A dispersion scale effect in model calibrations and field tracer experiments

Journal of Hydrology, 70 (1984) 123--132

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Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

[2]

A DISPERSION SCALE EFFECT IN MODEL CALIBRATIONS AND FIELD TRACER EXPERIMENTS

P.A. DOMENICO and G.A. ROBBINS

Department of Geology, Texas A & M University, College Station, TX 77843 (U.S.A.) (Received January 24, 1983;accepted for publication May 12, 1983)

ABSTRACT Domenico, P.A. and Robbins, G.A., 1984. A dispersion scale effect in model calibrations and field tracer experiments. J. Hydrol., 123--132. Field tracer experiments and model calibrations indicate that the magnitude of dispersivity increases as a function of the scale at which observations are made. Calculations presented in this study suggest that some part of this scaling may be explained as an artifact of the models used. Specifically, a scaling-up of dispersivity will occur whenever an (n -- 1)-dimensional model is calibrated or otherwise employed to describe an n-dimensional system. The calibrated coefficients for such models will depend not only on size of the contaminant plume or tracer experiment at the time of calibration, but will exhibit a size dependency beyond the calibration period. The magnitude of scaling appears to be sufficient to encompass the range of differences between laboratory measurements of dispersivity and model calibrations.

INTRODUCTION

The scale effect in dispersion literature refers to observations, calculations, or calibrations that suggest that the magnitude of measured coefficients of dispersion depend on the scale at which the measurements are taken (Cherry et al., 1975; Bredehoeft et al., 1976). This effect can be divided into two related parts: the increase in magnitude of dispersion coefficients as a function of distance of field measurement, and the many orders of m a g n i t u d e difference among measurements taken in the field, the laboratory, and values employed in calibrated models. Scale dependency in field measurements has been determined largely by tracer studies. Fried {1975), for example, suggests four scales of measurement ranging from local to regional where, presumably, the dispersivity value obtained reflects the scale of measurement. This has been the experience of Fried (1975), Molinari et al. {1977), and Pickens el~ al. (1977), who observed a five-fold increase in dispersivity when going from the local to the global scale. With regard to the second part of the problem, noted differences between field, laboratory, and model calibrations are clearly

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evident in Anderson's (1979) review where, in general, up to five orders of magnitude separate the laboratory measurements from the model calibrations, with tracer measurements being intermediate between these extremes. Several reasons have been offered to explain these scale effects, most of which rely on heterogeneity arguments or modeling requirements. The heterogeneity argument relates to the number, size, and arrangement of heterogeneities in the porous medium (Schwartz, 1977; Anderson, 1979). At larger scales, a greater number of heterogeneities may be encountered, and a higher dispersivity would be expected. The modeling arguments include such factors as dispersivity dependence on nodal spacing in calibration procedures (Robson, 1974), or profile vs. areal model selection (Robson, 1978). Thus, the magnitude of a calibrated coefficient can depend as much on the character of the model adopted as on the quality of the calibration itself. These arguments notwithstanding, there is y e t another, perhaps more common cause that can account for some of the observed scale effect associated with model calibrations and field tracer experiments. Namely, when field analysis procedures and model calibration techniques do not fully take into account the three4iimensional nature of dispersion, an observed scale effect results as an artifact of the models employed. Results of this study indicate a scaling-up of dispersivity will occur when the dimensionality of a calibrated model fails to match that of the natural system.

AN EXAMPLE OF THE SCALING EFFECT

As mentioned above, a scaling-up of dispersivity will occur where the dimensionality of a calibrated model fails to reflect that of the system it is supposed to represent. To demonstrate this conclusion, calculations have been performed using analytical models describing instantaneous and continuous sources. These models were chosen because they are computationally simple, they express the multidimensional nature of dispersion, and they permit examining longitudinal and transverse dispersion contributions to the overall dispersion process. As an example of such a model, consider a parallelepiped having initial source dimensions X, Y and Z, instantaneously injected into a unidirectional flow field (Fig. 1). At the start, the concentration within the parallelepiped is at a constant value, designated as Co, with its center of mass at X = Y = Z = 0. By convention, flow takes place parallel to the X dimension of the source. The plume that evolves in the x
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/j

/•

j J"

Z

./Z~'¸" j j-J"

jJ

J y~_$

J

~

Y

!

J

/-

~

J

j

Fig. 1. Parallelepiped source.

(z). Thus, the plume gets wider in the y-direction with distance from the source and, geology permitting, it gets thicker in the z-direction. The solution to this problem is given b y Carslaw and Jaeger (1959, p. 56), and Hunt (1978). For the geometry of Fig. 1 in a moving medium, this solution is:

C(x, y, z, t) = (C0/S) (erf [(x -- vt + ( X / 2 ) } / 2 ( D x t ) 1/2 ]

--

erf [(x -- vt -- ( X / 2 ) } / 2 ( D x t ) ~/2 ])

× (eft [{y + ( Y / 2 ) } / 2 ( D ~ t ) 1/2 ] - - e r r [(y - - ( Y / 2 ) } / 2 ( D y 0 `/2 ]) (eft [(z + (Z/2)}/2(Dzt) 1/2 ] err [ (z -- (Z/2)}/2

(Dz t)1/2 l)

(1)

In the usual convention, x, y and z represent a spatial coordinate system; v is the seepage velocity, Dx is the longitudinal dispersion coefficient; Dy and Dz are the transverse dispersion coefficients; t is time; and X, Y and Z refer to the dimensions of the source on Fig. 1. In this formulation, ( x - vt) is defined as the distance from the center of mass, which is originaUy located at x = y = z = 0. Two useful forms of eq. 1 will be given here. The first gives the concentration along what is referred to as the centerline of spreading where y = z = 0 for any x:

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C(y=~=o,x)

= (C0/2) (eft [{x-- vt + (X/2)}/2(Dxt) 1/2 ] -- erf [{x -- vt -- (X/2)}/2 (Dx t) 1/2 ]) [erf {Y/4 (Dy t) v2 } ] × [erf {Z/4(Dzt) 1/2} ]

(2)

Because of the spreading geometry, maximum concentrations will occur along this centerline of the plume. A second useful form of eq. I is for some point along the spreading center where x = vt, i.e. the center of mass coincides with the point of observation. For this case, eq. 1 or eq. 2 become: C(y =z=0, x =vt) = (Co) [erf {X/4(Dx t) vz} ] [erf {Y/4(Dy t) 1/2 } ] × [erf {Z/4(Dzt) An} ]

(3)

where the source dimensions come into play. Eqs. 1--3 provide an exact solution in which longitudinal and transverse dispersion contributions to the overall dispersion process may be examined. As an example of these contributions, consider the following. The dimensions, X, Y and Z of an instantaneous source are taken as 200, 100 and 10 cm, respectively. The dispersion coefficients are taken as more or less indicative of laboratory scale values, 10 -4 cm 2 s -1 for longitudinal dispersion and 10 -s cm 2 s-1 for transverse dispersion. With a groundwater velocity of 10 -4 c m s - 1 , the center of mass of the contaminant plume reaches a point 104 cm from the source in 10 s s. From eq. 3 the concentration is determined to be 0.03C0. For longitudinal dispersion alone, the concentration would have been 0.52C0, but transverse spreading reduces this to the calculated 0.03C0 value. Obviously, the calculated 0.03C0 value was obtained for a geometry that provided for maximum transverse spreading. Consider now that this three
127 of mass (travel time increases from 10 s to 101° s), the three-dimensional calculation with laboratory scale coefficients predicts a concentration of 3.58" 10 -s Co. To achieve this concentration with the two-dimensional model will once more require an increase in D~ or Dy, or both. If the absent z spreading is lumped into transverse spreading in y, the calibrated transverse dispersion coefficient is determined to be on the order of 0.10 cm: s -1 . If lumped into longitudinal dispersion, the calibrated value for D~ is l c m : s -1 . If lumped into both D~ and Dy, one workable nonunique combination is 10 -2 and 10-3 cm: s-1, respectively• These are reasonably close to typical values obtained in model calibrations (Anderson, 1979). From this example, it follows that calibrated models that fail to incorporate the three-dimensional nature of a dispersion process will be characterized by coefficients that depend not only on size at the time of calibration, but that require an increase in magnitude with increasing size beyond the calibration period. This conclusion will be demonstrated in more quantitative terms in the following section.

SIZE DEPENDENCY IN MODEL CALIBRATIONS

As shown in the previous calculations, if an ( n - 1)-dimensional model is calibrated or otherwise employed to describe an n-dimensional dispersive system, the calibrated coefficients increase appreciably over normal laboratory scale values. They also exhibit a dependence on distance. This particular feature of calibrated models is examined below for both instantaneous and continuous sources. I n s t a n t a n e o u s source m o d e l s

In this section, it is convenient to employ the equations presented by Baetsle (1969) for a one- and two~limensional instantaneous point-source. Assuming the system is actually two-dimensional, the equation for maximum concentration along the centerline of flow after some elapsed time t is given by: Cmax2

-~

M 2/4nt(DxDy)l/2

(4)

where M: is m L-1 so that C is m L-a • The maximum concentration for a one
= M 1/2(Dx~rt) 1/:

(5)

where M 1 is m L -2 so that concentration is m L -a. If a one-dimensional calibration (eq. 5) is forced on a two-dimensional condition (eq. 4), the observed concentration Cmax2 must be totally accounted for by a single scaled-up longitudinal dispersion coefficient in eq. 5. This is a generally accepted procedure in tracer experiments. Equating the right-hand sides

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of eqs. 4 and 5 and solving for this calibrated D* of eq. 5 gives: (6)

D* = M~ 47rDxDyt/M~

where Dx and D~ are the field coefficients as expressed in eq. 4. Recognizing that M~ is m L - : and M 2 is m L -1 eq. 6 becomes D* = [ 4 ~ ( l e n g t h ) 2 / ( a r e a ) 2 ] v x

(7)

= Avx

In this equation, time t has been replaced by the distance traveled by the center of mass divided by the velocity, and the dispersion coefficients have been replaced b y their conventional expressions as the product of dispersivity ~ and velocity. The calibrated coefficient is thus seen to be proportional to the velocity v and the distance traveled by the contaminant plume at the time of calibration. The scaling proportionality coefficient A includes the product of the actual coefficients and a length--area relationship that pertains to the dimensions over which the mass is distributed at the source. For a more or less constant velocity, the calibrated D* must accordingly reflect the size of the plume at the time of calibration, and must increase thereafter in proportion to the increase in plume size. As an example of the magnitude of values obtained in a hypothetical calibration, consider a wedge-shaped column with a circular opening at its apex. The column is designed so that transverse spreading in y is not impeded whereas spreading in z does not take place. That is, the column represents a two
' I0 s • 106 • 107 • 10 s • 109 • 10 l°

Distance (cm) 50 i00 1,000 10,000 100,000 1,000,000

D~ ( c m 2 s -1 ) 1 2 2 2 2 2

• 10 -4 • 10 - 4 • 10 -3 • 10 -2 • 10 - I

coefficient 0t~ (cm) 1 2 20 200 2,000 20,000

as a f u n c t i o n o f

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Continuous source models The concepts outlined above apply to the continuous source as well, at least during transients prior to reaching a steady state at time equal to infinity. The one
= (M1/2) [erfc {(x-- vt)/2(Dxt) 1/2 } + exp (xv/Dx)erfc {(x + vt)/2(D~t) 1/2 }]

(8)

where erfc is the complementary error function and M1 is m L -3, which is the initial concentration Co. As noted by Ogata and Banks {1961) for large xv/D~-values ( ~ 500) the second term on the right-hand side may be ignored. The two-dimensional line-source can be obtained from the equation described by Wilson and Miller (1978) for the condition y equals zero and xv/2D~ ~ 10, which corresponds to the centerline of a twodimensional plume: C2 =

[M 214(zrD~

)1/2 ] [(1/X7))1/2

erfc {(x -- vt)/2(Dxt) 1/2 }]

(9)

where M2 is m L-1 t-1 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 b y Hunt (1978, p. 77): C3 = (Ma/8xTrDT)[erfc {(x -- vt)/2(D~t) 1/2} + exp (xv/Dx) erfc {(x + vt)/2(D~t) 1/2 }]

(10)

where transverse dispersion coefficients in y and z are assumed equal and simply designated DT, and M3 is m t -1 . The procedures employed are similar to those used for the instantaneous source. To simplify matters, it is expedient to substitute the one-dimensional solution into the two- and three
C3 -- ~CIM3/M1DTXTr

(11)

where C 1 is given by eq. 8. A similar procedure may be employed for C2 provided the second term of the Ogata--Banks solution is ignored, as discussed previously. This gives, for eq. 9: C 2 = [CaM 2/2M 1 (TrDy)1/2 ]

(1/XV)I/2

(12)

where C1 is the abbreviated Ogata--Banks solution of eq. 8. The question posed is exactly the same as with the instantaneous source model, i.e. what is the effect of using a two-dimensional model to describe a three~limensional condition. From experience with instantaneous sources, it is expected that ,in order to achieve the field concentration C3, some

130

scaled-up coefficients will be required in the two
=

Axv

(13)

where the dispersion coefficient of eq. 11 has been replaced by the product of dispersivity and velocity. This result is virtually identical to the one developed for an instantaneous source (eq. 7) where a calibrated coefficient is proportional to velocity and scale. The scaling proportionality coefficient A includes the actual formation parameters and some length parameter that pertains to the dimensions over which the mass is distributed at the source. As an example of the magnitude of values obtained in a hypothetical calibration consider the column-type experiment and values for the parameters as discussed previously. For these values, the calibrated coefficient becomes D~ = 1.27" 10-Tx. For purposes of comparison, x is taken as equal to v t so that the position examined is consistently at the center of mass of a growing plume. Interestingly, for a value of x less than 1 0 0 c m (travel time of 106 s), the calibrated coefficient takes on an order of magnitude of 10 -6 cm 2 s -1 . This value is somewhat smaller than the initial value of 10 -s cm 2 s -1 . This apparent reverse scaling may result from two factors. First, the second term of the Ogata--Banks solution has been ignored. More importantly, the three
CONCLUSIONS

The main conclusion of this study is that the observed scale effect in dispersion modeling may be explained in part as an artifact of the models employed. With calibrated areal models, the following two points appear pertinent: (1) Because the scale effect is not limited to the calibration period, predictions beyond this period will tend to overestimate future concentrations. The degree of reliability of the model will thus depend on how

131

sensitive the solution is to one or more orders of magnitude increase in the model coefficients. This assumes, of course, that the dimensionality of the model fails to match that of the field, which is an accurate assessment for most areal model studies. (2) Model scale coefficients should not be confused with rigorously derived field values; they have little meaning- b e y o n d that of fitted scale parameters. The more the model departs from the field condition, and the larger the scale of the problem, the larger the coefficients. Tracer studies can also provide artificially inflated field values, but to a lesser extent. Unlike the laboratory column, such tests cannot be designed in geologic mediums so as to eliminate transverse dispersion. Hence, if an investigator is interested in longitudinal dispersion and fits tracer data to a one
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conditions, and on the capacity of a model to incorporate three-dimensional dispersion. If the latter condition is not satisfied, the coefficients must be scaled up. This, in turn, detracts from their physical meaning.

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