MATHEMATICAL MODEL
FOR DETROIT RIVER
USER MANUAL
ENVIRONMENTAL CONTROL TECHNOLOGY CORPORATION
ANN ARBOR, MICHIGAN
-------
MATHEMATICAL MODEL
FOR DETROIT RIVER
USER MANUAL
BY
Environmental Control Technology Corporation
3983 Research Park Dr.
Ann Arbor, Michigan 48104
Project Officer
Mr. Howard Zar
Region V
U.S. Environmental Protection Agency
-------
TABLE OF CONTENTS
Page
INTRODUCTION ]
MODELING THEORY 2
MODEL DEVELOPMENT H
PROGRAM IMPLEMENTATION 15
EXAMPLE 21
ADDITIONAL INFORMATION 30
REFERENCES 31
APPENDIX A . 32
Driving Program - SSMPX
APPENDIX B 35
Linear System - Flow 200,000 cfs
APPENDIX C 60
Linear System Flow 175,000 cfs
APPENDIX D 85
Subroutine SLE2
-------
INTRODUCTION
A steady state mathematical model was developed for the
Detroit River from its head at Lake St. Glair to its mouth
at Lake Erie. The basic computer program used for this study
is the Steady State Modeling Program (SSMP) developed by
Canale and Nachiappan . The model discussed herein was
developed as part of a contract with the U. S. Environmental
Protection Agency under contract number 68-01-1570,
This manual is a supplement to the final report submitted
under contract 68-01-1570, entitled "Water Quality Investi-
gations of the Detroit and St. Glair Rivers", and is intended
to serve as a basic guide for computer application of the
Detroit River Model. Discussions of modeling theory,
program input requirements, and a listing of the driving
program and basic input deck are included in.this report.
Discussion of the verification procedure, and results of
numerous simulations are not included in this manual. The
user should obtain a copy of the main report in order to
understand the step by step procedure which was utilized for
verification of the model for parameters such as chloride,
phenol, iron, ammonia nitrogen and phosphorus. It should be
noted at this point that while the model presented here is a
complete model for the river, it has not been verified for
the Canadian side of the river. Thus results from the model
can only be considered reliable for segments located in the
United States waters. It is hoped that the necessary
Canadian information can be obtained and incorporated into
the model in the near future.
-------
Modeling Theory
A general methodogy for modeling biological production in
aquatic systems has been described in an earlier report by
Canale (1970). This work emphasized the transient behavior
of relatively complex ecosystems in single homogeneous
zones. The ultimate goal of the project was to suggest how
the annual cylce of phytoplankton 'and nutrient behavior
might be simulated mathematically. The report described the
complex kinetics of the system as characterized by nonlinear
reaction terms and time-variable rate coefficients.
A relatively simpler class of problems is also of interest.
These problems are concerned with the steady state distri-
bution of species whose decay or production'tendency can be
decribed by first order kinetic formulations .' The following
list- of water quality variables has been traditionally
analyzed using such assumptions:
1) Total dissolved solids or conductivity
2) Chlorides
3) Any Conservative Chemical Species
4) Total Coliform Bacteria
5) Dissolved Oxygen
6) Biological Oxygen Demand
7) Radioactive Isotopes
8) Nitrogen (Nitrification Process)
This report describes a general user-oriented program capable
of calculating the three-dimensional steady state distribu-
tion of the above water quality variables in aquatic systems.
Mathematical models which can be useful for informed manage-
ment of water resources must be based on the diverse
chemical, physical, and biological mechanisms active in the
system. These mechanisms are recognized by appropriate
terms in equations of continuity for each chemical or
-------
biological element of interest, Essentially, two distinct
types of mechanisms are recognized. First, are those pro-
cesses xtfhich alter the concentration of material within a
closed system due to departures of the state from a chemi-
cal equilibrium state. The study of the rate of changes
toward or away from this equilibrium state is called kine-
tics. The kinetic expressions may be dependent on species
concentration, temperature, light intensity, and pHt
A second process which can bring about changes in the con-
centration of species results from the mechanical action of
the fluid circulation and the subsequent dilution of con-
centration gradients. The bulk behavior of the circulation
Is characterized by gross convective transfer, while the
random small-scale fluid movement is accounted for by
dispersion coefficients and transfer due to concentration ~
gradients alone. The above ideas are summarized by Equation
1'which is a descriptive statement of the continuity law
for any material. Equation 2'expresses this same law in
mathematical form for a general three-dimensional system.
A direct solution of Equation 2"for natural systems is not
possible. Therefore in practice it is necessary to use
approximations which are equivalent to considering a con-
tinuous body of water as a series of finite interconnected
segments as shown in Figure 1' In this case, the steady-
state continuity equation with first order kinetics reduces
to the following;
«» 'C _/>__ f, r _.f\ f .. fl J_ r> <"
where:
C, = concentration of water quality variable in segment
k, (nig/1)
V, - volume of segment k, (eft)
-------
Rate of Change of
i with time within
a cell :
Rate of Input of
i by advection,
dispersion, sed-
imentation, or
migration from
adjacent cells
Rate of Output of
i by advection,
dispersion, sedi- +
mentation, or
migration from
adjacent cells
Rate of Production
of i by growth,
excretion, or
dissolution within
cell
Rate of disap-
pearance of i by
uptake, predatioi
respiration, deal
or precipitation
Composition Change
Hydrodynamic Mechanism
Reaction Mechanism
V-(EVC)-V-CUC)
ACCUMULATION
DISPERSION BULK FLOW
REACTION
-------
Figure 1 - Uniform Rectangular Volume
i f.
ko
C,
k4
'k2
Jk3
'k5
'kl
-------
Q, . - net flow from segment k to segment j (positive:
^ ward) (cubic feet per second)
a, . = finite difference weight given by ratio of flow
.*' to dispersion, 0
-------
akk - '
-------
obtain the distribution of BOD and then use these results
in a continuity equation for D.O.
Thus, the mass balance equation for D,0, Is:
(10')
v?here C , is the saturation value of D.O. , K . is the reaera-
tion coefficient in segment k, K,, is the deoxygenation
coefficient, L is the biochemical oxygen demand and +W is
now interpreted as sources and sinks of D,0, such as benthal
demands and photosynthetic prodrt tion or respiration. With
L, known from previous calculations, the final solution of
Equation 10' is similar to solution of Equation 31?,
As spatial approximations to derivatives have .been used in
Equations 3" and 10', some error are introduced into the' .
analysis. One of the errors is "psuedo or numerical disper-
sion." It appears due to the assumption of completely mixed
finite volumes. Numerical dispersion is defined by Equation
11-.
' "'
kj
When ak. = 1/2, E is zero.
On the other hand, it can be shown that for a positive solu-
tion the terms off the main diagonal in the left-hand side of
Equation 9 should be non-positive. This condition is
satisfied if '
Writing Equation 12 in another way, it is seen that Lj . must
be chosen such that,
EA
-------
For the case of zero numerical dispersion a, . = 1/2 and
KJ
2E..A, .
Lkj< kj kj
'If a, . is set equal to 1/2 and L, . is chosen in such a way
as to satisfy Equation 14', then it may be necessary to
handle many segments. However, if a, . differs very much
from 1/2, then numerical dispersion would be high. Further,
making a, . = 1/2 does not imply the best solution for the
case of unequal-sized segments.
In SSMP a, . is first set equal to
akj ~ ]
In other words, the segment whose center is nearer to the
interface would have more weightage in determining the con-
centration at the interface in Equation 3"t The value of
a, . is then, checked against Equation 12", If it is not
satisfied, then
F"
cu. is made equal to 1 -' k'j
2
Choosing proper spatial grids for approximations to the
differential equations is still very much an art, The more
numerous the segments 5.n a model, the more accurate the
resulting solution. However, in such cases the computer
costs may be very high, so a compromise is necessary, Con-
siderations of computer size, nature of problems, degree of
accuracy, simplicity of the resulting finite difference
equations, and availability of verifying field data all in-
-------
fluence the choice, For additional details concerning these
questions the reader is referred to Thomann (1971) ,
10
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MODEL DEVELOPMENT
Utilizing the basic theory outlined in the previous section,
a specific model was developed for the Detroit River. The
river was divided into 73 segments. The resulting system of
coupled segments is represented in Figures 2 and 3, Size,
number, and placement of these segments was based on an exami-
nation of available water quality data, location of major
wastewater inputs, and on the flow pattern of the river.
In analyzing this information, it was found that the Upper
Detroit River (Peach Island to Zug Island) was fairly uni-
form in the concentration of various pertinent, parameters and
contained few waste inputs. Therefore, large segments were
used in this section of the river with generally two segments
across the river and each segment approximately two miles in.
length. In the lower river (below Zug Island) the characteristic:
changed considerably. There are many waste inputs along the
banks of the river and large concentration gradients wero. found
between the U. S. shore and Grosse lie. This portion of the
river is only 1000-1500 feet wide, However, because of the
large concentration gradients, the river width was divided
into four segments. The segments were also much shorter than
the upper segements due to the large number of waste inputs.
This more detailed characterization was followed throughout
the Trenton Channel all the way to Lake Erie. The rest of the
river was divided using similar considerations as discussed
above (concentration gradients, waste inputs, flow routing).
Islands located in the river were handled by starting new
segments on each side of the Island and splitting the flow
according to the available flow routing information. A few
of the very small islands were incorporated within one segment.
Several segments in the river were designed to contain water-
sediment boundary conditions. The choice of these segments
was based on general characteristics of the river (depth,
water velocity, proximity to waste inputs, etc.) and on infor-
mation gained during the core sampling portion of the survey
11
-------
Figure n Model Se~n-.cnt-irior,
Upper Detroit River
LAKE ST. CLAIH
MICHIGAN
Note: Not Drawn to Scale
Approx. only
1
-------
Figure
. ' icion
Lower Dcciroii: River
f.COI'Si: KIVER
28-30
37-40
MONT-UAGON CRD'.'K
Four Scsxme
Across Channel
,;.!
Note; Not Drawn to Scr.lc
Approx. only
'67 68 69 70 71 72
,.' I .
cvi- M;
^Vi^
1.3
-------
program. The cross sectional area of the water-sediment
interface was calculated for these designated segments and
incorporated with the physical description in the model.
The schematization was prepared using U. S. Department of
Commerce, National Ocean Survey, Lake Survey Center, naviga-
tion chart No. 400, scale 1:15,000. These maps were used
to determine the cell widths, characteristic lengths,
depths, and volumes. Flow routing -for the river was obtained
from the U. S. Public Health Service report
The flow rate in the Detroit River is exceptionally steady.
The average discharge for the period of 1936 through 1973 was
approximately 185,000 cfs. The average flow during 1962
through 1964 was 170,000 cfs. Flow rates have been somewhat
higher in the last few years averaging 200,000 to 220,000
cfs."
Dispersion
Specific dye tracer studies were not part of this project and
consequently it was not possible to calculate dispersion coeffi-
cients from such data. However, the Public Health Service
report contained some dye tracer information on a qualitative
basis, and also included maps indicating zones of pollution
and the paths they traveled. Ex .ination of river water
quality information gathered since the Public Health report
supported the various zones and pathways which had been defined
earlier. This information, coupled with the. large flow rates
and swift velocities characteristic of the river, indicated
that advective (bulk) flow was the major type of mass transfer
in the river.
Several dispersion coefficients were tried ranging from .01 to
.25 sq. mile/day. A coefficient of .05 sq, miles/day for
lateral dispersion and a coefficient of .10 for longitudinal
dispersion gave good results. These values were tested using
chloride data for 1968 and 1969.
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PROGRAM IMPLEMENTATION
Once the testing of the physical characteristics and dispersion
coefficients was accomplished, the linear system for the. Detroit
Model was generated. Two versions of this systera are presented
in Appendix B and C. These systems are both based on the 73
segment representation and on dispersion coefficients of 0,05
sq. miles/day for lateral dispersion and 0.10 sq. miles/day
for longitudinal dispersion. The only difference between these
two versions is the flow rate. System A is based on a flow
rate of 200,000 cfs and systera B on 175,000 cfs.
The drive program SSMPX (Appendix A) is used to read the linear
system and a data file supplied by the user. The user's data
file contains additional information including reaction coeffi-
cients, boundary conditions input loadings and appropriate scale
factors.
The segments which require boundary conditions are listed, in
Table 1. With the present linear system only segments 1, 2,
and 67-73 will affect the model. The other segments are
boundary conditions to be used with the sediment-water interface
and are presently set equal to zero. They are included so that
in the future, the option of using sediment-Wilted interchange
data can be included in the model. The basic linear system
is designed at this time with the dispersion coefficient for
these segments set equal to zero. Thus the computation for
these boundaries will have no effect in model results.
The boundary conditions for segments 1 and 2 are the incoming
conditions for the model. These boundary concentrations should
be set equal to the incoming water quality levels. Segments
67-73 serve as the ending conditions and ?:eflect the water
quality of Lake Erie at the mouth of the river.
The model can be used with parameters that follow first order
reaction kinetics, or with coupled systems such as BOD-DO.
-------
TABLE 1
BOUNDARY SEGMENTS
Incoming Water Quality Boundaries
Segments 1 2
Bottom Sediment - Water Interface Boundaries
Segments
3
15
21
26
35
64
4
16
22
29
40
11
19
23
30
43
12
20
24
33
63
Ending Water Quality Boundaries Interface at Lake Erie
Segments 67 68 69 70
71 72 73
16
-------
This is accomplished by entering the appropriate first order
reaction coefficient. If the parameter is a conservative
substance such as chloride, the value of the reaction coeffi-
cients (K) is set equal to zero, For a coupled system
additional first order reaction coefficients (KA, KD) are used
along with a saturation concentration CCSJ, Thus to use the
BOD-DO system, four variables are required, a first order reaction
coefficient, a rearatlon coefficient, a dcoxygenation coeffi-
cient and the saturated dissolved oxygen concentration at the
temperature used in the run.
The matrices presented in Appendix B and C are for single system
parameters. The BOD-DO system was not modeled in this study.
A coupled system matrix can be generated in the future should
the need arise. Thus, for these systems, only a reaction
coefficient K is required for the input data file.
Loads can be entered into the model for any segment. This is
indicated by the appropriate segment-load pairs used in the
data file. It is important to note that the number of loadings
must be indicated in card 3 and that each segment-load pair
must be on a separate card.
An example run is provided at the end of this manual to illustrate
the use of the various user supplied options,
SSMPX is the driving program listed in Appendix A, This program
reads the appropriate information, performs necessary calcula-
lations and writes the results. The program at present: is
written for the Michigan TS and reads from two devices. Device
4 reads the linear system (see Appendix B and C) and Device 5
reads the user supplied data file. All results are written on
device 6.
17
-------
A library subroutine called SLEl is called by the main program,
This subroutine is part of the MTS library and an appropriate
substitute will be needed if the model is used on a different
computing system. A description of the method used by SLEl is
given in Appendix D.
18
-------
uata
Set
No.
1
2
3
4
5
Loltrnsns
on Card
1-5
1-20
1-5
6-10
11-15
16-25
26-35
36-45
1-10
(NOTE: The
11-20
21-30
31-40
1-6
7-19
Variable
Name
NP
TITLE
NL
NB
KEY
SCALE (1)
SCALE (2)
SCALE (3.)
X
following
KA
KD
CS
ISEG
LOAD
Input
Format
I
(Integer)
A
(Literal)
I
I
I
F
(Real)
F
F
F
values on this
. F.
F
F
I
G
Description of Input
Number of water quality parameters to be
rt*
Verbal description of water quality paramet,
Number of loadings for the current run.
Number of boundary conditions for the
current run.
A switch which indicates the parameter be
run is dissolved oxygen when KEYrO.
Scale factor for boundary concentrations .
Scale factor for loadings.
Scale factor for final concentration.
First order reaction coefficient.
card.-g.re included only if KEY^O)_
First order reaeration coefficient
First order deoxygenation coefficient
Saturated dissolved oxygen concentration
the temperature used in the run.
Number of segment receiving loading.
Load in p.
in
at
(Real B- or
F-type)
Include one card typo f> for ooch of t.1>a NT, 1 nii
-------
Data Columns Variable Input Description of Input
Set on Card Name Format
No.
1_6 ISEG I Number of segment
7_19 BC G Boundary concentration for that segment
Include one card type 6 for each of the NB boundary concentrations
Data cards 2 through 6 are to be repeated for each of. the NP water quality parameters
to be run.
to
O
-------
EXAMPLE
This example is an actual model run using chloride data avail-
able from the summer of 1972. Loading information was obtained
from monitoring programs under the direction of the Michigan
Water Resources Commission. Boundary conditions x^ere obtained
from river survey data taken by the Michigan WRC. The first
order reaction coefficient was set equal to zero as chloride
is considred a conservative substance. A listing of the neces-
sary data is given in Table 2 and a computer data file listing
of the actual deck is shown in Table 3.
The results of the computer output for all segments is repro-
duced in Table 4. In order to compare this data with survey
data, as was done during the verification, it is necessary to
calculate average concentrations for a segment based on the
point data available from the field surveys, This average can
then be compared with the model output which is the steady
state concentration for the cell. The results of the comparison
obtained is shox-m in Table 5 and Figures 4 and 5.
For the case of projecting future water quality, the user can
enter the appropriate loads; boundary conditions, etc., and
the results will give the anticipated concentrations under the
specified conditions. For additional examples the user is
referred to the main report. This report contains the results
of several simulations under varying conditions, given both
verification comparisons and projections based on anticipated
future loadings.
21
-------
TABLE 2
USER SUPPLIED DATA
Kinetics:
Chloride conservative K = 0.0
Boundaries:
Chloride concentration headwaters
8.0 mg/1 ave. Segments 1, 2
Chloride concentration Lake Erie Interface
28 mg/1 ave. Segment 67
23 mg/1 ave. Segment 68
15 mg/1 ave. Segment 69
10 mg/1 ave. Segment 70
10 mg/1 ave, Segment 71
18 mg/1 ave. Segment 72
37 mg/1 ave. Segment 73 .
Loads: 1000 of -///day Segment
Detroit Waste Treatment Plant 1180 15
Rouge River 120 15
Wyandotte Chemical - North 30 31
Wyandotte Chemical - South 100 37
Pennwalt Chemical 230 37
Wayne County Waste Treatment Plant 90 37
Scale Factors:
Boundary concentrations 0.623E-04
Loads 1000.0
Final Concentration . 161E+05
22
-------
> 1
> 2
..> 3
> 5
.> 6
> 7
> 8
> 9
> 10
> 11
> 12
> 13
>* 14
> IS
> 16
> 17
..> 18
> 19
> 20
>---- 21
> 22
> 23
> 24
> _ 25
> 26
> 27
..> 28
> 29
> 30
> 31
> 32
> 33
> 34
> 35
> 36
> 37
#ENO OF
ft
l
CHI. on
3
15
31
37
1
?.
3
4
11
12
15
16
IS
20
21
22
23
24
26
29
30
33
35
40
43
63
64
67
68
69
70
71
72
73
FILE
EXAMPLE - USER'S DATA FILE
)E 1972
30 0 «00006230 1000.C 16051.0
30.0
400*0
8*0
0»0
0.0
0.0
0«3
0.0
0*0
0.0
090
0.0
0.0
0»0
0.0
0*9
0*3
28«0
1000
18*0
23
-------
TABLE 4
EXAMPLE - OUTPUT CHLORIDE 1972
'UN NUf-'RFI 1:
SCALE FACTIHS:
a:.= 0.623E-0't
E 1972
LOAO= 1000.
CONCENTRATIONS^ C.161E 03
K= 0.0
15
31
37
SEGMENT
I
2
3
4
11
12
15
16
19
20
21
22
23
24
26
29
3D
33
35
^ 0
43
63
6^
67
53
69.
70
71
72
73
LOAD
0.1300 04
30.0
400.
B:)US'r)ARY CDMOITION'
R.OO
8,00
0.0
" 0.0
0.0
0.0
0.0
0.0
0.0
0.0"
0.0
' 0.0
0.0
0.0
0.0
0.0
0.0
0.0 -
0.0
0.0 -
' 0.0
0.0
0.0
23.0
23.0
15.0
. 10. 0
10.0
18.0
37.0
CHI.
ON:
O'UOE
rNTRA1
1
2
3
4
5
6
7
8
9
10
11
12
1972
m^s IN
7.
7.
7.
7.
7.
7.
7.
7.
7.
7.
8.
8.
ALL SEGMENTS:
999797
000808
999001
999B08
999801
199808
999801
990308
999006
999008
007073
000126
24
-------
TABLE 4 (coat.)
EXAMPLE - OUTPUT CHLORIDE 1972
13 7.999816
14 7.999003
1ft R. 095170
17 8.001443
18 7.99984D
IP 12.630595
20 B. 105992
21 8.002334
22 7. 999889
23 12.590081
24 10.470796
25 0.835679
26 8.007062
27 7.999889
28 12.532568
29 13.5C9169
3^ in. 470434
31 12.72T573
3? 10.563371
33 8.851317
34 8.834512
35 0.013762
36 7.999889
37 21.011988
38 12.0P2831
39 10. 595430
40 10.563878
41 8.833024
42 8.559042
43 9.002776
44 7. 999896
45 . 20.385526
'tf. 12.251011
47 10. 723785
48 10.567742
49 19.954847
50 12.365527
51 10.803593
52 10.575997
53 8.R26300
54 8.559315
55 8.420355
56 19.701094
57 12.437676
58 10.844643
59 10.585692
60 19.445501
61 12.529932
62 11.958186
63 10.625573
64 8.821658
65 8.540020
66 8.420455
67 19,300799
68 17.577567
S9 12.605703
70 11.118909
71 8.837070
72 8.498783
-------
12
i
o
Figure 4
MODEL VERIFICATION - Choride
Detroit River Dt 20.6 and 19.0 1972
Dt 20.6
U-
100
400
800
15
Dt 19.0
10
U
100
400
800
T2W
-------
15
FIGURE "5
MODEL VERIFICATION - Chloride
Detroit River Dt 14.6W and 12.OW - 1972
10 -
Dt 14, 6W
o
E
5 -
0
100
500
1000
1500
2000
24
Dt 12,OW
16
too
B
0
100
500
1000
-------
30
MODEL VERIFICATION - Chloride
Detroit River Dt 8.7W and 3.9 - 1972
20
Dt 8.7W
CJ
15
0
100
500
1000
1500
50
40
Dt 3.9
30
20
~
10
0
1000
L_
5000
_J
10000
9 9
_J
15000
20000
-------
Chloride Concentrations
DETROIT RIVER - 1972
Point Model Segment
Numbor3
20.6 9,11
r/.4 W . 15-17
U.6 W 31-33
12.0 V 37-39 '
8-7 W 56-59
3-9 67-70
3-9 (cont.) 71-73
AMC
MPC
AMC .
MPC
AMC
MFC
AMC
MFC
AMC
MPC
AMC
MPC
AMC
MPC
8
8
13
13
% 13
13
20
21
20
20
25+
19
9
9
8
8
8
8
11
11
12
12
13
12
22
18
17
16
8
8
8
8
9
9
10
11
11
11
15 .
13
30+
26
10
11
values as mg/1
- average measured concentration based on stations located
within corresponding model segment
t .
- model predicted concentration
29
-------
ADDITIONAL INFORMATION OR SYSTEMS
Additional information on this modeling program can be
obtained by contacting our office in Ann Arbor, Michigan..
Additional linear systems can also be obtained from our
office. Generation of matrices for different flow rates,
other than the two presented here, is relatively easy
and can be provided via a listing, punched cards or
magnetic tape. The charge for this service will be. based
in the time required to produce and process the required
systems and a quote can be obtained by contacting the
office.
30
-------
REFERENCES
1. Canale, R. P., and Nachiappan, S., "Steady State Modeling
Program", University of Michigan, Sea Grants Program, Tech.
Report No. 27, March, 1972.
2. Canale, R. P., and Squire, J. R., "An Application Manual for
SSMP", Dept. of Civil Engineering, University of Michigan,
September, 1974.
3. Thomnnn, R. V., "Time Variable Water Quality Models -
Estuaries, Harbors, and Off Shore Waters", from Advanced
Topics in Mathematical Modeling of Natural Systems, 16th
Summer Institute in Water Pollution Control (1971).
4. Thoraann, R. V. , Systems Analysis and Water Quality Management,
Environmental Science Services Division, Environmental
Research and Applications, Inc., New York, NY, 1972.
5. Canale, R.P., "A Methodology for Mathematical Modeling of
Biological Production", Report to the Univ. of Michigan
Sea Grant Program (1970).
31
-------
APPENDIX A
DRIVING PROGRAM - SSMPX
32
-------
F;9: ONE
MTS : /VNN ARBOR ( CCOl-015'j)...
rY'^EzTERMINALr PRIO = K!ORMALf CLASLr'JNI V/GO\/T
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" BGH" SIGNED ON AT 14:39.41 OH THU MOV 21/74
* OK
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AHO COMMUTE THE STEAOY STATE CONCENTRATIONS OP ONE OR
MORE WATER QUALITY PARAMETERS. ..._....._ . _. ..-_
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00 2 1=1 »M
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CONTINUE
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RFAO(5»4) NP ......:. _ - .. - - - -
00 31 Itrl'NP
00 5 I = lrN - --.. - - - - -
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IF"INL.NE.O) GO TO 15
"JRITE(6» 14) . . .- - --- -- _....._ -. -
GO TO 18
QO I/ 1-1 f l< . .. . .- * "" *"**
RFAO(5rl6) J.LOAO(J) .
WRITF(6»16) JrLOAO(J).,,. . .. - - ~- - - --
CONTINUE
00 59 Isl.N -- -- --
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GO TO 25
no or T - 1 Mf* -
WRITE (r>»?3) J»*C ._ -- - - '- -
R-( J):RC( J) *^':+SCALF (1)
DO 26 1^1 »N - - - -~ '
B(I» I);^(IrM-*KtV(I)
RC.(I_)rRC( i)+LOAO( I)*SCALF(?)
33
-------
> V)3 00 21 I'-'l , N .
> >'» 'H 1 , / ) r^ ( I . .f ) *- V( I ) + (KA-K)
> b , . ?7 SC{ I) rRC( I) +V( I )+KAT ?8 CALL 5LE1 ( N» 1 On t<3, 11 l fin t
- > ..- "J7 . ._ 00 ?° 1 = 1 tN . ... ....
> '^ ?9 C ( I) rC ( I ) *'-V:Al.r (3)
> '^ ^RITE(6r3H) HTLFr(.I»C(I
> 60 CSCALP:SCALE(3)
> 61 31 CONTINUE
> 62 C
> 63 c FORMATS FOR IN^JT AND OUTPUT
> 6<* i FORMAT (40?n. i ri)
> 6', i* f'OC _ '67 6 F'JRN'Af <5A<*)
> 60' " 7 POi^'-'AT(3
> 69 8 FORMAT < M'TJ'-'I i."J'.'.H(it? '» .
> 70 "" " + !?;' f'jAfi/'O SCALE ^ACTORS:'/' f).C= ' Gl f). 3 .r)X r ' LOAD: ',
> . 71 tGKJ.o^Xi 'CON':fMT^ATIO'-l5= ' r G] 0.3) . . . - -
> 7? Q FORMAT(usio.3)
> 73 jo rO^MATCO RPACTIO'^l COCFf 1C KENTS: /' Kr»,GlO,3/'0 SEGMENT'».
> 74 +6Xf»LOAD«)
> . 7l) _ i2_ roRMATt'ri ^EACTIO'^I COF.Ff-'lCIFHTS: V Kr '.» 010,3/ ^X r_' KAr «_» ol 0 , o »..
> " "?f, " "+[,X, tKO= ' »S10.3»r.X/«C5=" ',G10.3/'0 5E&MF NT .6X1 ' LOAD')
> 77 in FORMAT" (() NO LOADINGS'.) - -- - -
> 78 16 FORMAT<16,Gl3.3)
> 70 .go FOPi/AT ( '0 '5EGMFNT ROU'MOARY CONDI T I0^ll ) . -----
> BO ?j FO^MATCO NO ^OUNOARV CONDITIONS')
> _ 81 ?3 Fo^f>'AT ( J6»7X.513.3) ..... ...
> 0? '. sfi' FORMAT ('-' (5AI*/1 CONCENTRATIONS Ii-l ALL. SEGMENTS:'/
> 63 4-innC5X. !?»
> fi'* . CALL EXIT
> 65 END "
Fr:NQ OF FILE ' " ~'
34
-------
APPENDIX B
LINEAR SYSTEM DETROIT RIVER MODEL
FLOW - 200,00 cfs
35
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73
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APPENDIX C
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84
-------
APPENDIX D
DESCRIPTION OF SUBROUTINE SLE1
85
-------
COMPUTING CENTER MEMO #M46
LIBRARY SUBROUTINE DESCRIPTION
Simultaneous Linear Equations
and
Matrix Inversion
SIMULTANEOUS LINEAR EQUATIONS
The subroutine.SEL.solves a set of simultaneous linear
equations AX=B or X A=B , by Gaussian elimination, with partial
pivoting, and double back substitution. All floating point
numbers are presumed to be double precision (REALMS).
A is a general NxN matrix of coefficients. The dimensioned
column length of A is LENA
B is the right-hand matrix with N rows and NSOL columns
X is the solution matrix with N rows and NSOL columns
The dimensioned column length of B and X is LENBX,
METHOD*
For ease of presentation, the permutation of rows due to
partial pivoting will for the present be ignored.
The maxtrix A is decomposed into triangular matrices L
and U such that A=LU. L is a lower triangular matrix with ones
along the diagonal, and U is an upper triangular matrix. This
decomposition is performed by the subroutine LRD which is called
by SLE.
The system AX=B in the form (LU)X=B can be easily solved
via the two equations LY=B and UX=Y.. We have:
T V D ..
Lii -D y,,
IK i tr \
1 v 4- v - K )
21 Ik 2k "~ 2Y /
k = 1,NSOL
'"'For a full description of this method and related topics
see 'Computer Solution of Linear Algebraic Systems' by
George Forsythe and Cleve Moler, published by Prentice and
Hall.
86
-------
CCMEMO #M46
and
UX=Y:
ullxlk
U12x2k
U22x2k
u2NxNk
UNNXNk =
k=l,NSOL
The double back substitution expressed by these two sets
of equations is performed by the subroutine DBS which is called
by SLE. Similar1 ly, the subroutine DBST performs the double
back substitution for the system X A=B .
SUBROUTINE LRD
This routine computes the LU decomposition with partial
pivoting of an NxM matrix A. The triangular matrices are stored
in a single matrix T. Thus we have for a square matrix:
A=
L21
aNl aN2
12N
*NN
"T =
U
12
* U
1N
u
such that
LU =
32
0
U12 U13
U
33
o
U
1N
22 U23 ' * * U2N
U
3N
~ PA
where PA is the matrix A with some of its rows interchanged.
87
-------
CCMEMO 0M46
If the parameters for A and T in the calling sequence are
set equal, then A will be destroyed during execution and re-
placed by the matrix T. If they are not equal, then A will re-
main unchanged.
LRD uses Gaussian elimination with partial pivoting. Con-
sider a general NxN matrix A. The largest element of the first
column of the matrix A is found, and the row in which it occurs
is interchanged, if necessary, with .the first row of the maxtrix,
The elements of the first column are divided by the pivot ele-
ment:
and the remaining columns are processed according to the equa-
tion:
aj = a^ - l*alj i=2,N> j = 2,N
the superscript indicates processed elements i
The residual matrix A consisting of the elements (a.'. )
for i=2,N, j=2,N is treated in the same way as was the -1
original matrix A. A pivot is found in the first column, inter
change of the pivot row with the first row of A takes place
if necessary, elements of the first column are divided by the
pivot then the remaining columns are processed according to the
equation
a?, s'aj. - a?*a*. i=3,N, j=3,N
ij ij X£ £J
M 9
When we have processed A , a 2x2 matrix, the decomposi-
tion is complete.
For a rectangular NxM matrix our inner loop equation becomes
a.. = a.. - a. *a,. i=k+l,N, j=k+l,M k=l,min(N,M)
1] 13 1JC Kj
Slightly different steps are taken in the final stages of the
routine to cover the two cases HM.
All interchange information is stored in a permutation
vector during the decomposition. This integer vector IPERM
has length 2AN"t>ut can be thought of as N pairs of elements.
Thus
-------
CCMEMO #M46
IPERM
IPERM(l)
IPERMC2)
IPERMC3)
IPERMC4)'
!PERM(2*N-1)
»IPERM(2*N)
Interchange information is stored in IPERM(l),IPERM(3) ,... ,
IMPERM(2*N-1). These elements are originally set to zero, but
if at the ith stage, we interchange row i with row j, then we
set IPL'RM(i) = 8CJ-1). (j>i)
The permutation is stored in IPERMC2),IPERM(4),...,IPERM(2*N).
These elements are originally set to 0 ,.8,16,... ,8(N-1) and two
elements are interchanged as the corresponding rows are inter-
changed. Multiples of eight are used for both parts of the
permutation vector since these elements are used to reference
the double precision (8 bytes) elements of A,'T, B and X.
Since the final interchange element, IPERN(2AN-1), is never
needed for interchange informal* on, we use this element to store
the sign of the permutation, +1 or -1. However, if the matrix
A is found to be singular, then zero is stored in IPERM(2*N-1).
Since the produce of the diagonal elements of the T matrix and
IPERM(2*N-1) is equal to the determinant of the matrix A, then
a zero along this diagonal is the test used for singularity.
EXAMPLE Decomposition of a general 3x3 matrix.
Consider
4.
1.
2.
~4.
0.
.?
*4.
0.
-°-
"*4.
0.
J).
4.
0.
.°-
0
0
0
0
25
5
0
25
5
0
25
5
0
25
5
2.
2.
0.
2
2
0
2
1
-1
2
1
-0
2
1
-0
0 4
0 0
0 -3
.0
.0
.0 -
.0
.5 -
.0 -
.0
.5
.667
.0
.5
.667
A
.0
.0
°-
4.0
0.0
3.0
4
4.0
1.0
5.0
1.0 2.0 0.
4.0 2.0 4.
_2.0 0.0 -3.
max ( a .
0
0
0_
. ) = 4.0 for i=2,3, so
i-J
we interchange the first and
M
MM
1
second
a1 =
rl
P
w
4.0"
-1.0
-5.0
al
V
In the
rows
a. /a1 i=2,3
11
1 * 1
= a±. - ail*alj
i=2,3, j=2,3
residual matrix
1-1*0 -5*0 l» the Pivot element
L" * * J is 1.5, no interchange
_ is required and a00 = a00/l.S
4.0
-1.
0 Final
f. *J O f.
element is processed
5C C *7 ^ _ -*- ^ A d-
* ^'ji ^Q'a "" ^oo^-oo
89
-------
CCMEMO //M46
initially
The permutation vector for this decomposition would be
80-1
[
0 0 0
0 8
, and finally
8 0
16
Thus
LU
1.0 0.0 0.0
0.25 1.0 0.0
0.5 -0.667 1.0
4.0 2.0 4.0
1.0 2.0 0.0
2.0 0.0 -3.0
4.0 2.0 4.0
.O.'O 1.5 -1.0
0.0 0.0 -5.667
= matrix A with first and
second rows interchanged,
Rectangular matrices are also decomposed such that PA=LU,
For an NxM matrix with N>M we have:
A =
*
12M
aMl aM2
aNl aN2
T =
U12 ' ' ' U1M
U
21 22 *
U
2M
"-Ml XM2 * ' " U
1MM
NM
such that
PA = LU =
21 1
O
.I
U
12 '
' * U1M
U22 ' * * U2M
O
u
MM
90
-------
CCMEMO #MU6
EXAMPLE
A =
1.0
-2.0
0.0
L 2.0
The permutation
0.0
3.0
-1.0
4.0
* T =
-2.0 3.0
-1.0 7.0
o.o -0.143
-0.5 0.214
vector in this case is IPERM =
8 24 0 1
8 24 16 0
For the rectangualr matrix v;ith N
-------
CCMEMO #M46
SUBROUTINE DBS
This subroutine performs the double back substitution for
the system of equations LUX=PB (obtained from AX=B, where
PA=LU). The triangualr matrices L and U are stored in a matrix
T which was set up in a previous call to the subroutine LRD
together with the interchange information stored in the vector
IPERM.
LUX=PB is solved via the two equations LY=PB and UX=Y.
Thus out first step in solving these equations is to permute
the elements of B. We thus have:
LY = PB:
ylk = blk
an
= b., -
ik
l *y
.s
i=2,N, k=l,NSOL
where .the superscript on the b.. represents the permuted ele-
ments of B, *
UX = Y:
'Nk
and
x
ik
-------
CCMEMO
Let D =
H.O
6.0
1.0
therefore PB =
Solving LY=PB, we get
6.0
r«f.O
1.0
0.25y1 + 1.0
0.5 - 0.667
vi = 6'°
y2 = -5.5
y3 = -5.667
Solving UX=Y, we get
u.o
1.0
= 6.0
= -5.5
5.667x3 = -5.667
x.
= 2.0
= -3.0
x3 = 1.0
SUBROUTINE DBST
This subroutine performs the double back substitution for
the system of equations X p LU=B (obtatined from X A=B , where
PA=LU). The triangular matrices L and U are stored in a matrix
T which was set up in a previous call to the subroutine LRD
together with the interchange information stored in the vector
IPERM.
XtPtLU=Bt is solved via the two equations YtU=Bt and XtPtL=Yt.
We thus solve:
= B
and
(b -
ki - . Ujij
,, i=2,N,
yt.
ykN and
y]f. - ^ 1Tiv-xkj isN-l,l, k=l,NSOL
j=N
93
-------
CCKEMO //M46
The elements of x.. are members of the matrix X P , so the
final step in the solving of this system is to back permute the
t t
rows of X P and thus we obtain the final matrix X.
As in DBS, if the parameter for B and X are set equal in
the calling sequence, then B will be destroyed during execution
and replaced by X. If they are not set equal then B will re-
main unchanged.
MATRIX INVERSION
The subroutine INV computes the inverse of an NxM matrix
A. All floating point number! are presumed to be double pre-
cision (REAL*8).
METHOD
The matrix A is first decomposed into triangualr matrices
L and U such that PA=LU. L is a lower triangular matrix with
ones' along the diagonal and U is an upper triangular matrix.
PA is the matrix A with some of its rows interchanged due to
the partial pivoting employed during the decomposition.
We have PA = LU or A = P"1LU
Thus A"1 = (P'-'-LUr1 = iT1!/1?
INV calls the subroutine LRD to perform the -decomposition and
calls the subroutine INV1 to compute the inverse of L and U
and hence A" .
INV itself is just an organizational routine which calls
LRD and INV1 and provides scratch storage for INV1.
The subroutine LRD is described in an earlier section of
this write-up. The tirangular matrices are stored in a single
matrix T and the interchange information due to partial pivoting
is stored in the vector IPERM.
SUBROUTINE INV1
This subroutine inverts the matrices L and U and thus com-
putes the inverse of A. The triangular matrices L and U are
stored in a matrix T which was set up in a previous call to the
subroutine LRD together with the interchange information stored
in the vector IPERM.
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The following equations are used to invert the triangular
matrices:
For the matrix U:
3-1
j=i+l,N, 1=1,N-l
For the matrix L:
3*1.1-1,
The matrices L~ and U"" are stored in the matrix T replacing
L and U.
We have A" =U~ L" P. The following equations form the quotient
13
N
E
,-1 -
N
3=1,1-1.
This product U" L is stored back into T.
The final step of the routine is to permute the columns of
T using the information stored in the permutation vector IPERM
and this will give us A .
EXAMPLE
Inversion of a general 3x3 matrix.
Consider A =
2.0 1.0 0.0
-1.5 0.0 3.0
1.0 2.0 -1.0
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LRD decomposes A into triangualr matrices L and U and stores
these matrices in T:
2.0 1.0 0.0
0.5 1.5 -1.0
-0.75 0.5 3.5
where LU - PA = matrix A with
its second and third rows
interchanged.
INV1 inverts L and U:
0.5 -0.333 -0.095
-0.5 0.667 0.191
1.0 -0.5 0.286
-1 .
INV1 computes the quotient
where L
and U
and stores it in T:
MB
1.
-0.
o.s
0.0
0.0
1.
0
5
0
-0
0
0
0.
1.
-0.
.333
.667
.0
0
0
5
0
0
0
-0
0
0
.0
.0
.0_
.09
.IS
.28
T = 0.571 -0.286 -0.095
-0.1^2 0.572 0.191
0.286 -0.143 0.286
Finally, the second and third columns of T are interchanged and
we ontain A"1.
CALLING SEQUENCES
SLE has four entry points
SLE1 (N,LENA,A,NSOL,LENBX,B,X,IPERM)
- this solves the system AX=B, and SLE will supply the space
needed to store the T matrix. A remains unchanged.
SLE2 (N,LENA,A,NSOL,LENBX,B,X,IPERM)
- this solves the system XtA=B , and SLE will supply the
space needed to store the T matrix. A remains unchanged.
SLE3 (N,LENA,A,NSOL,LENB1,B,X,IPERM,LENT,T)
- this solves AX=B. The user supplies the space for the
matrix T.
SLE4 (N,LENA,A,NSOL,LENBX,B,X,IPERM,LENT,T)
- this solves XtA=B. The user supplies the space for the
matrix T.
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INV (N,LENA,A,IPERM,LENT,T)
- this inverts the matrix A and puts the result into T.
If A is set equal to T in the parameter list, then A will
be destroyed during execution. Similarly, if B is set equal to
X in any of the SLE calling sequences, then B will be destroyed
during execution and replaced by X,
SLE1 and SLE2 provide space for the matrix T. If, on a
subsequent call to one of these routines, less space is needed,
then some of the space that was previously obtained will be
released. If, on a subsequent call, more space is needed,
then more will be obtained.
$
The parameters in the calling sequences are represented by
the same symbols as were used in the preceding descriptions.
A - a double precision (8 bytes) NxN input matrix of coef-
ficients ,wnc^se~3imensioned size is LENA.
LENA>N, N >1
B - a double precision NxNSOL input matrix whose dimensioned
size is LETTBXT
LENBX^NjNSOL, NSOL>1
X - a double precision NxNSOL output matrix whose dimensioned
size is
IPERM - an integer vector of dimension 2*N.
For SLE3, SLE4, and INV only;
T - a double precision storage matrix whose dimensioned
size" is LENT.
LENT>N
To explain the dimensioning parameters further, consider
the dimension statement in FORTRAN which contains A(50,50). In
this case, A will be allocated 2500 consecutive doublewords,
and the elements of A will be stored columnwise starting with
A(l,l)» Thus the linear subscript of the (I,J) element is
(50(J-1)+(I-L)), and the last element of one column is stored
immediately before the first element of the next column. Sup-
pose, however', that the matrix of coefficients contains only
30 rows and columns. Then A(I,J) = A(50(J-1)+(I-1)) still,
but since I and J vary from 1 to 30, then the last element of
one colum will not be followed immediately by the first element
of the np.xt column. There will be 20 storage locations at the
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end of each column which will be ignored.
Hence to properly locate the elements of the matrix.A both
the current size 30, and the dimensioned column length 50, will
be required. This technique of matrix specification, however,
allows many other variations. For example, the parameters
N=2, LENA=100 and base element A(I,J) describes the matrix.
ai+I,j
M
As has been said, SLE and INV are organizational subroutines
and call LRD and DBS or DBST or INV1 to perform all the calcu-
lations.
The calling sequences for these routines (not required if
one calls SLE or INV directly) are:
LRD (N,M,LENA,A,IPERM,LENT,T)
DBS (NM,NSOL,LENBX,B,X,IPERM,LENT,T,
DBST (NM,NSOL,LENBX,B,X,IPERM,LENT,T)
INV1 (N,IPERM,LENT,T,Q)
where A is an NxM matrix and NM is the dimension of the square
matrix on which the double back substitution is performed. Al-
though LRD will successfully decompose a rectangular matrix,
DBS, DBST and INV1 will only work with square matrices. Q is
a double precision scratch vector of dimension N.
Thus a call to SLE3 say is equivalent to a call to LRD fol-
lowed immediately be a call to DBS, with N«M. And a call to
INV is equivalent to a call to LRD followed immediately by a
call to INV1 with N=M.
ACCURACY AND EFFICIENCY OF METHOD
For a complete error analysis and description of the dif-
ficulties encountered when using the Gaussian elimination method,
see the book previously mentioned, 'Computer Solution of Alge-
braic Systems1, by Forsythe and Moler.
Several tests were carried out on the subroutine LRD to
test its efficiency and speed. Figure 1 Illustrates the timing
data obtained from these tests.
For an NxM matrix the inner loop of the routine, where
most of the actual processing is done, must be executed approx-
imately N^/S times. Thus the graph of N3/3 afainst N should
coincide as nearly as possible with the graph of CPU time
elapsed during one decomposition against N. This is seen to
be the case, where values of N between 70 and 200 were taken.
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time in
seconds
120 140
160
180
Figure 1. CPU time required to decompose an NxN matrix,
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We have
t = aN3/3 where a - 15.8*10~6
Using the instruction times quoted in IBM manuals, we find that
t
3
3 3
the time taken to execute N /3 inner loops is &N /3 where
Hence the percentage of time spent in the iner loop is
14. U 100 * 91%
.3
The accuracy of the routines was tested by comparing the
product of A*X with the matr>ix B, where the matrix X was calcu-
lated using the routines, and the input matrices were generated
from random numbers. The relative error between elements of
Al1lX and B was about 10"" oh average, occasionally dropping to
10"16 or rising to 10"12.
Doe to rouding errors, singular matrices will not always
be recognized by the subroutine LRD. For example, consider thte
M . This will be decomposed into the matrix fQ 333 j
where a = 2-(6*0.333. ..). a will not be recorded as a true zero
but as a very small number. Thus the result of the test for
singularity will be negative, and unpredictable results will be
obtained if the double back substitution routine is subsequently
executed.
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