CORRECTION
For figures 4 and 5, pages 10-11, Bulletin 65
On pages 10 and 11, the figure captions are on the correct pages, but the
illustrations have been reversed.
The correct illustration for figure 4 appears
on page 11, and the correct illustration for figure 5 appears on page 10.
BULLETIN 65
A "Random-Walk" Solute Transport Model
for Selected Groundwater Quality Evaluations
by THOMAS A. PRICKETT, THOMAS G. NAYMIK, and CARL G. LONNQUIST
Title: A "Random-Walk" Solute Transport Model for Selected Groundwater Quality Evaluations.
Abstract: A generalized computer code is given that can simulate a large class of solute transport
problems in groundwater. The effects of convection, dispersion, and chemical reactions are included.
The solutions for groundwater flow include a finite difference formulation. The solute transport
portion of the code is based on a particle-in-a-cell technique for the convective mechanisms, and
a random-walk technique for the dispersion effects. The code can simulate one- or two-dimensional
nonsteady/steady flow problems in heterogeneous aquifers under water table and/or artesian or
leaky artesian conditions. The program covers time-varying pumpage or injection by wells, natural
or artificial recharge, the flow relationships of water exchange between surface waters and ground­
water, the process of groundwater evapotranspiration, the mechanism of possible conversion of
storage coefficients from artesian to water table conditions, and the flow from springs. It also
allows specification of chemical constituent concentrations of any segment of the model. Further
features of the program include variable finite difference grid sizes and printouts of input data, time
series of heads, sequential plots of solute concentration distributions, concentrations of water
flowing into sinks, and the effects of dispersion and dilution or mixing of waters having various
solute concentrations.
Reference: Prickett, Thomas A., Thomas G. Naymik, and Carl G. Lonnquist. A "Random-Walk"
Solute Transport Model for Selected Groundwater Quality Evaluations. Illinois State Water Survey,
Champaign, Bulletin 65, 1981.
Indexing Terms: Aquifers, case histories, computer codes, computer programs, contaminant
plumes, contamination, convection, dispersion, distribution coefficient, finite difference theory,
groundwater, groundwater flow, groundwater quality, groundwater tracers, models, pollution,
program listings, random walk, solute transport.
STATE OF ILLINOIS
HON. JAMES R. THOMPSON, Governor
DEPARTMENT OF ENERGY A N D N A T U R A L RESOURCES
MICHAEL B. W I T T E , B.A., Acting Director
BOARD OF N A T U R A L RESOURCES AND CONSERVATION
Michael B. Witte, B.A., Chairman
Walter E. Hanson, M.S., Engineering
Laurence L. Sloss, Ph.D., Geology
H. S. Gutowsky, Ph.D., Chemistry
Lorin I. Nevling, Ph.D., Forestry
William L. Everitt, E.E., Ph.D.,
University of Illinois
John C. Guyon, Ph.D.,
Southern Illinois University
STATE WATER SURVEY DIVISION
STANLEY A. C H A N G N O N , JR., M.S., Chief
CHAMPAIGN
1981
Printed by authority of the State of Illinois
(11-81-1500)
CONTENTS
PAGE
Abstract
Introduction
Acknowledgments
P a r t 1. Mathematical b a c k g r o u n d
Calculations of flow
Solute t r a n s p o r t calculations
Dispersion
Dilution, mixing, r e t a r d a t i o n , and
r a d i o a c t i v e decay
P a r t 2. Random-walk solute t r a n s p o r t program
Job s e t u p
P r e p a r a t i o n of d a t a
P a r a m e t e r card
Default value card
Pump p a r a m e t e r c a r d and pumping s c h e d u l e
cards
Variable grid c a r d s
Pollution initial conditions c a r d
Pollution p a r a m e t e r card
Sink location c a r d s
Source c o n c e n t r a t i o n c a r d s
Node c a r d d e c k
B a r r i e r b o u n d a r y conditions
Leaky a r t e s i a n conditions
I n d u c e d infiltration
C o n s t a n t head conditions
Storage factors
Evapotranspiration
Flow from s p r i n g s
P a r t 3. Program operational s e q u e n c e s
R e q u i r e d s u b r o u t i n e s and functions
F u n c t i o n ANORM(0)
Function V ( I , J , K )
S u b r o u t i n e INIT
S u b r o u t i n e CLEAR
S u b r o u t i n e ADD(XX,YY)
S u b r o u t i n e VELO(XX, YY, VX, V Y)
S u b r o u t i n e ADVAN(DELP)
F u n c t i o n MOVE(XX, YY, DEL)
Optional s u b r o u t i n e s t h a t p r o d u c e head
distributions
S u b r o u t i n e HSOLVE
S u b r o u t i n e HSOLV2
S u b r o u t i n e HSOLV4
Optional s u b r o u t i n e s t h a t g e n e r a t e p a r t i c l e s . . . . . .
S u b r o u t i n e GENP2
S u b r o u t i n e GENP3
S u b r o u t i n e GENP(PL)
(Concluded on next page)
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2
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5
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8
9
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45
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47
48
48
50
51
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51
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.63
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64
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65
CONTENTS
(Concluded)
PAGE
S u b r o u t i n e s GENP4 and GENP5
S u b r o u t i n e SORGEN
S u b r o u t i n e SOURCE
Optional s u b r o u t i n e for calculating p a r t i c l e
retardation
S u b r o u t i n e RDSOLV
Optional s u b r o u t i n e s t h a t p r i n t r e s u l t s
S u b r o u t i n e MAP
S u b r o u t i n e CONMAP
S u b r o u t i n e SUMMRY
S u b r o u t i n e SNKCON
P a r t 4. Example problems and comparisons with t h e o r y . .
D i v e r g e n t flow from an injection well in
an infinite aquifer without d i s p e r s i o n
or dilution
Pumping from a well n e a r a line source of
c o n t a m i n a t e d water, with dilution b u t
without d i s p e r s i o n
L o n g i t u d i n a l d i s p e r s i o n in uniform o n e dimensional flow with continuous injection
at X = 0.0
L o n g i t u d i n a l d i s p e r s i o n in uniform o n e dimensional flow with a slug of t r a c e r
injected at X = 0.0
L o n g i t u d i n a l d i s p e r s i o n in a radial flow system
p r o d u c e d by an injection well
Longitudinal and t r a n s v e r s e dispersion in
uniform one-dimensional flow with a slug of
t r a c e r injected at X = 0.0
P a r t 5. Field application; Meredosia, Illinois
G r o u n d w a t e r flow system
Quality o f n a t i v e and contaminated g r o u n d w a t e r . . .
Contaminant plume
The Meredosia solute t r a n s p o r t model
References
66
66
66
67
67
69
69
69
69
70
71
71
72
79
81
81
86
91
91
92
96
97
103
A "Random-Walk" Solute Transport Model
for Selected Groundwater Quality Evaluations
by Thomas A. Prickett, Thomas G. Naymik, and Carl G. Lonnquist
ABSTRACT
A generalized computer code is given t h a t can simulate a
l a r g e class of solute t r a n s p o r t problems in g r o u n d w a t e r . T h e
effects of convection, d i s p e r s i o n , and chemical r e a c t i o n s a r e
included.
T h e solutions for g r o u n d w a t e r flow include a fi­
nite difference formulation.
The solute t r a n s p o r t portion of
the code is based on a p a r t i c l e - i n - a - c e l l t e c h n i q u e for t h e
convective mechanisms, and a random-walk t e c h n i q u e for t h e
d i s p e r s i o n effects.
The code can simulate o n e - or two-dimensional n o n s t e a d y /
steady flow problems in h e t e r o g e n e o u s a q u i f e r s u n d e r water
table a n d / o r a r t e s i a n o r leaky a r t e s i a n c o n d i t i o n s .
Further­
more this program c o v e r s t i m e - v a r y i n g pumpage or injection by
wells, n a t u r a l or artificial r e c h a r g e , the flow r e l a t i o n s h i p s
of water e x c h a n g e between s u r f a c e w a t e r s and the g r o u n d w a t e r
r e s e r v o i r , the p r o c e s s of g r o u n d w a t e r e v a p o t r a n s p i r a t i o n , t h e
mechanism of possible conversion of s t o r a g e coefficients from
a r t e s i a n to water table conditions,
and the flow from
springs.
In addition, the program allows specification of chemical
c o n s t i t u e n t c o n c e n t r a t i o n s of any segment of the model in­
cluding, but not limited to, injection of contaminated water
by wells, vertically a v e r a g e d s a l t - w a t e r f r o n t s , leachate
from landfills, leakage from overlying source beds of differ­
ing quality t h a n the aquifer, and s u r f a c e water s o u r c e s s u c h
as contaminated lakes and s t r e a m s .
F u r t h e r f e a t u r e s of the program include v a r i a b l e finite
difference grid sizes and p r i n t o u t s of i n p u t data, time
s e r i e s of h e a d s , s e q u e n t i a l plots of solute c o n c e n t r a t i o n
d i s t r i b u t i o n s , c o n c e n t r a t i o n s of water flowing into s i n k s ,
and the effects of d i s p e r s i o n and dilution or mixing of
w a t e r s h a v i n g various solute c o n c e n t r a t i o n s .
The d i s c u s s i o n of the digital t e c h n i q u e i n c l u d e s the n e c ­
e s s a r y mathematical b a c k g r o u n d , the documented program l i s t ­
i n g , explanations of job s e t u p p r o c e d u r e s , sample i n p u t d a t a ,
theoretical v e r s u s computer comparisons, and one field a p p l i ­
cation.
1
INTRODUCTION
P r e s e n t l y , t h e r e are four classes of
problems of concern in s t u d i e s of
solute
transport
in
groundwater:
1) chemical problems such as p r e d i c t i n g
TDS, C l , n i t r a t e s , e t c . , a s when deal­
ing with sea water i n t r u s i o n , e x c e s s i v e
fertilizer
applications,
hazardous
w a s t e l e a c h a t e , and injection of chemi­
cal wastes into t h e s u b s u r f a c e using
disposal wells; 2) b a c t e r i a l problems
associated
with
cesspools,
artificial
recharge,
sanitary
landfills,
waste
injection wells, e t c . ; 3) t h e r m a l p r o b ­
lems involving injection of hot water
into t h e g r o u n d w a t e r r e s e r v o i r , d e v e l ­
opment of geothermal e n e r g y , and in­
duced
infiltration of
surface
water
having
varying
temperatures;
and
4) m u l t i - p h a s e problems a r i s i n g from
such
situations
as
development
of
steam-water systems, secondary recovery
of water by a i r injection, and possible
air-water
interfaces
due
to
overp u m p i n g . Although t h i s t r a n s p o r t p r o ­
gram may be e x t e n d e d , the emphasis of
t h i s r e p o r t is placed on t h e chemical
problems.
for
is
One form of t h e g o v e r n i n g equation
solute t r a n s p o r t in one dimension
DISPERSION - CONVECTION ± PRODUCTION = QUALITY
ACCUMULATION
where
V = i n t e r s t i t i a l velocity
D = coefficient of h y d r o d y n a m i c
dispersion
D = d x V + D*, where d x =
longitudinal d i s p e r s i v i t y
and D* = coefficient of
molecular diffusion ( n e g ­
lected in the following
development)
x = space dimension
R d = r e t a r d a t i o n factor
C S Q= source or sink function
h a v i n g a concentration C s
C = concentration
2
Problems including solute t r a n s p o r t in
g r o u n d w a t e r involve solving equation 1
in one, two, or ( r a r e l y ) t h r e e dimen­
s i o n s . For t h e derivation of t h e e q u a ­
tion and f u r t h e r explanation, refer t o
F r e e z e and C h e r r y (1979), B e a r (1972),
and Ogata (1970).
A p o p u l a r numerical t e c h n i q u e u s e d
for solving e q u a t i o n 1 is t h e method of
c h a r a c t e r i s t i c s (MOC), or t h e p a r t i c l e i n - a - c e l l m e t h o d . MOC t r e a t s t h e e q u a ­
tion in two p a r t s .
First, the convect i v e term c o n t a i n i n g the velocity (V)
is solved with an a d a p t a t i o n of t h e
usual finite difference flow t y p e of
model.
T h e n t h e d i s p e r s i v e term is
solved by u s i n g a n o t h e r finite differ­
ence grid a s s o c i a t e d with t h e c o n c e n ­
tration distribution.
A large number
of computer—generated p a r t i c l e s move
about b y t h e velocity v e c t o r s which a r e
solved in t h e flow model and which
carry
the
concentration
information
between the convection and d i s p e r s i o n
t e r m s d u r i n g t h e solution of t h e e q u a ­
tion.
The d e s c r i p t i o n of t h e MOC is
s t r a i g h t f o r w a r d , b u t the computer code
is highly involved and v e r y e x p e n s i v e
to o p e r a t e , and it r e q u i r e s a l a r g e
computer to effect a solution.
There­
fore, r e s e a r c h e r s h a v e been looking for
a more efficient and more d i r e c t way to
solve problems c o n c e r n i n g s o l u t e t r a n s ­
port in g r o u n d w a t e r .
The conceived
"random-walk" method follows.
The random-walk t e c h n i q u e is based
on the concept t h a t d i s p e r s i o n in p o ­
r o u s media is a random p r o c e s s . On a
microscopic b a s i s , d i s p e r s i o n may occur
as shown in f i g u r e 1.
As indicated in
figure 1C, d i s p e r s i o n can t a k e place in
two directions even t h o u g h t h e mean
flow is in one direction to t h e r i g h t .
A particle, r e p r e s e n t i n g t h e mass of a
specific chemical c o n s t i t u e n t contained
in a defined volume of w a t e r , moves
t h r o u g h an aquifer with two t y p e s of
motion.
One motion is with t h e mean
flow (along s t r e a m l i n e s d e t e r m i n e d by
finite d i f f e r e n c e s ) , and t h e o t h e r is
random motion, g o v e r n e d by scaled p r o b -
Figure 1. Basic concepts of "random-walk" computer program
ability c u r v e s r e l a t e d to flow length
a n d the longitudinal and t r a n s v e r s e
d i s p e r s i o n coefficients.
Finally, in
t h e computer code, enough p a r t i c l e s a r e
included so t h a t t h e i r locations and
d e n s i t y , as t h e y move t h r o u g h a flow
model, are a d e q u a t e to d e s c r i b e the
d i s t r i b u t i o n of t h e dissolved c o n s t i t u ­
ent of i n t e r e s t .
The a d v a n t a g e s of this random-walk
t e c h n i q u e over the MOC or, for t h a t
matter,
over many
other
numerical
schemes are many:
1) T h e r e is no d i s p e r s i o n equation
to solve. The d i s p e r s i o n part of
equation 1 is solved in t h e com­
p u t e r code by the addition of
only 11 F o r t r a n s t a t e m e n t s a t ­
tached to the solution of the
convection p a r t of e q u a t i o n 1.
2) T h e r e is only one finite differ­
ence grid involved in solving t h e
convective portion of e q u a t i o n 1.
The particle movement t a k e s place
in continuous s p a c e .
3) Concentration d i s t r i b u t i o n n e e d s
to be calculated only when it is
of i n t e r e s t .
In t h e MOC, after
each particle is moved, new con­
c e n t r a t i o n s a r e a s s i g n e d on t h e
basis of t h e solution of t h e d i s ­
4)
5)
6)
7)
8)
persion term in t h e above e q u a ­
tion for e v e r y time s t e p of t h e
sim ulation.
Computer CPU time is d r a s t i c a l l y
reduced.
The simulations in
Part 4 of t h i s bulletin took no
more t h a n a few s e c o n d s , i n c l u d ­
ing compiling and loading on a
CDC CYBER 175.
Particles a r e needed only w h e r e
water quality is
of
interest.
T h e s e p a r t i c l e s a r e not needed
e v e r y w h e r e in the model, as with
t h e MOC.
Solutions a r e a d d i t i v e .
If not
enough p a r t i c l e s a r e i n c l u d e d for
a d e q u a t e definition in one r u n , a
second r u n can be done a n d t h e
r e s u l t s a c c u m u l a t e d . T h i s is not
t r u e of t h e MOC, w h e r e possible
s p a r s e a r e a s of p a r t i c l e s may
occur, c a u s i n g loss of a c c u r a c y .
This method is p a r t i c u l a r l y s u i t ­
ed to t i m e - s h a r i n g s y s t e m s w h e r e
velocity fields can be s t o r e d and
manipulated in conjunction with
an on-line particle mover c o d e .
In the t r a d i t i o n a l s e n s e of t h e
w o r d s , finite d i f f e r e n c i n g p h e ­
nomena associated with
"over­
shoot" and "numerical d i s p e r s i o n "
a r e eliminated.
3
Although t h e r e a r e numerous a d v a n ­
t a g e s t o t h i s t e c h n i q u e , t h e r e a r e also
some d i s a d v a n t a g e s :
1) As with the MOC, c o n c e n t r a t i o n s
greater than
initial
conditions
are
possible,
especially
when
coarse discretizing is used.
2) A p r i n t o u t of c o n c e n t r a t i o n s may
not be pleasing to t h e eye when
t h e number of p a r t i c l e s is small.
(A Calcomp plotting and smoothing
r o u t i n e could be a d d e d to t h i s
model to eliminate t h i s problem;
however, this is beyond t h e scope
and objectives of t h i s r e p o r t . )
3) The method may t a k e an u n u s u a l l y
l a r g e number of p a r t i c l e s to p r o ­
duce an a c c e p t a b l e solution for
some p r o b l e m s .
No more t h a n
5,000 p a r t i c l e s a r e u s e d in t h i s
report, however.
4) Engineering judgment is an a b s o ­
l u t e r e q u i r e m e n t in a r r i v i n g at
an acceptable solution.
T h i s is
b e c a u s e of t h e "lumpy" c h a r a c t e r
of t h e o u t p u t .
Therefore, expe­
rience with t h i s t e c h n i q u e is
needed before one can apply t h e
code successfully to a field sit­
uation.
T h e main objective of t h i s r e p o r t is
to p r e s e n t a generalized computer code
t h a t will simulate a l a r g e class of
solute
transport
problems
involving
4
convection and d i s p e r s i o n .
In t h e
p r e s e n t v e r s i o n , t h e effect of d e n s i t y i n d u c e d convection is not i n c l u d e d .
T h i s complication is n e c e s s a r y only
when a vertically a v e r a g e d c o n c e n t r a ­
tion d i s t r i b u t i o n is i n a d e q u a t e .
The
class of problems t h a t r e q u i r e a v e r t i ­
cal a v e r a g e d c o n c e n t r a t i o n d i s t r i b u t i o n
is sufficiently l a r g e , and t h e effects
of d e n s i t y differences will
be a d ­
d r e s s e d in a l a t e r p u b l i c a t i o n .
Acknowledgments
T h i s r e p o r t was p r e p a r e d u n d e r t h e
g e n e r a l s u p e r v i s i o n of D r . William C.
Ackermann, former Chief, and Stanley A.
C h a n g n o n , J r . , p r e s e n t Chief of t h e
Illinois State Water S u r v e y . Richard J .
S c h i c h t , A s s i s t a n t Chief, and James P .
Gibb, Head of the G r o u n d w a t e r Section,
made p a r t i c u l a r l y useful
suggestions
with r e g a r d to this m a t e r i a l .
T h e helpful comments of Adrian P.
Visocky, H y d r o l o g i s t , and Robert A.
Sinclair, Head of the Data and Informa­
tion Management Unit, a r e a p p r e c i a t e d ,
as is t h e s u p p o r t from t h e Analytical
Chemistry Laboratory Unit.
T h a n k s a r e also due Pamela Lovett
for t y p i n g t h e m a n u s c r i p t and t h e
camera copy;
Gail T a y l o r for final
e d i t i n g ; and John W. B r o t h e r , J r . ,
Linda Riggin, and William Motherway,
J r . , for p r e p a r i n g t h e i l l u s t r a t i o n s .
PART 1.
MATHEMATICAL BACKGROUND
The computer code contained in t h i s
r e p o r t can be b r o k e n into two p a r t s .
T h e first p a r t r e l a t e s to the "flow
model" and t h e second p a r t to t h e p a r ­
ticle moving section which is termed
t h e "solute t r a n s p o r t model."
Some sort of model is needed to p r o ­
vide t h e velocity v e c t o r s of an aquifer
flow system in o r d e r to calculate t h e
c o n v e c t i v e movement of p a r t i c l e s .
In
most of t h e s i t u a t i o n s to follow,
a
finite difference scheme is used to
g e n e r a t e t h e head d i s t r i b u t i o n .
From
t h i s head d i s t r i b u t i o n a velocity field
is d e r i v e d .
The velocity field t h e n
p r o v i d e s the means of moving t h e p a r t i ­
cles a d v e c t i v e l y in t h e a q u i f e r .
Calculations of Flow
T h r e e methods for head calculations
a r e w r i t t e n into the computer code in
o r d e r to g e n e r a t e a head d i s t r i b u t i o n
which can then be u s e d to calculate a
velocity field.
Two of t h e s e head cal­
c u l a t i o n s a r e analytical and t h e t h i r d
is a finite difference method.
A brief
b a c k g r o u n d of the finite difference
scheme for p r o d u c i n g a head d i s t r i b u ­
tion follows.
The
partial
differential
equation
( J a c o b , 1950) g o v e r n i n g t h e n o n s t e a d y s t a t e two-dimensional flow of g r o u n d ­
water may be e x p r e s s e d as
h
b
t
Q
=
=
=
=
head above bottom of aquifer
s a t u r a t e d t h i c k n e s s of aquifer
time
source or sink functions e x ­
p r e s s e d as net flow r a t e s p e r
unit area
A numerical solution of equation 2
can be obtained t h r o u g h a finite dif­
ference a p p r o a c h .
T h e finite differ­
ence a p p r o a c h involves first replacing
t h e c o n t i n u o u s aquifer system p a r a m e ­
t e r s with an equivalent set of d i s c r e t e
elements.
Second, t h e e q u a t i o n s gov­
e r n i n g t h e flow of g r o u n d w a t e r in t h e
d i s c r e t i z e d model a r e w r i t t e n in finite
difference form.
Finally, t h e r e s u l t ­
ing set of finite difference e q u a t i o n s
is solved numerically with t h e aid of a
digital c o m p u t e r .
A finite difference grid is s u p e r ­
posed o v e r a map of an aquifer, as
i l l u s t r a t e d in f i g u r e 2.
The aquifer
is t h u s s u b d i v i d e d into volumes h a v i n g
dimensions bΔxΔy, w h e r e b is t h e s a t u ­
r a t e d t h i c k n e s s of t h e a q u i f e r .
The
differentials ∂x a n d ∂y a r e a p p r o x i ­
mated by t h e finite l e n g t h s Δx and Δy,
where
K x b = T x = aquifer t r a n s m i s sivity in t h e x direction
K y b = T y = aquifer t r a n s m i s sivity in t h e y direction
K x = aquifer h y d r a u l i c c o n d u c ­
tivity in t h e x direction
K y = aquifer h y d r a u l i c c o n d u c ­
tivity in t h e y direction
S = aquifer s t o r a g e coefficient
Figure 2. Finite difference g r i d
5
r e s p e c t i v e l y . The area ΔxΔy should be
small compared with t h e t o t a l area of
t h e aquifer t o the extent t h a t t h e d i s ­
c r e t e model is a r e a s o n a b l e r e p r e s e n t a ­
tion of t h e c o n t i n u o u s s y s t e m .
The
i n t e r s e c t i o n s of grid lines a r e called
n o d e s and a r e r e f e r e n c e d with a column
(i) and row (j) coordinate system colin e a r with t h e x a n d y d i r e c t i o n s , r e ­
spectively.
The general form of t h e finite dif­
ference equation g o v e r n i n g t h e flow of
g r o u n d w a t e r in the d i s c r e t i z e d model is
t h e n given by
where
Ti,j,1=
aquifer t r a n s m i s s i v i t y
between nodes i,j and
i,j + 1 calculated as
P E R M i , j , 1 t i m e s h where
PERM is nydraulic c o n d u c ­
tivity
Ti,j,2=
aquifer t r a n s m i s s i v i t y
between nodes i,j and
i+l,j calculated as
PERM i,j,2
times h w h e r e
PERM is h y d r a u l i c c o n d u c ­
tivity
hi,j=
calculated heads at nodes
i, j at t h e end of a time
increment measured from
an a r b i t r a r y r e f e r e n c e
level
= calculated heads at nodes
i,j at t h e end of t h e
p r e v i o u s time increment
m e a s u r e d from the same
reference level defining
hi,j
Δt = time increment elapsed
since last calculation of
heads
Si,j=
aquifer s t o r a g e coeffi­
cient at node i,j
Q i,j =
net withdrawal r a t e if
positive, or net a c c r e ­
tion r a t e if n e g a t i v e , at
node i,j
6
Since t h e r e is an e q u a t i o n of the
same form as equation 3 for e v e r y node
of the digital model, a l a r g e set of
simultaneous e q u a t i o n s must be solved
for the principal u n k n o w n s
hi,j.
The
modified i t e r a t i v e a l t e r n a t i n g
direc­
tion implicit method (MIADI) given by
P r i c k e t t and L o n n q u i s t (1971) is u s e d
to solve the set of simultaneous e q u a ­
tions.
Briefly, the MIADI method involves
f i r s t , for a given time i n c r e m e n t , r e ­
d u c i n g the l a r g e set of simultaneous
e q u a t i o n s down to a n u m b e r of small
s e t s . This is done by solving t h e node
e q u a t i o n s , by G a u s s elimination, of an
individual row of the model while all
t e r m s related to the nodes in the two
adjacent rows a r e held c o n s t a n t . After
all row equations have been p r o c e s s e d
row by row, t h e node e q u a t i o n s a r e
solved again by Gauss elimination for
an individual column while all t e r m s
r e l a t e d to the two adjacent columns a r e
held c o n s t a n t .
After all e q u a t i o n s
h a v e been solved column by column, an
"iteration" has been completed.
The
a b o v e process is r e p e a t e d until c o n v e r ­
gence is a c h i e v e d , completing t h e cal­
culations for the given time i n c r e m e n t .
T h e calculated h e a d s a r e t h e n used a s
initial conditions for t h e n e x t time
increment.
T h i s total p r o c e s s i s r e ­
p e a t e d for s u c c e s s i v e time i n c r e m e n t s
until the d e s i r e d simulation is com­
pleted.
Equation 3 may be r e w r i t t e n to
i l l u s t r a t e the g e n e r a l form for c a l c u ­
lations by r o w s .
As a first simplifi­
cation it is assumed t h a t t h e finite
difference grid is made up of s q u a r e s
s u c h t h a t Δy = Δx. (The case w h e r e Δy
does
not equal Δx is t r e a t e d by
P r i c k e t t and L o n n q u i s t , 1971.) E q u a ­
tion 3 is then e x p a n d e d , t h e s i g n s a r e
r e v e r s e d , and t e r m s of h i , j a r e g r o u p e d
t o g e t h e r to yield
Equation 3 is of t h e form
Since t h e model in t h i s r e p o r t i n ­
cludes nonhomogeneous a q u i f e r p r o p e r ­
ties and v a r i a b l e grid dimensions, t h e
calculation of the velocity d i s t r i b u ­
tion by equation 5 a c c o r d i n g l y t a k e s
t h i s into a c c o u n t .
Velocities a r e t h u s
s t o r e d on the basis of e i t h e r
V(I,J,1)
nodes
V(I,J,2)
nodes
= V
I,J
= V
I,J
defined midway between
and I , J + 1 , or
defined midway between
and I+1,J
T h e s e velocities a r e u s e d a s t h e
i n p u t to a r a t h e r elaborate i n t e r p o l a ­
tion scheme to provide velocity v e c t o r s
for the movement of each particle in
continuous s p a c e .
A similar set of e q u a t i o n s can be
written for calculations by c o l u m n s .
One of t h e flow calculations used in
this r e p o r t ,
which p e r t a i n s to the
above mathematical d e s c r i p t i o n , is the
"composite aquifer simulation program"
given by P r i c k e t t and L o n n q u i s t (1971).
The r e p o r t by Prickett and L o n n q u i s t is
a companion to this r e p o r t , as many
details of flow modeling a r e taken
d i r e c t l y from it for u s e in t h i s r e ­
port.
Let us emphasize again t h a t several
methods can be used to p r o d u c e head
d i s t r i b u t i o n s from which velocities can
be calculated.
As will be explained
l a t e r , heads can be e n t e r e d from t h e o ­
retical d i s t r i b u t i o n s ,
field d a t a ,
or
o t h e r t e c h n i q u e s involving totally dif­
ferent methods for numerically g e n e r a ­
ting h e a d s .
Once a head d i s t r i b u t i o n is defined
for all the nodes of t h e finite differ­
ence grid, then the velocity d i s t r i b u ­
tion is calculated according to
where
V =
K=
I =
n =
i n t e r s t i t i a l velocity
hydraulic conductivity
hydraulic gradient
effective porosity
Solute Transport Calculations
The basis for the t r a n s p o r t calcula­
tions of dissolved c o n s t i t u e n t s in t h i s
computer code is t h a t t h e d i s t r i b u t i o n
of the c o n c e n t r a t i o n of chemical c o n ­
s t i t u e n t s of the water in an a q u i f e r
can be r e p r e s e n t e d by t h e d i s t r i b u t i o n
of a finite n u m b e r of d i s c r e t e p a r t i ­
c l e s . Each of t h e s e p a r t i c l e s is moved
by g r o u n d w a t e r flow and is a s s i g n e d a
mass which r e p r e s e n t s a fraction of t h e
total mass of chemical c o n s t i t u e n t i n ­
v o l v e d . In t h e limit, as t h e n u m b e r of
p a r t i c l e s gets extremely l a r g e and a p ­
p r o a c h e s the molecular l e v e l , an exact
solution to t h e actual
situation is
obtained.
However, it is our e x p e r i ­
ence t h a t relatively few p a r t i c l e s a r e
needed to a r r i v e at a solution t h a t
will suffice for many e n g i n e e r i n g a p ­
plications .
T h e r e a r e two prime mechanisms which
can change contaminant c o n c e n t r a t i o n in
groundwater:
d i s p e r s i o n , and dilution
and mixing.
The effects of mechanical
d i s p e r s i o n a s the fluid s p r e a d s t h r o u g h
the pore space of t h e porous medium a r e
d e s c r i b e d by t h e first and second t e r m s
on t h e left side of equation 1.
The
effects of dilution and mixing a r e e x ­
p r e s s e d in t h e second and t h i r d t e r m s
on t h e left side of equation 1.
7
Dispersion
To i l l u s t r a t e the d e t a i l s of t h e
random-walk t e c h n i q u e as it relates to
d i s p e r s i o n , consider t h e p r o g r e s s of a
unit
slug
of
tracer-marked
fluid,
placed initially at x = 0, in an infi­
nite column of p o r o u s medium with
s t e a d y flow in t h e x d i r e c t i o n .
With
C s Q e q u a l to zero, equation 1 d e ­
s c r i b e s t h e c o n c e n t r a t i o n of the slug
as it moves d o w n s t r e a m .
Bear (1972)
d e s c r i b e s t h e solution a s
t r i b u t e d if its d e n s i t y function,
is given by
n(x),
where
σ = s t a n d a r d deviation of t h e
tribution
u = mean of t h e d i s t r i b u t i o n
dis­
Now, let us e q u a t e t h e following t e r m s
of e q u a t i o n s 6 a n d 7 as
where
C =
dL =
V =
t =
x =
concentration
longitudinal d i s p e r s i v i t y
i n t e r s t i t i a l velocity
time
d i s t a n c e along t h e x a x i s
The s h a p e s of t h e c u r v e s C( , t ) are
shown in figure 3 w h e r e
= x-Vt.
Based upon c o n c e p t s found in compre­
h e n s i v e s t a t i s t i c s books (for example,
see Mood and Graybill, 1963) a random
variable x is said to be normally d i s -
With t h e i d e n t i t i e s of e q u a t i o n s
8
t h r o u g h 10 t a k e n into a c c o u n t , it will
be seen t h a t equations 6 and 7 a r e
equivalent.
The key t o solute t r a n s p o r t a s d e ­
s c r i b e d in t h i s r e p o r t is t h e realiza­
tion t h a t d i s p e r s i o n in a p o r o u s medium
can be c o n s i d e r e d a random p r o c e s s ,
tending to the
normal
distribution.
Figure 3. Progress of a slug around the mean flow
8
T h e r e a r e no new c o n c e p t s h e r e , as v e r y
complete
discussions
of
statistical
models of dispersion a n d t h e way d i s ­
persion affects water q u a l i t y h a v e been
given by o t h e r s , including Bear (1972)
and Fried (1975). The method by which
t h i s statistical approach gets into t h e
computer code and is applied to t r a n s ­
port problems is new.
F i g u r e 4A r e p r e s e n t s the way p a r ­
ticles a r e moved in t h e computer code
when t h e flow is in t h e x direction and
one c o n s i d e r s only longitudinal d i s p e r ­
sion. During a time i n c r e m e n t , DELP, a
particle
with
coordinates
xx,yy
is
first moved from an old to a new posi­
tion in t h e aquifer by convection a c ­
cording to its velocity at t h e old
position V x . T h e n , a random movement
in t h e +x or -x d i r e c t i o n is a d d e d to
r e p r e s e n t the effects of d i s p e r s i o n .
T h i s random movement is given t h e mag­
nitude
where
ANORM(0) = a number between -6
and +6, drawn from a
normal d i s t r i b u t i o n
of n u m b e r s h a v i n g a
s t a n d a r d deviation of
1 and a mean of zero
T h e new position of the particle in
figure 4A is the old position plus a
c o n v e c t i v e term ( V x DELP) plus t h e
effect of the d i s p e r s i o n term,
If t h e above p r o c e s s is r e p e a t e d for
n u m e r o u s particles, all having t h e same
initial position and convective t e r m , a
map of t h e new positions of t h e p a r t i ­
cles can be created h a v i n g t h e d i s c r e t e
density distribution
where
d x = incremental d i s t a n c e s over
which N particles a r e found
N o = total number of p a r t i c l e s
in t h e experiment
Equations 6, 7, and 12 a r e e q u i v a ­
l e n t , with the exception t h a t e q u a ­
tions 6 and 7 a r e continuous d i s t r i b u ­
tions and equation 12 is d i s c r e t e .
As
i l l u s t r a t e d in figure 4A, the d i s t r i b u ­
tion of particles a r o u n d t h e mean posi­
tion, V X DELP, is made to be normally
d i s t r i b u t e d via the function ANORM(0).
T h e function ANORM(0) is g e n e r a t e d in
t h e computer code as a simple function
involving a summation of random num­
bers.
Probable locations of p a r t i c l e s ,
however, are c o n s i d e r e d only out to 6
s t a n d a r d deviations either side of t h e
mean. On a practical b a s i s , the p r o b a ­
bility is low of a particle moving
beyond t h a t d i s t a n c e .
One f u r t h e r emphasis is a p p r o p r i a t e
c o n c e r n i n g the so-called " d e n s i t y f u n c ­
tion" of equations 6, 7, and 12.
The
equivalent d e n s i t y functions C ( x , t ) and
N / d x provide the means for r e l a t i n g
the concentration of a contaminant in a
field problem to the c o n c e n t r a t i o n of
particles found in portions of a finite
difference model.
Various
density
functions will be defined l a t e r ,
by
example, as t h e y a r e needed for a p p l i ­
cation p u r p o s e s .
F i g u r e 4B i l l u s t r a t e s the e x t e n s i o n
of the random-walk method to account
for d i s p e r s i o n in a direction t r a n s ­
v e r s e to the mean flow. F i g u r e s 5A and
5B i l l u s t r a t e the a l g e b r a involved when
the flow is not aligned with t h e x - y
coordinate s y s t e m .
Finally, figure 6
shows both longitudinal and t r a n s v e r s e
d i s p e r s i o n t a k i n g place simultaneously,
and the a p p r o p r i a t e v e c t o r a l g e b r a .
Dilution,
Mixing,
Retardation,
and Radioactive Decay
Consider the one-dimensional
problem in figure 7A in which the
and c o n c e n t r a t i o n s of the s o u r c e s
given.
With d i s p e r s i o n set to zero
flow
flow
are
and
9
Figure 4. Computer code scheme for convection and longitudinal (A)
and transverse (B) dispersion along x axis
10
Figure 5. General scheme for convection and longitudinal (A)
and transverse (B) dispersion
11
Figure 6. General scheme for convection and dispersion
12
Figure 7. Mixing and dilution effects in water quality problems
13
r e t a r d a t i o n set t o one, the d i s t r i b u ­
tion of c o n c e n t r a t i o n s in the system is
simply a r e s u l t of p u r e mixing as
i l l u s t r a t e d in f i g u r e 7 B .
Now, let us assume t h a t in the com­
p u t e r model one particle r e p r e s e n t s 10
m g / l . F i g u r e 7C shows the time d e n s i t y
of p a r t i c l e s that would be used for in­
p u t d a t a in the computer model. F i g u r e
7D shows the s p a c e d e n s i t y of p a r t i c l e s
in t h e computer model t h a t would be
simulated.
Once the space d e n s i t y of
p a r t i c l e s is known, a multiplication by
the particle mass yields the c o n c e n t r a ­
tion of the flowing w a t e r .
In equation 1 t h e r e t a r d a t i o n factor
(Rd) i s used t o r e p r e s e n t t h e c h a n g e
in t h e solute concentration in t h e
fluid caused by chemical r e a c t i o n s with
t h e medium.
T h e s e r e a c t i o n s include
adsorption,
organic
fixation,
etc.
Chemical r e a c t i o n s between t h e d i s ­
solved c o n s t i t u e n t and t h e medium t e n d
to r e t a r d the movement of the c o n s t i t u ­
ent relative to t h e g r o u n d w a t e r veloci­
t y . T h e r e t a r d a t i o n of a c o n c e n t r a t i o n
front in g r o u n d w a t e r r e l a t i v e to the
bulk mass of water is d e s c r i b e d by t h e
relation
where
V = i n t e r s t i t i a l velocity of t h e
groundwater
14
V c = velocity of the C/C o =
0.5 in the c o n c e n t r a t i o n
front
= bulk mass d e n s i t y
n = effective porosity
K d = d i s t r i b u t i o n coefficient
R d = r e t a r d a t i o n factor
T h e t r a n s p o r t code herein can take
e i t h e r K d or R d as input in o r d e r
to i n c o r p o r a t e the effects of chemical
reactions.
L a t e r , in figure 42, exam­
ple o u t p u t will be shown for t = 1000
d a y s u s i n g R d = 1.0 and R d = 2 . 0 .
Determining K d or R d for a c o n s t i t ­
u e n t in g r o u n d w a t e r r e q u i r e s a great
deal of information r e g a r d i n g t h e com­
position of the g r o u n d w a t e r a n d t h e
i n t e r a c t i o n of the c o n s t i t u e n t with the
medium.
To p u r s u e t h i s , one should
c o n s u l t o t h e r r e f e r e n c e s , such as Helfferich (1962), Higgins (1959), Baetsle
(1967,1969), F r e e z e and C h e r r y (1979),
a n d Borg et al. (1976).
Radioactive decay may be a d d e d to
t h e model if d e s i r e d .
The s t a t e m e n t
"PM = PM*(0.5)**(TIME/HL)" should be
a d d e d to s u b r o u t i n e ADVAN(DELP) after
t h e statement "TMAP = TMAP+DELP." In
t h e s t a t e m e n t , HL is the half-life of
t h e isotope in d a y s .
The n u m b e r of
d a y s can either s u b s t i t u t e for HL or be
made an i n p u t p a r a m e t e r .
Figure 8.
Listing of the transport program
(See Part 2)
15
Figure 8. Continued
16
Figure 8. Continued
17
18
Figure 8. Continued
Figure 8. Continued
19
Figure 8. Continued
20
Figure 8. Continued
21
22
Figure 8. Continued
Figure 8.
Continued
23
24
Figure 8. Continued
F i g u r e 8. C o n t i n u e d
25
Figure 8. Continued
26
Figure 8.
Continued
27
Figure 8. Continued
28
Figure 8. Continued
29
Figure 8. Continued
30
Figure 8. Continued
31
Figure 8. Continued
32
Figure 8. Continued
33
Figure 8. Continued
34
Figure 8. Continued
35
Figure 8. Continued
36
Figure 8. Continued
37
Figure 8. Continued
38
Figure 8. Continued
39
Figure 8. Continued
40
Figure 8. Continued
41
Figure 8. Concluded
42
PART 2. RANDOM-WALK SOLUTE TRANSPORT PROGRAM
The solute t r a n s p o r t code listing
given in figure 8 was coded in FORTRAN
IV to solve the s e t s of e q u a t i o n s in­
volved.
T h i s section i l l u s t r a t e s what
p a r a m e t e r s are included and how to set
up a simulation model. T h e operational
s e q u e n c e s of the computer code a r e
given later in Part 3.
Figure 9 i l l u s t r a t e s some important
p a r a m e t e r s included in t h e p r o g r a m .
Briefly, t h e program can simulate o n e or
two-dimensional
nonsteady/steady
flow problems in h e t e r o g e n e o u s a q u i f e r s
u n d e r water table a n d / o r a r t e s i a n o r
leaky a r t e s i a n c o n d i t i o n s .
Further­
more, this program c o v e r s t i m e - v a r y i n g
pumpage or injection from
or into
wells,
n a t u r a l or artificial r e c h a r g e
r a t e s , the relationships of water e x ­
change between surface w a t e r s and the
g r o u n d w a t e r r e s e r v o i r , t h e p r o c e s s of
groundwater
evapotranspiration,
the
mechanism of possible conversion of
storage coefficients from a r t e s i a n to
water table conditions, and the mecha­
nism of flow from s p r i n g s .
In a d d i ­
tion, the program allows specification
of the w a t e r quality c o n c e n t r a t i o n s of
any part of the model i n c l u d i n g , but
not limited to, injection well w a t e r ,
s a l t - w a t e r f r o n t s , leachate from l a n d ­
fills, leakage from o v e r l y i n g s o u r c e
beds of different quality t h a n t h e
aquifer, and surface water s o u r c e s s u c h
a s lakes and s t r e a m s . F u r t h e r f e a t u r e s
of the code allow v a r i a b l e finite dif­
ference grid sizes and p r i n t o u t s of i n ­
put d a t a , time s e r i e s of aquifer h e a d s ,
and s e q u e n c e s of maps of aquifer water
c o n c e n t r a t i o n , c o n c e n t r a t i o n of water
flowing from s i n k s , and the effects of
d i s p e r s i o n and mixing of water of v a r i ­
ous c o n c e n t r a t i o n s .
The program listing of figure 8 is
written in such a way as to be in t h e
gallon-day-foot system of u n i t s .
How­
e v e r , t h i s code can be c o n v e r t e d to a
c o n s i s t e n t set of u n i t s by simply r e ­
placing t h e c o n s t a n t s ,
7.48 gallons/
cubic foot and 8.3453 p o u n d s / g a l l o n , by
the e q u i v a l e n t
desired.
measure
of
the
units
Job Setup
The computer job s e t u p will be e x ­
plained in general first, and t h e n in
more detail as n e c e s s a r y .
T h e aquifer system p r o p e r t i e s a r e
d i s c r e t i z e d by s u p e r p o s i n g a finite
difference grid over a map of t h e a q u i ­
fer system as shown in f i g u r e 10. T h e
total dimensions of the grid a r e d e ­
fined by NC, the number of columns of
the model, and by NR, t h e number of
rows of t h e model.
Next, a p a r a m e t e r
card (line) and default value c a r d
(line) a r e p r e p a r e d a c c o r d i n g to t h e
formats i l l u s t r a t e d in f i g u r e s 11A and
11B.
T h e default value card (line)
p r o v i d e s data for simulating an NC by
NR aquifer system model h a v i n g homo­
geneous p r o p e r t i e s with identical ini­
tial h e a d s and net withdrawal r a t e s .
T h e n , pumping and p a r a m e t e r s c h e d u l e
c a r d s (lines) are p r e p a r e d a c c o r d i n g t o
t h e card (line) format i l l u s t r a t e d in
f i g u r e s 11C and 11D. Next, a g r o u p of
c a r d s (lines) is p r e p a r e d which define
the sizes of the v a r i a b l e grid in t h e x
and y d i r e c t i o n s a c c o r d i n g to t h e for­
mat shown in figures HE and 11F. Fol­
lowing t h i s a r e the c a r d s (lines) p e r ­
taining to t h e water quality a s p e c t s of
the problem. Included a r e the c a r d for
initial
conditions
of
pollution
(see
format in figure 11G) and the card for
the particle information, aquifer d i s p e r s i v i t i e s , and porosity of t h e a q u i ­
fer (see format in f i g u r e 11H).
The
sink location c a r d s are t h e n p r e p a r e d
a c c o r d i n g to the format i l l u s t r a t e d in
figure H I , and the s o u r c e c o n c e n t r a ­
tion c a r d s are p r e p a r e d a c c o r d i n g to
figure 11J.
Finally, a node card deck
is p r e p a r e d according to the format
i l l u s t r a t e d in figure U K .
T h e node
card deck contains one card for each
node t h a t h a s any aquifer system p r o p ­
e r t i e s differing from t h o s e defined on
the default value c a r d .
43
Figure 9. Generalized aquifer cross section showing simulation
program parameters
Figure 10. Plan view of finite difference grid over majority
of sample aquifer system
The program deck and d a t a d e c k s a r e
assembled in t h e o r d e r i l l u s t r a t e d in
figure 12.
A p p r o p r i a t e control c a r d s
a r e included for the p a r t i c u l a r com­
p u t e r installation, and the program is
ready to r u n .
Preparation of Data
Parameter
Card
Enter numerical values for NSTEP,
DELTA, ERROR, and NPITS a c c o r d i n g to
the format given in figure 11A. A g e n ­
eral rule for choosing an initial DELTA
is to decide at what minimum time d r a w ­
downs or heads of i n t e r e s t occur and
then precede t h i s time by at least six
time i n c r e m e n t s .
Choose an initial
value of ERROR from the following for­
mula:
ERROR = Q x DELTA/(10 x DELX
x DELY x 7.48 x s)
(14)
where
Q = total net with­
drawal r a t e of
model, in gpd
DELTA = initial time in­
c r e m e n t , in d a y s
S = average storage
coefficient of
model, in g p d / f t
DELX and DELY= t y p i c a l grid
s p a c i n g in cen­
t e r of model
T h e computer program is t h e n " t e s t e d "
by making a few preliminary r u n s with
different values of ERROR, with a final
value being chosen at t h e point where
r e d u c t i o n in t h a t term does not signif­
icantly change the solution.
To p r o d u c e a steady s t a t e head d i s ­
t r i b u t i o n , set NSTEPS equal to 1, DELTA
equal to 1 0 1 0 , and ERROR equal to 0 . 1 .
A water balance s u b r o u t i n e may be a d d e d
h e r e to f u r t h e r check on t h e a c c u r a c y
45
46
F i g u r e 1 1 . Data deck s e t u p f o r a mass t r a n s p o r t p r o b l e m
Figure 12. Order of input cards for job setup
of a steady s t a t e s o l u t i o n .
Otherwise,
t h e solution should be t e s t e d by p r i n t ­
ing out t h e head d i s t r i b u t i o n e v e r y 10
i t e r a t i o n s or so to check to see if the
h e a d s have r e a c h e d a steady v a l u e .
NPITS is the number of times p a r t i ­
cles will be a d v a n c e d in the t r a n s p o r t
simulation d u r i n g each DELTA. T h e r e ­
fore NPITS x DELP = total simulation
time over which t r a n s p o r t o c c u r s in a
s t e a d y - s t a t e flow s i t u a t i o n .
Default Value Card
After e n t e r i n g values of NC and NR
on the card ( l i n e ) , e n t e r the most com­
monly o c c u r r i n g values of the aquifer
system p a r a m e t e r s .
Pump Parameter Card
Schedule
Cards
and
Pumping
A stepwise pumping s c h e d u l e , as il­
l u s t r a t e d in f i g u r e 13, is set up for
each well of the model.
T h e computer
program can manipulate positive ( p u m p ­
ing),
negative ( r e c h a r g e ) ,
o r zero
withdrawal r a t e s for the total number
of wells (NPUMP). Between c h a n g e s in
pumping r a t e , the computer program
o p e r a t e s with nonuniform time s t e p s
within each period of p u m p i n g .
How­
e v e r , the initial DELTA is r e s e t at
each change in pumping r a t e for the
r e a s o n s outlined in the nonuniform time
increment section of this r e p o r t .
As an example, figure 13 shows four
pumping r a t e c h a n g e s (NRT = 4 ) . Each
pumping r a t e is in effect for 11 time
i n c r e m e n t s (NSP = 11), and t h e total
number of time i n c r e m e n t s is 44 (NSTEPS
= 44).
T h e l e n g t h of time t h a t each
pumping r a t e is effective is the same
for all w e l l s .
Once t h e pumping s c h e d u l e s h a v e been
set u p , t h e pump parameter card is in­
cluded to define NP, NSP, and NRT ac­
c o r d i n g to the format shown in f i g u r e
47
Figure 13. Example variable pumping rate schedule
11C.
Next t h e r a t e and well location
c o o r d i n a t e s a r e e n t e r e d on pumping
s c h e d u l e c a r d s a c c o r d i n g to the format
shown in figure 11D.
detailed a r e a of i n t e r e s t will
simplify the i n t e r p r e t a t i o n of the
puter output.
Pollution Initial Conditions
Variable Grid Cards
T h e v a r y i n g l e n g t h s , DELX(I) and
D E L Y ( J ) , of t h e finite difference grid,
as i l l u s t r a t e d in f i g u r e s 2 and 10, a r e
p r e p a r e d in a c c o r d a n c e with the formats
shown in f i g u r e s HE and 11F. A l t h o u g h
it is not n e c e s s a r y , it is s u g g e s t e d
t h a t l a r g e jumps in grid sizes be held
to a minimum to maintain a c c u r a c y .
It
is recommended t h a t l e n g t h c h a n g e s from
one node to t h e n e x t be l e s s t h a n d o u ­
b l e . Having a uniform grid size in t h e
48
also
com­
Card
T h e r e a r e s e v e r a l mechanisms avail­
able in this computer program for g e n ­
e r a t i n g particles which r e p r e s e n t water
of
various
chemical
concentrations.
T h e r e a r e two p r i n c i p a l p a r a m e t e r s of
concern here:
the area and the time
period over which t h e p a r t i c l e s s p r e a d .
F i g u r e 14 i l l u s t r a t e s t h e t y p e s of
a r e a s for which t h e computer program
can g e n e r a t e p a r t i c l e s . T h e most com­
monly used area for field problems is
t h e r e c t a n g l e shown in figure 14C.
F i g u r e 14. D e f i n i t i o n of p a r t i c l e e m i t t i n g s u b r o u t i n e s
49
Since t h i s t y p e of area would be used
most f r e q u e n t l y , it is made p a r t of t h e
c a r d input d a t a as opposed to c h a n g e s
being made in some s u b r o u t i n e .
The
d a t a to be e n t e r e d on t h e pollution
initial conditions card a r e t h u s d e ­
fined in f i g u r e 14C.
At this point, it is e s s e n t i a l to
realize t h a t in this model t h e p a r t i icles are moving on the basis of an x - y
coordinate system c o n g r u e n t with and on
the same scale as the I-J n u m b e r s of
the finite difference g r i d .
In o t h e r
w o r d s , if a particle is g e n e r a t e d at
t h e location x = 15, y = 4, t h i s p a r t i ­
cle is at grid coordinates I = 15, J =
4 even t h o u g h the actual field d i s t a n c e
measured from the origin is something
else.
For i n s t a n c e , refer to figure 10
w h e r e the sample d i s c r e t i z e d a r e a of a
streambed is a s s i g n e d to node I = 7, J
= 6.
To g e n e r a t e particles within that
area you would e n t e r Xl = 6 . 5 , DX =
1.0, Yl = 5 . 5 , and DY = 1.0 on t h e pol­
lution initial conditions c a r d .
Of
c o u r s e , by a change in DX or DY the
size of the r e c t a n g l e can be adjusted
from a point (DX = DY = 0) on up to an
area the size of the r e m a i n d e r of the
model (DX = NC - Xl, DY = NR - Y l ) .
The last e n t r y on the pollution ini­
tial conditions card is the time i n t e r ­
val DELP o v e r which p a r t i c l e s will be
generated.
In general, DELP g o v e r n s
the time the particles a r e allowed to
move and goes hand in h a n d with the
time increment DELTA of the flow
model.
F u r t h e r r e f e r e n c e to figure 14 will
be made l a t e r in this r e p o r t when o t h e r
ways of g e n e r a t i n g p a r t i c l e locations
will be e x p l a i n e d .
Pollution
Parameter
Card
In most c a s e s , the total pollutant
load ( P L ) , which is the total mass of
pollutant at 100 p e r c e n t c o n c e n t r a t i o n ,
is calculated on the basis of t h e p a r ­
ticle mass (PM) and t h e flow r a t e s or
number of p a r t i c l e s d e s i r e d within vol­
umes of the finite difference
grid
chosen t o r e p r e s e n t t h e c o n c e n t r a t i o n
50
of the s o l u t e .
T h e usual way to p r o ­
ceed is the following:
Assume an ini­
tial c o n c e n t r a t i o n of total dissolved
solids (TDS) of 200 m g / l . You s u s p e c t
t h a t , b e c a u s e of t h e effects of d i s p e r ­
sion and dilution,
the c o n c e n t r a t i o n
will r e d u c e e v e n t u a l l y to no more t h a n
1 m g / 1 . T h e n , calculate t h e mass (PM)
of a single particle a c c o r d i n g to the
following formula:
In t e r m s of the computer model, e q u a ­
tion 15 is w r i t t e n
where
APOR = actual p o r o s i t y of
aquifer
62.4 = p o u n d s per cubic
feet of water
CONC= ppm ≈ mg/1
H-BOT = s a t u r a t e d t h i c k n e s s
of aquifer
NPART= number of particles
r e p r e s e n t i n g con­
c e n t r a t i o n of TDS
in t h e p e r t i n e n t
volume
DELX, DELY = dimensions of t y p i ­
cal finite differ­
ence grid
PM= p o u n d s of TDS per
particle
Good resolution of t h e d i s t r i b u t i o n of
t h e TDS as t h i s water moves t h r o u g h t h e
aquifer can be obtained if we a r r a n g e
to have 200 p a r t i c l e s per grid space in
t h e model.
(Choice of the number of
particles per grid d e p e n d s upon t h e
resolution d e s i r e d . )
The d i s t r i b u t i o n
of the TDS t h r o u g h t h e aquifer may be
obtained from the particle d i s t r i b u t i o n
in the model by t h e following formula:
The computer code has I , J s u b s c r i p t s
on all of the p a r a m e t e r s of equation 16
except PM and APOR such t h a t t h e v a r i ­
able n a t u r e of t h e aquifer p a r a m e t e r s
is a c c o u n t e d f o r .
The value of PL to be filled in then
is t h e total number of p a r t i c l e s wanted
times PM. Under most c i r c u m s t a n c e s , PM
is t h e only calculation to be made, and
the computer will determine a PL.
T h e maximum number of p a r t i c l e s ,
MAXP, should seldom exceed 5000. T h e r e
may be problems t h a t r e q u i r e more, but
e x p e r i e n c e indicates t h a t 5000 is u s u ­
ally a d e q u a t e .
Values of longitudinal d i s p e r s i v i t y ,
in feet, d L = DISPL; t r a n s v e r s e d i s ­
p e r s i v i t y , in feet, d T = DISPT; ef­
fective aquifer p o r o s i t y EPOR ( f r a c ­
t i o n ) ; actual porosity (APOR); t h e r e ­
t a r d a t i o n factor (RD1); the d i s t r i b u ­
tion coefficient (KD) for the solute in
the p a r t i c u l a r aquifer material;
and
the bulk mass d e n s i t y of t h e aquifer
material should t h e n be e n t e r e d on the
pollution parameter card a c c o r d i n g to
the format indicated in figure 11H.
Node
Card
Deck
It should be emphasized t h a t t h e
node deck (lines) contains one card
(line) for each node t h a t h a s any
aquifer
system
properties
differing
from t h o s e defined on the default value
card (line).
If a node card (line) is
i n c l u d e d , all values must be p u n c h e d on
it even if some of t h e values a r e equal
to the default v a l u e s .
It is e s s e n t i a l
to h a v e a copy of t h e r e p o r t of P r i c k ett and Lonnquist
(1971)
to fully
u n d e r s t a n d the details of s e t t i n g up
the flow model.
Barrier Boundary Conditions.
Bar­
r i e r b o u n d a r i e s a r e simulated by s e t ­
t i n g PERM ( I , J , 1 ) and PERM ( I , J , 2 )
e q u a l to zero on individual node c a r d s .
Leaky Artesian Conditions.
For
leaky a r t e s i a n conditions calculate the
r e c h a r g e factors ( R i , j ) i n gallons per
d a y p e r foot ( g p d / f t ) from the follow­
ing formula:
where
Sink
Location
P r e p a r e a card with t h e I , J coordi­
nate for each s i n k d e s i r e d , according
to the format of figure 11I. On each
of t h e s e cards is a location for e n t e r ­
ing an i n t e g e r , o t h e r t h a n zero, termed
MARK. All sinks with the same n u m b e r
in MARK will be t r e a t e d as a common
sink when r e p o r t i n g c o n c e n t r a t i o n s . If
s e p a r a t e r e p o r t s on all s i n k s
are
wanted,
e n t e r different i n t e g e r s on
each of the sink location c a r d s in the
field MARK.
Source
= vertical h y d r a u l i c c o n d u c t i v ­
ity of confining bed, in gal­
lons per day per s q u a r e foot
(gpd/ft2)
= t h i c k n e s s of confining b e d ,
in feet
Cards
Concentration
Cards
Each node of the model t h a t has a
source of water with a c o n c e n t r a t i o n
different from zero needs a source con­
c e n t r a t i o n card with t h e I , J c o o r d i ­
n a t e s and concentration CONC, in m g / l ,
listed according to the format shown in
figure 11J.
T h e r e c h a r g e factor defines t h e slope
of t h e line given in figure 9C.
A p p r o p r i a t e s o u r c e bed h e a d s ( R H ) ,
elevations of the t o p of the a q u i f e r
( C H ) , and r e c h a r g e factors a r e e n t e r e d
on t h e default value card, and a n y dif­
fering values a r e e n t e r e d on t h e node
cards.
Induced
Infiltration.
d u c e d infiltration calculate
c h a r g e factors (in g p d / f t )
following formula:
For
the
from
in­
re­
the
where
= h y d r a u l i c c o n d u c t i v i t y of t h e
stream bed, in g p d / f t 2
51
= t h i c k n e s s of s t r e a m b e d , in
feet
A s = fraction of the area of t h e
s t r e a m b e d a s s i g n e d to full
area node, a fraction
T h e r e c h a r g e factor defines the slope
of t h e line shown in f i g u r e 9E.
A p p r o p r i a t e v a l u e s of stream water
s u r f a c e elevations ( R H ) , elevations of
the bottom of t h e s t r e a m b e d ( R D ) , and
r e c h a r g e f a c t o r s calculated from the
a b o v e equation a r e e n t e r e d on node
cards.
Constant Head Conditions.
If a
c o n s t a n t head is d e s i r e d in t h e model,
and t h e c o n c e n t r a t i o n of the water
flowing into that node to maintain the
head is of zero c o n c e n t r a t i o n , t h e n set
S F I ( I , J ) at t h a t node equal to 1 0 3 0 .
If the c o n c e n t r a t i o n is o t h e r t h a n
z e r o , set R ( I , J ) equal to 1 0 1 0 for that
node c a r d . T h i s is done to maintain a
small
resistance
to flow
from
the
s o u r c e to the aquifer enabling t h e flow
r a t e to be m e a s u r e d .
Storage F a c t o r s .
In the program
d e s c r i b e d by P r i c k e t t and L o n n q u i s t
(1971), s t o r a g e f a c t o r s were e n t e r e d on
52
the node c a r d s . T h i s is not n e c e s s a r y
here.
J u s t e n t e r the a r t e s i a n and
water table storage
coefficients
di­
r e c t l y i n t o S F 1 ( I , J ) and SF2 ( I , J )
positions on the node and default value
cards.
Evapotranspiration.
Evapotrans p i r a t i o n is defined by t h e slope of
the line shown in f i g u r e 9D and is cal­
culated from field d a t a ( P r i c k e t t and
L o n n q u i s t , 1971).
The value of t h e
slope of t h a t line is then e n t e r e d on
node c a r d s in the space r e s e r v e d for
the r e c h a r g e f a c t o r .
Set v a l u e s of
both RD a n d RH e q u a l to one a n o t h e r .
RD is defined as t h e elevation of the
w a t e r t a b l e below which t h e effects of
evapotranspiration cease.
Flow from S p r i n g s .
Determine a
r e c h a r g e factor (slope of the line
shown in f i g u r e 9D) from field d a t a of
flow v e r s u s head c h a n g e s in the v i c i n i ­
ty of the s p r i n g .
A r e c h a r g e factor
may also be found empirically by m a t c h ­
ing simulated with o b s e r v e d s p r i n g
flows.
T h e elevation at which water
flows from t h e s p r i n g is r e c o r d e d in
both RH and R D .
PART 3.
PROGRAM O P E R A T I O N A L SEQUENCES
The basic operation of the computer
program is explained in t h e following
discussion
according to the listing
given in figure 8 and t h e flow c h a r t s
and associated e x p l a n a t i o n s given in
figures 15 t h r o u g h 36.
The computer program c o n s i s t s of a
MAIN section, 20 s u b r o u t i n e s , and 3
functions.
As i l l u s t r a t e d in figure
15, it first r e a d s and w r i t e s all i n p u t
data.
After t h i s , the MAIN program
calls a s e q u e n c e of s u b r o u t i n e s , as
n e e d e d , t h a t 1) solves for the h e a d s
for a p a r t i c u l a r time increment DELTA;
2) p r i n t s t h e h e a d s ; 3) g e n e r a t e s p a r ­
ticles and i n s e r t s them in t h e flow;
4) allows t h e p a r t i c l e s to move a time
s t e p DELP; and 5) p r i n t s maps and sum­
maries of particle locations a n d con­
centration d i s t r i b u t i o n s . T h e s e q u e n c e
of s u b r o u t i n e s called by t h e MAIN p r o ­
gram can be c h a n g e d a c c o r d i n g to the
p a r t i c u l a r flow
and mass
transport
problem u n d e r s t u d y . After the d e s i r e d
s e q u e n c e of s u b r o u t i n e s is called, the
MAIN program cycles to t h e second time
F i g u r e 15. Flow c h a r t f o r MAIN p r o g r a m
53
increment of t h e flow model and p e r ­
forms the above 5 s t e p s r e p e a t e d l y
until all time i n c r e m e n t s are p r o c ­
essed.
T h e r e a r e 5 r e q u i r e d s u b r o u t i n e s and
3 required functions.
T h e r e are also
15 optional s u b r o u t i n e s to choose from,
d e p e n d i n g on t h e t r a n s p o r t problem of
interest.
Of t h e s e , t h e r e a r e 3 s u b ­
r o u t i n e s for p r o d u c i n g t h e head d i s t r i ­
bution for the flow model, and 7 s u b ­
routines t h a t g e n e r a t e different con­
figurations and n u m b e r s of p a r t i c l e s .
T h e r e is 1 s u b r o u t i n e to calculate
retardation,
and 4 s u b r o u t i n e s t h a t
print out v a r i o u s maps of particle lo­
cation,
concentration
distributions,
and summaries of c o n c e n t r a t i o n of water
flowing into s i n k s .
Table 1 provides
brief d e s c r i p t i o n s of the s u b r o u t i n e s
and f u n c t i o n s .
T h e details on how to
mix a n d match t h e various s u b r o u t i n e s
will be given in t h e example problems
section (Part 4 ) .
Required Subroutines and Functions
Function
ANORM(0)
F i g u r e 16 gives the flow c h a r t for
this f u n c t i o n .
T h i s function p r o d u c e s
a number between -6 and +6 from a n o r ­
mal d i s t r i b u t i o n of n u m b e r s having a
mean of zero and a s t a n d a r d deviation
of o n e .
In f i g u r e 8 it can be seen
that this function u s e s yet another
function called RANF(O), which is a CDC
system s u b r o u t i n e t h a t p r o d u c e s a r a n ­
dom number between 0 and 1. If other
systems are used, an appropriate s u b ­
s t i t u t e random number g e n e r a t o r t h a t
p r o d u c e s n u m b e r s in the same r a n g e can
be u s e d .
Function
V(I,J,K)
As shown in f i g u r e 17, whenever
calls a r e made to this f u n c t i o n , it is
t h e n o d e - t o - n o d e i n t e r s t i t i a l velocity
t h a t is s u p p l i e d .
T h e s e velocities are
the basis for which an e l a b o r a t e i n t e r ­
polation scheme is applied to provide
velocities a n y w h e r e in x - y s p a c e .
54
Subroutine
INIT
T h i s s u b r o u t i n e could j u s t as easily
h a v e been placed as a p a r t of MAIN.
However, by its n a t u r e it can be useful
as a s u b r o u t i n e for some special appli­
cations.
As shown in f i g u r e 18, this
s u b r o u t i n e merely zeros out t h e TMAP
and NP v a r i a b l e s and t h e S O R ( I , J ) ,
C O N S O R ( I , J ) , and MARK(I,J) a r r a y s .
Subroutine
CLEAR
As i l l u s t r a t e d in t h e flow c h a r t in
figure 19, t h i s s u b r o u t i n e zeros out
the N P A R T ( I , J ) and TABLE(I) a r r a y s .
T h i s is like e r a s i n g t h e b l a c k b o a r d in
p r e p a r a t i o n for k e e p i n g t r a c k of p a r t i ­
cle locations each time p a r t i c l e s are
going to be m o v e d .
Subroutine
ADD(XX,YY)
Any time t h a t a new p a r t i c l e is
a d d e d to the simulation, you must call
ADD(XX, YY) and s u p p l y the XX, YY coor­
d i n a t e s of the particle t h a t you want
added.
As indicated in f i g u r e 20, the
particle c o u n t e r NP is i n c r e m e n t e d a n d ,
as long as NP is less t h a n t h e maximum
number of p a r t i c l e s allowed MAXP, the
particle XX, YY coordinates a r e s t o r e d
in the X and Y a r r a y s .
In the e v e n t
t h a t you a r e a t t e m p t i n g t o add p a r t i ­
cles beyond t h e limits of t h e X a n d Y
a r r a y s , the code is w r i t t e n in s u c h a
way as to eliminate an a d d r e s s i n g
e r r o r . For many c a s e s , you don't know
how many particles will be n e e d e d for
your simulation since,
for i n s t a n c e ,
the r a t e of flow from a n e a r b y pollu­
tion source may be u n k n o w n .
In t h i s
case, the flow may be sufficient to
r e q u i r e a number of p a r t i c l e s far in
e x c e s s of the dimensions of t h e X a n d Y
arrays.
If t h i s h a p p e n s , t h e program
is written to randomly remove a p a r t i ­
cle somewhere in the model and make
room for the new o n e .
It may be t h a t
even
the
new
particle
itself
gets
thrown out.
In any case t h e particle
mass (PM) of all o t h e r p a r t i c l e s is
then p r o p o r t i o n a t e l y i n c r e a s e d to main­
tain
conservation-of-mass
principles.
T h e v a r i a b l e s NP and PM should be
checked as the simulation p r o g r e s s e s to
Table 1.
ADD(XX,YY):
Srief Descriptions of S u b r o u t i n e s and Functions
a d d single p a r t i c l e a t c o o r d i n a t e s ( X X , Y Y )
ADVAN(DELP): a d v a n c e s all p a r t i c l e s DELP d a y s
ANORM(O):
p r o d u c e s a single n u m b e r from a normal d i s t r i b u t i o n of mean
r a n g e +6 to -6
CLEAR:
c l e a r s a r r a y s NPART and TABLE
CONMAP:
p r i n t s c o n c e n t r a t i o n s at all n o d e s of model
GENP(PL) :
p r o d u c e s PL/PM p a r t i c l e s r a n d o m l y in a r e c t a n g l e Xl, Y l . D X , D Y
a n d randomly in time i n c r e m e n t DELP
GENP2:
g e n e r a t e s 29 p a r t i c l e s along column 30 of model,
p e r node
GENP3:
g e n e r a t e s 51 p a r t i c l e s , all at c o o r d i n a t e XX = 1, YY = 2
GENP4:
g e n e r a t e s 360 p a r t i c l e s a r o u n d an R = 0.7 c i r c l e with c e n t e r
a t c o o r d i n a t e s ( 1 5 , 1 5 ) , randomly d u r i n g DELP
GENP5:
g e n e r a t e s 101 p a r t i c l e s a r o u n d an R = 0 . 7 c i r c l e with c e n t e r
at c o o r d i n a t e s ( 1 5 , 1 5 ) , at t h e onset of t h e call
HSOLVE:
ISVVS Bulletin 55 ( P r i c k e t t and L o n n q u i s t ,
program
HSOLV2:
c a l c u l a t e s l i n e a r h e a d s p e r g r i d left t o r i g h t
HSOLV4:
Thiem e q u a t i o n ,
d i n a t e s (15,15)
INIT:
initializes o r z e r o s o u t a r r a y s TMAP,
CONSOR, and MARK
MAP:
p r i n t s number of
model
r a d i a l flow
particles
from
an
residing
1971),
1 particle
composite
injection well at
NP,
in
ANC,
each
ANR,
coor­
SOR,
zone for whole
MOVE(XX, YY, DEL) : moves p a r t i c l e s a n d t a b u l a t e s in which zone p a r t i c l e
r e s i d e s a n d r e m o v e s p a r t i c l e s when c a p t u r e d by a sink
RDSOLV(EPOR,RHO,KD,RDl):
c a l c u l a t e s r e t a r d a t i o n f a c t o r from i n p u t d a t a
SNKCON:
for time DELP,
MARK
SORGEN:
p r o d u c e s p a r t i c l e s b a s e d o n e i t h e r h e a d d e p e n d e n t flow r a t e s
or injection wells r a n d o m l y in s p a c e in time d u r i n g DELP
SOURCE:
p r o d u c e s p a r t i c l e s when p a r t i c l e mass
m u l a t e d from a n y s o u r c e
SUMMRY:
p r i n t s out n u m b e r of
time DELP
V:
p r o v i d e s p r o p e r velocities t o s u b r o u t i n e VELO
VELO(XX, YY, VX, VY) :
prints
concentration
particles
at
sinks
specified
by
h a s sufficiently a c c u ­
captured
by a
sink
during
calculates i n t e r p o l a t e d velocities for p a r t i c l e movement
55
Figure 16. Flow chart for
Function ANORM(0)
Figure 19. Flow chart for
Subroutine CLEAR
Figure 17. Flow chart for
Function V ( I , J , K )
Figure 18. Flow chart for
Subroutine INIT
56
Figure 20. Flow chart for
Subroutine A D D ( X X , Y Y )
see if this is h a p p e n i n g .
If it i s ,
back up t h e simulation a n d make n e c e s ­
s a r y c h a n g e s such t h a t PM s t a y s con­
stant.
Suitable c h a n g e s might be to
i n c r e a s e PM or MAXP and t h e X,Y a r r a y
dimensions.
Subroutine
VELO(XX,YY,VX,VY)
This s u b r o u t i n e (see f i g u r e 21) p r o ­
vides velocity v e c t o r s for the p a r t i ­
cles as t h e y move a n y w h e r e within the
b o u n d s of the model. It is v e r y impor­
tant t h a t an a c c u r a t e scheme be em­
ployed to p r o d u c e a continuous velocity
field from d i s c r e t e velocities provided
from F u n c t i o n V ( I , J , K ) .
One of the
main d r a w b a c k s of the MOC is its ina­
bility to handle the s t r o n g l y d i v e r g e n t
problem.
For i n s t a n c e , in the vicinity
of injection wells the flow is entirely
d i v e r g e n t away from t h e c e n t e r of the
well.
This
divergent
flow
problem
is
avoided in t h e code by borrowing an
idea from t h e finite element method. A
set of Chapeau functions is defined for
the p u r p o s e of i n t e r p o l a t i n g velocities
a n y w h e r e in the model on the basis of
the d i s c r e t e velocities between n o d e s .
T h e explanation of how t h e s e C h a p e a u
functions w o r k is shown in f i g u r e s 22
t h r o u g h 24.
T h e interpolation of velocities is a
three-step procedure.
T h e first i n t e r ­
polation involves the u s e of the four
i n t e r - n o d e velocities in t h e immediate
vicinity of the x, y position of the
p a r t i c l e , as i l l u s t r a t e d in figure 22.
The second s t e p of the i n t e r p o l a t i o n is
i l l u s t r a t e d in figure 23,
where the
n e x t closest four i n t e r - n o d e velocities
are used.
The t h i r d i n t e r p o l a t i o n is
shown in figure 24, w h e r e t h e final
velocity v e c t o r s VX and VY a r e calcu­
lated.
In the event t h a t t h e particle is
near the model b o u n d a r i e s , the v e c t o r s
used for interpolation a r e as shown in
figure 25.
T h e r e s u l t of
this
interpolation
scheme is a c o n t i n u o u s velocity field
that p r o v i d e s a c c u r a t e velocities even
for the s t r o n g l y d i v e r g e n t problem (see
figure 42, for e x a m p l e ) .
F i g u r e 2 1 . Flow c h a r t f o r S u b r o u t i n e
VELO(XX,YY,VX,VY)
57
Figure 22. First step in calculating x and y velocity
vectors for particle
58
F i g u r e 23. Second step in c a l c u l a t i n g x and y v e l o c i t y
vectors for particle
59
Figure 24. T h i r d step in calculating x and y velocity
vectors for particle
60
Figure 25. Node to node velocity vectors used in
interpolation scheme
61
Subroutine
ADVAN(DELP)
This s u b r o u t i n e controls a n d a d ­
v a n c e s all p a r t i c l e s p r e s e n t l y in t h e
flow field and does some bookkeeping on
particle x, y c o o r d i n a t e locations as
some particles d i s a p p e a r in s i n k s . T h e
flow c h a r t in f i g u r e 26 i n d i c a t e s the
s e q u e n c e of e v e n t s .
The particles a r e
all allowed to move for a time period
defined by DELP. For most cases DELP
is equal to the flow model time i n c r e ­
ment DELTA.
However, if it is d e s i r e d to move
t h e particles more often between DELTA
time s t e p s , call ADVAN(DELP) more often
a n d such t h a t t h e sum of the DELPs
e q u a l s DELTA. S t e a d y s t a t e flow p r o b ­
lems a r e an example, as i l l u s t r a t e d in
the examples section of this r e p o r t .
Function
MOVE(XX,YY,DEL)
E v e r y time t h a t this function is
called, a particle is moved.
Calls
will be coming from s u b r o u t i n g ADVAN
(DELP) a n d certain particle g e n e r a t i o n .
As mentioned p r e v i o u s l y , the p a r t i c l e s
move on the basis of the I , J coordinate
system numbers.
T h e flow c h a r t for
function MOVE is given in figure 27 a n d
should be s t u d i e d along with t h e l i s t ­
ing in f i g u r e 8.
T h e function MOVE moves a particle
on t h e basis of its velocity v e c t o r s as
calculated from s u b r o u t i n g VELO. New
velocity v e c t o r s a r e computed at least
5 times d u r i n g t h e time a particle is
moving a c r o s s 1 grid of the model. This
is done to maintain high a c c u r a c y .
T h e r e are cases w h e r e this r e s t r i c t i o n
can be r e l a x e d , and some cases w h e r e it
can be t i g h t e n e d u p . However, 5 times
is a good number for most p r o b l e m s . It
is a d v i s e d t h a t t h e flow c h a r t in fig­
u r e 27, the listing in f i g u r e 8, and
the v e c t o r a l g e b r a in figure 6 be s t u d ­
ied to see how the s e q u e n c e of e v e n t s
o c c u r s when t h e function MOVE is
called.
T h e function MOVE performs two o t h e r
t a s k s besides moving a p a r t i c l e . F i r s t ,
it c h e c k s to see if t h e particle goes
into a sink; i . e . , within half a grid
62
Figure 26. Flow chart for
Subroutine ADVAN(DELP)
of M A R K ( I , J ) . If so, the mass of t h e
particle is t a b u l a t e d in t h e TABLE(I)
array.
Secondly, after t h e particle
moves i t s a s s i g n e d DELP time, if it
doesn't go into a s i n k , i t s x , y c o o r d i ­
nates a r e used to t a b u l a t e its position
in the NPART ( I , J ) a r r a y . T h e NPART and
TABLE a r r a y s are p r i n t e d when a map of
particle positions and s i n k h i s t o r i e s
is d e s i r e d .
Optional Subroutines That
Produce Head Distributions
A set of h e a d s must be obtained for
all nodes of the flow model from which
to compute the velocity field.
The
t h r e e s u b r o u t i n e s d i s c u s s e d h e r e can be
used to fill in the head a r r a y H ( I , J ) ,
However, it should be realized t h a t
almost any means can be used to do
this.
A new code could be w r i t t e n ,
someone else's code could be u s e d , or
field d a t a for the head a r r a y could be
inserted.
T h e t h r e e following methods
are j u s t e x a m p l e s .
Subroutine
HSOLVE
This s u b r o u t i n e comes directly from
SWS Bulletin 55 ( P r i c k e t t and L o n n quist, 1971) and is a p o r t i o n of t h e i r
" Composite Aquifer Simulation P r o g r a m . "
The section of P a r t 2 of this r e p o r t
entitled "Job S e t u p " is based upon
using this s u b r o u t i n e as t h e means to
fill in t h e H ( I , J ) a r r a y .
The flow
c h a r t in figure 28 is t h u s a b b r e v i a t e d .
A copy of SWS Bulletin 55 is needed to
fully u n d e r s t a n d t h i s . T h e "Job S e t u p "
section of this r e p o r t , however,
is
a d e q u a t e for a s t a r t .
Figure 27. Flow chart for
Function MOVE(XX, YY,DEL)
F i g u r e 28. Flow c h a r t f o r
S u b r o u t i n e HSOLVE
63
Subroutine
HSOLV2
This s u b r o u t i n e calculates heads as
a function of column number such that a
uniform flow of 1 foot p e r day in the x
direction is r e a l i z e d .
Refer to the
l i s t i n g in f i g u r e 8 and t h e flow chart
in figure 29 for the d e t a i l s .
This is one example of g e n e r a t i n g a
head d i s t r i b u t i o n d i r e c t l y without the
n e e d of solving some complex matrix
problem associated with finite differ­
ence formulation.
Subroutine
HS0LV4
This s u b r o u t i n e fills the H ( I , J )
a r r a y with a solution to the s t e a d y s t a t e Theim formula with t h e c e n t e r of
t h e well at x = 15 a n d y = 15. T h e flow
c h a r t in f i g u r e 30 a n d t h e s u b r o u t i n g
code in f i g u r e 8 show how t h i s was
done.
T h e idea h e r e is to waste no
time in p r o d u c i n g a head d i s t r i b u t i o n
when you know exactly what you w a n t .
Figure 30. Flow chart for
Subroutine HSOLV4
Optional Subroutines That
Generate Particles
Figure 29. Flow chart for
Subroutine HSOLV2
64
Seven
different
subroutines
are
included as examples of ways particles
can be g e n e r a t e d in v a r i o u s c o n f i g u r a ­
tions.
Configurations
available a r e
p o i n t s , l i n e s , r e c t a n g u l a r a r e a s , and
circles.
T h r o u g h o u t t h e following d i s ­
cussion,
r e p e a t e d r e f e r e n c e will be
made to figure 14, as this figure indi­
cates the needed i n p u t data for g e n e r ­
a t i n g particles for the v a r i o u s s u b ­
routines.
In general, the particle g e n e r a t i n g
s u b r o u t i n e s merely g e n e r a t e x, y c o o r d i ­
n a t e s in v a r i o u s configurations and
t h e n call the s u b r o u t i n e ADD(XX,YY) to
get the particle into the flow.
Subroutine
GENP2
The flow c h a r t in figure 31, the
l i s t i n g in figure 8, and the illus­
t r a t e d definition in figure 14B shows
how to place one particle at each node
of t h e flow model in a l i n e . Again x, y
c o o r d i n a t e s a r e g e n e r a t e d and t h e n the
s u b r o u t i n e ADD(XX,YY) is called.
Subroutine
flow c h a r t is given in f i g u r e 32. S u i t ­
able c h a n g e s in t h e xx, yy c o o r d i n a t e s
of this s u b r o u t i n e could e n a b l e the
g e n e r a t i o n of p a r t i c l e s at a n y point in
the model.
Subroutine
GENP(PL)
T h i s s u b r o u t i n e g e n e r a t e s a n y number
of particles randomly located within a
r e c t a n g u l a r area of configuration given
by f i g u r e 14C. T h e n e c e s s a r y Xl, DX,
Yl, DY, DELP coordinates may be s p e c i ­
fied by the initial pollution c o n d i ­
tions card or by a call from a n o t h e r
s u b r o u t i n e such as SORGEN. The total
pollutant load PL can also be specified
in the pollution p a r a m e t e r card or by a
call to a n o t h e r s u b r o u t i n e such as
GENP3
F i g u r e 14A and t h e listing in figure
8 i l l u s t r a t e how one or more p a r t i c l e s
can be placed at a p a r t i c u l a r point in
t h e model at one i n s t a n t of time.
The
F i g u r e 3 1 . Flow c h a r t f o r
S u b r o u t i n e GENP2
F i g u r e 32. Flow c h a r t f o r
S u b r o u t i n e GENP3
65
SORGEN. The idea here is that once PL
is s u p p l i e d , the s u b r o u t i n e b r e a k s up
t h e total into p a r t s PL/PM and s p r e a d s
particles randomly in t h e r e c t a n g u l a r
a r e a specified and randomly o v e r t h e
time increment DELP specified until the
total PL is a c c o u n t e d for.
T h e listing
in f i g u r e 8 and t h e flow c h a r t in fig­
u r e 33 give more d e t a i l s .
Subroutines
GENP4
and
GENP5
The flow c h a r t and listing in fig­
u r e s 34 and 8, r e s p e c t i v e l y , i l l u s t r a t e
how a circle of p a r t i c l e s is g e n e r a t e d
a c c o r d i n g to the configuration shown in
figure 14E.
Here a g a i n , the x and y
c o o r d i n a t e s are g e n e r a t e d and ADD(XX,
YY) is called to c r e a t e t h e p a r t i c l e s .
GENP5 g e n e r a t e s 101 p a r t i c l e s in a
circle uniformly s p a c e d at R = 0.7 at
the b e g i n n i n g of the time increment
DELP. S u b r o u t i n e GENP4 is the same as
GENP5 e x c e p t t h a t 360 p a r t i c l e s a r e
g e n e r a t e d randomly d u r i n g t h e time i n ­
crement DELP. T h e listing in figure 8
shows t h a t this is done by calling
RANF.
Subroutine
SORGEN
This s u b r o u t i n e is similar to SOURCE
in t h a t particles a r e emitted on the
basis of flow r a t e s from sources having
a c o n c e n t r a t i o n different t h a n z e r o .
A
value in the a r r a y C O N S O R ( I , J ) is all
t h a t is n e c e s s a r y to a c t u a t e a p a r t i c l e
when this s u b r o u t i n e is called.
Both
s u b r o u t i n e s SORGEN and SOURCE calcu­
late the total mass (PL) involved in
the water flowing from the v a r i o u s
s o u r c e s of the model on the basis of
the specified p a r t i c l e mass (PM) and
the flow r a t e . As shown in the listing
in f i g u r e 8, the flow c h a r t in figure
35, and the area definition in f i g u r e
14D, particles a r e emitted by calling
GENP(PL) on the b a s i s of calculated PL
v a l u e s . T h u s , p a r t i c l e s a r e emitted b y
s u b r o u t i n e SORGEN randomly in both
space and time (DELP) .
66
Figure 33. Flow chart for
Subroutine GENP(PL)
Subroutine
SOURCE
T h i s s u b r o u t i n e is similar to SORGEN
with t h e e x c e p t i o n t h a t particles a r e
emitted r i g h t at t h e I , J coordinates of
the source and by calling s u b r o u t i n e
ADD (XX, YY ) only when a sufficient q u a n ­
tity of water has left t h e source in
the amount of PM. In o t h e r words, the
particle is emitted at t h e beginning of
the front and t h e n no other one is
emitted until the specified PM h a s
flowed from t h e s o u r c e . T h u s , the p a r ­
ticles a r e emitted in uniform lumps at
v a r i o u s times d u r i n g t h e i n t e r v a l DELP
Figure 34. Flow chart for
Subroutine GENP5
when u s i n g t h e s u b r o u t i n e SOURCE. T h e
flow c h a r t , l i s t i n g , and area configu­
ration a r e shown in f i g u r e s 36, 8, and
14F, r e s p e c t i v e l y .
Figure 35. Flow chart for
Subroutine SORGEN
factor.
It solves the e q u a t i o n
RD1 = 1 + (RHO/EPOR) x KD
Optional Subroutine for Calculating
Particle Retardation
Subroutine
RDSOLV
RDSOLV is a simple s u b r o u t i n e which
can be used to calculate a r e t a r d a t i o n
(18)
only when RD1 is set equal to z e r o . If
RD1 is known and e n t e r e d as i n p u t , the
code will use t h a t v a l u e .
IF KD is
known t h e n RD1 will be calculated if
0.0 is the initial value of R D 1 .
67
Figure 36. Flow chart for Subroutine SOURCE
68
Optional Subroutines That
P r i n t Results
T h e r e are four s u b r o u t i n e s t h a t can
be called which will p r i n t summaries of
c o n c e n t r a t i o n s , maps of p a r t i c l e loca­
tions,
and
c o n c e n t r a t i o n s of
water
flowing into s i n k s .
Any of t h e s e can
be called at any time for a r e p o r t of
w h e r e and what is going o n .
Subroutine
Subroutine
CONMAP
On the basis of equation 16, the
concentration
distribution
is
calcu­
lated and p r i n t e d out in numeric t a b l e
form.
F i g u r e 38 shows the flow c h a r t
and equation 16 can also be found in
the listing in figure 8.
MAP
This s u b r o u t i n e p r i n t s out PM, NP,
a n d TMAP and the N P A R T ( I , J ) a r r a y so
t h a t the location of particles in the
g r i d s can be n o t e d . This map, h o w e v e r ,
does not r e p o r t t h e location of p a r t i ­
cles within each g r i d .
(It would be
n e c e s s a r y to p r i n t out t h e X,Y a r r a y s
to obtain this information.) T h i s is a
convenient map to contour to get an
o v e r a l l p i c t u r e of the s i t u a t i o n .
Fig­
u r e 37 gives the flow c h a r t for this
s u b r o u t i n e , and f i g u r e 8 gives the
listing.
F i g u r e 37. Flow c h a r t f o r
S u b r o u t i n e MAP
Subroutine
SUMMRY
T h e flow c h a r t for t h i s s u b r o u t i n e
is given in figure 39. Any time d u r i n g
t h e time increment DELP that a p a r t i c l e
falls within the c a p t u r e a r e a (half of
a grid either side of a specified
s i n k ) , the particle is removed from the
flow system and is summed into the
TABLE(I) a r r a y . T h i s s u b r o u t i n e p r i n t s
out a summary of this action in t e r m s
of sink number a n d total mass deleted
d u r i n g a n y p a r t i c u l a r time increment
DELP.
F i g u r e 38. Flow c h a r t f o r
S u b r o u t i n e CONMAP
F i g u r e 39. Flow c h a r t f o r
S u b r o u t i n e SUMMRY
69
Subroutine
SNKCON
This s u b r o u t i n e calculates the con­
c e n t r a t i o n of water e n t e r i n g s i n k s on
the basis of w h e t h e r the sink is p u m p ­
i n g (value in the Q a r r a y ) or is an
a r e a of runoff (value in R a r r a y ) . The
a p p r o p r i a t e formula for pumping s i n k s
is
DELP = time increment
over which p a r t i ­
cles were deleted,
days
The appropriate
runoff is
formula
for
area
where
TABLE(MOVE) = mass of p a r t i c l e s
d e l e t e d by t h i s
s i n k , mg/l, d u r i n g
DELP
Q ( I , J ) = pumping r a t e of
s i n k , in gpd
F i g u r e 40 i l l u s t r a t e s t h e flow c h a r t
for t h i s s u b r o u t i n e .
T h e listing in
figure 8 contains t h e code for e q u a ­
tions 19 and 20.
Figure 40. Flow chart for Subroutine SNKCON
70
PART 4. EXAMPLE PROBLEMS AND COMPARISONS WITH THEORY
The following g r o u p of example p r o b ­
lems was chosen to i l l u s t r a t e how t h e
computer code is assembled and a p p l i e d .
Where possible,
theoretical solutions
a r e compared with the computer o u t p u t
so the r e a d e r can o b s e r v e t h e a c c u r a c y
of the t e c h n i q u e and gain c o n f i d e n c e .
Each of the examples includes a d e ­
scription of the problem, a list of t h e
r e q u i r e d data input c a r d s , the chosen
s e q u e n c e of s u b r o u t i n e calls, and a
comparison of the r e s u l t s with t h e o r y .
Divergent Flow from an Injection
Well in an Infinite Aquifer without
Dispersion or Dilution
T h i s example problem is a convection
problem chosen to d e m o n s t r a t e how well
t h e velocity VELO works for a s e v e r e l y
d i v e r g e n t problem. T h e problem is also
chosen to demonstrate how a c i r c u l a r
d i s t r i b u t i o n of particles a p p e a r s when
r e p o r t e d on t h e basis of the s q u a r e
grid configuration p r o d u c e d by t h e s u b ­
r o u t i n e MAP.
F i g u r e 41 gives the i n p u t d a t a n e c ­
e s s a r y to do t h i s problem.
In a c t u a l ­
i t y , the only d a t a u s e d in t h e i n p u t
c a r d s come from the p a r a m e t e r c a r d
(NSTEPS = 1 ) , the default value card
(NC = 30, NR = 29), t h e v a r i a b l e grid
c a r d s [DELX(I) and DELY(J) = 1000
f e e t ] , the pollution initial conditions
c a r d w h e r e DELP = 100 d a y s , and the
pollution parameter c a r d w h e r e MAXP =
5000, DISPL = 1E-10, DISPT = 1E-10, and
EPOR = 0 . 2 .
All o t h e r data are r e a d
Figure 4 1 . Input data for divergent flow problem (A) and
subroutine sequence (B)
71
a n d written b u t not u s e d since all this
information is s u p p l i e d by s u b r o u t i n e s
HSOLV4 and GENP5. Refer to f i g u r e s 8
a n d 42 for t h e s e other d a t a . The e x t r a
d a t a c a r d s a r e included t o avoid r e ­
w r i t i n g the code in MAIN .
F i g u r e 41B gives the s e q u e n c e of
s u b r o u t i n e calls t h a t will solve this
problem.
The s u b r o u t i n e calls in fig­
u r e 15A and B are merely r e p l a c e d by
t h o s e in 41B. S u b r o u t i n e HSOLV4 p r o ­
d u c e s the s t e a d y - s t a t e head d i s t r i b u ­
tion, GENP5 g e n e r a t e s 101 p a r t i c l e s in
a circle a r o u n d the injection well, and
the DO 620 loop p r i n t s out t h e particle
d i s t r i b u t i o n as the
particles
follow
along the velocity v e c t o r s e v e r y 100
days.
F i g u r e 42 s h o w s some of t h e r e s u l t s
of the particle positions as p r o d u c e d
by s u b r o u t i n e MAP as a function of
time.
The t h e o r e t i c a l solution of p a r ­
ticle location
and
density
is
also
given in figure 42.
One should recall
t h a t MAP is r e p o r t i n g on t h e basis of
particles found in an area 1/2 grid
e i t h e r side of the I , J c o o r d i n a t e s .
If
t h e x, y c o o r d i n a t e s of the particles
themselves were p r i n t e d , a smooth c u r v e
d r a w n t h r o u g h them would still be a
c i r c l e . A Calcomp s u b r o u t i n e would be
useful h e r e , b u t that is left to the
u s e r to implement.
Pumping from a Well Near a Line
Source of Contaminated Water, with
Dilution but without Dispersion
Consider pumping 1 million gallons
of water per day from a well located
5000 feet from a c o n s t a n t line source
of contaminated water of c o n c e n t r a t i o n
200 mg/l g r e a t e r t h a n t h e r e s i d e n t
groundwater.
Assume a semi-infinite
a q u i f e r and s t e a d y - s t a t e flow.
Also
a s s u m e that t h e line s o u r c e becomes
polluted at time zero and t h a t you a r e
i n t e r e s t e d in
the
time-concentration
c u r v e for the pumped well.
A crosssectional view
of this s i t u a t i o n is
shown in f i g u r e 43 along with o t h e r
data.
S u b r o u t i n e HSOLVE and t h e i n p u t data
(figure 44) were used to solve this
72
problem. An NC = 30 by NR = 29 v a r i ­
able grid model was used and a pumping
well located at I = 25, J = 15 was
specified.
A time increment DELTA =
1E10 was u s e d to p r o d u c e t h e s t e a d y s t a t e head d i s t r i b u t i o n ; a s i n k was
specified at I = 25, J = 15 and MARK =
1; and s o u r c e c o n c e n t r a t i o n d a t a were
p r e p a r e d along column 30.
Node lines
w e r e p r e p a r e d specifying r e c h a r g e f a c ­
t o r s R ( I , J ) of 1E10 to obtain a con­
s t a n t head along column 30 and a means
of measuring flow from t h a t s o u r c e .
T h e particle mass PM was calculated
from equation 15 as 500 so that 50 p a r ­
ticles found in one of the 1000' x
1000' grids would r e p r e s e n t 200 m g / 1 .
T h i s was j u d g e d to be sufficient for
resolution of the d i s t r i b u t i o n of the
concentration.
Figure 44B shows the s e q u e n c e of
s u b r o u t i n e s t h a t was chosen to solve
t h i s problem.
Here a g a i n , the s u b r o u ­
tine calls in f i g u r e 15A and B a r e r e ­
placed by t h o s e in f i g u r e 4 4 B . S u b r o u ­
t i n e HSOLVE p r o d u c e s the s t e a d y - s t a t e
h e a d d i s t r i b u t i o n , s u b r o u t i n e SORGEN
g e n e r a t e s the c o r r e c t number of p a r t i ­
cles on a n o d e - f o r - n o d e basis a c c o r d i n g
to their individual flows,
a n d the
p r i n t o u t s u b r o u t i n e s give r e p o r t s on
concentration a n d particle locations as
t h e simulation p r o g r e s s e s DELP d a y s at
a time.
Figure 45A shows the r e s u l t s of the
s i n k c o n c e n t r a t i o n as a function of
time as compared to t h e o r y . (See func­
tion MOVE in figure 8 for the defini­
tion of D . )
Note t h e lumpy c h a r a c t e r
of the o u t p u t . E n g i n e e r i n g judgment is
n e e d e d in d r a w i n g a c u r v e t h r o u g h the
computer d a t a . From a simple knowledge
of the aquifer conditions, it should
follow t h a t the c o n c e n t r a t i o n c u r v e for
t h i s well should be a smooth monotonically
increasing
function
of
time
a p p r o a c h i n g 200 m g / 1 . Knowing in gen­
e r a l what the solution should look like
g r e a t l y aids d r a w i n g c u r v e s t h r o u g h the
computer d a t a . If no idea is known of
what the solution should look like,
simply d e c r e a s e the particle mass and
time increment DELP until the computer
a n s w e r s achieve the d e s i r e d s m o o t h n e s s .
F i g u r e 4 2 . O u t w a r d r a d i a l flow of p a r t i c l e s f r o m an i n j e c t i o n well
73
Figure 43. Cross section view of well pumping near a
line source of pollution
T h i s may be an e x p e n s i v e method to u s e ,
however.
With practice a n d some e x p e r i e n c e ,
it may be possible to obtain a valid
solution from v e r y few p a r t i c l e s and
r a t h e r l a r g e time i n c r e m e n t s DELP. For
i n s t a n c e , the a b o v e problem was r u n
with PM = 500 a n d DELP = 100 d a y s , as
shown in figure 4 5 B . Another r u n was
made with PM = 1500 and DELP = 100
d a y s , as shown in f i g u r e 45C, and a n ­
o t h e r was made w i t h PM = 2497 a n d DELP
= 200 d a y s (figure 4 5 D ) . An examina­
tion of the differences between the
computed r e s u l t s of figures 45A t h r o u g h
45D indicates what is really n e e d e d to
define t h e c o n c e n t r a t i o n c u r v e .
F i g u r e 45E shows t h a t some loss of
a c c u r a c y o c c u r s when velocities of p a r ­
ticles a r e computed only 2.5 times p e r
grid movement.
F i g u r e 46 shows a plan view of the
same aquifer condition as would be d e ­
picted by t h e s u b r o u t i n e MAP just at
t h e time of b r e a k t h r o u g h to t h e well.
T h e map was p r o d u c e d with the same data
i n p u t as was shown in f i g u r e 44A, b u t
with a different s e q u e n c e of s u b r o u ­
t i n e s , as shown in f i g u r e 47.
74
In the plan view t h e difference b e ­
tween the actual x, y location of the
p a r t i c l e s and the method of p r i n t i n g
out t h e NPART matrix on a s q u a r e p a t ­
t e r n must be r e a l i z e d .
This explains
most of the p r i n t e d n u m b e r s less t h a n
50 along t h e front d e p i c t e d in f i g u r e
46. Here again a Calcomp r o u t i n e would
eliminate the need of i n t e r p r e t a t i o n .
A t h i r d example of the same condi­
tion is given by calling the s e q u e n c e
of s u b r o u t i n e s shown in figure 48. The
q u e s t i o n of i n t e r e s t a n s w e r e d in this
example is the s h a p e of the interface
as t h e front moves from the s o u r c e to
the well as a function of time.
The
s u b r o u t i n e GENP2 is called, which g e n ­
e r a t e s a single p a r t i c l e per node along
the line s o u r c e . T h e n t h e s e q u e n c e of
calling s u b r o u t i n e s MAP, CLEAR, ADVAN
(DELP) p r o d u c e s a s e r i e s of maps as
shown in figure 4 9 . T h e front is e a s i ­
ly d e s c r i b e d in t h i s fashion with only
29 p a r t i c l e s .
An excellent definition
of t h e front can be obtained by a d d i n g
more particles t h r o u g h calls to GENP
(PL) with Xl = 30, DX = 0 , Yl = 1, DY =
29, PL = 5000, a n d PM = 1 and p r i n t i n g
out t h e X,Y c o o r d i n a t e h i s t o r y of the
particles.
F i g u r e 44. I n p u t data for pumping in the v i c i n i t y of a polluted
line source ( A ) and s u b r o u t i n e sequence ( B )
75
Figure 45. Concentration of water derived from a well pumping
near a line source of pollution
76
Figure 46. Distribution of particles in the vicinity of a well
pumping near a line source of pollution
77
Figure 47. Subroutine sequence to
produce f i g u r e 46
Figure 48. Subroutine sequence
to produce f i g u r e 49
Figure 49. Advance of a f r o n t toward a well pumping
near a line source
78
Longitudinal Dispersion in Uniform
One-Dimensional Flow w i t h C o n t i n u o u s
I n j e c t i o n at X = 0.0
The data deck for this situation is
shown in figure 50A. An NC = 30 a n d NR
= 3 flow model is used for this p r o b ­
lem.
Longitudinal d i s p e r s i v i t y is set
at 4.5 feet and t h e t r a n s v e r s e d i s p e r ­
sivity at z e r o .
T h e total pollutant PL
is set at 100. The s e q u e n c e of s u b r o u ­
tine calls for this problem is shown in
figure 50B. The s u b r o u t i n e HSOLV2 is
used to provide t h e head d i s t r i b u t i o n
s u c h t h a t the x - d i r e c t i o n velocity of
flow is 1 f o o t / d a y . A grid i n t e r v a l of
10 feet is used to map the d i s t r i b u t i o n
of particles as they a r e g e n e r a t e d at x
= l , y = 2 randomly over the time i n c r e ­
ment DELP. This approximates the con­
t i n u o u s injection scheme d e s i r e d and
fully randomizes the flow.
The r e s u l t i n g set of maps p r o v i d e d
the n e c e s s a r y d a t a to plot t h e g r a p h s
of figure 51.
The theoretical c u r v e s
were derived from e q u a t i o n s given by
Fried (1975).
Again, note the lumpin e s s of the r e s u l t s .
Remedies could
include using more particles t h a n 100
p e r time increment, and smaller grid
sizes.
Likewise, application of e n g i ­
n e e r i n g judgment would allow fewer p a r ­
ticles.
Notice in figure 51 t h a t t h e r e
is no "overshoot" in the t r a d i t i o n a l
F i g u r e 50. I n p u t data f o r d i s p e r s i o n in a u n i f o r m one-dimensional flow
w i t h c o n t i n u o u s i n j e c t i o n at x=0 ( A ) a n d s u b r o u t i n e sequence ( B )
79
Figure 5 1 . Longitudinal dispersion in uniform one-dimensional flow
with continuous injection at x=0
80
mathematical sense of t h e w o r d .
An
i n c r e a s e in the number of p a r t i c l e s of
the simulation will r e d u c e t h e magni­
t u d e of t h e C/C o > 1 phenomenon shown
in figures 51B and 51C. No number of
additional particles would c u r e such a
phenomenon in the MOC.
Statisticians note t h a t it is p o s s i ­
ble to e x p r e s s the p r o b a b i l i t y of t h e
variation in magnitude of t h e number of
particles either side of the number
actually p r i n t e d .
With t h i s t y p e of
simulation, the possible r a n g e in t h e
number
of
particles
printed varies
a c c o r d i n g to a Poissan d i s t r i b u t i o n ,
and t h e magnitude of t h a t possible
variation would v a r y a c c o r d i n g to t h e
s q u a r e root of the number p r i n t e d .
T h u s , if the number 10 shows u p , i t s
actual value could r a n g e from (10-10 ½ )
to (10+10 ½ ), or 6.8 to 1 3 . 2 .
T h i s fact
helps in deciding what may be signifi­
cant when i n t e r p r e t i n g p r i n t o u t in t h i s
t y p e of simulation.
Use of t h i s con­
cept, applied to the t h e o r e t i c a l c u r v e ,
shows t h a t nearly all simulated v a l u e s
of figure 51C fall within the e x p e c t e d
range.
Longitudinal Dispersion in Uniform
One-Dimensional Flow with a Slug of
Tracer Injected at X = 0.0
The data for t h i s problem are t h e
same as those used in figure 50.
The
only difference between t h e p r e v i o u s
continuous injection scheme and t h i s
problem of an injected slug is in t h e
s e q u e n c e of s u b r o u t i n e s called. F i g u r e
52 shows that GENP3 is called once to
get the slug injected a n d t h e remainder
of the s u b r o u t i n e calls advance t h e
particles and p r i n t out the r e s u l t s as
the s l u g moves d o w n s t r e a m .
F i g u r e 53 gives t h e r e s u l t s of t h i s
simulation compared with t h e t h e o r e t i ­
cal r e s u l t s d e r i v e d by an equation
given by Bear (1972).
The effects of
injecting slugs of i n c r e a s i n g n u m b e r s
of particles are i l l u s t r a t e d by compar­
ing f i g u r e s 53A, 53B, and 53C. F i g u r e s
53A t h r o u g h 53C can actually be p r o ­
duced with a single simulation by call­
ing GENP3 applicable to t h r e e s e p a r a t e
rows of the flow model. T h e n , the con­
c e n t r a t i o n c u r v e of figure 53A would be
plotted on t h e basis of the first row
d a t a . F i g u r e 53B would be plotted by
s u p e r p o s i n g data from rows 1 a n d 2.
And finally, figure 53C would be plot­
ted on t h e basis of the sum of all
three rows.
Longitudinal Dispersion in a
Radial Flow System Produced
by an Injection Well
F i g u r e 54A gives the i n p u t data for
an NC = 30 by NR = 29 model with l o n g i ­
tudinal d i s p e r s i v i t y set to 450 feet (a
high d i s p e r s i v i t y like t h i s is u n h e a r d
of in the traditional s e n s e of d i s p e r ­
sion t h e o r y ; however, it could be a t ­
t r i b u t e d to the
effects of
aquifer
stratification).
A grid i n t e r v a l of
1000 feet is used to give a s e v e r e t e s t
to the d i s c r e t e particle t e c h n i q u e .
F i g u r e 54B i l l u s t r a t e s t h e choice of
s u b r o u t i n e calls to solve t h i s problem.
The s u b r o u t i n e HSOLV4 defines the head
d i s t r i b u t i o n by t h e Theim formula, and
GENP4 g e n e r a t e s the p a r t i c l e s in t h e
vicinity of the injection well.
F i g u r e 55 gives the r e s u l t s of this
problem compared with t h e o r y d e r i v e d
from an equation given by Bear ( 1 9 7 2 ) .
A plan view map of t h e c o n c e n t r a t i o n
d i s t r i b u t i o n at time = 350 d a y s is
given in figure 56 along with the
t h e o r y for comparison.
If more infor-
Figure 52. Subroutine sequence for
longitudinal dispersion in a uniform
one-dimensional flow field with a
slug of tracer injected at x=0
81
Figure 53. Longitudinal dispersion in a uniform one-dimensional flow
with a slug of tracer injected at x=0
82
F i g u r e 54. I n p u t d a t a f o r l o n g i t u d i n a l d i s p e r s i o n in a r a d i a l
flow p a t t e r n ( A ) and s u b r o u t i n e sequence ( B )
83
Figure 55. Radial dispersion from an injection well
84
F i g u r e 56. Map of t h e o r e t i c a l v e r s u s computed r a d i a l d i s p e r s i o n
from an i n j e c t i o n well
85
mation is desired to b e t t e r define the
concentration
distribution
in
figure
56, the simulation should be r e r u n with
a finer grid i n t e r v a l and more p a r t i ­
cles .
L o n g i t u d i n a l and T r a n s v e r s e
Dispersion in Uniform OneDimensional Flow w i t h a Slug of
T r a c e r I n j e c t e d at X = 0.0
F i g u r e 57A gives t h e i n p u t data for
this problem with a longitudinal d i s p e r s i v i t y of 4.5 ft and a 1.125-ft
transverse dispersivity.
The subrou­
tine calls a r e given in figure 57B,
showing t h a t HSOLV2 is used to p r o d u c e
the head d i s t r i b u t i o n and GENP3 is
86
called once to
g e n e r a t e the
slug.
Before this problem is r u n , GENP3 h a s
to be modified to t h e following:
1) DO
10 I = 1,200; 2) YY = 15; and 3) XX =
5.
F i g u r e 58 gives the computer r e ­
s u l t s compared with a theoretical solu­
tion d e r i v e d from an equation given by
Fried (1975). In t h i s example, c o n c e n ­
t r a t i o n is replaced simply by t h e d i s ­
t r i b u t i o n of p a r t i c l e s found in t h e
model. The a g r e e m e n t between computer
r e s u l t s and t h e o r y i s e x c e l l e n t .
F i g u r e s 59 a n d 60 i l l u s t r a t e in plan
view t h e o u t p u t for this problem as
p r o d u c e d by s u b r o u t i n e MAP when t h e
number of particles is v a r i e d between
500 a n d 2000.
Figure 57. Input data for longitudinal and transverse
dispersion (A) and subroutine sequence (B)
87
Figure 58. Dispersion of a slug injected in a uniform
one-dimensional flow in the x direction
88
Figure 59. Map of the number of particles of the model
g r i d 20 days after injection of a slug into a
uniform flow compared to theory
89
Figure 60. Map of the number of particles of the model
g r i d 70 days after injection of a slug into a
uniform flow compared with theory
90
PART 5.
FIELD A P P L I C A T I O N ; MEREDOSIA,
The field problem is located in
Morgan C o u n t y , Illinois, south of Meredosia, n e a r the Illinois River (figure
61).
T h e area is i n d u s t r i a l i z e d , with
petroleum
and
agricultural
chemical
companies o c c u p y i n g most of t h e p r o p e r ­
ty.
T h e contamination problem was
first d e t e c t e d in 1978 at a d i s t r i b u ­
tion terminal for liquid ammonia f e r t i ­
lizer.
This facility r e q u i r e s 1000 gpm
(5420 m 3 / d a y ) g r o u n d w a t e r supply to
o p e r a t e c o n d e n s e r s in the r e f r i g e r a t i o n
s y s t e m . The water is fed directly into
the system from high capacity p r o d u c ­
tion wells. D u r i n g 1978 the pump bowls
of the production wells s t a r t e d binding
d u r i n g operation because of calcium
carbonate precipitation.
The purpose
of t h i s s t u d y was to identify and c h a r ­
ILLINOIS
acterize t h e newly developed g r o u n d ­
water condition in o r d e r to recommend
corrective measures.
G r o u n d w a t e r Flow
System
T h e site outlined in figure 62 is
about 850 ft (257.4 m) by 625 ft
(189.2 m ) .
The major s t r u c t u r e s a r e
the terminal, the office b u i l d i n g , and
two l a r g e s t o r a g e t a n k s .
Underlying
the site is 90 ft (27.2 m) of u n c o n ­
solidated sand and g r a v e l on t o p of
P e n n s y l v a n i a n age b e d r o c k (Heigold a n d
R i n g l e r , 1979).
T h e r e is a gentle
slope (.006 ft/ft) in land surface from
east to west across t h e s i t e .
The
water t a b l e , contoured from d a t a for 12
wells (figure 62), reflects a g e n t l e ,
F i g u r e 6 1 . Location o f f i e l d s i t e
91
Figure 62. Water table contours and control wells at the field site
(The zero datum for the contours is the water
table level inferred at the source)
n a t u r a l gradient toward t h e Illinois
River (west) and i l l u s t r a t e s t h e cone
of d e p r e s s i o n from the p r o d u c t i o n well,
PW4.
Aquifer t r a n s m i s s i v i t i e s r a n g e
from 150,000 g p d / f t (1863 m 2 / d a y ) to
300,000 g p d / f t (3727 m 2 / d a y ) in t h i s
p a r t of the Illinois River
bottom­
lands .
T h e pump b i n d i n g problem first d e ­
veloped in two wells housed in the west
end of the office building (figure 6 2 ) .
L e a c h e d chemical c o n s t i t u e n t s from an
uncovered
chemical
fertilizer
bin
(SOURCE) migrated d o w n - g r a d i e n t , t o t h e
w e s t - n o r t h w e s t , a n d were d r a w n into the
cone of d e p r e s s i o n of t h e p r o d u c t i o n
wells in t h e office b u i l d i n g , which a r e
now a b a n d o n e d . T h e source c o n s i s t e d of
ammonium sulfate, n i t r a t e s , and t r a c e s
of i n o r g a n i c p h o s p h a t e s salvaged from
fire-damaged
fertilizer
processing
92
plants.
It had been in place for at
l e a s t a y e a r p r i o r to the development
of p u m p - b i n d i n g problems and n e a r l y
t h r e e y e a r s before t h i s s t u d y w a s in­
itiated .
Quality of Native and
Contaminated Groundwater
T h e n a t i v e g r o u n d w a t e r at t h e site
is of good chemical q u a l i t y .
The par­
tial chemical analysis in table 2 shows
t h a t the water is only moderately min­
eralized a l t h o u g h levels of n i t r a t e are
somewhat h i g h e r t h a n t h e USEPA p r i m a r y
s t a n d a r d of 10 m g / 1 .
On the s i t e , ten s a n d - p o i n t w e l l s - SP1 t h r o u g h SP10 (figure 6 2 ) - - w e r e
d r i v e n to check g r o u n d w a t e r q u a l i t y and
l e v e l s . Wells S P 1 , SP4, and SP10 were
Table 2. Chemical Composition of
Uncontaminated G r o u n d w a t e r
(mg/l)
Calcium
Magnesium
Sodium
Potassium
Iron
Manganese
Ammonia
68.0
24.0
7.5
2.0
0.4
0.14
0.04
Bicarbonate
Sulfate
Chloride
Nitrate
Fluoride
278.0
33.0
11.0
13.4
0.10
permanently installed for the p u r p o s e s
of t r a c e r e x p e r i m e n t s and m o n i t o r i n g .
Two production wells, PW3 a n d PW4, were
also used as monitoring w e l l s . PW4 was
p r o d u c i n g 1000 gpm (5420 m 3 / d a y ) d u r ­
ing t h e s t u d y , while PW3 was pumped
only for sampling a n d a g r o u n d w a t e r
t r a c e r e x p e r i m e n t . On the basis of the
chemical analysis of g r o u n d w a t e r taken
from 11 wells (table 3 ) , the g r o u n d ­
water quality was classified into t h r e e
g r o u p s : 1) i n t e r i o r plume, 2) marginal
plume, and 3) distal or native g r o u n d ­
water.
Table 3.
T h e g r o u n d w a t e r in the plume i n t e ­
rior had ammonia c o n c e n t r a t i o n s r a n g i n g
from 285 mg/1 to 2100 mg/1 ( f i g u r e 6 3 ) .
Nitrate r a n g e d from 570 mg/1 to 1885
mg/1 in the i n t e r i o r , while calcium and
magnesium
concentrations
were
well
below c o n c e n t r a t i o n s of native g r o u n d ­
water.
Sulfate, potassium, c h l o r i d e ,
p h o s p h a t e , and iron were found in con­
c e n t r a t i o n s above b a c k g r o u n d (figure
64).
The pH of the i n t e r i o r plume
water r a n g e d from 8.6 to 8 . 9 . The pH
of the native g r o u n d w a t e r is 8.0 or
slightly l e s s .
T h e g r o u n d w a t e r on the margins of
the plume was identified as h a v i n g con­
centrations
noticeably
above
back­
g r o u n d , b u t c o n s i d e r a b l y lower t h a n
i n t e r i o r c o n c e n t r a t i o n s ( f i g u r e s 63 and
64).
O b s e r v a t i o n wells to the n o r t h ,
f a r t h e s t from the s o u r c e , had the low­
e s t c o n c e n t r a t i o n s and were termed
distal.
A. chemical analysis was made on the
scale deposit which bound t h e p u m p .
Calcium c a r b o n a t e was the major con­
s t i t u e n t (95 p e r c e n t ) , and magnesium
c a r b o n a t e comprised 0.5 to 5 p e r c e n t of
the scale sample by w e i g h t . The rapid
precipitation of C a C O 3 on t h e pump
bowls was p r o b a b l y caused by the high
pH and by d e g a s s i n g due to the a g i t a ­
tion of the g r o u n d w a t e r in the pump
bowls.
Chemical Composition of G r o u n d w a t e r Samples from S t u d y Site (mg/1)
8.00
25.0
SP3
SP4
SP5
SP6
SP7
60.0
34.0
42.0
62.0
22.0
65.0
31.0
68.0
104.0
10.8
84.0
64.0
68.0
104.0
108.0
34.0
39.0
27.0
27.0
24.0
150.0
150.0
K
Calcium
Ca
2.80
6.40
Magnesium
Mg
3.90
5.10
Iron
Fe
0.60
2.00
Chloride
Cl–
Sulfate
SO
Phosphate
PO
Nitrate
NO–
1380.
680.
1280.
570.
480.
1180.
390.
Ammonia
(total)
NH°
2114.
457.
801.
282.
187.
967.
191.
59.0
196.
20.7
26.0
141.
12.3
4.40
11.0
0.55
29.0
140.
8.80
0.20
47.0
112.
3.60
0.30
14.0
89.0
<0.10
8.20
0.25
49.0
159.
16.5
0.27
12.0
103.
<0.10
6.00
SP9
PW3
SP2
Potassium
SP8
SP10
SP1
2.00
0.40
0.40
0.30
<0.02
5.00
7.00
6.00
13.0
49.0
33.0
42.0
90.0
<0.10
<0.10
<0.10
<0.10
60.0
55.0
1.30
0.04
220.
11.6
440.
137.
93
Figure 63. Partial chemical analysis from the
11 non-pumping control wells
94
Figure 64. Partial chemical analysis from the
11 non-pumping control wells
95
A plume configuration was developed
b a s e d on the chemical d a t a . C o n s i d e r a ­
tions of chemical equilibria involved
then provided insight into the t r a n s ­
port and transformation of ammonia and
minor c o n s t i t u e n t s of the chemical f e r ­
tilizer.
Contaminant Plume
T h e ammonia c o n c e n t r a t i o n s were used
to determine a two-dimensional configu­
r a t i o n for the plume.
By c o n t o u r i n g
ammonia values (figure 65) two conclu­
sions were d r a w n :
1) the plume was
migrating in a generally w e s t e r n d i r e c ­
tion, and 2) p a r t of the plume was
being d r a w n into the cone of d e p r e s s i o n
of PW4. Wells S P 1 , SP4, SP10, PW3, and
PW4 were sampled periodically o v e r nine
m o n t h s . C o n c e n t r a t i o n s at SP1 a v e r a g e d
2020 mg/l with a s t a n d a r d deviation of
79 mg/1 (N = 6 ) . C o n c e n t r a t i o n s in the
other
wells
showed
no
significant
change (<10 p e r c e n t ) in ammonia concen­
t r a t i o n d u r i n g the same p e r i o d .
Peri­
odic flushing e v e n t s are p r o b a b l e , b u t
at the scale of the s t u d y s i t e , a
steady-state groundwater quality condi­
tion p r e v a i l e d .
Two additional i n v e s t i g a t i o n s were
c a r r i e d out to improve the c h a r a c t e r i ­
zation of the plume:
1) a s t u d y of the
s o u r c e solubility, and 2) t r a c e r e x p e r ­
iments n e a r t h e two p r o d u c t i o n wells,
PW3 and PW4. To assimilate t h e ammonia
c o n c e n t r a t i o n in t h e g r o u n d w a t e r b e ­
n e a t h t h e s o u r c e , samples of t h e solid
fertilizer were b r o u g h t into t h e l a b o ­
r a t o r y a n d washed with distilled water,
a n d s u c c e s s i v e washes were analyzed for
t o t a l ammonia.
Leachate c o n c e n t r a t i o n s
a v e r a g e d from 2000-2200 m g / 1 . A more
e l a b o r a t e leaching
experiment
would
Figure 65. Ammonia concentrations (mg/l) in the
groundwater at the field site
96
h a v e revealed no more information b e ­
c a u s e the v a r i a b l e s affecting t h e leach
r a t e a t the source a r e u n k n o w n s .
Therefore, concentrations encountered
in the o b s e r v a t i o n well n e a r e s t the
s o u r c e (SP1) were used as the source
concentration.
Two t r a c e r e x p e r i m e n t s were p e r ­
formed to determine g r o u n d w a t e r flow
r a t e s near t h e p r o d u c t i o n wells:
one
n e a r PW4 at an injection distance of 50
ft (15.2 m) and one n e a r PW3 at 82 ft
(25 m) injection d i s t a n c e .
Rhodamine
WT, a fluorescent dye developed specif­
ically for t r a c i n g w o r k ,
was
used
b e c a u s e of i t s low adsorption on miner­
al and o r g a n i c materials (Smart and
Laidlaw, 1977; A u l e n b a c h , Bull, and
Middlesworth, 1978). T h e dye was in­
t r o d u c e d and flushed into SP4 a n d SP10
while PW3 a n d PW4 were p r o d u c i n g 1000
gpm (5420 m 3 / d a y ) e a c h .
Wells PW3
a n d PW4 were sampled at 10-minute in­
tervals.
The initial b r e a k t h r o u g h time
of the dye was 150 minutes at well PW3
a n d 40 minutes at PW4. The p e a k con­
c e n t r a t i o n at PW4 o c c u r r e d
at
65
m i n u t e s , and the t r a c e r concentration
w a s still i n c r e a s i n g slightly in PW3
a f t e r the termination of the experiment
(200 m i n u t e s ) .
Assuming t h a t the in­
t e r s t i t i a l velocity (V) can be calcu­
lated from t h e a v e r a g e a r r i v a l time of
a n o n r e a c t i v e t r a c e r , then V equals
0.77 ft/min (0.23 m/min) between SP10
a n d PW4.
The Meredosia Solute Transport Model
The model developed for t h e s t u d y
a r e a south of Meredosia, Illinois, cov­
e r e d an area of 56.7 a c r e s (22.95 h a ) .
T h e model (figure 66) was 1770 ft
(539.6 m) along the x - a x i s , r o u g h l y
n o r t h - s o u t h , and 1395 ft (425.3 m)
along the y - a x i s .
It contained 61
cells in t h e x - d i r e c t i o n and 36 cells
in the y - d i r e c t i o n , p r o g r e s s i v e l y i n ­
c r e a s i n g in size beyond the site toward
t h e n o r t h a n d west b o u n d a r i e s of the
model.
The b o u n d a r y conditions for the
g r o u n d w a t e r flow portion of t h e model
consisted of fixed head and flux c o n d i ­
t i o n s . Head conditions were t a k e n from
the water table map which was con­
s t r u c t e d from field d a t a , and flux c o n ­
ditions used along the n o r t h and west
b o u n d a r i e s were calculated from g r a d i ­
e n t s in the
unconsolidated
aquifer
material. A flux withdrawal of 1000 gpm
(5420 m 3 / d a y ) was used at PW4.
Cali­
bration of the flow portion of the
model was not difficult b e c a u s e of the
high d e n s i t y of data in this small
area.
From the contoured d a t a , fixed
head values were used on the model
boundaries.
The flow model was cali­
b r a t e d and used in the s t e a d y - s t a t e
mode.
The s t e a d y - s t a t e flow computa­
tion was chosen b e c a u s e : 1) t h e r e was
no a p p r e c i a b l e change in t h e c o n c e n t r a ­
tion of contaminants in t h e g r o u n d w a t e r
t h r o u g h o u t the s t u d y time (9 m o n t h s ) ,
t h u s r e d u c i n g t h e importance of f l u s h ­
ing e v e n t s at t h e s o u r c e , and 2) the
h y d r a u l i c g r a d i e n t remained c o n s t a n t
d u r i n g t h e s t u d y time. The second con­
dition seems r e a s o n a b l e , with only 5 ft
(1.54 m) of t o p o g r a p h i c relief within
the site and a h y d r a u l i c c o n d u c t i v i t y
of about 3000 g p d / f t (1.42 x 1 0 – 3
m / s e c ) in the unconsolidated material.
T h e b o u n d a r y conditions for solute
transport were:
1) a c o n s t a n t influx
at t h e source of 2000 m g / l , as cali­
b r a t e d against o b s e r v e d ammonia c o n c e n ­
t r a t i o n s ; and 2) sink conditions at PW4
a n d along t h e d o w n - g r a d i e n t b o u n d a r i e s
( n o r t h and w e s t ) , in o r d e r to o b s e r v e
the
ammonia
concentrations
in
the
g r o u n d w a t e r e x i t i n g from the model
boundaries.
Ammonia was used as the contaminant
plume indicator a n d for t h e p u r p o s e of
transport calibration.
F i g u r e 67 dem­
o n s t r a t e s the n a t u r e of computer o u t p u t
after 400 d a y s of t r a n s p o r t simulation.
The n u m b e r s within the model b o u n d a r i e s
r e p r e s e n t the n u m b e r of p a r t i c l e s r e ­
siding in each cell, with one particle
per cell equaling a c o n c e n t r a t i o n of
200 mg/1 ammonia. As shown, the plume
h a s migrated mainly in t h e downg r a d i e n t d i r e c t i o n , but it also has
been d r a w n toward p r o d u c t i o n well PW4.
T h e l a r g e r n u m b e r s of p a r t i c l e s near
97
Figure 66. Particle cells superposed over the study area
the west b o u n d a r y reflect t h e i n c r e a s e
in cell size.
The model was calibrated
to known ammonia concentrations with a
series of simulations in which the
longitudinal and t r a n s v e r s e d i s p e r s i v ity were v a r i e d .
Solute Transport Simulations
The a p p r o a c h used in the t r a n s p o r t
simulations w a s :
1) to calibrate the
model and to i n v e s t i g a t e the unknowns-longitudinal and t r a n s v e r s e d i s p e r s i v i t y , 2) to obtain the s t e a d y plume con­
figuration within the b o u n d a r i e s of the
model, and 3) to i n v e s t i g a t e t h e most
feasible
remedial
action
scheme
to
flush t h e a q u i f e r of the u n d e s i r a b l e
water.
98
The first s t e p in the t r a n s p o r t
model calibration involved s o r t i n g out
the p a r a m e t e r s to which t h e model was
sensitive.
T h i s was done within t h e
r a n g e of u n c e r t a i n t y of the p a r a m e t e r s
at this p a r t i c u l a r s i t e .
Most of t h e
p a r a m e t e r s were well-known b e c a u s e of
the e x i s t i n g d a t a and field e x p e r i ­
m e n t s . The model was most s e n s i t i v e to
longitudinal and transverse dispersivity.
Even the effects from v a r y i n g t h e
r e c h a r g e r a t e were small when compared
to those c a u s e d by d i s p e r s i v i t y v a r i a ­
tion.
Longitudinal d i s p e r s i v i t y ( d L )
was v a r i e d from 0.0 to 11 ft (3.35 m)
and t r a n s v e r s e d i s p e r s i v i t y ( d T ) from
0.0 to 6.5 ft (1.98 m). T h e most r e a ­
sonable r a n g e s for calibration to t h e
field-measured plume geometry were 7 ft
Figure 67. Distribution of particles residing in the
model at 400 days of simulation
(2.13 m) < d L < 11 ft (3.35 m) and 2
ft (0.61 m) < d T < 3 ft (0.915 m ) .
T h e second issue a d d r e s s e d was
w h e t h e r or not the plume had reached a
s t e a d y - s t a t e configuration within the
b o u n d a r i e s of the model or was i n c r e a s ­
ing in concentration with time toward
the d o w n - g r a d i e n t b o u n d a r i e s .
Longterm (1500 d a y s ) simulations were made
within t h e r a n g e of r e a s o n a b l e d L and
d T v a l u e s , and the c o n c e n t r a t i o n of
ammonia e n t e r i n g t h e sink b o u n d a r i e s
was plotted against time.
A steady
c o n c e n t r a t i o n a p p e a r e d after about 360
days at the down-gradient boundaries
a n d p r o d u c t i o n well u s i n g t h e smallest
r e a s o n a b l e d L , and d T v a l u e s , 7 ft
( 2 . 1 3 m) a n d 2 ft (0.61 m ) , r e s p e c t i v e ­
l y . This r e p r e s e n t s t h e u p p e r limit or
l o n g e s t time it would t a k e t h e contami­
n a n t s to establish a c o n s t a n t plume
geometry within the b o u n d a r i e s of the
model. In other w o r d s , the contamina­
tion problem will remain the same at
t h e site until a source or sink b o u n d a ­
ry condition is a l t e r e d .
The most likely change in t h e b o u n d ­
a r y conditions and the one t h a t would
p r o d u c e t h e most d e s i r a b l e r e s u l t s is
removal of t h e s o u r c e . T h i s was i n v e s ­
tigated u s i n g the model and evaluated
100
in t e r m s of ammonia c o n c e n t r a t i o n at
the d o w n - g r a d i e n t b o u n d a r i e s and t h e
p r o d u c t i o n well (PW4).
With a s t e a d y s t a t e plume condition e x i s t i n g at 360
d a y s from the s t a r t of t h e simulation,
the s o u r c e flux was removed from the
model at t h e n e x t time s t e p , 390 d a y s .
The ammonia c o n c e n t r a t i o n s at the s i n k s
diminished to nearly zero at 810 d a y s
(figure 6 8 ) .
T h e r e f o r e , 420 d a y s is
approximately t h e time it would take
for t h e s t u d y a r e a and p r o d u c t i o n well
to n a t u r a l l y flush itself of the con­
taminant.
T h e application of solute t r a n s p o r t
modeling to this g r o u n d w a t e r contamina­
tion problem h a s been useful in d e t e r ­
mining t h e e x t e n t of the p r o b l e m .
It
seems unlikely t h a t the p r e s e n t c o n d i ­
tion will worsen at the s t u d y s i t e , and
n a t u r a l p r o c e s s e s should
flush
the
plume within fourteen months of s o u r c e
removal.
As p a r t of continuing i n v e s t i g a ­
t i o n s , the effect of the contaminant
plume at d o w n - g r a d i e n t wells will be
emphasized.
Additional sampling and
modeling efforts will s e r v e to f u r t h e r
define controlling s u b s u r f a c e p r o c e s s e s
and to improve the u s e f u l n e s s of the
approach.
Figure 68. Ammonia concentration at the production well and
down-gradient boundary after source removal
101
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