients kr (20°C): (a) 

RM 86.5-55.2, kr = 0.06/d, (b) RM 55.2-5.0, kr = 0.03/d. For kr at temperatures other than 20°C the adjustment formula 

was Kr = K 20 oc x 1.047<T-2ooc). 

2. Nitrification. Simulated with first-order decay kinetics. Significant oxygen-demanding nitrification was found to occur 

only in the shallow surface-active segment RM 86.5-55.2. The effective rate of nitrification (kn) calculated from the rate 

of appearance of nitrate-N = 0.7/d. No temperature adjustment was necessary because water temperatures in the affected river segment were essentially 20°C. 

3. Benthic-oxygen demand. Simulated as an oxygen-demanding load distributed over the segment RM's 12.8-5.2. The total 

load in the reach was estimated to be approximately 27,000 lb/d. As discussed in Circular 715-1, only part of this demand 

is thought to result from "in place" benthic-oxygen demand. The remainder probably results from several additional 

causes. 

4. Photosynthesis and respiration. DO produced and consumed by algae were taken to be in balance over the 81.5 mi stretch 

of river. This assumption is generally supported by primary production data and by DO mass balance computations 

made with preliminary versions of the model. See Circular 715-1 for details. 

5. Atmospheric reaeration. Calculated on segment-by-segment basis using the Velz (1970) method. The reaeration at Willamette Falls (RM 26.5) was simulated by adding 13,400 lb/d of DO to the profile calculation. This increment was 

consistent with measurements made above and below the Falls during 1974low-flow conditions. 

6. Streamflow. Boundary condition discharge at RM 86.5 = 6760 ft3fs as measured at the Salem gage. Below RM 86.5, 

inflows from tributaries and waste-water outfalls were added to flow in the mainstem to produce a cumulative total at 

any site. Water losses due to evaporation and diversions· were assumed to be equalized by seepage and inflows of small 

unmeasured tributaries. 

7. Water temperature. Boundary condition water temperature at RM 86.5 = 20°C. Water temperature showed a gradual 

increase in the downstream direction to 21 °C at RM 52.6, to 22°C at RM 45.0, to 23°C at RM 37.0, and to 23.5°C at RM 

25.6 and below. These data were used directly into the model. 

8. Channel geometry. See section in text entitled "Channel Geometry" for description of how channel geometry characteristics were calculated and used in the model. 

9. Waste loading. See summary below. 

Location Flow DO BODutt load NOD load 

Description of waste input (RM) (ft3/s) (percent saturation) (lb/d) (lb/d)1 

Residual load in Willamette River _____ 86.5 6,760 92 87,300 25,200 

Boise Cascade Corp ------------- 85.0 24.9 10 13,100 93,100 

Salem municipal STP ----------- 77.9 40.4 50 11,700 7,500 

Yamhill River _________________ 55.0 75 90 1,400 

Newberg municipal STP ---------- 50.5 1.2 20 2100 

Publishers Paper Co _____________ 49.8 18.0 10 7,300 

Wilsonville municipal STP -------- 39.0 .2 20 220 

Molalla River _________________ 35.8 151 100 1,500 

Canby municipal STP ____________ 33.0 .5 20 270 

Tualatin River ---------------- 28.4 39 100 5,300 

Publishers Paper Co _____________ 28.0 19.9 10 11,000 

West Linn municipal STP #1 ------- 27.8 .7 20 2100 

Crown Zellerbach Corp ----------- 27.6 20.7 40 4,900 

Oregon City municipal STP -------- 25.0 4.3 50 2140 

Clackamas River -------------- 24.8 1,290 96 9,600 

West Linn municipal STP #2 ------- 24.1 1.0 50 2170 

Tryon Creek municipal STP -------- 20.2 5.7 50 2220 

Oak Lodge municipal STP --------- 19.9 3.1 50 2120 

Kellogg Creek _________________ 18.5 1.0 70 2260 

Johnson Creek ---------------- 18.4 2.0 80 240 

Milwaukie municipal STP --------- 18.4 2.1 50 2220 

1 Only those NOD loads subject to nitrification are listed. See Circular 715-1, Table 8 for summary of nitrogen loading. See page J27 of this circular for the 

method used to compute NOD from ammonia-N data. 

2 Estimate based on samples collected in 1973. Sensitivity analysis showed that these small BODult loads had minimal impact on DO concentration. 

J9 
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-----Portland Harbor-----

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w 

u 

a: 

w 

a. 

z 

z' 

w 

(!) 

>-

X 

0 

0 

w 

> ...J 

0 

(I) 

(I) 

0 

50 

100 

90 

80 

70 

60 

50 

0 

EXPLANATION 

Max.! 

Avg. Measured DO saturation 

Min. 

Predicted DO profile 

State DO standards 

4 

Discharge at Salem = 6760 ft3 /s 

6 8 10 

TIME OF TRAVEL, IN DAYS 

FIGURE 3.-Comparison of WIRQAS Model predictions with percent DO saturation measured during 

calibration conditions, August 197 4. 

RIVER MILES ABOVE MOUTH 

Measured loading 

Predicted in-river load 

Discharge at Salem = 6760 tt3 /s 

TIME OF TRAVEL, IN DAYS 

12 

FIGURE 4.-Comparison of predicted BODult loads with loads measured during calibration conditions, August 1974. 

JlO 
er Portland Harbor (RM's 12.8-5.0). In contrast, 

the carbonaceous demand is spread over the entire 81.5 mi. In assessing the relative importance 

of carbonaceous deoxygenation, the reader 

should note (as explained in Circular 715-1) that 

about one-half of the exerted demand originates 

from near natural, nonpoint-source loads of tributary streams. Such loads are not amenable to 

treatment. 

VERIFICATION 

Verification, by definition, implies the "proof 

of truth" of the model. However, Bella (1969) 

cautioned that model verification should not be 

thought of in these absolute terms. All models 

have limitations and specific domains of applicability. Thus, model verification is more realistically described as a means for validating or 

substantiating the model's predictive power under a specific set of environmental conditions. In 

practice, verification should involve the use of a 

calibrated model (that is, the same model parameters developed during calibration) and a new set 

of observed data. Some of the new data establish 

the initial and boundary conditions necessary to 

"start" and "run" the model. The remainder 

serve as an independent set of observations for 

comparison with model predictions. 

Verification data for the WIRQAS Model were 

obtained from synoptic studies conducted during a prolonged low-flow period of late July to 

mid August 1973 (Circular 715-1, fig. 10 and table 4). As was the case for the calibration period, 

stable hydrologic and waste-water loading conditions made the Velz-Lagrangian modeling approach compatible with its underlying steadystate assumptions. 

Table 3 summarizes the model parameters, input data, and other information used in verification of the WIRQAS Model. Note that in keeping 

with the described requirement for verification, 

the model parameter values for items 1-5 are 

identical with those used for calibration (see table 1). 

Based on the information in table 3, a predicted DO profile was generated by the WIRQAS 

Model. The predicted profile is compared in figure 5 with average measured DO values from the 

synoptic studies of July 24-26 and August 15-18, 

1973. The two profiles are in good agreement 

throughout the 81.5 mi segment. The largest difference occurs at RM 28.6 where the predicted 

DO saturation is 4 percent lower than the daily 

average. 

The in-river 1973 data for BODult and ammonia-N were considered too poor to be used as a 

basis for additional model verification. However, 

we consider the model to be fully verified for future use by the closeness of fit between the predicted and measured DO profiles (fig. 5). 

Circular 715-1 describes the difficulties encountered with the 1973 data for BODult and nitrogen. 

SENSITIVITY ANALYSIS 

Sensitivity analysis is concerned with changes 

in model outputs that result from variations in 

model parameters and data inputs. In most 

cases, the primary concern is with the identification of those factors that are most important in 

controlling model outputs. For example, a pertinent question to ask in the sensitivity analysis of 

the WIRQAS Model is "What is the impact,·in 

terms of predicted DO concentrations, if (with 

all other variables held constant) water temperatures were 3°C higher or lower than those used in 

calibration?" By inserting the changed values of 

water temperature into the model and making a 

run on the computer, one can observe whether 

the impact on predicted DO concentrations is 

small or large. If small, the model is said to be 

"insensitive" to water temperature (at least in 

the analyzed range). If the impact is large, it suggests that water temperature is an important 

control of the DO regimen. 

Sensitivity analyses such as the one described 

above have at least two general categories of use. 

First, they help to identify those factors that deserve most attention in model formulation. This 

is to say, sensitivity analysis can lead the investigator to spend more time and effort on those 

data and parameters that are most important to 

the model's simulatory and predictive capability. 

Perhaps of equal importance is a second use. 

Sensitivity analysis can lead to a better recognition of the management alternatives that are 

most efficient and practical for controlling riverquality problems. Only the first type of sensitivity analysis is addressed here, because a subseJ11 
TABLE 2.-Reach-by-reach account of DO gains 

[Flow at Salem gage= 6,760 ft%. Note that as discharge increases in the downstream direction, 

DO losses 

Modeled Amount Percentage 

river Factor causing of loss of total 

segment loss (lb-02/d) loss in 

(RM's) reach 

86.5-55.2 Nitrification ------------------- 69,300 90 

(traveltime= Carbonaceous deoxygenation __________ 7,600 10 

0.5 days) To~l ____________________ 76,900 100 

55.2-26.5 

(traveltime= Carbonaceous deoxygenation __________ 28,400 100 

3.7 days) 

26.5-12.8 

(traveltime= Carbonaceous deoxygenation __________ 25,900 100 

3.1 days) 

12.8-5.0 Carbonaceous deoxygenation __________ 24,200 47 

(traveltime= Benthic demand ----------------- 27,200 53 

3.7 days) Total ____________________ 51,400 100 

86.5-5.0 Nitrification ------------------- 69,300 38 

(traveltime= Carbonaceous deoxygenation __________ 86,100 147 

11.0 days) Benthic demand ----------------- 27,200 15 

Grand total --------------- 182,600 100 

TABLE 3.-Summary and explanation of parameters, major input data, and information used in verification of WIRQAS 

DO model 

[Data collected July 24-August 7, 1973] 

1. Carbonaceous deoxygenation _ _ _ _ _ _ _ See ~ble 1. made at Salem gage for the period July 15-August 15, 

2. Nitrification _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Do. 1973. See table 1, item 6, for further discussion. 

3. Benthic oxygen demand ______________ Do. 7. Water temperature-boundary condition water tem4. Photosynthesis and respiration __________ Do. perature at RM 86.5 = 20°C. Water temperature 

5. Atmospheric reaeration _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Do. showed a gradual increase in the downstream direc-

(12,200 lb/d DO added at Willamette Falls (RM 26.5) tion to 21 °C at RM 52.6; and to 22°C at RM 20.9 and 

based on observations during 1973 low-flow condi- below. 

tions). 8. Channel geometry _______________ See table 1. 

6. Streamflow-boundary condition discharge at RM 86.5 = 9. Waste loading ___________ See summary below. 

6,000 ft3/s as determined from average of measurements 

Description of waste input 

Residual load in Willamette River ____ _ Boise Cascade Corp ____________ _ 

Salem municipal STP __________ _ Yamhill River ________________ _ 

Newberg municipal STP _________ _ Publishers Paper Co ____________ _ 

Wilsonville municipal STP _______ _ Molalla River ________________ _ 

Canby municipal STP ------------ Tualatid River _______________ _ Publishers Paper Co ____________ _ West Linn municipal STP #1 ______ _ Crown Zellerbach Corp. _________ _ Oregon City municipal STP _______ _ Clackamas River _____________ _ 

West Linn municipal STP #2 ______ _ Tryon Creek municipal STP _______ _ Oak Lodae municipal STP ________ _ Kellogg reek ________________ _ 

Johnson Creek ---------------- Milwaukie municipal STP ________ _ 

1 Only those NOD loads subject to nitrification are listed. 

Location 

(RM) 

86.5 

85.0 

77.9 

55.0 

50.5 

49.8 

39.0 

35.8 

33.0 

28.4 

28.0 

27.8 

27.6 

25.0 

24.8 

24.1 

20.2 

19.9 

18.5 

18.4 

18.4 

6,000 

25 

49 

50 

1.2 

16 

0.2 

118 

0.5 

6.0 

19 

0.7 

20 

4.3 

910 

1.0 

5.7 

3.1 

1.0 

2.0 

2.1 

J12 

DO 

(percent saturation) 

93 

10 

50 

90 

20 

10 

20 

100 

20 

80 

25 

20 

25 

50 

96 

50 

50 

50 

70 

80 

50 

BODu1tload 

(lb/d) 

106,600 

11,600 

13,100 

1,600 

110 

9,500 

20 

1,270 

70 

320 

6,050 

100 

1,410 

140 

9,200 

170 

220 

120 

260 

40 

220 

NOD load 

(lb/d) 1 

16,800 

69,100 

9,900 
and losses for calibration conditions 

it is possible for the mass of DO (in lb-02/d) to increase while the DO concentration (in mg/L) decreases] 

DO gains Net DO changes 

Factor causing Gain or Amount Percentage of Loss Gain loss in gain of gain total gain {lb-02/d) Ob-Oid> DO concentration (lb-02/d) in reach (mg/L) 

Tributary inflows -------------- 1,100 8 -- -- --

Atmospheric reaeration ---------- 12,000 92 -- -- -- To~l __________________ 13,100 100 63,800 -- -1.8 

Tribu~ry inflows 12,800 27 -- -- --

Atmospheric reaeration ---------- 20,900 44 -- -- -- Reaeration at Willamette Falls _______ 13,400 29 -- -- -- To~l __________________ 47,100 100 -- 18,700 +.2 

Tributary inflows -------------- 57,500 85 -- -- --

Atmospheric reaeration ---------- 10,000 __1Q -- -- -- To~l __________________ 67,500 100 -- 41,600 -.1 

Atmospheric reaeration ---------- 14,700 100 36,700 -- -.80 

Tributary inflows -------------- 71,400 50 -- -- --

Atmospheric reaeration ---------- 57,600 40 -- -- -- Reaeration at Willamette Falls _______ 13,400 10 -- -- --

Grand to~l ----------------- 142,400 100 40,200 -- -2.5 

1 About one-half of the carbonaceous-oxygen demand entering the Willamette River originated from non point-source loads of tributary streams. As explained in 

Circular 715-1 and in Rickert, Hines, and McKenzie, 1975, the tributary loads represent near natural conditions and are not amenable to reduction through wastewater treatment. 

Albany 

120 

Salem 

86 

Newberg 

50 

RIVER MILES ABOVE MOUTH 

Willamette Falls 

26.5 

1-------Portland Harbor-------

17 

~ 110r---~----~----~-----r-----.--L-,-----,--L--,-----~----.-----.-----r-L-~~ 

i= 

<( 

a: 

::> ·100 

1-

<( 

en 

1-

ffi 90 (.) 

a: 

UJ 

c.. 

z 80 

z· 

UJ 

(!) 

> 70 X 

0 

0 

UJ 

> 60 .....1 

0 

en 

en 

0 50 

0 

EXPLANATION 

Max. i 

Avg. Measured DO saturation Discharge at Salem= 6000 ft3/s Min. 

Predicted DO profile 

State DO standards 

TIME OF TRAVEL, IN DAYS 

FIGURE 5.-Comparison of WIRQAS Model predictions with percent DO saturation measured during verification 

conditions, July-August, 1973. 

J13 
quent publication (Circular 715-K) will focus on 

management implications derived from analysis 

of numerous production runs of the WIRQAS 

Model. 

Figures 6-15 provide a graphical depiction of 

the sensitivity of the WIRQAS DO Model to 

changes in selected model parameters and data. 

During the model runs, one parameter or data input was varied within a specified range of values 

while all other variables were held constant. All 

sensitivity analyses were based on a "standard 

condition" comprised of flows, temperatures, 

rate constants, and waste loadings identical to 

those measured during early August 1974 (the 

calibration period), except for ammonia loading 

at Boise Cascade Corp. (RM 85.0). Because, during this period, the ammonia loading from Boise 

Cascade was well above normal (for the summers 

of 1973-74), the measured value of 93,000 lb/d 

was decreased to 70,000 lb/d to create a more 

standard river condition. To permit comparisons, the same standard profile is included in 

each of the sensitivity analysis figures (figs. 6-

15). The results of the analyses, as presented in 

the 10 figures, are summarized and com pared in 

Table 4. 

RIVER KILOMETERS ABOVE MOUTH 

z 

0 

i= 

<( 

a: 

:::> 

1-

<( 

(/) 

1-

z 

UJ 

u 

a: 

UJ 

a.. 

~ 

z' 

UJ 

t:) 

>-

X 

0 

0 

UJ 

> 

.....J 

0 

(/) 

(/) 

0 

Salem 

80 

60 

40 

20 

100 

Newberg 

9000 ft3 /s 

5000 ft3 /s 

3260 ft3 /s 

EXPLANATION 

--- State DO standards 

80 60 40 

RIVER Ml LES ABOVE MOUTH 

Willamette 

Falls 

20 

!

_Portland -I 

Harbor 

0 

0 

FIGURE 6.-DO profiles of the Willamette River for selected levels of streamflow. For the predicted 

profiles, BODult and ammonia-nitrogen were handled as follows: (1) At RM 86.5 the concentrations were held constant at standard conditions. (2) Below RM 86.5 the loadings were held constant at standard conditions. 

J14 
DISCUSSION AND SUMMARY 

A chronic problem with the configuration of 

river DO models appears to be a failure to develop models that are both simple and conceptually 

satisfying. In an earlier paper, Hines (Hines and 

others, 1975) suggests that this situation is at 

least partly attributable to the proliferation of 

the "general case" model. That is, in an attempt 

to make DO models capable of handling all conceivable river conditions (presumably in the 

name of conceptual satisfaction), complex mathematical configurations have been used. Such 

models are often proposed for steady-state application. However, the conceptual simplicity and 

explanatory power offered by the steady-state 

concept has all too often been lost in dealing with 

the complexity of the "general case" model. 

A second, chronic problem with DO models 

lies in the methods and data that are used for 

calibration and verification. In applying steadystate models, it has been common practice 

among modelers having little field experience to 

RIVER KILOMETERS ABOVE MOUTH 

!

_Portland -I 

Harbor 

z 

0 

1-

<( 

a: 

::> 

1-

<( 

C/) 

1-

z 

w 

u 

a: 

w 

~ 

~ 

z' 

w 

<--' 

>-

X 

0 

0 

w 

> _J 

0 

C/) 

C/) 

0 

Salem 

90 

80 

70 

60 

100 80 60 

Newberg 

Willamette 

Falls 

EXPLANATION 

- --State DO standards 

40 20 

RIVER Ml LES ABOVE MOUTH 

0 

0 

FIGURE 7.-DO profiles of the Willamette River assuming different percent DO saturations at RM 

86.5. All other factors held constant at standard conditions. 

J15 
RIVER KILOMETERS ABOVE MOUTH 

Salem Newberg 

Willamette 

Falls !

_Portland -I 

Harbor 

z 

0 90 

1-

<( 

a: 

::> 

1-

<( 

en 

1-

z 

LU 

u 

a: 80 LU 

a.. 

~ 

z' 

LU 

(!} 

>-

X 

0 

0 70 LU 

> 

....1 

0 

en 

en 

0 

60 

100 

EXPLANATION 

---- State DO standards 

80 60 40 

0 

20 0 

RIVER Ml LES ABOVE MOUTH 

FIGURE 8.-DO profiles of the Willamette River for a temperature range from 3°C above to 3°C below 

standard conditions. All other factors held constant. 

calibrate and (supposedly) verify with data that, 

even in the most optimistic sense, were not collected during steady-state conditions. This is 

evident in that many discussions of river DO 

models deal with steady-state only in the context 

of the river's transport and waste-loading regimes. Thus, for example, a 2- or 3-day stability 

in average daily streamflow and waste loads is 

commonly erroneously cited as proof of a steadystate condition. Worse yet is the case wherein 

numerous measurements of streamflow and 

waste loads have been made during a hydrologically variable 1- or 2-month period, and then, 

averaged and used as data for calibration and 

verification of a steady-state model. These exampies reflect a lack of fundamental understanding 

as to what constitutes a steady-state DO regimen. 

What does determine a steady-state condition 

for rivers? In our view, steady state involves 

nothing less than a short-term ecological stability of the river. In addition to a stable transport 

and waste-loading regimen, this ecological 

steady state ~ust reach a day-to-day constancy 

in biochemical processes and reaction rates at 

any given cross section. The attainment of such a 

steady state entails, in turn, an antecedent stability of the river's "immediate environment" 

(Hines and others, 1976) as reflected primarily 

by streamflow, water temperature, and channelJ16 
RIVER KILOMETERS ABOVE MOUTH 

!

_Portland -I 

Harbor Salem Newberg 0 

z 

Q 90 

1-

<( 

a: 

::J 

1-

<( 

(/) 

1-

z 

w 

(.) 

a: 80 w 

~ 

~ 

z 

w 

(!) 

>-

X 

0 

0 70 w 

> ..J 

0 

(/) 

(/) 

0 

EXPLANATION 

- --- State DO standards 

100 80 60 40 20 0 

RIVER MILES ABOVE MOUTH 

FIGURE 9.-DO profiles of the Willamette River between RM's 86.5 and 26.5 using four methods of 

calculating reaeration. No comparison is presented below RM 26.5 because only the Velz method 

is applicable to tidal rivers. The equations are: Bennett and Rathbun (1972), k 2 = (20.17) 

(V-607fDL689) (1.024<T-20)); Churchill, Elmore, and Buckingham, k 2 = (11.58) (V·969fD1.673) (1.024 

(T-20)); Isaacs and Gaudy (1968), k 2 = (5.62) (V/DL5) (1.024 (T-20)). For the Velz method, see Velz, 

1970, p. 184-197. All factors other than k2 held constant at standard conditions. 

morphology conditions. This antecedent constancy is necessary for the river's chemical and 

biological subsystems to adjust to the surrounding environment-that is, to get "used to" (or to 

come to a dynamic equilibrium wiih) the immediately surrounding environment. 

Without the antecedent stability of the river 

environment, measurements used for calibration 

and verification are unlikely to reflect a steadystate DO regimen. Consequently, even if model 

predictions "fit" the observed calibration or verification data, the model is likely to be deterministically erroneous. Invariably, such a model will 

be a poor explanatory and predictive tool. 

The WIRQAS modeling effort was designed to 

overcome the problems described above. With 

regard to the problem of extraneous mathematical complexity, simple algorithms were devised. 

The algorithms incorporate the Velz (1970) 

bookkeeping-type DO accounting system in conjunction with a Lagrangian moving-reference 

frame. The resulting configuration is extremely 

simple, yet applicable to simulation of the lowflow DO regimen of the Willamette River. 

J17 
RIVER KILOMETERS ABOVE MOUTH 

Salem Newberg 

Willamette 

Falls !

_Portland -I 

Harbor 

z 

0 90 ~ <( 

cc 

::> 

1-

<( 

en 

1-

z 

w 

(..) 

cc 80 w 

~ 

~ 

z 

w 

<.::) 

r 

X 

0 

Cl 70 w 

> 

....J 

0 

en 

en 

Cl 

0 

EXPLANATION 

---- State DO standards 

60 

100 80 60 40 20 0 

RIVER MILES ABOVE MOUTH 

FIGURE 10.-DO profiles of the Willamette River for a fourfold range of BODult loading. Loading of 

BODult at RM 86.5 and at each downstream source was varied by the indicated percentage of the 

standard conditions. All other factors held constant. 

To provide reliable data for calibration and 

verification, a series of intensive field and laboratory studies was conducted during the summers 

of 1973 and 1974 (Circular 715-1). Importantly, 

in keeping with the notion of steady state, the 

two independent sets of data were collected under extended low-flow, high-temperature conditions. During the study periods, the Willamette 

River DO regimen was in a state of relative ecological stability and, thus, compatible with the 

underlying assumptions of the steady-state concept. Moreover, DO depletion was maximum 

during these low-flow, steady-state conditions, 

thus making the periods the "critical condition" 

for basing the design of waste treatment and 

river-management plans. 

Calibration and verification of the WIRQAS 

DO Model involved comparison of model predictions with measured data. With minor exceptions, good agreement was found between 

predictions and observations. Agreement was 

particularly good between predicted and measured percent DO saturation (figs. 3 and 5). Nowhere over the modeled 81.5 mi of river were 

there differences of more than 5 percent saturation. 

J18 
RIVER KILOMETERS ABOVE MOUTH 

Salem Newberg 

Willamette 

Falls I

_Portland -I 

Harbor 

z 

0 90 ~ 

~ 

a: 

:J 

~ 

<( 

en 

~ 

z 

UJ 

~ 80 UJ 

0.. 

~ 

z 

UJ 

(.!) 

>-

X 

0 

0 70 UJ 

> 

....J 

0 

en 

en 

0 

60 

100 

EXPLANATION 

------- k,= .02 

-----Standard conditions 

-----k,= .04 

-----k,= .06 

-----State DO standards 

80 60 

0 

40 20 0 

RIVER Ml LES ABOVE MOUTH 

FIGURE 11.-DO profiles of the Willamette River for selected rates of carbonaceous deoxygenation. All 

other factors held constant at standard conditions. 

Based on the DO calibration and verification 

results, the WIRQAS Model appears to be a valid 

mathematical description of the summertime, 

steady-state DO regimen of the Willamette River 

between RM's 86.5-5.0. 

Sensitivity analysis suggests that the WIRQAS DO Model is relatively insensitive to 

changes in water temperature (fig. 8), BOD loading (fig. 10), carbonaceous deoxygenation rate 

(fig. 11), and nitrification rate (fig. 13). The 

model is relatively sensitive to changes in 

streamflow (fig. 6), the initial DO concentration 

at RM 86.5 (fig. 7), ammonia-N loading upstream of RM 55 (fig. 12), and benthic-oxygen 

demand in Portland Harbor (fig. 15). Based on 

comparative reaeration computations (fig. 9), 

the model is also sensitive to the method of calculating reaeration. The WIRQAS Model employs the Velz (1970) reaeration calculation 

method. This method resulted in good agreement between predicted and observed data (figs. 

3 and 5), while the other methods shown_ in figure 

9 did not. Reasons for the differences in predicted DO profiles based on the various reaeration 

computation methods await further research. 

Based on the results discussed in this report 

and in Circular 715-1, the WIRQAS DO Model is 

considered to be a reliable simulatory and predictive tool, subject to the following conditions 

and limitations: 

J19 


~ 

~ 

~ 

RIVER Kl LOMETERS ABOVE MOUTH 

I 

Portland Harbor 

40 30 20 10 I I 

2 85 

0 

..... 

<( 

a: 

::> 

..... 

~ 80 

..... 

2 

w 

u 

a: 

w 

Q.. 

2 75 - EXPLANATION ' 2 

w 2 feet 

(.!) ----5 feet >-

X 10 feet 0 

0 70 

w 

> _, 

0 

(/) 

(/) 

0 

65 

25 20 15 10 5 

RIVER MILES ABOVE MOUTH 

FIGURE 14.-DO profiles of the Willamette River between RM's 24.0 and 5.0 for various conditions of backwater. Two feet was the average 

for 1973 conditions; 5 feet was the average for 197 4 conditions. Depths are referenced to gage at Morrison Street Bridge, RM 12.8. All 

factors other than river stage held constant at standard conditions. 
z 

0 

~ 

<{ 

a: 

:::J 

~ 

<{ 

en 

~ 

z 

w 

u ~ a: t-:) w CJj CL 

z 

-

z 

w 

(!) 

>-

X 

0 

Cl 

w 

> 

...J 

0 

en 

en 

Cl 

RIVER KILOMETERS ABOVE MOUTH 

I Portland Harbor 

40 30 20 10 

85~--------------.---~--~------.-------~------r-----------~-. 

80 

75 

70 

Without oxygen demand 

------ ~flj o...,.Y. 

.9(91) 

OE>'>) 

qi)Q' 

65~--------------~~--------------~----------------~--------------~ 25 20 15 10 5 

RIVER MILES ABOVE MOUTH 

FIGURE 15.-DO profiles of the Willamette River with and without an assumed oxygen demand between RM's 12.8 and 5.2. All other factors 

held constant at standard conditions. 
TABLE 4.-Sensitivity analysis summary of the WIRQAS DO Model 

[Based on figs. 6-15 which show the impacts on percent DO saturation which result from varying selected model parameters and data inputs. 

See page J14 for explanation of standard conditions] 

Variable tested Applicable river Figure showing 

mile segment results 

Streamflow _________________ 86.5-5.0 

Percent DO 

saturation at 

boundary point (RM 86.5) _______ do 

Water temperature _____________ do 

Reaeration calculation 

method _________________ 86.5-26.5 

BOD loading ________________ 86.5-5.0 

Rate of carbonaceous 

deoxygenation(kr) ____________ do 

Ammonia-N loading ____________ do. 

Rate of nitrogenous deoxygenation (kn) ____________ do 

Variation in water depth 

owing to backwater or tidal influences _____________ 24.8-5.0 

Benthic demand ______________ 12.8-5.0 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

J24 

Comments 

Model is sensitive to flow, particularly at values less 

than 6,760 ft3/s. At 5,000 ft3/s, predicted percent 

DO saturations are as much as 10 percent less than 

those at 6,760 ft3/s. At 3,260 ft3/s (estimated natural 

low flow for July, 1973), predicted values are as 

much as 30 ~ercent lower than standard conditions. 

At 9,000 ft /s, predicted values are higher by S-8 

percent saturation. 

Model is sensitive to changes in initial percent of DO 

saturation. The major impact is near the boundary 

point; differences between profiles become smaller 

with downstream distance. 

For the reasonably expected range of summertime wa- ter temperatures, the model is insensitive totemperature changes. Maximum predicted deviation 

from standard conditions are ± 3 percent of DO 

saturation. 

Model is sensitive to the method used to calculate 

reaeration. Only the Velz method gave segment-bysegment reaeration inputs which resulted in good 

agreement of predicted and observed DO profiles. 

Model is relatively insensitive to BODult load variations. A doubling of 1974 loads (from each point 

source) results in deviations of 5-9 percent DO 

saturation from the standard profile. Reducing the 

point-source BOD load by 50 percent causes insignificant changes in predicted DO levels. 

Model is relatively insensitive to changes in kr over a three-fold range of 0.02-0.06. Predicted DO concentrations deviate no more than 6 percent saturation 

from standard profile. 

Model is sensitive to variations in ammonia-N loading. A doubling of loads (from outfalls in the nitrifying segment RM 86.5-55) results in as much as a 14 

percent reduction in percent DO saturation values 

from the standard profile. Reducing the ammonia 

loading by 50 percent increases the predicted DO 

values by up to 8 percent saturation. 

Model is insensitive to changes in kn over a range of 

0.5-0.9. Predicted DO concentrations differ from 

standard profile (kn = 0. 7) by less than 3 percent. 

Note that differences decrease with downstream 

distance. 

Model is insensitive to expected range of changes in 

summertime water depth in the Tidal Reach. Predicted DO values differ from standard profile by 

less than 3 percent saturation. 

The model is sensitive to benthic-oxygen demand ex- erted between RM's 12.8-5.2. If the demand is re- moved, the predicted DO value at RM 5.0 is 8 

percent higher than the standard condition. 
REFERENCES CITED 

Bella, D. A., 1969, A further note on verification: In modeling 

the eutrophication process, Proc. workshop sponsored 

by Federal Water Quality Adm. and Univ. Florida, St. 

Petersburg, Fla., November 19-21, 1969, p. 21-25. 

Bella, D. A., and Dobbins, W. E., 1968, Difference modeling 

of stream pollution: Am. Soc. Civil Engineers Proc., 

Jour. Sanitary Eng. Div., v. 94, SA5, p. 995-1016. 

Bennett, J.P., 1971, Convolution approach to the solution 

for dissolved oxygen balance equation in a stream: Water Resources Research, v. 7, no. 3, p. 580-590. 

Bennett, J. P., and Rathbun, R. E., 1972, Reaeration in 

open-channel flow: U.S. Geol. Survey Prof. Paper 737, 

75 p. 

Churchill, M. A., Elmore, H. L., and Buckingham, R. A., 

1962, The prediction of stream reaeration rates: Internat. Jour. Air and Water Pollution, v. 6, p. 

467-504. 

Columbia Basin Interagency Committee, Hydrology Subcommittee, 1963, River mile index Willamette River 

and tributaries: Columbia Basin Interagency Committee, 57 p. 

Gleeson, G. W., 1972, The return of a river, the Willamette 

River Oregon: Advisory Comm. on Environmental Sci. 

and Technology and Water Resources Inst., Oregon 

State Univ., Corvallis, 103 p. 

Harris, D. D., 1968, Travel rates of water for selected streams 

in the Willamette River basin, Oregon: U.S. Geol. Survey Hydrol. Inv. Atlas HA-273. 

Hines, W. G., McKenzie, S. W., Rickert, D. A., and Rinella, 

F. A., 1977, Dissolved-oxygen regimen of the Willamette 

River, Oregon under conditions of basinwide secondary 

treatment: U.S. Geol. Survey Circ. 715-I, 52 p. 

Hines, W. G., Rickert, D. A., McKenzie, S. W., and Bennett, 

J.P., 1975, Formulation and use of practical models for 

river-quality assessment: U.S. Geol. Survey Circ. 715-B, 

13 p. 

Hines, W. G., Rickert, D. A., and McKenzie, S. W., 1976, Hydrologic analysis and river-quality data programs: U.S. 

Geol. Survey Circ. 715-D, 20 p. 

Isaacs, W. P., and Gaudy, A. F., Jr., 1968, Atmospheric oxygenation in a simulated stream: Am. Soc. Civil Engineers Proc., Jour. Sanitary Eng. Div., v. 94, SA2, p. 

319-344. 

Rickert, D. A., Hines, W. G., and McKenzie, S. W., 1975, 

Planning implications of dissolved-oxygen depletion in 

the Willamette River, Oregon, in Whipple, William, Jr., 

ed., Urbanization and water-quality control: Minneapolis, Am. Water Resources Assoc., p. 70-84. 

Rickert, D. A., Hines, W. G., and McKenzie, S. W., 1976, 

Project development and data programs for assessing 

the quality of the Willamette River, Oregon: U.S. Geol. 

Survey Circ. 715-C, 31 p. 

Streeter, H. W ., and Phelps, E. B., 1925, A study of the pollution and natural purification of the Ohio River: U.S. 

Public Health Service, Public Health Bull. 146, 75 p. 

Velz, C. J ., 1970, Applied stream sanitation: New York, John 

Wiley and Sons, 619 p. 

J25 
EXPLANATION OF WIRQAS 

MODEL PRINTOUT 

Table 5 is an example of the computer printout of the WIRQAS DO Model. The illustrated 

printout includes 29 columns of information that 

were used to calibrate the model at the first five 

river cross sections between RM's 86.5 and 84.0. 

Columns 1 through 11 together with input coefficients for kr and kn provide the information necessary to initiate and drive the model. Columns 

12-16 summarize model calculations of carbonaceous deoxygenation, whereas columns 17-21 

summarize similar calculations for nitrification. 

Columns 22-29 complete the printout by listing 

reaeration calculations and a summary of DO 

gains and losses. 

To aid the reader, column 1, which shows the 

section boundary stations, has been included at 

the left of each part of the table. The reader 

should note that the first seven columns of the 

printout includes double spacing for each station, whereas the rest of the columns have single 

spacing. The double spacing is necessitated by 

the river segment averaging calculations listed in 

columns 2, 5, 6, and 7. 

The model begins at RM 86.5 and the startup 

is handled by treating the conditions at this station as those from an inflowing tributary. Values 

in each column are exactly as observed on the 

original printout sheets; there has been no 

rounding to selected significant figures. 

A complete explanation of each column is presented on the following pages. 

Column 1. STATION is the station location in 

river miles as measured from the mouth. These 

values correspond with the Willamette River 

Mile Index as established by the Hydrology 

Subcommittee, Columbia Basin Interagency 

Committee (June, 1963). 

Column 2. WATER TEMP is water temperature 

in degrees Celsius. The temperature values are 

used in adjusting self-purification process rate 

coefficients. Note that the model calculates 

and uses an average water temperature for 

each river segment. 

Column 3. TIME PASS is the cumulative timeof-passage in days from the first station to the 

downstream station of interest. Time-of-pasTABLE 5.-Example of the computer printout of the WIRQAS DO model 

1 2 3 4 5 6 7 8 9 10 11 

STATION WATER TIME TRIB EFF RIV AREA IM DIS BOD SLUDGE NBOD 

TEMP PASS DIS DEP VOL DEM 

86.50 20.00 0.0 6,760 5.27 2,400 0 363,912 0 105,000 

20.00 6.85 66 2,400 

85.80 20.00 0.01519 0 8.42 2,400 0 0 0 0 

20.00 7.45 57 2,400 

85.20 20.00 0.02821 0 6.49 2,400 0 0 0 0 

20.00 6.18 19 2,375 

85.00 20.00 0.03250 25 5.88 2,350 0 54,400 0 388,000 

20.00 5.09 93 2,350 

84.00 20.00 0.05367 0 4.31 2,350 0 0 0 0 

12 13 14 15 16 17 18 19 20 21 

STATION k, COD DIS COD TOT COD TCOD kn NOD DIS NOD TOT NOD TNOD 

RES US SAST RES US SAST 

86.50 0.0 363,912 0 363,912 0 0.0 105,000 0 105,000 0 

85.80 0.0600 0 363,149 363,149 763 0.7000 0 102,461 102,461 2,539 

85.20 0.0600 0 362,497 362,497 1,415 0.7000 0 100,333 100,333 4,667 

85.00 0.0600 54,400 362,282 416,682 1,630 0.7000 388,000 99,641 487,641 5,359 

84.00 0.0600 0 415,465 415,465 2,847 0.7000 0 471,286 471,286 21,715 

1 22 23 24 25 26 27 28 29 

STATION TOT TOTRAST PEAT SAT NET REAERATION DOBAL PCT MGILCONC 

SAST SAT 

86.50 0 1,282,114 1,393,603 1,282,114 0 1,282,114 92.00 8.45 

85.80 3,302 1,282,114 1,393,603 1,278,812 470 1,279,281 91.80 8.44 

85.20 6,082 1,282,114 1,393,603 1,276,032 367 1,276,868 91.62 8.42 

85.00 6,989 1,282,627 1,398,736 1,275,637 161 1,276,635 91.27 8.39 

84.00 24,562 1,282,627 1,398,736 1,258,065 1,133 1,260,196 90.10 8.28 

J26 
sage between two successive stations is calculated by the equation: 

. 0.0611(A) (x) (8) Time of passage (days) = Q , 

where 

A = cross-sectional area (ft2) 

x = length of segment (mi) 

Q = discharge (ft3/s) 

Column 4. TRIB DIS is a listing of tributary and 

waste-water discharges (ft3/s) into Willamette 

River. The model can use discharge values to 

the nearest 0.01 ft3/s, whereas the printout values in column 4 are rounded to the nearest 

whole ft3/s. The value shown, for station 86.50 

is the inflowing discharge of the mainstem 

Willamette River (6,760 ft3/s). The model uses 

the discharge at Salem as the index flow and 

routes flow downstream accounting for inflqwing tributary and waste-water discharges. 

Column 5. EFF DEP is the effective depth (ft) at 

each cross section. Effective depth is calculated by dividing the cross-sectional area, column 

7, by the water-surface width (required as input to the program but not printed). Note that 

the model calculates and uses an average effective depth for each river segment. 

Column 6. RIV VOL is the volume of water between stations, in millions of gallons. The river 

volume may be calculated from 

RIV VOL = 0.0394(A)(x) (9) 

where A and x are as defined for equation 8. 

Note that the model calculates and uses an 

average volume for each river segment. 

Column 7. AREA is the cross-sectional area in 

ft2. Areas were determined by field measurement or from existing channel maps provided 

by the U.S. Army Corps of Engineers and the 

U.S. Coast Guard. Note that the model calculates and uses an average area for each river 

segment. 

Column 8. IM DEM provides for a listing for inflowing loads of "immediate" oxygen demand 

(see Velz, 1970) in units of population equivalents. One population equivalent (PE) is defined as 

PE = BODult(lb/d) 

0.24 (10) 

No immediate demands occur in the modeled· 

river segment. 

Column 9. DIS BOD is the load, in PE, of the 

ultimate carbonaceous biochemical-oxygen 

demand for each inflowing tributary or wastewater discharge. For station 86.50, the tributary discharge is equal to the Willamette inriver load of 363,912 PE. This value is calculated from 

PE =(discharge) (BODult concentration) 

(conversion factor) (

34 7) 

= (6,760 ft3/s) (2.4 mg/L) (1.547) (11) 

= 363,912 

Column 10. SLUDGE is the oxygen demand, in 

PE's, at specific stations due to benthic deposits. Benthic demand was found to be significant only in the Portland Harbor area (not 

included in table 5). 

Column 11. NBOD is the inflowing nitrogenous 

oxygen demand, in PE's, from each tributary 

or waste-water discharge. Nitrification occurs 

only between stations 86.50 and 55.20. The 

oxygen demand of ammonia is calculated as 

follows: 

PE = (unit 0 2 demand of NH4-N) (NH4-N 

concentration) (discharge) 

(conversion factor) (12) 

For example, at station 85.00, the NBOD is 

as follows: 

NBOD(PE) = (4.33)(160 mg/L)(25.0 ft3/s) (i~~;~) 

= 388,000 

Column 12. kr is the temperature corrected 

in-river rate of carbonaceous deoxygenation 

(day-1.) Values of kr at 20°C are input to the 

model and corrected for actual water temperature by the expression: 

where Tis the average temperature between 

stations as listed in column 2. 

Column 13. COD DIS is the inflowing load of 

BODult from each tributary or waste-water 

discharge, in PE's. The river was sectioned. so 

there would be no more than one inflow1ng 

load for each designated segment. COD DIS is 

J27 
calculated by adding the values for IM DEM 

(column 8) and DIS BOD (column 9). 

Column 14. COD RESUS is the residual BODult 

load, in PE's, remaining at each station. COD 

RES US is equal to the BODult load remaining 

after the total load at the upstream station 

(TOT COD, column 15) has been satisfied at 

the specified deoxygenation rate (kr column 

12) for the incremental time-of-passage between successive stations (calculated from column 3). For further details, see Circular 715-1, 

p. 40. 

Column 15. TOT COD is the total BODult load, 

in PE's, remaining at each station. TOT COD 

is equal to the residual load remaining in the · 

river (COD RESUS, column 14) plus the inflowing load (COD DIS, column 13). 

Column 16. T COD SAST is the cumulative load 

of satisfied BODult , in PE's. 

Column 17. kn is the in-river rate of nitrification 

( day-1) at 20° C. Significant nitrification· occurs only between stations 86.50 and 55.20. 

During calibration and verification conditions, 

the average daily water temperatures in this 

river segment were essentially 20°C; therefore, 

no temperature corrections were made for kn. 

Column 18. NOD DIS is the inflowing nitrogenous-oxygen demand, in PE's, from each tributary or waste-water discharge. This column 

is identical to column 11 (NBOD). The data 

are repeated here so all computations related 

to nitrogenous demand appear as a unit in the 

printout (columns 17-21). . 

Column 19. NOD RESUS is the residual nitrogenous-oxygen demand, in PE's, remaining at 

each station. NOD RESUS is equal to the nitrogenous-oxygen demand remaining after the 

total demand at the upstream station (TOT 

NOD, column 20) has been satisfied at the 

specified nitrification rate (kn,column 17) for 

the incremental time-of-passage between 

successive stations (calculated from column 3). 

For further details, see Circular 715-1, p. 42. 

Colu~n 20. TOT NOD is the total nitrogenousoxygen demand, in PE's, remaining at each 

station. TOT NOD is equal to the residual load 

remaining in the river (NOD RESUS, column 

19) plus the inflowing load (NOD DIS, column 

18). 

Column 21. T NOD SAST is the cumulative load 

of satisfied nitrogenous-oxygen demand, in 

PE's. 

Column 22. TOT SAST is the cumulative total 

satisfied oxygen demand, BODult and nitrogenous, in PE's. For a specific station, TOT 

SAST is the sum ofT COD SAST (column 16) 

plus T NOD SAST (column 21). 

Column 23. TOT RAST is the cumulative total 

DO, in PE's, resulting from inflowing tributaries and waste-water discharges. 

Column 24. PE AT SAT is the amount of DO, in 

PE's, that would be in the river if the water 

were oxygen saturated at the temperature listed in column 2. 

Column 25. NET is the DO, in PE's, remaining 

after the BODult and nitrogenous demands 

have been subtracted. NET equals TOT 

RAST (column 23) minus TOT SAST (column 

22). 

Column 26. REAERATION is the DO added, in 

PE's, due to atmospheric reaeration within 

each river segment. 

Column 27. DO BAL is the amount of oxygen, in 

PE's, in the river after reaeration has been 

added. DO BAL equals NET (column 25) plus 

the cumulative reaeration which is found by 

summing successive values of REAERA TION 

(column 26). 

Column 28. PCT SAT is the DO concentration in 

percent saturation. PCT SAT equals DO BAL 

(column 27) divided by PEAT SAT (column 

24) times 100. 

Column 29. MG/L CONC is DO concentration in 

mg/L. MG/L CONC is calculated as follows: 

Do (DO BAL (PE) • mg/L (d" h ) (converSIOn factor){14) 1sc arge 

For example, at station 86.50 the DO in mg/L 

is as follows: 

DO _ (1,282,114) (1.547) 

mg/L - (6760) (34.7) 

= 8.45 

J28