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Determining dispersion coefficients and sources of model-dependent scaling
Journal o f Hydrology, 75 (1984/1985) 195--211 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
195
[2] DETERMINING DISPERSION COEFFICIENTS AND SOURCES OF MODEL-DEPENDENT SCALING
G.A. ROBBINS and P.A. DOMENICO
Department o f Geology, Texas A & M University, College Station, T X 77843 (U.S.A.) (Received July 18, 1983; revised and accepted April 12, 1984)
ABSTRACT Robbins, G.A. and Domenico, P.A., 1984. Determining dispersion coefficients and sources of model-dependent scaling. J. Hydrol., 75: 195--211. This paper deals primarily with methods for determining dispersion coefficients in multidimensional dispersion problems. The methods developed are applicable to field and laboratory tracer tests, contamination spills, or long-term contamination histories. The analysis of three-dimensional data from instantaneous sources is simplified by embedding two u n k n o w n coefficients into an experimentally determined variable, namely the maximum observed concentration. The analysis of multidimensional continuous point-source data takes advantage of centerline solutions that possess the same form as the simple one-dimensional dispersion--convection problem. Importantly, all coefficients can be determined independently of initial concentration and source volumetric flow rate, two parameters that are seldom known in contamination problems. Finally, some aspects of model-dependent scaling are discussed. Modeling factors found to produce scaling in the measured dispersion coefficients include finite-source influences on pointsource methods and a nonconcurrence between the dimensionality of a particular model and the field dispersion condition it is supposed to represent.
INTRODUCTION
Among the major problems associated with predicting contaminant t r a n s p o r t is t h e d e t e r m i n a t i o n o f d i s p e r s i v i t y . I n t h i s r e g a r d , A n d e r s o n (1979, p . 1 0 0 ) has stated " . . . as yet, there is no standard technology for measuring longitudinal and transverse dispersivities." T h e v a s t m a j o r i t y o f s u c h d e t e r m i n a t i o n s t h a t d o e x i s t are f o r l o n g i t u d i n a l dispersion from laboratory column experiments. By-and-large, these experim e n t s h a v e b e e n c o n d u c t e d o n m a n - m a d e (e.g., glass b e a d s ) or r e m o d e l e d n a t u r a l materials and the a p p l i c a t i o n of their results to field p r o b l e m s can be questioned. Few laboratory experiments have been conducted to measure transverse dispersivity owing to the elaborateness of experimental procedures
0022-1694/84-85/$03.00
O 1984 Elsevier Science Publishers B.V.
196 and limited analysis techniques. Such experiments, as exemplified by those o f Bruch and Street {1967) and Simpson (1962), have employed large flow tanks. Although t he y provide insight into dispersion phenomena, t hey are impractical from the point of view of measuring transverse dispersion coefficients on unmolded samples representative of field conditions. Relative to laboratory experiments, only a limited num ber of field tests have been conducted. These studies entail pulse-type injection methods that generally do not include transverse dispersion in the calculations. One notable exception to this is the work of Sauty (1980}, who provided type-curve solutions to a two-dimensional point-source. However, a more c o m m o n procedure is to design a radial flow experiment and apply the point-source models developed by Fried (1975, p.66), or simply fit a one-dimensional dispersion model to field data and assume the dimensionality of the problem matches that of the model. If the assumptions of one-dimensional dispersion break down, an inevitable result is the lumping of transverse spreading into the calibrated longitudinal dispersion coefficient, i.e. a scaling effect (Domenico and Robbins, 1984). On the o t her hand, even if these various assumptions hold, most methods do not address the problem of multidimensional dispersion. Continuous-source m e t hods present yet a different set. of problems. Although such m et hods are not often used in field tests, continuous-source data provided by contaminant plumes have been used extensively in transport model calibrations. In such calibrations, however, the dispersion coefficients are seldom, if ever, the only unknowns in the problem. For example, the initial concentration and the volumetric flow rate of the source are seldom known. Other problems involve elements of the scaling effect already mentioned for pulse-type models, as well as those that occur when pointsource assumptions are forced on finite-source problems. Given these factors, the physical basis for calibrated field dispersion coefficients is highly suspect. Although it does not seem possible at this stage to standardize a technology o f measurement, there does appear to be a need for the design of tracer experiments whereby three-dimensional dispersion can be measured and modeled. This applies to the laboratory as well as the field. Further, there is a need to extract physically based coefficients from contamination histories. The purpose of this paper is to suggest a variety of procedures and appropriate models to achieve these ends and to demonstrate their application in field and laboratory studies.
INSTANTANEOUS SOURCE MODELS As discussed earlier, laboratory experiments to measure transverse dispersion coefficients are generally conducted in large rectangular flow tanks. The necessity of using rectangular tanks appears to be dictated largely by the analytical model c o m m o n l y used to analyze such tests (see e.g.,
197
H a r l e m a n and R u m e r , 1963). In this section, t h e o r e t i c a l m e t h o d s will be p r e s e n t e d t h r o u g h which it is possible t o calculate longitudinal and transverse dispersion c o e f f i c i e n t s in l a b o r a t o r y c o l u m n e x p e r i m e n t s or field tracer experiments. Finite-source m o d e l s T h e parallelepiped m o d e l as given b y H u n t ( 1 9 7 8 ) is o f the f o r m : C =
(C0/8) [ e r f { [ x - - v t -
-
-
-
-
-
+ ( X / 2 ) ] / 2 ( D : , t ) 1/2}
erf{ [x - - vt -- ( Z / 2 ) ] / 2 ( D x t) 1/2 }] [ e r f { [ y + ( Y / 2 ) ] / 2 ( D y t) 1/2} erf{ [ y -- ( Y / 2 ) ] / 2 ( D y t ) l / 2 } ] [erf ([z + ( Z / 2 ) ] / 2 ( D z t ) 1/2 } erf{ [z -- ( Z / 2 ) ] / 2 ( D z t ) l / 2 } ]
(1)
In t h e usual c o n v e n t i o n , x, y, z r e p r e s e n t a spatial c o o r d i n a t e system; v is the seepage velocity; Dx is the longitudinal dispersion coefficient; D y and Dz are the transverse dispersion coefficients; t is time; X, Y and Z r e f e r t o the dimensions o f the source o n Fig. 1; and erf is the e r r o r f u n c t i o n . In this f o r m u l a t i o n , ( x - vt) is d e f i n e d as the distance f r o m the c e n t e r o f mass, which is originally l o c a t e d at x = y = z = 0. T w o useful f o r m s o f eq. 1 are r e q u i r e d here. T h e first gives t h e c o n c e n t r a t i o n along w h a t is r e f e r r e d t o as t h e c e n t e r l i n e o f spreading w h e r e y = z = 0 f o r any x : C =
(C0/2) [ e r f { [ x - - v t -
-
+ ( X / 2 ) ] / 2 ( D ~ t ) 1/2}
erf{ [x -- vt -- ( X / 2 ) ] / 2 ( D ~ t ) l / 2 } ] [erf{ Y / 4 ( D y t)1/2}]
× [erf{Z/4(nzt)]/2}]
(2) ,JJ
J ~ 1 "/,!
.
I
•
Zl
.
i
-~ ~
-~J](o,o,oT
T_z ! i I i
2
Fig. 1. P a r a l l e l e p i p e d f i n i t e s o u r c e .
•
X
~
7
198
j~f Y
INJECTION ORIGIN
//
" - "~"'~3
OBSERVATION POINTS
Fig. 2. Coordinate identification of observation points. Because of the spreading g e o m e t r y , m a x i m u m c o n c e n t r a t i o n s will o c c u r along this centerline o f the plume. A second useful f o r m of eq. 1 is for some p o i n t along the spreading c e n t e r where x = vt, i.e. the c e n t e r of mass coincides with the p o i n t o f observation. F o r this case, eq. 1 or eq. 2 becomes: C' = (Co) [ e r f { X / 4 ( D x t ' ) l / 2 } ]
[erf{Y/4(Dyt')i'2}]
[erf{Z/4(D~t')12}]
(3)
where C' defines the m a x i m u m c o n c e n t r a t i o n at time t'. Fig. 2 is the m o s t c o m p l e t e scheme o f m e a s u r e m e n t for application o f these e q u a t i o n s to a three-dimensional dispersion p r o b l e m . T h e e x p e r i m e n t is c o n d u c t e d SO t h a t c o n c e n t r a t i o n is m o n i t o r e d at all f o u r points simult a n e o u s l y until Cl(x,O,O,t)
= C'~(x,O,O,t')
i.e. the c o n c e n t r a t i o n at measuring p o i n t 1 is described by eq. 3. Substituting eq. 3 into eq. 2 yields one e q u a t i o n with one u n k n o w n : C4 = (C'/2) [erf{ Ix4 -- vt' + ( X / 2 ) ] / 2 ( D x t ' ) ~ 2 } --erf{[x4--vt'--(X/2)]/2(Dxt')12}]/[erf{X/4(Dxt'~J
2}}
(4)
where C4 and C' are c o n c e n t r a t i o n s m e a s u r e d at points 4 and 1, respectively; x 4 refers to the distance to p o i n t 4; and time t' c o r r e s p o n d s to the time of m a x i m u m c o n c e n t r a t i o n b r e a k t h r o u g h at p o i n t 1. T h e p a r a m e t e r Dx m a y t h e n be d e t e r m i n e d by iteration. Similar e q u a t i o n s m a y be d e v e l o p e d for Dy and D~ by substituting eq. 3 into eq. 1 for the a p p r o p r i a t e conditions. Thus, for x = vt and z = 0: C~ = (C'/2) [ e r f { [ y + ( Y / 2 ) ] / 2 ( D y t ' ) l J 2 } - - e r f { [ y -- ( Y / 2 ) ] / 2 ( D y t ' ) ~ 2}] / [erf{ Y / 4 ( D y t ' ) l
2}]
(5)
where C2 and C' are m e a s u r e d c o n c e n t r a t i o n s at points 2 and 1, respectively; and time t' is again the b r e a k t h r o u g h time for C'. F o r x = t't and y = 0, the a p p r o p r i a t e e q u a t i o n becomes: C.~ -
(C'/2) [eft{ [z + ( Z / 2 ) ] / 2 ( D z t ' ) 1/2 } -- erf{ [z -- ( Z / 2 ) ] / 2 ( D ~ t ' ) 12 }l /
199
[erf { (ZI4 (Dz t') t/2 }]
(6)
where the variables are interpreted as above. The equations described above have been applied to a series of laboratory column experiments by Robbins (1983). The column was 10 cm in diameter, 93 cm in length, and contained glass beads with a diameter of 0.48 mm. For the glass bead experiment, it was assumed that Dy = Dz = DT where DT is the transverse dispersion coefficient. Tracer concentrations were measured at four locations in the column using calibrated conductivity probes. Other measurements were made to assure no tracer impingement on the column walls. Full details of the experimental apparatus and procedures are given by Robbins (1983). Fig. 3 illustrates relative concentration--time curves attained for a threedimensional pulse test at two probe locations in the column. These breakthrough curves were analyzed as described above in order to determine the dispersion coefficients. The dashed-line drawings in Fig. 3 illustrate a reproduction of these breakthrough curves based on the determined dispersion parameters. For this case, Dx = 2.03" 10 -4 and DT = 5.72" 10 -s cm 2 s- 1. For a series of such tests, the transverse dispersivity c~w ranged from 2 . 1 0 - 10 -3 to 6.14" 10 -3 cm. These values compare favorably with flow tank determinations of transverse dispersivity in similar type materials reported by others (Grane and Gardner, 1961; Harleman and Rumer, 1 9 6 3 ; Lawson and Elrick, 1972). Identical procedures can be applied to field distributions of concentration, such as contaminant spills or controlled tracer experiments of the type reported by Sudicky et al. (1983).
IO 0.8 06 c C' 0.4 0.2 0.0 4
1.0 0.8
~li°t%nc e .9cm
I
f
,
T
l
I
1
l
~/
,
,
~cm
I
I
,
1
Flow Distance
0.6
_c C' 0.4
0.2 // 6
8
TIME(rain)
lO
.
12
76
78
.
80
.
.
.
82
.
.
84
.
.
86
.
88
90
TiME (min)
• . . . .
Actual Model
curve curve
Fig. 3. Comparison of observed and predicted relative concentration--time curves for three-dimensional pulse test.
a
200 Point-source models
The point-pulse model as discussed by Baetsle (1969) is of the form: C = [ M 3 / 8 ( T r t ) 3/2 ( D x D y D z ) l " 2
'/, [exp { - - (x - - v t ) 2 / 4 D x
] t - - y2/4Dy t -- z 2 / 4 D z t}]
(7)
where M 3 is the mass of the tracer introduced at a point; and all other variables are as defined previously. The maximum concentration is defined at the point x = vt for y and z equal to zero by: (8)
C' = M 3 / 8 ( T r t ' ) 3'2 ( D x D y D z ) 1'2
where t' corresponds to the time when the maximum concentration is observed. Fig. 2 again describes the scheme of measurement, with the exception that measuring point 4 can be ignored. The experiment is conducted so that the concentration is monitored at all three points simultaneously until Cltx,O,O,t)
= C'(x,O,O,t')
i.e. the concentration at point I is described by eq. 8. At point 1, for any t, eq. 7 becomes: CI
= [m3/8(Trt)3/2(DxDyDz)l;2
] [exp{-- (x -- v t ) Z / 4 D x t } ]
(9)
Combining this equation with eq. 8 gives, eventually: -- (x -- v t ) 2 / 4 t
Dx
(10)
In { ( C z / C ' ) / ( t ' / t ) 3/2 }
Here, time t corresponds to the measurement time of C1, and t' to the measurement time of C'. Similar procedures for points 2 and point 3 yield: D y
=,
-- Y 2 / 4 t - -
(11)
In { ( C z / C ' ) / ( t ' / t ) 3 1 2 } + tx -- v t ) 2 / 4 D x t
and -- z 2 / 4 t
Dz
:: l n { ( C 3 / C ' ) / ( t ' / t ) 3 2 } + (x -- v t ) 2 / 4 D x t
(12)
The determination of Dx can be simplified through graphical procedures. This requires a plot of C / C ' vs. time for the point-/ data. Given such a plot, the standard deviation of the C / C ' time plot ( a t ) is described by: o t = F/2.354
(13)
where F is the full curve width at C / C ' = 0.5. The relationship between the standard deviation in time (Or) and the standard deviation in distance (ax) is given by: (axt 2 = v2ot 2
(14)
where v is the seepage velocity. For the point-source distribution, the longitudinal dispersion coefficient is then simply determined from:
201
Dx
(15)
= 0~212t '
It is emphasized that the graphical method is valid only for a true pointsource in that the full curve width for a finite source will reflect in part the initial source dimension. An alternate method of tracer test analysis for the instantaneous pointsource is given by Sauty (1980). Sauty's (1980) method is applicable to a two-dimensional spreading field where Oc= and ocT are determined by a typecurve procedure. Within certain ranges of the coefficients the similarity of the type curves introduces uncertainty in parameter determinations. The methods developed above incorporate the complete three-dimensional spreading field, and permit a more direct approach to the problem. M o d e l - d e p e n d e n t scaling
Investigations using instantaneous source tests have often reported the observation of scaling of dispersion coefficients with distance (Molinari et al., 1977; Pickens et al., 1977). This effect has been attributed to formation heterogeneity (Pickens and Grisak, 1981). This argument notwithstanding, there is yet another, perhaps more c o m m o n cause that can account for some of the observed scale effect. This type of scaling is of a model-dependent form. Fig. 4 has been generated to examine various forms of model-dependent A
--FINITE SOURCE --POINT SOURCE
x=lSm
B x=Srn I0
,°I
O8
08
C o6
cO6
C04
C 04I 02
O2
O0
6
8
I0
12
14 16 18 20 TIME (xlO4sec)
22
24
''
0.0
26
59
, o
b7
59
61
O6
C' 0 4
C'04
02
02
94
55
08
o
90
47 49 5[ 55 T I ME (x 104sec)
×=25m
O8
86
45
~o
O6
82
45
D
C
x =lOrn
c
41
98
102
106
TIME (* 105see)
I10
114
118
O0 23
24 25 26 TIME (x 105sec)
27
Fig. 4. Relative c o n c e n t r a t i o n - - t i m e curves for three-dimensional finite- and point-source models.
202
TABLE
I
Model calculation
d a t a u s e d in F i g s . 4 a n d 5
Three-dimensional
Three-dimensional
parallelepiped
point-source
X 100 cm Y - 100 cm Z = 100 cm
Co = 1 , 0 0 0 p p m
('0 - 1 , 0 0 0 p p m ~" .... l ' 1 0 - 3 c m s Dx :
injection mass = 10,000 mg
l
l't0-3em; s -1 D v :::: 5 • 1 0 - 4 e m 2 s -1 D~ -: 5 " 1 0 - 4 e m 2 s - '
V - 1"10
~cms
D x -- 1 • 10
Dy -
~cm: 5"104cm2s
D z = 5.10-4cm:s
i S-1
-~ -j
scaling. The figure shows a series of relative concentration--time data for several observation points along the centerline of flow, using the threedimensional parallelepiped and point-source models discussed earlier. The curves were generated using the data listed in Table I. The inner and outer curves are for the point and parallelepiped models, respectively. As reported in a previous paper (Domenico and Robbins, 1984), a scalingup of dispersivity will occur whenever an ( n - 1)-dimensional model is calibrated or otherwise e m pl oyed to describe an n-dimensional system. The scale effect associated with a dimensionality mismatch is shown as curve A in Fig. 5. This effect results from the fitting of three-dimensional dispersion data as shown in Fig. 4 to a one-dimensional finite-source model. As noted, the calibrated coefficient is approximately a function of the square of the distance traveled by the center of the mass. In Domenico and Robbins (1984) the dispersion coefficient was found to increase linearly with distance when two-dimensional data axe fitted to a one-dimensional model. Interestingly, from Fig. 5, observation points close to the source approach the actual value for the longitudinal dispersion coefficient. This occurs because, for three-dimensional finite sources, the error functions containing Y and Z in eq. 3 approach 1 for small times. Scaling of the type discussed above is evident in the processing of data collected by Sudicky and Cherry (1979). These investigators conduct ed what is likely the most comprehensive field test to date, employing five injection wells and 69 bundle-type samplers. The data have been analyzed with a onedimensional parallelepided model (8udicky and Cherry, 1979) and with a three-dimensional parallelepiped model (Sudicky et al., 1983). For their slow-zone data, the one-dimensional model predicts a piecewise linear increase in longitudinal dispersivity from ~ 2 to ~ 1 8 c m over an 11-m distance of observation from the source. Fast-zone data examined by Pickens and Grisak (1981) from the viewpoint of the one-dimensional analysis also exhibit scaling. The three-dimensional analysis (Sudicky et al., 1983) suggests a constant longitudinal dispersivity of ~ 8 cm over the observation
203
o 12
'
I .,/Y ,
l
4.0
A
2.o
m4! N z
IO
~ °
...... / __
0 O
I 5
, ,0
Actual Dx , 15
B 1 __ 2'0
25
30
Fig. 5. Scaling patterns for dimensionality mismatches (A and A') and point-source assumptions for finite-source data (B and B').
distance 4 - - 1 1 m from the source. The type of scaling associated with the one-dimensional analysis is clearly an artifact of the model employed. Additional scaling effects arise when finite-source data are analyzed with point-source models. Such procedures have been employed in both laboratory and field studies (Klotz and Moser, 1974; Ivanovich and Smith, 1978). As demonstrated in Fig. 4, the concentration distributions predicted by both models would eventually match with increasing distance. This is illustrated in Fig. 5 as curve B where at 30 m or so finite-source data can be analyzed with a point-source equation with no appreciable error in the calibrated coefficient. At distances less than 30 m, curve B demonstrates the scaling effect associated with analyzing finite-source data with a point model. As one might expect from the concentration distributions shown in Fig. 4, the longitudinal dispersion coefficient would have to be scaled up near the source, and would asymptotically approach the actual value with distance. The a m o u n t of scaling is dependent on the initial source size and observation distance. This same effect is anticipated when continuous finitesource data are analyzed with continuous point-source models. Such an observation has been made by Lawson and Elrick {1972) who used a pointsource model to analyze data from a large flow tank experiment. A laboratory demonstration of this effect is shown in Fig. 6, where finitesource data o f Robbins {1983) were analyzed with the point-source model.
204 0 55
Longitudinal
Pulse
- • -- ---* --. ---
030
Dimension
(cm)
325 2.93 210
~ ' 0 25
~ 0 20 E
~
0~5 0 bO
\
005
oi 0
I
5
.'-l
io
_ I
~5
I
I
zo
Flow
25 distance
#
1
1
30
35
x
(cm)
t
40
I,
45
I
50
I
55
Fig. 6. Apparent scaling associated with applying a point-source model to finite-source laboratory data. F o r the three tests shown, the a p p a r e n t dispersivity decreases to a finite value with distance. Also, the larger the pulse dimension (i.e. source size) the m o r e p r o n o u n c e d the scaling e f f e c t in the near field. T o b e t t e r visualize the relations shown in Fig. 5, the data are r e p l o t t e d in the f o r m o f Crx2 vs. observation distances in the same figure. T h e Gaussian relation b e t w e e n the variance Ox 2 and observation distance is expressed as: a~ 2 = 2 D x t = 2D,,x/v and is shown as the straight line in the figure. T h e c o n s t a n t slope o f the line reflects a c o n s t a n t Dx-value. Curve A' is for the case where a one- or twodimensional m o d e l is used to describe a three-dimensional dispersion c o n d i t i o n . This is perhaps a c o m m o n o c c u r r e n c e in field tracer studies where investigators are interested exclusively in longitudinal dispersion. U n f o r t u n a t e l y , for any d i m e n s i o n a l i t y mismatch, the e f f e c t o f a missing spreading d i r e c t i o n or directions will be l u m p e d into the calculation. As the observation points b e c o m e m o r e widely spaced, total dispersion increases, resulting in an a p p a r e n t increase in longitudinal dispersion with scale of the e x p e r i m e n t . This is e x e m p l i f i e d in curve A' where the variance--distance curve is c h a r a c t e r i z e d by a n o n l i n e a r segment at small distances f o l l o w e d by a relatively linear segment at larger travel distances. This e x a c t shape o f the variance--distance curve has been r e p o r t e d by Pickens and Grisak ( 1 9 8 1 ) to be c o m m o n in n u m e r o u s surface w a t e r dispersion studies. It is also a c o m m o n shape in n u m e r o u s tracer studies for g r o u n d w a t e r systems. As n o t e d b y Pickens and Grisak {1981), typical examples include the s t u d y o f Molinari et al. ( 1 9 7 7 ) and Pickens et al. {1977), a two-dimensional dispersion analysis b y P e a u d e c e r f and S a u t y (1978), and the one-dimensional analysis o f S u d i c k y and Cherry (1979). In all cases, the observed scale e f f e c t has been a t t r i b u t e d to heterogeneities.
205 Curve B' describes the case where a point-source model is used to describe a finite-source dispersion condition. As noted in Fig. 5, curve B' exhibits an initial increase in slope until a terminal value is reached, reflecting thereafter a constant dispersion coefficient. If the dispersion coefficient is determined on the basis of the o~--distance curve, there would be an apparent increase in the coefficient until a constant value is reached. Hence, we again have an apparent scaling of the t y p e discussed above.
CONTINUOUS POINT-SOURCE MODELS As a general rule, continuous-source methods are not often used in field testing procedures. On the other hand, continuous-source data provided by contaminant plumes have been useful in transport model calibration (Bredehoeft and Pinder, 1973; Wilson and Miller, 1978). In this section the structure of a few analytical models will be examined for the purpose of describing some methods for easy determination of field parameters. Although n o t designed for field application, the one-dimensional pointsource solution of Ogata and Banks (1961) will be the focal point of these discussions. This equation is: C = (C0/2) [ e r f c { ( x - - vt)/2(Dxt) 1/2} + exp (xv/Dx) erfc {(x + vt)/2(Dxt)w2}]
(16)
where erfc is the complementary error function; and Co is the initial concentration. As noted by Ogata and Banks (1961) for large xv/Dx-values ( ~ 500) the second term on the right-hand side may be ignored. A two-dimensional line-source solution can be obtained from the equation described by Wilson and Miller (1978) for the condition y = 0 and xv/2Dx ~ 10, which corresponds to the centerline of a two-dimensional plume:
C = [M2/4(~rD s )1/2 (1/xv)l/2] [erfc {(x -- vt)/2(Dx t) 1/2 }]
(17)
where M 2 is a flux term having the dimensions of M/Lt, in which L is length. This equation includes the series expansion given by Wilson and Miller (1978, p.505). As with instantaneous-source models, maximum concentrations will occur along this centerline of spreading. A three-dimensional centerline solution (y - z = 0) can be developed from the equation given by H u n t (1978, p.77):
C = [M3/8xTr(D~Dz) 1/2] [erfc {(x -- vt)/2(Dxt) 1/2 } + exp (xv/D~) erfc {(x + vt)/2(Dx t)1/2}]
(18)
where M3 = m/t. From the form of these equations, it is seen that the Ogata--Banks solution can be embedded in both the two- and three-dimensional centerline
206 results. T h a t is, considering the a b b r e v i a t e d solution, eq. 17 can be e x p r e s s e d as:
form
of 1;he O g a t a - - B a n k s
C = [Q/2(TrDyxz,) l;2] [(C0/2) e r f c { ( x -- vt)/2(D~t)12}]
(19)
where Q is a v o l u m e t r i c injection rate (L2/t) so t h a t CoQ = m / L t . T h e b r a c k e t e d q u a n t i t y on the right c o n t a i n s the a b b r e v i a t e d f o r m of the oned i m e n s i o n a l O g a t a - - B a n k s solution. Similarly, the t h r e e - d i m e n s i o n a l centerline solution o f H u n t ( 1 9 7 8 ) can be e x p r e s s e d as:
C
[Q/4xTr(DyDz) 12 ] [(C0/2) erfc {(x -- vt)/2(D:,t~ l 2} f e x p (xv/Dx) e r f c { ( x + v t ) / 2 ( D x t ) 12/1
(20)
w h e r e Q is again a v o l u m e t r i c injection rate (L3/t) so t h a t C0Q - m / t and the b r a c k e t e d q u a n t i t y c o n t a i n s the e x a c t O g a t a - - B a n k s solution. T h e s e interesting results indicate t h a t the centerline c o n c e n t r a t i o n s o f a two- or t h r e e - d i m e n s i o n a l p o i n t - s o u r c e are described by the w e l l - k n o w n oned i m e n s i o n a l solution, m o d i f i e d o f course by the t e r m s Q/2(rrDyxvl l z and Q/4xTr(DyDz) 1/2, which c o n t a i n the transverse dispersion coefficients. It is clear t h a t these t e r m s m u s t be less t h a n one, i.e. with transverse dispersion it is p a t e n t l y impossible for the field c o n c e n t r a t i o n at a n y x > 0 to be greater t h a n t h a t p r e d i c t e d by the o n e - d i m e n s i o n a l O g a t a - - B a n k s solution. This means, a m o n g o t h e r things, t h a t high side errors in the source flow rate Q or initial c o n c e n t r a t i o n Co will result in an artificial scaling o f the transverse coefficients. F u r t h e r , it is i m p o r t a n t to recognize t h a t eq. 19 is valid o n l y for xv/Dx > 10. T h e t h r e e - d i m e n s i o n a l result suffers f r o m this same p r o b l e m , b u t its c o n d i t i o n o f validity is n o t k n o w n . F o r x ~ vt, the O g a t a - - B a n k s solution will c o n s i s t e n t l y p r e d i c t a concent r a t i o n a p p r o x i m a t e l y equal to the initial c o n c e n t r a t i o n C 0. Thus, f o r x ~ vt, eqs. 1 6 - - 1 8 can be a p p r o x i m a t e d as: C' -" Co
C' -~ Co[Q/2(TrDyxv) 1 2]
(21)
C' ~ Co [ Q / 4 x n ( D y D ~ ) 121 which are all m a x i m u m c o n c e n t r a t i o n s . I n t e r e s t i n g l y , if a m a x i m u m concent r a t i o n is k n o w n at o n e p o i n t along the centerline, a c o r r e s p o n d i n g m a x i m u m can be d e t e r m i n e d at a n y o t h e r p o i n t along the centerline. F o r the Wilson--Miller t w o - d i m e n s i o n a l result:
C'1/C'2 = ( x 2 / x l ) 1'2
(22)
w h e r e C'~ is a k n o w n m a x i m u m at x~; and C' 2 is a p r e d i c t e d m a x i m u m at s o m e p o i n t x2. T h e t h r e e - d i m e n s i o n a l result yields a similar r e l a t i o n s h i p e x c e p t t h a t the c o n c e n t r a t i o n ratio is just equal to a distance ratio. T h e a b o v e cited e q u a t i o n s p r o v i d e a m e a n s by which dispersion coefficients m a y be d e t e r m i n e d . An e x a m p l e is p r o v i d e d b y the L o n g Island,
207
New York, U.S.A., simulation of Wilson and Miller (1978), which was depicted as a line source with longitudinal dispersion in x and transverse dispersion in y. Fig. 7 illustrates the approach. Curve A is a centerline concentration profile based o n their field data, curve B is a steady-state centerline concentration profile generated by eq. 22, and curve C is a relative concentration profile (C/C') developed by taking the ratio of curves A to B. Using any steady-state concentration from curve A, Dy is readily determined with eq. 21. For this calculation, values for seepage velocity and source flow rate are given by Wilson and Miller (1978), and Dy was found to vary between 0.23 and 0.25 cm 2s -1. The Wilson and Miller study assumed a transverse coefficient of 0 . 2 2 9 c m 2 s -1. The longitudinal coefficient is determined by taking any relative concentration off curve C for input into eq. 19. It is noted that in eq. 19, QCo/2(lrDyxv) 1t2 is equal to C'. For this case, Dx was determined to be 1.2 cm 2 s -1, which compares favorably with the value of 1.13 cm 2 s- 1 employed by Wilson and Miller (1978). Plots such as given in Fig. 7 are useful for other purposes. For example, the center of mass (vt) o f a contaminant will always be located at a unique distance x where the relative concentration ratio C/C' equals 0.5. For this case, the center of mass is located at x = 1.28" 10 s cm. Hence if one of the quantities in the product vt is unknown, it may be easily determined from this plot. A chemically retarded species would also have its center of mass located at some point x where C/C' = 0.5. The distance x 5O
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-
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z
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~
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I
-I
--
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0.8
40
\
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I--z W 0
f
\ \
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*
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0
20 I--Z
,J
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8
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CENTERLINE FLOW DISTANCE ( x l 0 4 c m )
Fig. 7. Centerline c o n c e n t r a t i o n s for the L o n g Island plume. Curve A s h o w s field concentrations, curve B s h o w s steady-state concentrations, and curve C is the relative concentration distribution.
208 here would be less than t h a t c o r r e s p o n d i n g to a conservative species. The r e t a r d a t i o n f a c t o r is easily f o u n d b y taking the ratio o f the respective distances. T h e s t r u c t u r e o f the few analytical p o i n t - s o u r c e models discussed above can be f u r t h e r e x p l o i t e d in the analysis o f field i n f o r m a t i o n . F o r example, it would be m o s t helpful to have a means by which the dispersion coefficients can be d e t e r m i n e d f r o m s o u r c e - i n d e p e n d e n t forms o f the above cited equations. This follows f r o m the fact t h a t b o t h Co and Q are seldom, if ever, k n o w n in a simulation study. In this regard, eq. 19 for the Wilson--Miller ( 1 9 7 8 ) m o d e l can be expressed as:
C/C' = ½erfc{(x -- vt)/2(Dxt) 1 2}
(23)
which has the basic f o r m o f the Ogata--Banks solution, lts application requires a centerline p o i n t o f observation in f r o n t of the c e n t e r o f mass where C is any c o n c e n t r a t i o n less than C' t h a t occurs at time t, and C' is the m a x i m u m t h a t w o u l d o c c u r at t h a t point, as d e t e r m i n e d f r o m eq. 22. The longitudinal dispersion c o e f f i c i e n t can t h e n be easily f o u n d f r o m eq. 23. As with the i n s t a n t a n e o u s cases, the d e t e r m i n a t i o n o f the transverse c o e f f i c i e n t which is i n d e p e n d e n t of source i n f o r m a t i o n requires an observation o f f the centerline o f flow. T h e c o n c e n t r a t i o n for any p o i n t in the flow field is expressed by the full f o r m o f the Wilson--Miller model: C = [CoQ/47r(D~Dy)1/2] exp (x/B) (T:B/2r) 1 2 ,, exp (--r/B) e r f c { - - ( r / B - - 2u)/2u I 2}
(24)
where B = 2D x/v; u - rZ/4Dxt; and r = ( x 2 Jr- yZD~/Dy )1,2
(25)
Substituting eq. 21 into eq. 24 gives:
C/C' = ( x / r ) l ' 2 e x p ( x v / 2 D ~ ) e x p ( - - r v / 2 D ~ ) ~ e r f c { ( r - - vt)/2(Dxt) 12}
(26)
where C c o r r e s p o n d s to the c o n c e n t r a t i o n at some o f f c e n t e r p o i n t ( x , y ) at time t; and C' is again the m a x i m u m c o n c e n t r a t i o n at the centerline p o i n t x. The centerline p o i n t x is the same as the selected x distance in the measurem e n t o f the o f f - c e n t e r c o n c e n t r a t i o n C(x,y), i.e. points 1 and 2 o f Fig. 2. It follows t h a t eq. 26 can be iterated for r, providing an estimate o f the transverse coefficient. T h e p r o b l e m simplifies considerably if the o f f - c e n t e r c o n c e n t r a t i o n is also at a steady state, which reduces eq. 26 to:
C/C . . . .
(x/r) 1'2 exp (xv/2D~) exp (-- rv/2D~ )
(27)
I t e r a t i o n will provide a value f o r r and, u l t i m a t e l y , the transverse coefficient. Given the coefficients, the mass flow rate CoQ can be easily d e t e r m i n e d . This, however, is n o t necessary in that the spatial distribution o f concentration can be r e p r o d u c e d i n d e p e n d e n t o f the mass flow rate by e m p l o y i n g eqs. 22, 23 and 26.
209 It follows that a similar set of equations can be derived for the threedimensional point-source solution of Hunt {1978). The expression equivalent to eq. 23 becomes:
C/C' = ½[erfc{[x - - v t ] / 2 { D x t ) lj2 } + exp(xv/Dx)erfc{[x + vt] /2{Dxt)l/2}] (28) which will provide a source-independent determination of D~. The equivalent to eq. 26 is:
C/C' = (x/r) exp (xv/2Dx)½ [exp (-- rv/2Dx) erfc {(r -- vt)/2(Dxt) lj2 } ÷ exp (rv/2Dx) erfc {(r + vt)/2(D~t)l/2}]
(29)
If the concentration C is measured at some point (x,y,0), r is expressed by eq. 25. If the concentration is measured at some point (x, 0,z), r = (x 2 +z2Dx/Dz) 1/2 The maximum concentration is, as usual, measured at (x,0,0). Given three such points, both transverse coefficients may be determined. If steady state is achieved at all three points of observation, we have the equivalent of eq. 27:
C/C' = (x/r) exp (xv/2Dx) exp (-- rv/2D~)
(30)
where C corresponds to the point (x, y , 0 ) or (x,0,z). The difference between this equation and its two-dimensional equivalent (eq. 27) lies merely in the square root of the first term. As with the two-dimensional result, the spatial distribution of concentration can be reproduced independently of the source mass flow rate. CONCLUDING STATEMENT The above techniques permit simplified determinations of dispersion coefficients using data from field or laboratory tracer studies, contaminant spills, or long-term contamination histories. Importantly, they do require knowledge of the flow field in order to define a centerline of flow. Further, the use of such techniques presupposes a rather simple geology in the region of interest. Clearly, a complex geology will produce complex plumes that can only be analyzed with site-specific numerical models. The analysis given for instantaneous sources is reasonably straightforward and merely consists of replacing one equation with three unknowns b y one with a single unknown. This is accomplished by embedding two of the unknowns into an experimentally determined variable, namely the maximum observed concentration. The analysis for multidimensional continuous pointsources does essentially the same by taking advantage of centerline solutions that possess the same form as the simple one-dimensional dispersion convection solution. In these relations, the multidimensional maximum
210 c o n c e n t r a t i o n C ' w h i c h is a f u n c t i o n o f mass flow rate, d i s t a n c e and transverse d i s p e r s i o n , p l a y s t h e s a m e r o l e as t h e o n e - d i m e n s i o n a l m a x i m u m c o n c e n t r a t i o n Co. I m p o r t a n t l y , t h e t o t a l p r o b l e m can be a d d r e s s e d i n d e p e n d e n t l y o f mass f l o w r a t e s a n d initial c o n c e n t r a t i o n , w h i c h are s e l d o m k n o w n . In a d d i t i o n , s o m e a s p e c t s o f scaling are discussed. D i m e n s i o n a l i t y mism a t c h e s a n d f i n i t e - s o u r c e i n f l u e n c e s o n p o i n t - s o u r c e a s s u m p t i o n s a p p e a r to have a r a t h e r d r a m a t i c i m p a c t on c a l i b r a t e d c o e f f i c i e n t s . A l t h o u g h it is reasonable to accept the physically based premise that heterogeneity i n f l u e n c e s t h e m a g n i t u d e o f t h e d i s p e r s i v i t y , several o f t h e m e a s u r e m e n t s p u r p o r t i n g t h e d e t e c t i o n o f this c o n d i t i o n m a y , i n s t e a d , r e f l e c t m o d e l dependence.
ACKNOWLEDGEMENTS This s t u d y has b e e n f u n d e d in p a r t by g r a n t s f r o m t h e C e n t e r f o r E n e r g y a n d Mineral R e s o u r c e s at T e x a s A & M U n i v e r s i t y , t h e A l u m i n u m C o m p a n y of A m e r i c a and the Texas Municipal Power Agency.
REFERENCES Anderson, M.P., 1979. Using models to simulate the movement of plumes through groundwater flow systems. Crit. Rev. Environ. Control, 9: 97--156. Baetsl~, L.H., 1969. Migration of radionuclides in porous media. In: A.M.F. Duhamel (Editor), Progress in Nuclear Energy, Series XII, Health Physics. Pergamon, Elmsford, N.Y., pp. 707--730. Bredehoeft, J.D. and Pinder, G.F., 1973. Mass transport in flowing groundwater. Water Resour. Res., 9: 194--210. Bruch, J.C. and Street, R.L., 1967. Two-dimensional dispersion. J. Sanit. Eng. Div., Proc. Am. Soc. Civ. Eng., 93: 17--39. Domenico, P.A. and Robbins, G.A., 1984. A dispersion scale effect in model calibrations and field tracer experiments. J. Hydrol., 70: 123--] 32. Fried, J.J., 1975. Groundwater Pollution. Elsevier, Amsterdam, 330 pp. Grane, F.E. and Gardner, G.H.F., 1961. Measurements of transverse dispersion in granular media. J. Chem. Eng. Data, 6: 283--287. Harleman, D.R.F. and Ruiner, R.R., 1963. Longitudinal and lateral dispersion in an isotropic porous medium. J. Fluid Mech., 16: 385--394. Hunt, B., 1978. Dispersive sources in uniform groundwater flow. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng., 104: 75--85. Ivanovich, M. and Smith, D.B., 1978. Determination of aquifer parameters by a two-well pulsed method using radioactive tracers. J. Hydrol., 36: 35--45. Klotz, D. and Moser, H., 1974. Hydrodynamic dispersion as aquifer characteristic, model experiments with radioactive tracers. In: Isotope Techniques in Groundwater Hydrology, Vol. II. International Atomic Energy Agency (I.A.E.A.), Vienna, pp. 341--355. Lawson, D.W. and Elrick, D.E., 1972. A new method for determining and interpreting dispersion coefficients in porous media. In: Proceedings of the Second Symposium on Fundamentals of Transport Phenomena in Porous Media, Vol. II, University of Guelph, Guelph, Ont., pp. 753--777.
211
Molinari, J.P., Peaudecerf, B.G. and Launay, M., 1977. Essais conjoints en laboratoire et sur le terrain en rue d'une approche similif6e de la pr6vision des propagations des substances miscibles dans les aquifbres r6els. Proc. Symp. on Hydrodynamic Diffusion and Dispersion in Porous Media, Int. Assoc. Hydraul. Res., Pavia, pp. 89--102. Ogata, A. and Banks, R.B,, 1961. Solution of the differential equation of longitudinal dispersion in porous media. U.S. Geol. Surv., Prof. Pap. 411-A, 7 pp. Peaudecerf, P. and Sauty, J.P., 1978. Application of a mathematical model to the characterization of dispersion effects of groundwater quality. Prog. Water Technol., 10: 443--454. Pickens, J.F. and Grisak, G.E., 1981. Modeling of scale-dependent dispersion in hydrogeologic systems. Water Resour. Res., 17: 1701--1711. Pickens, J.F., Merritt, W.F. and Cherry, J.A., 1977. Field determination of the physical contaminant transport parameters in a sandy aquifer. Proc. Int. At. Energy Agency, Adv. Group Meet., Cracow. Robbins, G.A., 1983. Determining dispersion parameters to predict groundwater contamination. Ph.D. Dissertation, Texas A & M University, College Station, Texas, 226 pp. Sauty, J.P., 1980. An analysis of hydrodispersive transfer in aquifers. Water Resour. Res., 16: 145--148. Simpson, E.S., 1962. Transverse dispersion in liquid flow through porous media. U.S. Geol. Surv., Prof. Pap., 411-C, 30 pp. Sudicky, E.A. and Cherry, J.A., 1979. Field observations of tracer dispersion under natural flow conditions in an unconfined sandy aquifer. 14th Can. Syrup. on Water Pollution Research, Cent. Div., Victoria College, Toronto, Ont., 17 pp. Sudicky, E.A., Cherry, J.A. and Frind, E.O., 1983. Migration of contaminants in groundwater at a landfill: a case study, 4. A natural-gradient dispersion test. In: J.A. Cherry (Guest-Editor), Migration of Contaminants in Groundwater at a Landfill: A Case Study. J. Hydrol., 6 3 : 8 1 - - 1 0 8 (special issue). Wilson, J.L. and Miller, P.J., 1978. Two dimensional plume in uniform groundwater flow. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng., 104: 503--514.
195
[2] DETERMINING DISPERSION COEFFICIENTS AND SOURCES OF MODEL-DEPENDENT SCALING
G.A. ROBBINS and P.A. DOMENICO
Department o f Geology, Texas A & M University, College Station, T X 77843 (U.S.A.) (Received July 18, 1983; revised and accepted April 12, 1984)
ABSTRACT Robbins, G.A. and Domenico, P.A., 1984. Determining dispersion coefficients and sources of model-dependent scaling. J. Hydrol., 75: 195--211. This paper deals primarily with methods for determining dispersion coefficients in multidimensional dispersion problems. The methods developed are applicable to field and laboratory tracer tests, contamination spills, or long-term contamination histories. The analysis of three-dimensional data from instantaneous sources is simplified by embedding two u n k n o w n coefficients into an experimentally determined variable, namely the maximum observed concentration. The analysis of multidimensional continuous point-source data takes advantage of centerline solutions that possess the same form as the simple one-dimensional dispersion--convection problem. Importantly, all coefficients can be determined independently of initial concentration and source volumetric flow rate, two parameters that are seldom known in contamination problems. Finally, some aspects of model-dependent scaling are discussed. Modeling factors found to produce scaling in the measured dispersion coefficients include finite-source influences on pointsource methods and a nonconcurrence between the dimensionality of a particular model and the field dispersion condition it is supposed to represent.
INTRODUCTION
Among the major problems associated with predicting contaminant t r a n s p o r t is t h e d e t e r m i n a t i o n o f d i s p e r s i v i t y . I n t h i s r e g a r d , A n d e r s o n (1979, p . 1 0 0 ) has stated " . . . as yet, there is no standard technology for measuring longitudinal and transverse dispersivities." T h e v a s t m a j o r i t y o f s u c h d e t e r m i n a t i o n s t h a t d o e x i s t are f o r l o n g i t u d i n a l dispersion from laboratory column experiments. By-and-large, these experim e n t s h a v e b e e n c o n d u c t e d o n m a n - m a d e (e.g., glass b e a d s ) or r e m o d e l e d n a t u r a l materials and the a p p l i c a t i o n of their results to field p r o b l e m s can be questioned. Few laboratory experiments have been conducted to measure transverse dispersivity owing to the elaborateness of experimental procedures
0022-1694/84-85/$03.00
O 1984 Elsevier Science Publishers B.V.
196 and limited analysis techniques. Such experiments, as exemplified by those o f Bruch and Street {1967) and Simpson (1962), have employed large flow tanks. Although t he y provide insight into dispersion phenomena, t hey are impractical from the point of view of measuring transverse dispersion coefficients on unmolded samples representative of field conditions. Relative to laboratory experiments, only a limited num ber of field tests have been conducted. These studies entail pulse-type injection methods that generally do not include transverse dispersion in the calculations. One notable exception to this is the work of Sauty (1980}, who provided type-curve solutions to a two-dimensional point-source. However, a more c o m m o n procedure is to design a radial flow experiment and apply the point-source models developed by Fried (1975, p.66), or simply fit a one-dimensional dispersion model to field data and assume the dimensionality of the problem matches that of the model. If the assumptions of one-dimensional dispersion break down, an inevitable result is the lumping of transverse spreading into the calibrated longitudinal dispersion coefficient, i.e. a scaling effect (Domenico and Robbins, 1984). On the o t her hand, even if these various assumptions hold, most methods do not address the problem of multidimensional dispersion. Continuous-source m e t hods present yet a different set. of problems. Although such m et hods are not often used in field tests, continuous-source data provided by contaminant plumes have been used extensively in transport model calibrations. In such calibrations, however, the dispersion coefficients are seldom, if ever, the only unknowns in the problem. For example, the initial concentration and the volumetric flow rate of the source are seldom known. Other problems involve elements of the scaling effect already mentioned for pulse-type models, as well as those that occur when pointsource assumptions are forced on finite-source problems. Given these factors, the physical basis for calibrated field dispersion coefficients is highly suspect. Although it does not seem possible at this stage to standardize a technology o f measurement, there does appear to be a need for the design of tracer experiments whereby three-dimensional dispersion can be measured and modeled. This applies to the laboratory as well as the field. Further, there is a need to extract physically based coefficients from contamination histories. The purpose of this paper is to suggest a variety of procedures and appropriate models to achieve these ends and to demonstrate their application in field and laboratory studies.
INSTANTANEOUS SOURCE MODELS As discussed earlier, laboratory experiments to measure transverse dispersion coefficients are generally conducted in large rectangular flow tanks. The necessity of using rectangular tanks appears to be dictated largely by the analytical model c o m m o n l y used to analyze such tests (see e.g.,
197
H a r l e m a n and R u m e r , 1963). In this section, t h e o r e t i c a l m e t h o d s will be p r e s e n t e d t h r o u g h which it is possible t o calculate longitudinal and transverse dispersion c o e f f i c i e n t s in l a b o r a t o r y c o l u m n e x p e r i m e n t s or field tracer experiments. Finite-source m o d e l s T h e parallelepiped m o d e l as given b y H u n t ( 1 9 7 8 ) is o f the f o r m : C =
(C0/8) [ e r f { [ x - - v t -
-
-
-
-
-
+ ( X / 2 ) ] / 2 ( D : , t ) 1/2}
erf{ [x - - vt -- ( Z / 2 ) ] / 2 ( D x t) 1/2 }] [ e r f { [ y + ( Y / 2 ) ] / 2 ( D y t) 1/2} erf{ [ y -- ( Y / 2 ) ] / 2 ( D y t ) l / 2 } ] [erf ([z + ( Z / 2 ) ] / 2 ( D z t ) 1/2 } erf{ [z -- ( Z / 2 ) ] / 2 ( D z t ) l / 2 } ]
(1)
In t h e usual c o n v e n t i o n , x, y, z r e p r e s e n t a spatial c o o r d i n a t e system; v is the seepage velocity; Dx is the longitudinal dispersion coefficient; D y and Dz are the transverse dispersion coefficients; t is time; X, Y and Z r e f e r t o the dimensions o f the source o n Fig. 1; and erf is the e r r o r f u n c t i o n . In this f o r m u l a t i o n , ( x - vt) is d e f i n e d as the distance f r o m the c e n t e r o f mass, which is originally l o c a t e d at x = y = z = 0. T w o useful f o r m s o f eq. 1 are r e q u i r e d here. T h e first gives t h e c o n c e n t r a t i o n along w h a t is r e f e r r e d t o as t h e c e n t e r l i n e o f spreading w h e r e y = z = 0 f o r any x : C =
(C0/2) [ e r f { [ x - - v t -
-
+ ( X / 2 ) ] / 2 ( D ~ t ) 1/2}
erf{ [x -- vt -- ( X / 2 ) ] / 2 ( D ~ t ) l / 2 } ] [erf{ Y / 4 ( D y t)1/2}]
× [erf{Z/4(nzt)]/2}]
(2) ,JJ
J ~ 1 "/,!
.
I
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.
i
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•
X
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7
198
j~f Y
INJECTION ORIGIN
//
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OBSERVATION POINTS
Fig. 2. Coordinate identification of observation points. Because of the spreading g e o m e t r y , m a x i m u m c o n c e n t r a t i o n s will o c c u r along this centerline o f the plume. A second useful f o r m of eq. 1 is for some p o i n t along the spreading c e n t e r where x = vt, i.e. the c e n t e r of mass coincides with the p o i n t o f observation. F o r this case, eq. 1 or eq. 2 becomes: C' = (Co) [ e r f { X / 4 ( D x t ' ) l / 2 } ]
[erf{Y/4(Dyt')i'2}]
[erf{Z/4(D~t')12}]
(3)
where C' defines the m a x i m u m c o n c e n t r a t i o n at time t'. Fig. 2 is the m o s t c o m p l e t e scheme o f m e a s u r e m e n t for application o f these e q u a t i o n s to a three-dimensional dispersion p r o b l e m . T h e e x p e r i m e n t is c o n d u c t e d SO t h a t c o n c e n t r a t i o n is m o n i t o r e d at all f o u r points simult a n e o u s l y until Cl(x,O,O,t)
= C'~(x,O,O,t')
i.e. the c o n c e n t r a t i o n at measuring p o i n t 1 is described by eq. 3. Substituting eq. 3 into eq. 2 yields one e q u a t i o n with one u n k n o w n : C4 = (C'/2) [erf{ Ix4 -- vt' + ( X / 2 ) ] / 2 ( D x t ' ) ~ 2 } --erf{[x4--vt'--(X/2)]/2(Dxt')12}]/[erf{X/4(Dxt'~J
2}}
(4)
where C4 and C' are c o n c e n t r a t i o n s m e a s u r e d at points 4 and 1, respectively; x 4 refers to the distance to p o i n t 4; and time t' c o r r e s p o n d s to the time of m a x i m u m c o n c e n t r a t i o n b r e a k t h r o u g h at p o i n t 1. T h e p a r a m e t e r Dx m a y t h e n be d e t e r m i n e d by iteration. Similar e q u a t i o n s m a y be d e v e l o p e d for Dy and D~ by substituting eq. 3 into eq. 1 for the a p p r o p r i a t e conditions. Thus, for x = vt and z = 0: C~ = (C'/2) [ e r f { [ y + ( Y / 2 ) ] / 2 ( D y t ' ) l J 2 } - - e r f { [ y -- ( Y / 2 ) ] / 2 ( D y t ' ) ~ 2}] / [erf{ Y / 4 ( D y t ' ) l
2}]
(5)
where C2 and C' are m e a s u r e d c o n c e n t r a t i o n s at points 2 and 1, respectively; and time t' is again the b r e a k t h r o u g h time for C'. F o r x = t't and y = 0, the a p p r o p r i a t e e q u a t i o n becomes: C.~ -
(C'/2) [eft{ [z + ( Z / 2 ) ] / 2 ( D z t ' ) 1/2 } -- erf{ [z -- ( Z / 2 ) ] / 2 ( D ~ t ' ) 12 }l /
199
[erf { (ZI4 (Dz t') t/2 }]
(6)
where the variables are interpreted as above. The equations described above have been applied to a series of laboratory column experiments by Robbins (1983). The column was 10 cm in diameter, 93 cm in length, and contained glass beads with a diameter of 0.48 mm. For the glass bead experiment, it was assumed that Dy = Dz = DT where DT is the transverse dispersion coefficient. Tracer concentrations were measured at four locations in the column using calibrated conductivity probes. Other measurements were made to assure no tracer impingement on the column walls. Full details of the experimental apparatus and procedures are given by Robbins (1983). Fig. 3 illustrates relative concentration--time curves attained for a threedimensional pulse test at two probe locations in the column. These breakthrough curves were analyzed as described above in order to determine the dispersion coefficients. The dashed-line drawings in Fig. 3 illustrate a reproduction of these breakthrough curves based on the determined dispersion parameters. For this case, Dx = 2.03" 10 -4 and DT = 5.72" 10 -s cm 2 s- 1. For a series of such tests, the transverse dispersivity c~w ranged from 2 . 1 0 - 10 -3 to 6.14" 10 -3 cm. These values compare favorably with flow tank determinations of transverse dispersivity in similar type materials reported by others (Grane and Gardner, 1961; Harleman and Rumer, 1 9 6 3 ; Lawson and Elrick, 1972). Identical procedures can be applied to field distributions of concentration, such as contaminant spills or controlled tracer experiments of the type reported by Sudicky et al. (1983).
IO 0.8 06 c C' 0.4 0.2 0.0 4
1.0 0.8
~li°t%nc e .9cm
I
f
,
T
l
I
1
l
~/
,
,
~cm
I
I
,
1
Flow Distance
0.6
_c C' 0.4
0.2 // 6
8
TIME(rain)
lO
.
12
76
78
.
80
.
.
.
82
.
.
84
.
.
86
.
88
90
TiME (min)
• . . . .
Actual Model
curve curve
Fig. 3. Comparison of observed and predicted relative concentration--time curves for three-dimensional pulse test.
a
200 Point-source models
The point-pulse model as discussed by Baetsle (1969) is of the form: C = [ M 3 / 8 ( T r t ) 3/2 ( D x D y D z ) l " 2
'/, [exp { - - (x - - v t ) 2 / 4 D x
] t - - y2/4Dy t -- z 2 / 4 D z t}]
(7)
where M 3 is the mass of the tracer introduced at a point; and all other variables are as defined previously. The maximum concentration is defined at the point x = vt for y and z equal to zero by: (8)
C' = M 3 / 8 ( T r t ' ) 3'2 ( D x D y D z ) 1'2
where t' corresponds to the time when the maximum concentration is observed. Fig. 2 again describes the scheme of measurement, with the exception that measuring point 4 can be ignored. The experiment is conducted so that the concentration is monitored at all three points simultaneously until Cltx,O,O,t)
= C'(x,O,O,t')
i.e. the concentration at point I is described by eq. 8. At point 1, for any t, eq. 7 becomes: CI
= [m3/8(Trt)3/2(DxDyDz)l;2
] [exp{-- (x -- v t ) Z / 4 D x t } ]
(9)
Combining this equation with eq. 8 gives, eventually: -- (x -- v t ) 2 / 4 t
Dx
(10)
In { ( C z / C ' ) / ( t ' / t ) 3/2 }
Here, time t corresponds to the measurement time of C1, and t' to the measurement time of C'. Similar procedures for points 2 and point 3 yield: D y
=,
-- Y 2 / 4 t - -
(11)
In { ( C z / C ' ) / ( t ' / t ) 3 1 2 } + tx -- v t ) 2 / 4 D x t
and -- z 2 / 4 t
Dz
:: l n { ( C 3 / C ' ) / ( t ' / t ) 3 2 } + (x -- v t ) 2 / 4 D x t
(12)
The determination of Dx can be simplified through graphical procedures. This requires a plot of C / C ' vs. time for the point-/ data. Given such a plot, the standard deviation of the C / C ' time plot ( a t ) is described by: o t = F/2.354
(13)
where F is the full curve width at C / C ' = 0.5. The relationship between the standard deviation in time (Or) and the standard deviation in distance (ax) is given by: (axt 2 = v2ot 2
(14)
where v is the seepage velocity. For the point-source distribution, the longitudinal dispersion coefficient is then simply determined from:
201
Dx
(15)
= 0~212t '
It is emphasized that the graphical method is valid only for a true pointsource in that the full curve width for a finite source will reflect in part the initial source dimension. An alternate method of tracer test analysis for the instantaneous pointsource is given by Sauty (1980). Sauty's (1980) method is applicable to a two-dimensional spreading field where Oc= and ocT are determined by a typecurve procedure. Within certain ranges of the coefficients the similarity of the type curves introduces uncertainty in parameter determinations. The methods developed above incorporate the complete three-dimensional spreading field, and permit a more direct approach to the problem. M o d e l - d e p e n d e n t scaling
Investigations using instantaneous source tests have often reported the observation of scaling of dispersion coefficients with distance (Molinari et al., 1977; Pickens et al., 1977). This effect has been attributed to formation heterogeneity (Pickens and Grisak, 1981). This argument notwithstanding, there is yet another, perhaps more c o m m o n cause that can account for some of the observed scale effect. This type of scaling is of a model-dependent form. Fig. 4 has been generated to examine various forms of model-dependent A
--FINITE SOURCE --POINT SOURCE
x=lSm
B x=Srn I0
,°I
O8
08
C o6
cO6
C04
C 04I 02
O2
O0
6
8
I0
12
14 16 18 20 TIME (xlO4sec)
22
24
''
0.0
26
59
, o
b7
59
61
O6
C' 0 4
C'04
02
02
94
55
08
o
90
47 49 5[ 55 T I ME (x 104sec)
×=25m
O8
86
45
~o
O6
82
45
D
C
x =lOrn
c
41
98
102
106
TIME (* 105see)
I10
114
118
O0 23
24 25 26 TIME (x 105sec)
27
Fig. 4. Relative c o n c e n t r a t i o n - - t i m e curves for three-dimensional finite- and point-source models.
202
TABLE
I
Model calculation
d a t a u s e d in F i g s . 4 a n d 5
Three-dimensional
Three-dimensional
parallelepiped
point-source
X 100 cm Y - 100 cm Z = 100 cm
Co = 1 , 0 0 0 p p m
('0 - 1 , 0 0 0 p p m ~" .... l ' 1 0 - 3 c m s Dx :
injection mass = 10,000 mg
l
l't0-3em; s -1 D v :::: 5 • 1 0 - 4 e m 2 s -1 D~ -: 5 " 1 0 - 4 e m 2 s - '
V - 1"10
~cms
D x -- 1 • 10
Dy -
~cm: 5"104cm2s
D z = 5.10-4cm:s
i S-1
-~ -j
scaling. The figure shows a series of relative concentration--time data for several observation points along the centerline of flow, using the threedimensional parallelepiped and point-source models discussed earlier. The curves were generated using the data listed in Table I. The inner and outer curves are for the point and parallelepiped models, respectively. As reported in a previous paper (Domenico and Robbins, 1984), a scalingup of dispersivity will occur whenever an ( n - 1)-dimensional model is calibrated or otherwise e m pl oyed to describe an n-dimensional system. The scale effect associated with a dimensionality mismatch is shown as curve A in Fig. 5. This effect results from the fitting of three-dimensional dispersion data as shown in Fig. 4 to a one-dimensional finite-source model. As noted, the calibrated coefficient is approximately a function of the square of the distance traveled by the center of the mass. In Domenico and Robbins (1984) the dispersion coefficient was found to increase linearly with distance when two-dimensional data axe fitted to a one-dimensional model. Interestingly, from Fig. 5, observation points close to the source approach the actual value for the longitudinal dispersion coefficient. This occurs because, for three-dimensional finite sources, the error functions containing Y and Z in eq. 3 approach 1 for small times. Scaling of the type discussed above is evident in the processing of data collected by Sudicky and Cherry (1979). These investigators conduct ed what is likely the most comprehensive field test to date, employing five injection wells and 69 bundle-type samplers. The data have been analyzed with a onedimensional parallelepided model (8udicky and Cherry, 1979) and with a three-dimensional parallelepiped model (Sudicky et al., 1983). For their slow-zone data, the one-dimensional model predicts a piecewise linear increase in longitudinal dispersivity from ~ 2 to ~ 1 8 c m over an 11-m distance of observation from the source. Fast-zone data examined by Pickens and Grisak (1981) from the viewpoint of the one-dimensional analysis also exhibit scaling. The three-dimensional analysis (Sudicky et al., 1983) suggests a constant longitudinal dispersivity of ~ 8 cm over the observation
203
o 12
'
I .,/Y ,
l
4.0
A
2.o
m4! N z
IO
~ °
...... / __
0 O
I 5
, ,0
Actual Dx , 15
B 1 __ 2'0
25
30
Fig. 5. Scaling patterns for dimensionality mismatches (A and A') and point-source assumptions for finite-source data (B and B').
distance 4 - - 1 1 m from the source. The type of scaling associated with the one-dimensional analysis is clearly an artifact of the model employed. Additional scaling effects arise when finite-source data are analyzed with point-source models. Such procedures have been employed in both laboratory and field studies (Klotz and Moser, 1974; Ivanovich and Smith, 1978). As demonstrated in Fig. 4, the concentration distributions predicted by both models would eventually match with increasing distance. This is illustrated in Fig. 5 as curve B where at 30 m or so finite-source data can be analyzed with a point-source equation with no appreciable error in the calibrated coefficient. At distances less than 30 m, curve B demonstrates the scaling effect associated with analyzing finite-source data with a point model. As one might expect from the concentration distributions shown in Fig. 4, the longitudinal dispersion coefficient would have to be scaled up near the source, and would asymptotically approach the actual value with distance. The a m o u n t of scaling is dependent on the initial source size and observation distance. This same effect is anticipated when continuous finitesource data are analyzed with continuous point-source models. Such an observation has been made by Lawson and Elrick {1972) who used a pointsource model to analyze data from a large flow tank experiment. A laboratory demonstration of this effect is shown in Fig. 6, where finitesource data o f Robbins {1983) were analyzed with the point-source model.
204 0 55
Longitudinal
Pulse
- • -- ---* --. ---
030
Dimension
(cm)
325 2.93 210
~ ' 0 25
~ 0 20 E
~
0~5 0 bO
\
005
oi 0
I
5
.'-l
io
_ I
~5
I
I
zo
Flow
25 distance
#
1
1
30
35
x
(cm)
t
40
I,
45
I
50
I
55
Fig. 6. Apparent scaling associated with applying a point-source model to finite-source laboratory data. F o r the three tests shown, the a p p a r e n t dispersivity decreases to a finite value with distance. Also, the larger the pulse dimension (i.e. source size) the m o r e p r o n o u n c e d the scaling e f f e c t in the near field. T o b e t t e r visualize the relations shown in Fig. 5, the data are r e p l o t t e d in the f o r m o f Crx2 vs. observation distances in the same figure. T h e Gaussian relation b e t w e e n the variance Ox 2 and observation distance is expressed as: a~ 2 = 2 D x t = 2D,,x/v and is shown as the straight line in the figure. T h e c o n s t a n t slope o f the line reflects a c o n s t a n t Dx-value. Curve A' is for the case where a one- or twodimensional m o d e l is used to describe a three-dimensional dispersion c o n d i t i o n . This is perhaps a c o m m o n o c c u r r e n c e in field tracer studies where investigators are interested exclusively in longitudinal dispersion. U n f o r t u n a t e l y , for any d i m e n s i o n a l i t y mismatch, the e f f e c t o f a missing spreading d i r e c t i o n or directions will be l u m p e d into the calculation. As the observation points b e c o m e m o r e widely spaced, total dispersion increases, resulting in an a p p a r e n t increase in longitudinal dispersion with scale of the e x p e r i m e n t . This is e x e m p l i f i e d in curve A' where the variance--distance curve is c h a r a c t e r i z e d by a n o n l i n e a r segment at small distances f o l l o w e d by a relatively linear segment at larger travel distances. This e x a c t shape o f the variance--distance curve has been r e p o r t e d by Pickens and Grisak ( 1 9 8 1 ) to be c o m m o n in n u m e r o u s surface w a t e r dispersion studies. It is also a c o m m o n shape in n u m e r o u s tracer studies for g r o u n d w a t e r systems. As n o t e d b y Pickens and Grisak {1981), typical examples include the s t u d y o f Molinari et al. ( 1 9 7 7 ) and Pickens et al. {1977), a two-dimensional dispersion analysis b y P e a u d e c e r f and S a u t y (1978), and the one-dimensional analysis o f S u d i c k y and Cherry (1979). In all cases, the observed scale e f f e c t has been a t t r i b u t e d to heterogeneities.
205 Curve B' describes the case where a point-source model is used to describe a finite-source dispersion condition. As noted in Fig. 5, curve B' exhibits an initial increase in slope until a terminal value is reached, reflecting thereafter a constant dispersion coefficient. If the dispersion coefficient is determined on the basis of the o~--distance curve, there would be an apparent increase in the coefficient until a constant value is reached. Hence, we again have an apparent scaling of the t y p e discussed above.
CONTINUOUS POINT-SOURCE MODELS As a general rule, continuous-source methods are not often used in field testing procedures. On the other hand, continuous-source data provided by contaminant plumes have been useful in transport model calibration (Bredehoeft and Pinder, 1973; Wilson and Miller, 1978). In this section the structure of a few analytical models will be examined for the purpose of describing some methods for easy determination of field parameters. Although n o t designed for field application, the one-dimensional pointsource solution of Ogata and Banks (1961) will be the focal point of these discussions. This equation is: C = (C0/2) [ e r f c { ( x - - vt)/2(Dxt) 1/2} + exp (xv/Dx) erfc {(x + vt)/2(Dxt)w2}]
(16)
where erfc is the complementary error function; and Co is the initial concentration. As noted by Ogata and Banks (1961) for large xv/Dx-values ( ~ 500) the second term on the right-hand side may be ignored. A two-dimensional line-source solution can be obtained from the equation described by Wilson and Miller (1978) for the condition y = 0 and xv/2Dx ~ 10, which corresponds to the centerline of a two-dimensional plume:
C = [M2/4(~rD s )1/2 (1/xv)l/2] [erfc {(x -- vt)/2(Dx t) 1/2 }]
(17)
where M 2 is a flux term having the dimensions of M/Lt, in which L is length. This equation includes the series expansion given by Wilson and Miller (1978, p.505). As with instantaneous-source models, maximum concentrations will occur along this centerline of spreading. A three-dimensional centerline solution (y - z = 0) can be developed from the equation given by H u n t (1978, p.77):
C = [M3/8xTr(D~Dz) 1/2] [erfc {(x -- vt)/2(Dxt) 1/2 } + exp (xv/D~) erfc {(x + vt)/2(Dx t)1/2}]
(18)
where M3 = m/t. From the form of these equations, it is seen that the Ogata--Banks solution can be embedded in both the two- and three-dimensional centerline
206 results. T h a t is, considering the a b b r e v i a t e d solution, eq. 17 can be e x p r e s s e d as:
form
of 1;he O g a t a - - B a n k s
C = [Q/2(TrDyxz,) l;2] [(C0/2) e r f c { ( x -- vt)/2(D~t)12}]
(19)
where Q is a v o l u m e t r i c injection rate (L2/t) so t h a t CoQ = m / L t . T h e b r a c k e t e d q u a n t i t y on the right c o n t a i n s the a b b r e v i a t e d f o r m of the oned i m e n s i o n a l O g a t a - - B a n k s solution. Similarly, the t h r e e - d i m e n s i o n a l centerline solution o f H u n t ( 1 9 7 8 ) can be e x p r e s s e d as:
C
[Q/4xTr(DyDz) 12 ] [(C0/2) erfc {(x -- vt)/2(D:,t~ l 2} f e x p (xv/Dx) e r f c { ( x + v t ) / 2 ( D x t ) 12/1
(20)
w h e r e Q is again a v o l u m e t r i c injection rate (L3/t) so t h a t C0Q - m / t and the b r a c k e t e d q u a n t i t y c o n t a i n s the e x a c t O g a t a - - B a n k s solution. T h e s e interesting results indicate t h a t the centerline c o n c e n t r a t i o n s o f a two- or t h r e e - d i m e n s i o n a l p o i n t - s o u r c e are described by the w e l l - k n o w n oned i m e n s i o n a l solution, m o d i f i e d o f course by the t e r m s Q/2(rrDyxvl l z and Q/4xTr(DyDz) 1/2, which c o n t a i n the transverse dispersion coefficients. It is clear t h a t these t e r m s m u s t be less t h a n one, i.e. with transverse dispersion it is p a t e n t l y impossible for the field c o n c e n t r a t i o n at a n y x > 0 to be greater t h a n t h a t p r e d i c t e d by the o n e - d i m e n s i o n a l O g a t a - - B a n k s solution. This means, a m o n g o t h e r things, t h a t high side errors in the source flow rate Q or initial c o n c e n t r a t i o n Co will result in an artificial scaling o f the transverse coefficients. F u r t h e r , it is i m p o r t a n t to recognize t h a t eq. 19 is valid o n l y for xv/Dx > 10. T h e t h r e e - d i m e n s i o n a l result suffers f r o m this same p r o b l e m , b u t its c o n d i t i o n o f validity is n o t k n o w n . F o r x ~ vt, the O g a t a - - B a n k s solution will c o n s i s t e n t l y p r e d i c t a concent r a t i o n a p p r o x i m a t e l y equal to the initial c o n c e n t r a t i o n C 0. Thus, f o r x ~ vt, eqs. 1 6 - - 1 8 can be a p p r o x i m a t e d as: C' -" Co
C' -~ Co[Q/2(TrDyxv) 1 2]
(21)
C' ~ Co [ Q / 4 x n ( D y D ~ ) 121 which are all m a x i m u m c o n c e n t r a t i o n s . I n t e r e s t i n g l y , if a m a x i m u m concent r a t i o n is k n o w n at o n e p o i n t along the centerline, a c o r r e s p o n d i n g m a x i m u m can be d e t e r m i n e d at a n y o t h e r p o i n t along the centerline. F o r the Wilson--Miller t w o - d i m e n s i o n a l result:
C'1/C'2 = ( x 2 / x l ) 1'2
(22)
w h e r e C'~ is a k n o w n m a x i m u m at x~; and C' 2 is a p r e d i c t e d m a x i m u m at s o m e p o i n t x2. T h e t h r e e - d i m e n s i o n a l result yields a similar r e l a t i o n s h i p e x c e p t t h a t the c o n c e n t r a t i o n ratio is just equal to a distance ratio. T h e a b o v e cited e q u a t i o n s p r o v i d e a m e a n s by which dispersion coefficients m a y be d e t e r m i n e d . An e x a m p l e is p r o v i d e d b y the L o n g Island,
207
New York, U.S.A., simulation of Wilson and Miller (1978), which was depicted as a line source with longitudinal dispersion in x and transverse dispersion in y. Fig. 7 illustrates the approach. Curve A is a centerline concentration profile based o n their field data, curve B is a steady-state centerline concentration profile generated by eq. 22, and curve C is a relative concentration profile (C/C') developed by taking the ratio of curves A to B. Using any steady-state concentration from curve A, Dy is readily determined with eq. 21. For this calculation, values for seepage velocity and source flow rate are given by Wilson and Miller (1978), and Dy was found to vary between 0.23 and 0.25 cm 2s -1. The Wilson and Miller study assumed a transverse coefficient of 0 . 2 2 9 c m 2 s -1. The longitudinal coefficient is determined by taking any relative concentration off curve C for input into eq. 19. It is noted that in eq. 19, QCo/2(lrDyxv) 1t2 is equal to C'. For this case, Dx was determined to be 1.2 cm 2 s -1, which compares favorably with the value of 1.13 cm 2 s- 1 employed by Wilson and Miller (1978). Plots such as given in Fig. 7 are useful for other purposes. For example, the center of mass (vt) o f a contaminant will always be located at a unique distance x where the relative concentration ratio C/C' equals 0.5. For this case, the center of mass is located at x = 1.28" 10 s cm. Hence if one of the quantities in the product vt is unknown, it may be easily determined from this plot. A chemically retarded species would also have its center of mass located at some point x where C/C' = 0.5. The distance x 5O
Z 0
-
I " ~"
z
O
~
1
I
-I
--
I
1.0
0.8
40
\
rr
I--z W 0
f
\ \
50
*
0.6
0
20 I--Z
,J
0.4 _
0.2
I0
< X W
-c
I
0 0
I
4
I
I
8
L
I
12
I
I
16
m~
20
OD
CENTERLINE FLOW DISTANCE ( x l 0 4 c m )
Fig. 7. Centerline c o n c e n t r a t i o n s for the L o n g Island plume. Curve A s h o w s field concentrations, curve B s h o w s steady-state concentrations, and curve C is the relative concentration distribution.
208 here would be less than t h a t c o r r e s p o n d i n g to a conservative species. The r e t a r d a t i o n f a c t o r is easily f o u n d b y taking the ratio o f the respective distances. T h e s t r u c t u r e o f the few analytical p o i n t - s o u r c e models discussed above can be f u r t h e r e x p l o i t e d in the analysis o f field i n f o r m a t i o n . F o r example, it would be m o s t helpful to have a means by which the dispersion coefficients can be d e t e r m i n e d f r o m s o u r c e - i n d e p e n d e n t forms o f the above cited equations. This follows f r o m the fact t h a t b o t h Co and Q are seldom, if ever, k n o w n in a simulation study. In this regard, eq. 19 for the Wilson--Miller ( 1 9 7 8 ) m o d e l can be expressed as:
C/C' = ½erfc{(x -- vt)/2(Dxt) 1 2}
(23)
which has the basic f o r m o f the Ogata--Banks solution, lts application requires a centerline p o i n t o f observation in f r o n t of the c e n t e r o f mass where C is any c o n c e n t r a t i o n less than C' t h a t occurs at time t, and C' is the m a x i m u m t h a t w o u l d o c c u r at t h a t point, as d e t e r m i n e d f r o m eq. 22. The longitudinal dispersion c o e f f i c i e n t can t h e n be easily f o u n d f r o m eq. 23. As with the i n s t a n t a n e o u s cases, the d e t e r m i n a t i o n o f the transverse c o e f f i c i e n t which is i n d e p e n d e n t of source i n f o r m a t i o n requires an observation o f f the centerline o f flow. T h e c o n c e n t r a t i o n for any p o i n t in the flow field is expressed by the full f o r m o f the Wilson--Miller model: C = [CoQ/47r(D~Dy)1/2] exp (x/B) (T:B/2r) 1 2 ,, exp (--r/B) e r f c { - - ( r / B - - 2u)/2u I 2}
(24)
where B = 2D x/v; u - rZ/4Dxt; and r = ( x 2 Jr- yZD~/Dy )1,2
(25)
Substituting eq. 21 into eq. 24 gives:
C/C' = ( x / r ) l ' 2 e x p ( x v / 2 D ~ ) e x p ( - - r v / 2 D ~ ) ~ e r f c { ( r - - vt)/2(Dxt) 12}
(26)
where C c o r r e s p o n d s to the c o n c e n t r a t i o n at some o f f c e n t e r p o i n t ( x , y ) at time t; and C' is again the m a x i m u m c o n c e n t r a t i o n at the centerline p o i n t x. The centerline p o i n t x is the same as the selected x distance in the measurem e n t o f the o f f - c e n t e r c o n c e n t r a t i o n C(x,y), i.e. points 1 and 2 o f Fig. 2. It follows t h a t eq. 26 can be iterated for r, providing an estimate o f the transverse coefficient. T h e p r o b l e m simplifies considerably if the o f f - c e n t e r c o n c e n t r a t i o n is also at a steady state, which reduces eq. 26 to:
C/C . . . .
(x/r) 1'2 exp (xv/2D~) exp (-- rv/2D~ )
(27)
I t e r a t i o n will provide a value f o r r and, u l t i m a t e l y , the transverse coefficient. Given the coefficients, the mass flow rate CoQ can be easily d e t e r m i n e d . This, however, is n o t necessary in that the spatial distribution o f concentration can be r e p r o d u c e d i n d e p e n d e n t o f the mass flow rate by e m p l o y i n g eqs. 22, 23 and 26.
209 It follows that a similar set of equations can be derived for the threedimensional point-source solution of Hunt {1978). The expression equivalent to eq. 23 becomes:
C/C' = ½[erfc{[x - - v t ] / 2 { D x t ) lj2 } + exp(xv/Dx)erfc{[x + vt] /2{Dxt)l/2}] (28) which will provide a source-independent determination of D~. The equivalent to eq. 26 is:
C/C' = (x/r) exp (xv/2Dx)½ [exp (-- rv/2Dx) erfc {(r -- vt)/2(Dxt) lj2 } ÷ exp (rv/2Dx) erfc {(r + vt)/2(D~t)l/2}]
(29)
If the concentration C is measured at some point (x,y,0), r is expressed by eq. 25. If the concentration is measured at some point (x, 0,z), r = (x 2 +z2Dx/Dz) 1/2 The maximum concentration is, as usual, measured at (x,0,0). Given three such points, both transverse coefficients may be determined. If steady state is achieved at all three points of observation, we have the equivalent of eq. 27:
C/C' = (x/r) exp (xv/2Dx) exp (-- rv/2D~)
(30)
where C corresponds to the point (x, y , 0 ) or (x,0,z). The difference between this equation and its two-dimensional equivalent (eq. 27) lies merely in the square root of the first term. As with the two-dimensional result, the spatial distribution of concentration can be reproduced independently of the source mass flow rate. CONCLUDING STATEMENT The above techniques permit simplified determinations of dispersion coefficients using data from field or laboratory tracer studies, contaminant spills, or long-term contamination histories. Importantly, they do require knowledge of the flow field in order to define a centerline of flow. Further, the use of such techniques presupposes a rather simple geology in the region of interest. Clearly, a complex geology will produce complex plumes that can only be analyzed with site-specific numerical models. The analysis given for instantaneous sources is reasonably straightforward and merely consists of replacing one equation with three unknowns b y one with a single unknown. This is accomplished by embedding two of the unknowns into an experimentally determined variable, namely the maximum observed concentration. The analysis for multidimensional continuous pointsources does essentially the same by taking advantage of centerline solutions that possess the same form as the simple one-dimensional dispersion convection solution. In these relations, the multidimensional maximum
210 c o n c e n t r a t i o n C ' w h i c h is a f u n c t i o n o f mass flow rate, d i s t a n c e and transverse d i s p e r s i o n , p l a y s t h e s a m e r o l e as t h e o n e - d i m e n s i o n a l m a x i m u m c o n c e n t r a t i o n Co. I m p o r t a n t l y , t h e t o t a l p r o b l e m can be a d d r e s s e d i n d e p e n d e n t l y o f mass f l o w r a t e s a n d initial c o n c e n t r a t i o n , w h i c h are s e l d o m k n o w n . In a d d i t i o n , s o m e a s p e c t s o f scaling are discussed. D i m e n s i o n a l i t y mism a t c h e s a n d f i n i t e - s o u r c e i n f l u e n c e s o n p o i n t - s o u r c e a s s u m p t i o n s a p p e a r to have a r a t h e r d r a m a t i c i m p a c t on c a l i b r a t e d c o e f f i c i e n t s . A l t h o u g h it is reasonable to accept the physically based premise that heterogeneity i n f l u e n c e s t h e m a g n i t u d e o f t h e d i s p e r s i v i t y , several o f t h e m e a s u r e m e n t s p u r p o r t i n g t h e d e t e c t i o n o f this c o n d i t i o n m a y , i n s t e a d , r e f l e c t m o d e l dependence.
ACKNOWLEDGEMENTS This s t u d y has b e e n f u n d e d in p a r t by g r a n t s f r o m t h e C e n t e r f o r E n e r g y a n d Mineral R e s o u r c e s at T e x a s A & M U n i v e r s i t y , t h e A l u m i n u m C o m p a n y of A m e r i c a and the Texas Municipal Power Agency.
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