Journal of Contaminant Hydrology, 4 (1989) 93-107 93 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

VERTICAL TRANSPORT PROCESSES IN UNCONFINED AQUIFERS

DAVID W. OSTENDORF, DAVID A. RECKHOW and DAVID J. POPIELARCZYK

Environmental Engineering Program, Civil Engineering Department, University of Massachusetts, Amherst, MA 01003, U.S.A.

(Received August 18, 1987; revised and accepted April 12, 1988)

ABSTRACT

Ostendorf, D.W., Reckhow, D.H. and Popielarczyk, D.J., 1989. Vertical transport processes in unconfined aquifers. J. Contam. Hydrol., 4: 93-107.

We derive simple two-dimensional mathematical models describing the unsteady transport of conservative contaminants through an unconfined aquifer with a gently sloping aquiclude subject to advection, recharge, and vertical dispersion. The inclusion of vertical transport terms permits the proper nonreactive analysis of closed and open chemical systems, with the latter allowing dispersion of volatile constituents across the water table. These systems exhibit conservative and pseudoreactive behavior respectively when the pollution is analyzed on a depth-integrated basis, as is common in present one-dimensional models of groundwater contamination. Vertical and longitudinal chloride and total inorganic carbon observations at the well-documented Babylon, Long Island sanitary landfill plume are used to calibrate and test the analyses with a modest level of accuracy, using the vertical dispersivity as a calibration factor in this testing process. The parameter is important in the determination of reaeration rates across the water table and nutrient mixing from below in the related problem of biological transformations near the free surface.

INTRODUCTION

W e m o d e l t h e t w o - d i m e n s i o n a l u n s t e a d y t r a n s p o r t of a n o n r e a c t i v e con- s t i t u e n t t h r o u g h a n u n c o n f i n e d a q u i f e r w i t h a g e n t l y s l o p i n g p l a n a r b o t t o m . T h e a n a l y s i s p r o c e e d s u n d e r s t e a d y h y d r a u l i c s a n d i n c l u d e s a n a p p r o x i m a t e a c c o u n t o f t h e c o u p l e d v e r t i c a l t r a n s p o r t p r o c e s s e s o f r e c h a r g e a n d d i s p e r s i o n , s u b j e c t to c l o s e d ( n o n d i s p e r s i v e ) a n d o p e n ( d i s p e r s i v e ) b o u n d a r y c o n s t r a i n t s a t t h e w a t e r t a b l e . T h e l a t t e r c a s e g o v e r n s t h e loss o f v o l a t i l e c o n s t i t u e n t s t h r o u g h t h e f ree s u r f a c e . T h e r e s u l t i n g d e c r e a s e of c o n c e n t r a t i o n a p p e a r s as a r e a c t i v e loss t e r m , o r s ink , in c o n v e n t i o n a l d e p t h - i n t e g r a t e d m o d e l s of s u b s u r f a c e p o l l u t i o n ( W i l s o n a n d M i l l e r , 1978; B e a r , 1979; P r a k a s h , 1982). H e n c e t h e n e w a n a l y s i s d i s t i n g u i s h e s t h e n o n - r e a c t i v e d e g a s s i n g o f s u c h s p e c i e s as c a r b o n d i o x i d e o r n i t r o g e n f rom t r u l y r e a c t i v e p h e n o m e n a in w h i c h t h e l o s se s a r e i n t r i n s i c to t h e c h e m i s t r y a n d m i c r o b i o l o g y o f t h e s u b s u r f a c e e n v i r o n m e n t . T h e r e c h a r g e a n d d i s p e r s i v i t y a r e f u n c t i o n s o f t h e f low field, a n d a c c o r d i n g l y m a y be u s e d to c h a r a c t e r i z e t h e r e l a t e d a n d i m p o r t a n t p r o c e s s e s of a q u i f e r r e a e r a t i o n a n d n u t r i e n t m i x i n g w i t h d e p t h b e l o w t h e w a t e r t ab l e . T h e s e

0169-7722/89/$03.50 © 1989 Elsevier Science Publishers B.V.

94

latter phenomena are of interest to investigators of subsurface microbiology and in situ hazardous waste treatment.

Our model builds upon the existing analytical modeling efforts of Gelhar and Wilson (1974) and Ostendorf et al. (1984), who derive the recharge hydraulics and source term behavior for the present study, as discussed below. The advec- tire-dispersion solutions of Ogata and Banks (1961) and Parker and Van Genuchten (1984) describe the open- and closed-system profiles respectively, although the transport is horizontal in the original contexts. Additional, less directly related, depth-integrated models may be cited as well: Lenau (1972) postulates a steady-state, conservative injection of pollution from a recharge well in a confined aquifer, while Wilson and Miller (1978) consider the transient response of a simply reactive contaminant from a similar source configuration. Chen (1987) derives a model including radial hydraulics and transport of a conservative species in a confined aquifer. Bear (1979) summarizes unsteady contaminant migration due to a series of one-dimension- al reactive sources, and Prakash (1982) models steady-state reactive pollution due to instantaneous point, line, and volume sources. The method of charac- teristics adopted here follows one-dimensional analytical applications in the unsaturated and saturated zones by Wilson and Gelhar (1981) and Charbeneau (1981), respectively. Valocchi (1985, 1986) examines the validity of the local equilibrium assumption for simply adsorbing solutes in one-dimensional and radial flow fields on an analytical basis as well.

The present simple analytical modeling approach complements existing more complex numerical contaminant transport codes. In this regard, we may distinguish two (Konikow and Bredehoeft, 1978; Borden and Bedient, 1986) and three (Huyakorn et al., 1986) -dimensional numerical solute transoirt models of single and coupled (Cederberg et al., 1985) constituents. These site-specific programs, with their at tendant documentation requirements, are appropriate for the detailed analysis of fully measured plumes in complex geological settings. As such, they contrast with the present analytical effort, which is intended to provide a readily used, generic account of two-dimensional conta- mination of an idealized or sparsely measured flow field by a single species.

HYDRAULICS

We begin by considering the two-dimensional conservation of water mass in the unconfined aquifer, wherein a small (near)-vertical, average linear velocity component w is imposed by recharge upon a vertically uniform, (near)- horizontal, average linear velocity u (Gelhar and Wilson, 1974):

~u 0w - - + - 0 ( 1 ) ~x ~z

with distance z below the water table and downgradient distance x from the source of pollution, as sketched in Fig. 1. Following Ostendorf et al. (1984), the unconfined aquifer flow field is perturbed by small changes due to a gently

landf i l l

x x x x × x

It ~,w ~ ----,

hs L _ _ source I x,u

] - plane ] h

Fig. 1. Definition sketch.

95

sloping aquiclude, head loss, and recharge:

[ X(n )1 u = us 1 + ~ s t anf l (2a)

t an f l = tanf l ' - n v u s (2b) kg

with aquifer depth h, kinematic viscosity v, permeability k, gravitational ac- celeration g, recharge e, and porosity n. The s subscript denotes pollutant source conditions at the downgradient boundary of the landfill, while the slope tan fl relative to the water table includes the horizontally based slope tan fl'.

The corresponding vertical average linear velocity follows from eqns. (1) and (2a). To leading order, we have:

w - - us tan (3) n h s

Equation (3) suggests that w equals e / n at the water table and yields a velocity parallel to the aquielude where z equals ha, the first-order estimate of the actual aquifer thickness h.

We are interested in frames of reference moving at speed u from the down- gradient end of the landfill, taken as the (far field) source of pollution, as indicated by Fig. 1. The velocity and path xf of the frames are given by Ostendorf et al. (1984) as:

dxf - - U

dt

x f [ x f ( ~ z = - - 1 -- U~ ~ss rtUs

--tan )l (4a)

(4b)

96

= t - ts (4c)

with t ime r in the moving re ference frame which leaves the source at t ime t~. A (near field) l inear rese rvo i r analysis under the landfill has been der ived by Os tendor f et al. (1984) in order to specify source plane co n cen t r a t i o n as a func t ion of ts and user populat ion. Fol lowing the concept of Ge lha r and Wilson (1974), this simple approach t rea t s the complex nea r field flow region as a fully mixed chemos ta t rou t ing pol lu t ion input from the landfill to the ou tpu t plane at the downgrad ien t end of the facili ty, where it serves as a ver t ica l ly uni form source of con tamina t ion . Thus, the frames ca r ry known pol lu t ion levels into the far field with presumed uni form init ial ver t ica l profiles.

OPEN- AND CLOSED-SYSTEM CONCENTRATION PROFILES

The conserva t ion of nonreac t ive con t aminan t mass concen t r a t i on c can be conven ien t ly s tudied in the moving re ference frame as a ba lance of s torage, ver t ica l advect ion, and ver t ica l dispersion:

e~c ~ ~c ~2c ~-~ + -n--~z - D--0z 2 = 0 (5a)

D = ~us (5b)

Longi tud ina l dispersion is neglected in this t r anspor t equa t ion on the premise of an appreciable longi tudina l ve loci ty and a con t inuous pol lu t ion source (Os tendorf et al., 1984); ~ accord ingly represents the ver t ica l dispersivi ty (Bear, 1979). The use of sin in place of w in the advect ive t r anspor t term reflects our in te res t in condi t ions near the wate r table [see eq. (3)], where the bulk of t r anspor t occurs in the form of di lu t ion and degassing th rough the r echa rge lens.

The govern ing conserva t ion equa t ion (5a) is solved subject to a presumedly uni form source p lane concen t r a t i on cs and a deep aquifer, i.e.:

c = cs (T = 0) (6a)

c = cs (z = ~ ) (6b)

The l a t t e r cons t ra in t suggests tha t most of the t r anspor t occurs nea r the free surface and is unaf fec ted by the bottom; an assumption in keeping wi th the use of E/n as the ver t ica l ve loc i ty est imate. The remain ing wate r table boundary condi t ion differs for open and closed chemical systems:

c = c a (z = 0, open) (7a)

0c ~ - Dc~-- ~ + -nC = -nCa (z = 0, closed) (7b)

with cons tan t ambient concen t r a t i on c~. The open-system condi t ion (7a) rests on the premise of a rapidly diffusing

un sa tu r a t e d zone, as discussed for inorganic carbon in the Appendix.

97

Equations (5a), (6) and (7a) constitute the classical advective-diffusion problem with solution (Ogata and Banks, 1961):

c~ ca 2 erfc L 2 ~ - - ~ J + exp ~-~ erfc L 2(Dr),. 2 j j (8a)

(open)

erfc y - it,/2 exp ( -y '2)dy ' (8b) Y

with complementary error function erfc y (Abramowitz and Stegun, 1972). Asymptotic behavior of this function may be used to advantage, whence:

(z - ~ / n ) 2 7

c - Ca -- 1 -- i { rz__ ~,/n 1 ( _ ~ y , 2 e x p -4D-~T] ]1 ca - ca ~ erfcL2(Dr)l/2_j + 2 z + ~ j (9a)

z + ~r/n > 5(Dr) '/2 (9b)

The closed-system condition (7b) is predicated on a no-flux constraint at the water table, so that the advective and dispersive mechanisms are in balance at that location. A more complicated solution follows, as derived by Parker and Van Genuchten (1984):

c - C a - ] + exp(~'Z ) [Z +---~.1[~ ~(z + 8"In) 1 cs -- ca ~ erfc L 2(Dr)1/2 J + 2On ]

( ~ ~1,,~ [ (z - ~/n) ~] 1 [~_ : ~ /n l n \ ~ ) exp ] ~ - -j - ~ erfc L 2(Dr) '/2 J (10)

This too simplifies for a large argument:

(z - ~r/n)'- ' ]

exp JlerfcrZ- /n 1 (lea) C--Ca -- 1 + c~ - ca z + ~:~/n 2 L2(-D~-~J

z + ~r/n > 5(Dr) 1/2 (l lb)

Figure 2 displays typical profiles for open and closed systems under dominant dispersion and recharge conditions for a pure ambient aquifer. The strong dispersion case features deeper mixing of open constituents and an appreciable free-surface closed-system concentration. Strong recharge on the other hand, establishes a pure overlying lens of water regardless of the open or closed nature of the pollutant. The area under the contaminant profiles is proportion- al to the depth averaged concentration: Fig. 2a indicates that open and closed systems exhibit values below their source levels. The open-system decrease is biggest since it includes dilution and degassing effects. The dilution reduction is predominant in Fig. 2b and is common to both systems.

98

Z

h s

0 0

0.5

I.O

~ " - - " - , ~ s e d "~, ope n ~

\

Z

hs 0.5

open and closed

(°) I ,.o (b) I o o.5 I.O 0 0,5

c / c s c~ c s 1.0

Fig. 2. Typical open and closed c o n t a m i n a n t sys tem profiles for v s = 10-Sms -1, n = 0.3, and = 10Ss. S t rong d ispers ion (e = 10 -gms -1, a = 10-~m) is ske tched in Fig. 2a, whi le s t rong

r echa rge (~ = 10-Sins -~, ~ = 10-3m) is shown in Fig. 2b.

CALIBRATION AND TESTING

The total inorganic carbon and chloride observations of Kimmel and Braids (1974, 1980), supplemented by raw data supplied by the USGS may be used to calibrate and test the open and closed models, respectively. Vertical and longitudinal profiles of contamination from a sanitary landfill on Babylon, Long Island provide ideal and independent data bases, since the hydraulics are well understood and the unconfined aquifer geometry is appropriately simple. Indeed, the existing depth-integrated model of Ostendorf et al. (1984) is calibrated with this plume, based on a source average linear velocity (us) of 3.37 × 10 6ms-l , recharge (e) of 3.25 × 10 9ms 1, aquifer depth (ha) of 22.5m at the source plane, permeability of 6.34 × 10-11m ~, viscosity of 1.1 × 10-6m2s -1, and a porosity of 0.27.

Source plane concentrations are computed from monitoring well data located at the downgradient boundary of the landfill, following the near field procedure proposed by Ostendorf et al. (1984). The source plane wells are used to calibrate a per capita contaminant generation rate Scl equals 1.40 × 10 _8 kg cap-is -1 for chloride in the earlier analysis, and this value is adopted here to estimate the c a values in closed system testing. A total inorganic carbon value of Sc equals 7.90 × 10-gkgcap-ls -1 generates the open-system source term concentrations in the present study, based on 1973 observations in the source plane monitoring wells. The new Sc value is needed since Ostendorf et al. (1984) studied bicarbonate and not total inorganic carbon in their prior work.

We compare data and theory using statistics of the error 5 defined by:

c (measured) - c (predicted) 5 = (12a)

c (measured)

= _1. Z5 (12b) ]

~ . t I/2 = Z& 2 - ~ (12c)

99

The mean e r ro r ~ and s t andard devia t ion a are computed in accordance with Benjamin and Cornel l (1970). The sign of 3 indicates model over or underpredic- t ion and is accord ing ly useful in ident i fy ing systemat ic model errors. The e r ror s t andard dev ia t ion is based on the absolute va lue of individual 3's and conse- quent ly measures the magn i tude of the error. In this regard, about 2/3 of our predic t ions lie wi th in a of the i r measured values for a zero mean error .

The (far field) longi tudina l to ta l inorganic carbon profile da ta observed in 1973 by Kimmel and Braids (1980) are used to ca l ibra te the ver t ica l dispersivi ty in eqns. (5b) and (8a) with the resul ts summarized in Table 1 and Fig. 3a. The

TABLE 1

Longi tudinal profile of total inorganic carbon: model cal ibrat ion

x ~ c s z c (meas) c (pred) (m) (10 s s) (kg m- 3) (m) (kg m 3) (kg m- 3) (%)

360 1.05 0.136 1.75 0.017 0.022 - 32 9.05 0.087 0.100 - 15

25.11 0.141 0.136 4 630 1.80 0.117 23.16 0.091 0.114 24 900 2.54 0.091 3.08 0.007 0.015 - 115

8.97 0.136 0.044 68 18.45 0.137 0.077 44

920 2.59 0.089 11.74 0.035 0.055 56 21.50 0.060 0.081 - 36

1550 3.89 0.057 12.17 0.055 0.029 47 1890 5.02 0.045 12.69 0.067 0.020 69 2180 5.69 0.038 8.75 0.018 0.011 42

19.33 0.037 0.024 36 2230 5.80 0.037 17.66 0.010 0.021 99 2260 5.87 0.036 21.38 0.040 0.024 40 2810 7.05 0.020 11.89 0.010 0.007 32

18.26 0.010 0.010 - 6

Notes: 1973 data (Kimmel and Braids, 1980), supplemented by Source condit ions from Ostendorf et al. (1984). Concent ra t ions expressed as carbon. Calibrated dispersivity ~ 0.073 m, zeros mean error. Cal ibrat ion er ror s tandard deviation a = 54%.

USGS raw data.

0.15

0.10

c (kg m -3) 0.05__

(0) 0

0

el I I I ©15-25m o 5 - 1 5 m e O - 5 m

"u'x"~. ~ "a = 0.073 m o

500 1000 1500 2000 2500 3000 x (m)

I0

Z (rn)

2 0

5 0 (b) I xaXt l 0 0.025 0.050 0.0"75

¢(kg m -3)

II

0.0[0

Fig. 3. Total carbon cal ibrat ion (a) and test (b) data (circles) and predict ions (lines). 1973 observa- t ions of Kimmel and Braids (1974, 1980), supplemented by raw data supplied by USGS.

100

more distant wells have larger travel times (~), corresponding to frames leaving the landfill at earlier times with smaller user populations hence lower source concentrations. The vertical location refers to distance below the average water table elevation, and the open system is chosen for calibration purposes since it possesses a steeper gradient than its closed counterpar t and is accord- ingly more sensitive to ~. The calibration is a Fibonacci search (Beveridge and Schechter, 1970) for the dispersivity value zeroing the mean error. The operation yields a standard deviation of 54% associated with the optimal dispersivity:

= 0.073 m (13)

This dispersivity value is close to the first-order decay based value of 0.10 m put forth by Ostendorf et al. (1984) on an ad hoc basis in their depth-integrated analysis of bicarbonate data at the site. The newer version reflects a more rigorous estimate of the transport mechanisms: the "decay" is in reali ty a mass t ransport process constrained by vertical mixing and recharge. Table 1 suggests no systematic dependence of the error upon horizontal or vertical distance, so that the relatively high standard deviation may in part be at tr ibuted to sampling errors. In any event, additional independent testing of the calibrated model is certainly warranted.

The vertical inorganic carbon, longitudinal chloride, and vertical chloride observations of Kimmel and Braids (1974, 1980) provide these necessary tests, as summarized in Tables 2-4 and Figs. 3b and 4. The accuracy of the vertical inorganic carbon test is encouraging, as evidenced by the 28% mean and 21% standard deviation error statistics. The longitudinal chloride data provide an additional measure of support, part icularly if one neglects two data points immediately beneath the water table. Most of the mean error ( - 4 6 % ) and standard deviation (126%) for this test is created by the two near-surface data points 360m and 900m downgradient: we compute quite accurate values of - 2°./o and 25% for the statistics without these samples. The vertical chloride

TABLE 2

Vertical profile of total inorganic carbon: model test

z c (measured) c (predicted) 3 (m) (kgm ~) (kgm -3) (%)

5.58 0.0152 0.0113 25 12.01 0.0382 0.0249 34 14.27 0.0405 0.0294 27 19.30 0.0921 0.0378 59 24.17 0.0420 0.0442 - 5

Notes: USGS raw data, except 24.17 m value (Kimmel and Source condit ion from Ostendorf et al. (1984). Concen t ra t ions expressed as carbon. Profile location x = 1570m, r = 4.26 × 108s. Test s ta t is t ics ~ = 28%, a = 21%.

Braids, 1974).

T A B L E 3

L o n g i t u d i n a l profile of chlor ide: model t es t

101

x ~ c~ z c (meas) c (pred) (m) (10Ss) ( k g m 3) (m) ( k g m -3) ( k g m 3) (%)

360 1.05 0.258 1.75 0.054 0.213 - 294* 9.05 0.190 0,247 - 30

25.11 0.270 0.258 5 630 1.80 0.224 23.16 0.180 0.223 - 24 900 2.54 0.177 3.08 0.024 0.134 - 459*

8.97 0.200 0.156 22 18.45 0.360 0.172 52

920 2.59 0.174 11.74 0.130 0.159 - 22 21.50 0.180 0.171 5

1550 3.89 0.117 12.17 0.109 0.102 6 1890 5.02 0.096 12.69 0.150 0.081 46 2180 5.69 0.087 8.75 0.060 0.067 - 12

19.33 0.057 0.078 - 38 2230 5.80 0.080 17.66 0.057 0.071 - 25 2260 5.87 0.079 21.38 0.067 0.072 - 8 2810 7.05 0.051 11.89 0.042 0.042 0

18.26 0.045 0.045 0

Notes: 1973 da t a (Kimmel and Braids , 1980), supp l emen ted by USGS raw data . Source cond i t ions f rom Os t endo r f et al. (1984). A m b i e n t c o n c e n t r a t i o n of 0.020 kg / m 3 inc luded. Tes t s t a t i s t i c s wi th nea r - su r face po in t s ~ = - 46%, ~ = 126%. * Tes t s t a t i s t i c s w i t h o u t nea r - su r f ace po in t s ~ = - 2%, ~ = 25%.

TABLE 4

Ver t ica l ch lor ide profile: model t es t

z c (measured) c (predicted) (m) ( k g m -3) ( k g m -3) (%)

5.58 0.088 0.082 6 12.01 0.140 0.094 33 14.27 0.170 0.097 43 19.30 0.200 0.102 49 24.17 0.170 0.106 39

Notes: USGS raw data , except 24.17m d a t a (Kimmel and Braids, 1974). Source cond i t ions f rom Os t endo r f et al. (1984). Profile loca t ion x = 1570m, ~ = 4.26 × 108s. Tes t s t a t i s t i c s ~ ~ 34%, ~ = 15%.

t e s t s u m m a r i z e d i n T a b l e 4 a n d F i g . 4 b d o e s n o t f a r e a s w e l l h o w e v e r , b a s e d o n

t h e 3 4 % m e a n a n d 1 5 % s t a n d a r d d e v i a t i o n . B o t h c h l o r i d e t e s t s r e f l e c t a n

a m b i e n t c h l o r i d e c o n c e n t r a t i o n o f 0 . 0 2 0 k g m -3 , a s i n d i c a t e d b y w e l l s o u t s i d e

t h e p l u m e . T h e c l o s e d - s y s t e m e r r o r s d o s u g g e s t a t e n d e n c y t o o v e r p r e d i c t n e a r

102

0.40

0.30

c ( kg m -3)

0.20

0. I0

I 01 I , ,5!25m o5-15m • O - 5 m

o % . ct= 0.073 m

%.

(o)

0

I

0

- - I 0

Z (m)

2O

I "i 30 500 I000 1 5 0 0 2000 2500 3000

1 I I

X (m]

(b)

0 I I

0.05 O.lO 0.15 c (kg m -3)

I

0.20

Fig. 4. Longitudinal (a) and vertical (b) chloride test data (circles) and predictions (lines). 1973 observations of Kimmel and Braids (1974, 1980), supplemented by raw data supplied by USGS.

the free surface and underpred ic t a t deeper locat ions in the profile, perhaps in response to a nonun i fo rm source behav io r not included in the present analysis. Taken as a whole then, the ca l ib ra t ion and tes t resul ts provide a modest level of suppor t for this analysis , a l though a depth-vary ing source term charac ter iza- t ion could well improve the model accu racy for the closed-system const i tuents .

RESULTS

The ca l ib ra ted and tes ted open-system model may be compared with its one-dimensional coun t e rpa r t to e luc ida te the physical basis of the ad hoc decay cons tan t A used in conven t iona l models. The depth-averaged concen t r a t i on follows upon in t eg ra t ion of eqn. (8a), as der ived in the Appendix:

ca - 1 ierfc - e r f (14a) cs - c. h, ' : 2 ] k ]

ierfc y = ~ erfc(y')dy" (14b) Y

e r f y = 1 - i e r f c y (14c)

with e r ro r func t ion in tegra l ierfc y defined by Abramowitz and S tegun (1972). Equa t ion (14) may be used with depth- in tegra ted da ta to ca l ibra te a ver t ica l d ispers ivi ty va lue in the absence of more definit ive ver t ica l open-system con- cen t r a t i on profiles.

The " t r u e " one-dimensional model represented by eqn. (14) cont ras t s with the conven t iona l ba lance of advec t ion and first-order decay (Bear, 1979):

- ca = (cs - c , ) e x p ( - / l ~ ) (15)

in tha t r echa rge and ver t ica l dispers ivi ty pa ramete r s appear expl ici t ly in the former express ion as a consequence of the pr ior depth-vary ing analysis . The ad

1 0 3

hoc decay constant can be related directly to the responsible vertical transport properties for small ~ values, whence:

- - C a (1 - ~tr) (ad hoc decay) (16a)

C s - - C a

- C, 1 - 2 I__~l I/2 cs - Ca ~ (true mixing) (16b)

Thus dispersion induces pseudoreactive decay of a degassing constituent, with an ad hoc constant approximately given by:

- - (17a) )+ ~ h+ \ ~ X c /

xc = u+~ (17b)

with the plume length scale Xc in the longitudinal direction specified by advection in the unconfined aquifer.

We may also estimate the flux J of open system contaminant through the water table once the concentration profile is specified, since J is the combined result of vertical recharge advection and dispersion (Bear, 1979):

Oc J = n w ( c - c . ) - n D - ~ z (z = 0) (18)

In view of eqns. (7a) and (8), we find:

{i/D'~l/2 [- (~.c/n)2_l ~ -/" ~ . 1 / 2 x )

g = - ( c + - ca) n[~)__ exp|_ 4D+ J - ~erfc[~)~ (19) The contrasting roles of the vertical transport mechanisms may be noted for limiting cases of strong recharge and strong dispersion

(O~ ']2 J ~ - n ( c ~ - c . ) \-~-~/ ( D = ~ ) (20a)

J ~ 0 (s = ~ ) (20b)

Dispersion mixes the carbon towards the water table at a rate that decreases with travel time due to a flattening vertical gradient. The vertical recharge flow flushes pollution back into the plume by downward advection, and so reduces J.

C O N C L U S I O N S

We derive simple two-dimensional mathematical models describing the unsteady transport of conservative contaminants through an unconfined aquifer with a gently sloping aquiclude subject to advection, recharge, and vertical dispersion. Open (dispersive) and closed (nondispersive) chemical

104

systems are considered by imposition of appropriate boundary conditions at the water table, and the pseudoreactive depth-integrated behavior exhibited by the volatile open-system constituents is explained in terms of fundamental vertical mixing parameters. Total inorganic carbon and chloride data at the Babylon, Long Island landfill are used to calibrate and test the open- and closed-system models with a modest level of accuracy. The observed longitudinal distribution of total inorganic carbon is used for calibration, an exercise yielding a vertical dispersivity estimate of 0.073m with a standard deviation of 54%. Vertical profiles of inorganic carbon and chloride as well as a longitudinal chloride data set are used to test the theory, an exercise resulting in means and standard deviations ranging from - 2 to 34% in magnitude. The statistics do not include two chloride data points immediately below the water table, which are substan- tially overpredicted by the model.

Future research may proceed along various related fronts. The present analysis of groundwater contamination rests on a sequential understanding of the following phenomena, both in terms of data and theory: - - groundwater hydraulics - - pollution source history - - depth-integrated contamination, conservative contaminants - - depth-varying contamination, conservative contaminants The first extension of the modeling effort should be the inclusion of a depth- varying source condition in response to the systematic errors noted in chloride testing. Such a modification should preserve the simplicity inherent in an analytical model approach while maintaining the accuracy of the open-system profile tests in the present study. Additional extensions of the modeling effort might include the depth-varying transport of truly reactive pollutants, first for a single species and then for coupled contaminants. One is certainly drawn to dissolved oxygen as a reactive constituent of interest, and the speciation of inorganic carbon at Babylon seems a logical starting point for the coupled work. The present model should be modified to account for the influence of a nondispersive bottom boundary to expand its applicability, with the possible inclusion of an appropriately simple allowance for the vertical variation of the vertical velocity w. Additional testing of the present theory is certainly called for, at other sites and for other contaminants. In the latter regard, total nitrogen in a sewage effluent plume seems a likely candidate for open system testing, due to degassing associated with the denitrification process.

ACKNOWLEDGMENTS

Partial funding for this research was provided by the Massachusetts Division of Water Pollution Control under Grant Number 86-30765; the Authors acknowledge and appreciate their support. We also thank the Water Resources Division of the United States Geological Survey for providing the raw data base for the Babylon landfill.

REFERENCES

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Abramowitz, M. and Stegun, I.A., 1972. Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, 1046 pp.

Bear, J., 1979. Hydraulics of Groundwater. Wiley, New York, NY, 567 pp. Benjamin, J.R. and Cornell, C.A., 1970. Probability, Statistics, and Decision for Civil Engineers.

McGraw-Hill, New York, NY, 684 pp. Beveridge, G.S.G. and Schechter, R.S., 1970. Optimization: Theory and Practice. McGraw-Hill,

New York, NY, 773 pp. Borden, R.C. and Bedient, P.B., 1986. Transport of dissolved hydrocarbons influenced by oxygen

limited biodegradation 1. Theoretical development. Water Resour. Res., 22: 1973-1982. Cederberg, G.A., Street, R.L. and Leckie, J.O., 1985. A groundwater mass transport and equilibrium

chemistry model for multicomponent systems. Water Resour. Res., 21:1095 1104. Charbeneau, R.L., 1981. Groundwater contaminant transport with adsorption and ion exchange

chemistry: method of characteristics for the case without dispersion. Water Resour. Res., 17: 705 713.

Chen, C.S., 1987. Analytical solutions for radial dispersion with Cauchy boundary at injection well. Water Resour. Res., 23:1217 1224.

Gelhar, L.W. and Wilson, J.L., 1974. Groundwater quality modeling. Ground Water, 12: 399~408. Hillel, D., 1982. Introduction to Soil Physics. Academic Press, New York, NY, 364pp. Huyakorn, P.S., Jones, B.G. and Andersen, P.F., 1986. Finite element algorithms for simulating

three-dimensional groundwater flow and solute transport in multilayer systems. Water Resour. Res., 22:361 374.

Kimmel, G.E. and Braids, O.C., 1974. Leachate plumes in a highly permeable aquifer. Ground Water, 12: 388-393.

Kimmel, G.E. and Braids, O.C., 1980. Leachate plumes in groundwater from Babylon and Islip landfills, Long Island, New York. U.S. Geol. Surv., Prof. Pap. 1085, 38 pp.

Konikow, L.F. and Bredehoeft, J.D., 1978. Computer model of two-dimensional solute transport and dispersion in groundwater. Techn. Water Resour. Invest. USGS, USGS, Reston, VA.

Lenau, C,W., 1972. Dispersion from recharge well. J. Eng. Mech. Div., ASCE, 98: 331-334. Ogata, A. and Banks, R.B., 1961. A solution of the differential equation of longitudinal dispersion

in porous media. U.S. Geol. Surv., Prof. Pap. 411-A, 7 pp. Ostendorf, D.W., Noss, R.R. and Lederer, D.O., 1984. Landfill leachate migration through shallow

unconfined aquifers. Water Resour. Res., 20: 291-296. Parker, J.C. and Van Genuchten, M.T., 1984. Flux averaged and volume averaged concentrations

in continuum approaches to solute transport. Water Resour. Res., 20: 866-872. Prakash, A., 1982. Groundwater contamination due to vanishing and finite size continuous sources.

J. Hydraul. Div., ASCE, 108: 572-590. Stumm, W. and Morgan, J.J., 1981. Aquatic Chemistry. Wiley, New York, NY, 780 pp. Valocchi, A.J., 1985. Validity of the local equilibrium assumption for modeling sorbing solute

transport through homogeneous soils. Water Resour. Res., 21: 808-820. Valocchi, A.J., 1986. Effect of radial flow on deviations from local equilibrium during sorbing

solute transport through homogeneous soils. Water Resour. Res., 22: 1693-1701. Wilson, J.L. and Gelhar, L.W., 1981. Analysis of longitudinal dispersion in unsaturated flow 1. The

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Hydraul. Div., ASCE, 104: 503-514.

APPENDICES

Inorganic carbon boundary condition

T h e i n o r g a n i c c a r b o n c o n c e n t r a t i o n c o n s i s t s o f b i c a r b o n a t e cl a n d d i s s o l v e d

c a r b o n d i o x i d e c2 c o n t r i b u t i o n s ( S t u m m a n d M o r g a n , 1981) in t h e pH r a n g e

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(5-7) common to sanitary landfill plumes like that at Babylon, Long Island:

c = cl + c2 (21a)

cl = c2 ~ (21b)

K~ = 10-63mol L -1 (21c)

The equilibrium equation (21b) includes the hydrogen ion activity [H + ] and equilibrium constant K1 expressed in moles/liter. The latter parameter is tem- perature and concentration dependent to a modest degree, but our intent here is an order of magnitude discussion of mass fluxes and a representative value is sufficient. The inorganic carbon and its species are expressed in terms of kgm -3 carbon, for consistency with the main analysis.

The carbon dioxide species governs the total inorganic carbon degassing rate, since it is in gaseous equilibrium with the unsaturated zone air monolayer at the water table. Henry's law is appropriate (Stumm and Morgan, 1981) at this location, and may be expressed as:

c2 = KHRTp (z = 0) (22a)

R = 0.0821 atm L m o l - ' K 1 (22b)

KH = 0.045 mol L- ' a tm- ' (22c)

with Henry's constant KH, universal gas constant R, temperature T, and gaseous carbon dioxide density p, in kgm -3 carbon. For typical groundwater temperatures of 290 K, eqn. (22) suggests that the dissolved concentration and gaseous density are of equal magnitude on either side of the free surface:

c2 ~ p (z = 0) (23)

The carbon dioxide (hence total inorganic carbon) flux J through the un- saturated zone is a gaseous dispersion process which may be estimated in accordance with (Hillel, 1982):

P (24) J ~ - D ~

with unsaturated zone thickness b and gaseous dispersivity Da of order 10- 6 m2s-, in magnitude. This crude estimate may be compared with its liquid side counterpart:

c J ~ - D - - (25)

h~

in order to assess the magnitude of c at the water table. Collecting eqns. (21)-(25), we deduce:

c(z = 0) D( K1) c D~ 1 + ~ (26a)

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C(Z = 0) ~ C (pH < 7.3) (26b)

where the sa tura ted and unsa tu ra t ed zone thicknesses are the same order of magni tude in size. Physica l ly speaking, the unsa tu ra t ed zone efficiently carries off whatever to ta l carbon the sa tura ted zone can deliver due to the smallness (1/10) of the dispersivity ratio; the simple boundary condi t ion (7a) is the result.

Open system The depth-integrated open-system concent ra t ion is defined as:

hs

1 f cdz (27) - hs

so that , recal l ing eqn. (8a):

c s - c a = c~ 2hs (I + II) (28a)

i L [z - ~ln] I = erfc b/--n~--~/dz (28b) 0

i (sn)~ L[z+~z/nl-2~D~-~ J II -- exp erfc . - d z (28c) 0

The use of the infinite upper l imit of in tegra t ion in eqn. (28) reflects the assumpt ion of an infinite bot tom boundary condi t ion in eqn. (6b). The first in tegral I is s t ra ightforward, while the second II requires an in tegra t ion by parts:

i [ (D~ 'i2 -_ aUn)~.~ II - nD~ erfc + \~--~] exp 4D~ j dz (29a)

[ ( n i p = , ] ~''cli2"~ \2nD'121#_] (e'rli2"~] II - nD erfc 2 ~ i ~ . - erfc (29b)

This last re la t ion simplifies with the aid of an identi ty:

e r f c ( - y ) = 2 - e r f c y (30)

giving rise to eqn. (14a).