HYDROLOGICAL PROCESS, VOL. 4, 45-57 (1990)

ANALYSIS OF SOLUTE TRANSPORT WITH A HYPERBOLIC SCALE-DEPENDENT DISPERSION MODEL

S. MISHRA* AND J. C. PARKER

Blacksburg, VA 24061-0YDY, U.S.A. Center for Environmental and Hazardous Materials Studies, Virginia Polytechnic Institute and State University

ABSTRACT

An empirical hyperbolic scale-dependent dispersion model, which predicts a linear growth of dispersivity close to the origin and the attainment of an asymptotic dispersivity at large distances, is presented for deterministic modelling of field-scale solute transport and the analysis of solute transport experiments. A simple relationship is derived between local dispersivity, which is used in numerical simulations of solute transport, and effective dispersivity, which is estimated from the analysis of tracer breakthrough curves. The scale-dependent dispersion model is used to interpret a field tracer experiment by nonlinear least-squares inversion of a numerical solution for unsaturated transport. Simultaneous inversion of concentration-time data from several sampling locations indicates a linear growth of the dispersion process over the scale of the experiment. These findings are consistent with the results of an earlier analysis based on the use of a constant dispersion coefficient model at each of the sampling depths.

KEY WORDS Solute transport Scale-dependent dispersion Heterogeneous porous media Unsaturated zone

INTRODUCTION

Concerns with groundwater contamination have focussed attention on the development of accurate models for the prediction of subsurface pollutant transport. Current modelling practice typically involves solving the convection-dispersion (CD) equation for prescribed initial and boundary conditions using analytical, semianalytical, and/or numerical techniques (Javandel et a f . , 1984). For the simple case of one-dimensional transport of a non-reactive chemical tracer, the CD equation is given as

where c is the solution phase concentration [ML-3], q is Darcian fluid flux density [LT-'1, 8 is volumetric water content [L3LL-3], x is distance [L], t is time [TI, and D is the hydrodynamic dispersion coefficient [L2T-'] commonly taken to be of the form

D = Do + E IvI ( 2 )

Here Do is a porous medium diffusion coefficient [L'T-'1 which can often be neglected for field-scale tracer transport, IvI = Iq/0I is the magnitude of the macroscopic pore water velocity [LT-'1, and E is the dispersivity [L] which is classically assumed to be a scale-invariant characteristic of the porous medium (Bear, 1972).

Recent field investigations, both in the saturated zone and in the unsaturated zone, have indicated that dispersivity in fact is not constant but generally increases with the scale of observation, approaching a constant value only after the tracer plume has moved large distances (e.g. Jury et al., 1982; Gelhar et al. ,

*Present Address-INTERA Inc., 6850 Austin Center Blvd., Suite 300, Austin, TX 78731, USA

0885-6087/90/01OO45-13$O6.50 0 1990 by John Wiley & Sons, Ltd

Received 12 January 1989 Revised I0 August 1989

46 S. MISHRA AND J . C . PARKER

1985; Freyberg, 1986). Such behaviour has been attributed to variations in hydraulic conductivity which cause the pore water velocity to be spatially variable. Consequently, when convective transport in the spatially variable velocity field is modelled as a convective-dispersive process with some average velocity, it yields a scale-dependent dispersivity function. Results from stochastic analyses of dispersion in heterogeneous media have also shown that dispersivity increases linearly with distance when the mean distance of travel, X, is much smaller than the correlation length scale of hydraulic conductivity variations, h, and becomes asymptotically constant at large distances when X B h (Matheron and de Marsily, 1980; Gelhar and Axness, 1983; Dagan, 1984).

In Figure 1, we present data from several experiments (summarized in Table 3-4 of Gelhar et al., 1985) showing dispersivity as a function of observation scale for the unsaturated zone. The trend clearly indicates a growth in dispersivity with increasing displacement length. Although detailed dispersivity- distance data is not available for each site, it is reasonable to postulate that dispersivity would initially increase with distance in a linear fashion, stabilizing at some limiting value at large distances. These observations suggest that longitudinal dispersivity , E, may be approximated by an empirical hyperbolic distance-dependent model of the form

1 E=-

1 1 - + - Em P X

(3)

where E, is an asymptotic dispersivity [L] attained at large distances, p is a scale factor [LO] describing the linear growth of the dispersion process near the origin, and x is distance from an injection point. As will be shown later in the paper, Equation 3 is a reasonable representation of scale-dependent dispersion

l o o

A

E v -

10

>. t

> m

H

H

[r w a m H

cI1

-3 10

0

D O

0

0

0

0

0

0

l o o 10 l o 2 -1 10

DISTANCE (m>

Figure 1 . Data from Gelhar ef al. (1985) showing scale-dependent dispersion observed in unsaturated zone experimcnts

SOLUTE TRANSPORT ANALYSIS 47

phenomena observed in field experiments. Notice that when p -+ M , Equation 3 reduces to the classical model of constant dispersivity (i.e. E = E ~ ) ; whereas E, + ~0 results in a linear increase of dispersivity with distance, i.e. E = fix.

We emphasize here that the scale-dependent dispersion model introduced via Equation 3 is but a useful modelling tool. It is more general than the classical model of constant dispersivity (e.g. Bear, 1972), or models predicting purely linear growth of dispersivity with distance (e.g. Jury et al., 1982). Equation 3 may thus be utilized for deterministic modelling of scale-dependent dispersion in conjunction with numerical solutions of the CD equation. The hyperbolic form of Equation 3 also makes it a flexible model for fitting tracer BTC data which exhibit scale-dependent and/or asymptotic dispersivity behaviour.

LOCAL VERSUS EFFECTIVE DISPERSIVITY

When Equation 3 is used in a numerical simulator for modelling scale-dependent dispersion, the dispersivity, E, reflects spreading characteristics of the porous medium at the local scale. We will refer to this attribute as the local dispersivity. However, this is not the same as the dispersivity which might be obtained from the analysis of a tracer breakthrough curve (BTC). Such a value would reflect spreading characteristics of the entire soil profile up to the location at which the tracer BTC was monitored. We refer to the latter attribute as the effective dispersivity. Since the effective dispersivity is essentially an integrated measure of the variable local dispersivity, we postulate that the two definitions can be related by simple averaging, e.g. the arithmetic average

E ( X ) = 1; E(X) dx / 1; dx (4)

where E is the effective dispersivity and E is the local dispersivity. For the hyperbolic scale-dependent dispersion model, Equation 3, integration of Equation 4 yields

In(1 + fix/&,)

pX/ECC - Em

- + - E m px

where E% is the asymptotic dispersivity and p the dispersion scale factor, as introduced earlier. Numerical simulations of 1 -D solute transport were carried out to verify this postulated relationship

between local and effective dispersivities. A fully implicit Galerkin-type mass-lumped linear finite element code (van Genuchten, 1982) was used to solve Equation 1 with appropriate modifications to incorporate the empirical dispersion model given by Equation 3. Numerical dispersion was suppressed by introducing dispersion correction factors to the temporal discretization procedure (van Genuchten, 1978). The system considered was a Sm long soil column with a constant volumetric water content, 0 = 0.4. -4 0.1 pore volume slug of a non-reactive tracer was injected at a velocity of v = 0.01 m d-'. Assumed parameters for the scale-dependent dispersion model were E, = 0.2m and fi = 0.1. The spatial and temporal discretizations used were Ax = 0.05 m and At = 0.001 d , respectively. Tracer breakthrough curves were computed at depths of 1.2.5, 2.5 and 3.7Sm. For each of these depths, the effective dispersivity, E, was calculated using Equation 5, which was then used to predict tracer BTC behaviour assuming a constant effective dispersivity up to that depth. Comparison between these two sets of BTC data indicates excellent agreement (Figure 2) thus validating use of the arithmetic average in converting local dispersivity values to effective dispersivities via Equation 5 .

In order to demonstrate that the empirical hyperbolic scale-dependent dispersion model, Equation 3, is consistent with field observations, we use data from a natural gradient tracer experiment conducted in the Borden sand aquifer (Freyberg, 1986). Using spatial moment analysis, Freyberg calculated dispersivities as a function of the elapsed travel time (see his Table 3 and Figure l la ) . Note that his estimation of dispersivities from the temporal rate of change of spatial covariance is equivalent to fitting tracer BTC data to a solution of the CD equation with constant v and D (or E). Furthermore, these dispersivities

48 S. MISHRA AND J . C. PARKER

30 -

25 -

20 -

15 -

10 -

05 -

0.

0.

‘7 I: ‘r( 0.

u

‘ 0. u

0.

0.

x = 1 - 2 5 m

00 with local e

w i t h effective e

x = 2 . 5 0 m

x = 3 . 7 5 m

0.00 1 I I I I 25 75 125 175 225

T I M E (days) Figure 2. Comparison of breakthrough curves computed using local and effective dispersivity values

represent spreading characteristics of the solute between the origin and the monitoring location, and hence are effective dispersivities, i.e. E values. The E-t data of Freyberg’s Table 3 are readily converted to E-x values since the horizontal displacement of the solute plume’s centre of mass is observed to be a linear function of travel time (Figure 8 in Freyberg). A least-squares fit of the E-x data to Equation 5 yields E, = 0.385m and p = 0.222. Good agreement between effective dispersivities estimated by Freyberg and those computed from Equation 5 using the fitted values of ern and /3 are indicated in Figure 3. Note also that the fitted asymptotic dispersivity, E, = 0-385m, compares well with the asymptotic dispersivity value of 0.49m obtained by fitting the Dagan (1984) model to the spatial covariance data (Freyberg, 1986).

To reiterate then, our empirical scale-dependent dispersion model is consistent with field and theoretical observations which indicate a linear growth of the dispersion process close to the origin and the attainment of an asymptotic dispersivity at large distances. The local dispersivity, E, is appropriate for use with numerical models, whereas the effective dispersivity, E, corresponds to the dispersivity estimated from tracer BTC data.

Next, we examine the application of these concepts in the interpretation of tracer breakthrough curve data obtained from a field experiment conducted in the unsaturated zone.

APPLICATION TO UNSATURATED ZONE TRACER EXPERIMENT

Description of experiment We consider the analysis of a field solute transport experiment described in detail by Butters (1987).

Here we present only information pertinent to our analysis. The experimental site was located at the

SOLUTE TRANSPORT ANALYSIS 49

0. 45

n E

W

0. 30

>- I-

> Ln IY w 0.15 a Ln

n

H

H

n

0. 00

n

0 0

o b s e r v e d

f i t t e d

I I

0 20 40 60

D I STANCE (m> Figure 3. Comparison of dispersivity-distance data estimated by Freyberg (1986) and fitted using Equation 5

Etiwanda Field Station near Riverside, California, on a nearly level, deep, well-drained loamy sand soil with intermittent gravel lenses common at depths below 0.15 m (USDA Soil Survey, 1980). The 0.64 ha monitoring area consisted of 16 sampling locations arranged in a 4 x 4 square grid with 20m spacing between sampling points. Each location was instrumented with soil solution samplers at depths of 0.3, 0.6,0.9, 1.2, 1.8, and 3.05 m, with six of the locations having an additional sampler at 4.5 m. The site was irrigated by a sprinkler system capable of delivering 0-005 m h-' at 80-90 per cent uniformity.

Initially, the experimental field was irrigated regularly for several weeks to leach the soil of residual salts from previous experiments, as well as to achieve a reasonably time-invariant water content distribution in the soil profile. Areally averaged steady state volumetric water contents calculated from the measured gravimetric water contents using an average value of 1560 kg mW3 for bulk density are shown in Figure 4a as a function of depth. A narrow pulse of NaBr was introduced at a concentration of 58.9 mol m-3 in 0.0024 m of irrigation water. A field average of 2-32 m of water was added over a period of 265 days to leach the tracer beyond the deepest solution sampler at 4.5 m. The net applied water (i.e. cumulative throughput into the system) was estimated by subtracting evapotranspiration (ET) from the total applied water. A modified Penman combination equation (Doorenbos and Pruitt, 1975) with site meteorological data as well as data from a weighing lysimeter located at the centre of the field was used to estimate ET for the field. An estimate of the hydraulic flux (i.e. the flux responsible for tracer movement) was obtained by differentiating the net applied water versus time data, and is shown in Figure 4b. Over the course of approximately nine months, the soil solution bromide concentration was sampled 91 times at each of the 16 sampling sites and seven depths. The field scale areally averaged breakthrough curve (BTC) for each depth was constructed by arithmetically averaging the 16 individual BTCs for that depth.

0

-1 n E v

-2

I CL W 0

I-- -3

-4

-5

E

1 I I I

0. 0 0. 1 0. 2 0. 3 0. 4

WATER CONTENT ( m 3 m-3>

I

0. 008 v

0. 006 X 3 _1

0. 0 0 4

u H

_I -l 0 . 0 0 2

0 >- 0 . 0 0 0 T

0 50 100 150 200 _L

T I M E <days> Figure 4. Field average steady-state volumetric water content profile (a) and estimated hydraulic flux as a function of time (b)

SOLUTE TRANSPORT ANALYSIS 51

Analysis of tracer BTC data Assuming that the areally averaged behaviour of the NaBr tracer can be modelled as a one-dimensional

process, our objective is to estimate the unknown transport parameters, E, and 6, from measured concentration-time data by solving a non linear least-squares problem formulated as

min J = [c* - k(~-,@)] [c* - c ( E , , @ ) ] ' ~ (6)

where J is the least-squares objective function, c * = c* (x , t ) is the vector of concentration measurements at different spatial and/or temporal locations, and i5 = k(x,t;Em,f3) is the corresponding vector of model predicted concentrations for given values of the unknown parameters, E, and 0, obtained by solving Equation 1 with appropriate initial and boundary conditions. The nonlinear nature of the direct problem with respect to the parameters E, and necessitates the adoption of an iterative solution methodology for the inverse problem. Starting from an initial parameter vector, bo = b(Em.@)", the procedure involves computing a correction vector, Abi, at every iteration, i, such that

J(b' + Ah') d J(b') (7)

until changes in the objective function, J , and/or the elements of the correction vector, Ab, are sufficiently small. The Levenberg-Marquardt modification to the Gauss-Newton minimization scheme (Beck and Arnold, 1977) was used to calculate Ab. Details of this method as applicable to unsaturated flow and transport problems have been discussed by Kool et al. (1987) and will not be repeated here. The direct problem for unsaturated transport is solved using the numerical code mentioned earlier (van Genuchten, 1982) with a time step of 0.001 d and a spatial increment of 0.05 m.

Measured water content versus depth data (Figure 4a) and estimated hydraulic fluxes (Figure 4b) were used in the inversion analysis, together with approximately 10 concentration measurements from each of the seven depths. From the simultaneous inversion of the pooled data for all depths and times the dispersion scale factor was estimated to be = 0.19 and the asymptotic dispersivity was estimated to be E, = 3-67 m. This indicates that dispersivity, E, essentially increases linearly with distance over the scale of the experiment. The tracer BTCs predicted with these values of E, and @ for each of the seven depths are compared with observed data in Figures 5-11. The agreement between predicted and observed data is quite reasonable at 0.3m, 1*2m, and 1 4 m depths (Figures 5 , 8 and 9, respectively). The magnitude of the concentration peak is overestimated at 0.6 and 0.9 m depths (Figures &7), and underestimated at 3.05 and 4.5m depths (Figures 10-11). The arrival time of the concentration peak is also overestimated at these depths.

Discussion of results A major reason for the discrepancy between observed and predicted concentration data (as evidenced

by a regression R2 = 0.78) is believed to be uncertainty associated with the estimation of hydraulic flux. The numerical model used to solve the direct problem assumes hydraulic flux to be depth-independent, although the results suggest possible variability in water flux with depth. Integration of the area under the tracer BTCs yields hydraulic fluxes ranging from 0.0078 m d-' at 0.3 m to 0.0045 m d-' at 4-5 m (Butters, 1987) suggesting deviations from the assumption of one-dimensional flow. An estimate of mass recovery at each of the seven sampling depths indicated that tracer mass recovery varied from a low of 88 per cent at 4.5m to an anomalously high 120 per cent at 1.8m (Butters, 1987).

'The data set analysed here was also interpreted by Butters (1987), who fitted the BTC data at each depth to an analytical solution of the CD equation for pulse injection of a solute, which is given by

X (x- vt)2 - - - c(x, t )

cinJ 2 d ( x D ? )

where c(x,t) is the observed concentration and cinJ, is the injected concentration, with the dispersion coefficient, D , and local velocity, v = q/0, taken as independently adjustable parameters at each depth.

A m

I E

r-l

0 f

W

z 0

I- .( K t z W c) z 0 0

+-I

n m I E

0 E

4

v

z 0

I- < [r t Z w 0 Z 0 0

H

1. 50

0. 7 5

DEPTH - 0.3 rn 0 I ” \ fitted

0. 00

0 5 10 15 20 2s 30 35 T I M E <days>

Figure 5 . Comparison of observed and fitted breakthrough curves at 0.3 m

1. 2

0. 8

0. 4

0. 0

0

0 4 \O

DEPTH - 0 . 6 rn

0

fitted observed

0

60 10 20 30 40 50

T I M E (days) Figure 6. Comparison of observed and fitted breakthrough curves at 0.6m

n m I E

0 E

4

W

Z 0

I- < CT I- Z W L) Z 0 u

H

n

I m E 4

0 E

W

Z 0

t < IY I- Z W L) Z 0 0

H

0. 8

0. 6

0. 4

0. 2

0. 0

0. 6

0 . 5

0. 4

0. 3

0. 2

0. 1

0. 0

0

_ _ _ _ _ ~ ~

DEPTH - 0.9 rn

o observed

I I I

0 20 4 0 60 80

TIME (days) Figure 7. Comparison of observed and fitted breakthrough curves at 0.9m

f itted observed

100 20 40

TIME 60 80

Cdays) Figure 8. Comparison of observed and fitted breakthrough curves at 1.2m

n rn I E

0 E

d

W

Z 0

t < oi t- Z w 0 Z 0 0

H

n rn

I E

0 E

d

W

Z 0

I- < OL I- Z w 0 Z 0 0

H

0. 5

0. 4

0. 3

0. 2

0. 1

0. 0

0. 3

0 . 2

0. 1

0. 0

-0. 1

0

DEPTH - 1. 8 rn

- fittad observed

I I I I I d

I I I I I

0 20 40 60 80 100 120

TIME (days) Figure 9. Comparison of observed and fitted breakthrough curves at 1 4 m

0

DEPTH - 3.05 m fitted

0 obse rved I I I I I I

20 40 60

TIME 80 100 120 140

(days) Figure 10. Comparison of observed and fitted breakthrough curves at 3.05 m

SOLUTE TRANSPORT ANALYSIS 55

n

I m

E

0 E

r - l

v

Z 0

I- < [41 t Z W 0 Z 0 u

H

0. 3

0. 2

0. 1

0. 0

-0. 1

0

0

0

DEPTH - 4 .5 rn

obsarvad f i ttad

30 7 0 110 150 190

T I M E (days) Figure 11. Comparison of observed and fitted breakthrough curves at 4.5m

Estimates of D and v at each of the seven sampling locations, which were evaluated by fitting Equation 8 to observed concentration data with a least-squares procedure (Butters, 1987) are given in Table I. Two interesting observations may be made from this data. The first is that neither the local velocity, v, nor the flux, q = v8, is constant with depth-here 8 is used as depth-averaged volumetric water content rather than the local value. The second observation is that the effective dispersivity, E = D / v , varies with depth. Dispersivity appears to increase linearly with distance until about 3 m (excepting the anomalous behaviour at 0.9m) and seems to stabilize thereafter (see also Figure 12). However, the apparent

Table I . Transport parameters estimated by Butters (1987)

X 0* D V 9+ e* (m) (m3m-3) (m'd-') (md-') (md-') (m)

0.30 0.17 0.00103 0.028 0.00476 0.037 0.60 0.18 0.00239 0.027 0.00486 0.089 0.90 0.17 0.00715 0.023 0.00391 0.321 1.20 0.15 0.00495 0.027 0.00405 0.183 1.80 0.16 0.00730 0.031 0.00496 0.235 3.05 0.19 0.01329 0.036 0.00684 0.369 4.50 0.22 0.00892 0.035 0-00770 0.255

*Depth-averaged volumetric water content.

'Apparent dispersivity, E = Dh. +Local flux, y = v 8.

56 S. MISHRA AND J . C. PARKER

0.75

n E

W

0 . 50 > I-

> (I] [li 0 . 2 5 W CL (I]

0

H

H

H

0. 00

0

Butters (1987)

E f factive

0’

1 2 3 4 5

DEPTH ( m > Figure 12. Local and effective dispersivity-distance relationships estimated from this work, and effective dispersivity-distance

relationship estimated by Butters (1987)

stabilization observed at 3.05 m and 4.5 m can also be interpreted as short-range perturbations in the transport mechanism.

Based on the estimated values of E, and fi from our sirnulation-optimization analysis, the local dispersivity-distance (E-X) relationship can be calculated using Equation 3 . Moreover, the effective dispersivity-distance (E-X) relationship, which governs tracer breakthrough behaviour at a given location, can be computed from Equation 5 . These are shown in Figure 12 as solid and dashed lines, respectively. Also shown for comparison are dispersivity values estimated from the analysis of BTC data at several depths by Butters (1987). The latter are also effective dispersivities, albeit calculated using an analytical model with depth-dependent hydraulic fluxes. Notwithstanding these differences, there appears to be reasonable agreement between the effective dispersivities predicted from the hyperbolic local dispersivity model and those calculated by Butters (1987). Our approach, which is based on the integrated use of concentration data from different depths and times, can thus be viewed as yielding transport parameters representative of the entire soil profile. The approach of Butters (1987), on the other hand, which relies on the successive use of the constant coefficient CD equation at different depths, yields parameters averaged over the depth at which the BTC data were obtained.

SUMMARY

In this work, we have presented an empirical scale-dependent dispersion model which predicts a linear growth of dispersivity with distance close to the origin with dispersivity becoming constant at large distances. This model has been explicitly incorporated into the CD equation for deterministic numerical

SOLUTE TRANSPORT ANALYSIS 57

modelling of scale-dependent dispersion and for interpreting tracer transport data from field experiments. A simple relationship is derived between the spatially variable local dispersivity used in the numerical code, and the effective dispersivity. The scale-dependent dispersion model is used in the analysis of a solute transport experiment conducted in the unsaturated zone. A comprehensive analysis of tracer breakthrough data from seven depths, in conjunction with a nonlinear regression approach and a numerical simulator for solving the one-dimensional C D equation, indicates that dispersivity increases linearly with distance over the scale of the experiment. These results are consistent with earlier analyses of Butters (1987), who used an analytical solution to the CD equation under the assumption of a constant (depth averaged) dispersion coefficient.

The hyperbolic form of Equation 3 is consistent with field observations and theoretical findings, and provides a simple methodology for representing scale-dependent dispersion in numerical codes of solute transport. Furthermore, the relationship derived between local and effective dispersivities allows conventional interpretation of tracer BTC data (using a constant dispersivity model or a model predicting purely linear growth of dispersivity with distance) to be related to the results of simulation-optimization analyses using the hyperbolic model as in this study.

ACKNOWLEDGEMENTS

We thank Bill Jury of the University of California at Riverside for providing us with the experimental data, and our anonymous reviewers for their comments which helped clarify certain portions of the manuscript. Financial support was provided by the Electrical Power Research Institute, Solid Waste Environmental Studies Program under contract number RP2485-06.

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