Solute transport through a heterogeneous coupled vadose-saturated zone system with temporally random rainfall

WATER RESOURCES RESEARCH, VOL. 37, NO. 6, PAGES 1577-1588, JUNE 2001

Solute transport through a heterogeneous coupled vadose- saturated zone system with temporally random rainfall

X. Foussereau, 1 W. D. Graham, and G. A. Akpoji, 2 Interdisciplinary Hydrologic Science Program, University of Florida, Gainesville, Florida

G. Destouni

Department of Civil and Environmental Engineering, Royal Institute of Technology, Stockholm, Sweden

P.S. C. Rao 3

Interdisciplinary Hydrologic Science Program, University of Florida, Gainesville, Florida

Abstract. The transport of nonreactive solutes through a coupled, two-dimensional, randomly heterogeneous vadose-saturated zone system subject to temporally random rainfall is predicted by Monte Carlo simulation and compared with previously published analytic results for three different rainfall patterns. The relative contributions of the uncertain inputs (i.e., rainfall and saturated conductivity) to the prediction uncertainty of solute transport are quantified in terms of the statistical moments of the pore water velocity, the plume spatial moments, and solute flux breakthrough curves at downstream control planes. Results show that the mean and variance of the saturated zone pore water velocity were approximately equivalent for the cases of uniform and random rainfall and were well predicted by the analytical relationships developed by Rubin and Bellin [1994]. As a result, the mean plume displacement, estimated by the trajectory of the mean plume center of mass, was found to be nearly identical for these cases. In the temporally random rainfall case, the saturated zone mean plume experienced more spread in the direction of mean flow at early times. However, the asymptotic rates of spatial spreading of the mean solute plumes were found to be approximately equivalent for the uniform and random rainfall cases and well predicted by the approximate expressions for longitudinal macrodispersivity in nonuniform flow proposed by Destouni and Graham [1995]. Random rainfall and random soil properties increased prediction uncertainty of the solute plume behavior in the vadose zone by an order of magnitude when compared with the uniform rain and random soil case. This effect was reduced considerably when the solute entered the saturated zone, where random rainfall produced only slightly larger prediction uncertainty than the uniform rainfall case. The analytic model developed by Destouni and Graham [1995] accurately predicted the temporal breakthrough of the mean solute plume at saturated zone control planes for all cases, if transport through the unsaturated zone accounted for the effects of temporally random rainfall using the methodology developed by Foussereau et al. [2000a, 2000b]. Results of this work indicate that for the humid climates studied here, uncertain rainfall patterns dominate transport prediction uncertainty in the shallow unsaturated zone, while uncertain solute breakthrough to the saturated zone and uncertain hydraulic conductivity dominate prediction uncertainty in the saturated zone.

1. Introduction

Contamination of groundwater occurs in many cases through surface application of chemicals, particularly agricultural fertilizers and pesticides. These contaminants are then transported by infiltrating water, first into the vadose zone and

•Now at Camp Dresser and McKee, Maitland, Florida. 2Now at South Florida Water Management District, West Palm

Beach, Florida. 3Now at School of Civil Engineering, Purdue University, West

Lafayette, Indiana.

Copyright 2001 by the American Geophysical Union.

Paper number 2000WR900389. 0043-1397/01/2000WR900389509.00

then into the saturated zone. There is a need for the development of efficient quantitative tools capable of predicting the transport of contaminants through integrated vadose-saturated zone systems past subsurface compliance boundaries and/or into lakes and streams. As total maximum daily loads (TMDL) are established for receiving surface and groundwater bodies in the United States, these predictive tools will be useful for evaluating compliance with these standards.

Natural variations in weather patterns and soil properties lead to complex flow and transport problems that are difficult to solve analytically. Numerical methods are therefore often used to allow for time and space variability and nonlinearity in the input variables. Furthermore, input variables, such as weather patterns and formation properties, vary irregularly in space and time. Thus many investigators have treated weather

1577

1578 FOUSSEREAU ET AL.: COUPLED VADOSE-SATURATED ZONE TRANSPORT

patterns and soil hydraulic properties as stochastic variables, for example, Dagan [1989], Gelhat [1993], and Jury and Roth [1990].

Analysis of coupled vadose-saturated zone flow poses problems because the governing equation is nonlinear, nonuniform, and transient owing to interrelated flow processes, random soil properties, and time-dependent rainfall. Simplifications of the problem have been studied by decoupling the integrated processes in a theoretical, stochastic flamework. Bresler and Dagan [1981], Yeh et al. [1985a, 1985b], Jury and Gruber [1989], Destouni and Cvetkovic [1991], Russo [1993a, 1993b], Russo et al. [1994], Hatter and Yeh [1996], and Foussereau et al. [2000a, 2000b] have used this technique to investigate flow and con- taminant transport in the unsaturated zone. Common simplifications proposed by the above studies (e.g., steady water flow and/or noninteracting, vertical stream tubes) have led to useful analytic expressions for the moments of the advective solute travel time and solute flux breakthrough in the unsaturated zone. Dagan et al. [1992], Cvetkovic and Dagan [1994, 1996], and Cvetkovic et al. [1992] developed similar expressions for saturated zone transport. Destouni and Graham [1995] extended the Lagrangian stochastic approach of Cvetkovic and Dagan [1994] to predict the transport of a solute applied at the surface subject to uniform steady (i.e., deterministic) rainfall. Their study focused on the quantification of the expected field- scale solute breakthrough at subsurface control planes (CP) in the system.

In this study, numerical Monte Carlo simulations of nonlinear, nonuniform, transient water flow and solute transport through a coupled vadose-saturated zone system are performed for three rainfall patterns. The saturated hydraulic conductivity is represented as an autocorrelated random field. Rainfall is introduced as a temporally random flux boundary condition at the surface, following a first-order discrete Markov process. The uncertainty in solute transport prediction is quantified in terms of the pore water velocity statistics, the spatial moments of the ensemble mean plume, and the ensemble moments of solute flux breakthrough curves (BTC) at specified control planes located normal to mean flow directions. The objective of this work is to evaluate the relative contributions of the uncertain input variables to the prediction uncertainty of solute transport in an integrated, randomly heterogeneous vadose-saturated zone system and to compare the numerical results with approximate analytic solutions where applicable.

The unique aspect of this work is that the problem examined is flow and transport through an integrated heterogeneous soil-aquifer system replenished by random rainfall. A conservative, nonreactive solute is assumed to be instantaneously introduced at the soil surface into an initially solute-free heterogeneous porous formation. Solute transport is predominantly vertical as it moves downward through the vadose zone. The solute then begins to move horizontally, in the regional flow direction, as it enters the capillary fringe and saturated zone. Some factors that influence water flow (e.g., hysteresis of the water retention curve, plant root uptake, and evaporation) and solute transport (e.g., sorption and transformations) are neglected in the present study. A brief description of the setup of the Monte Carlo simulations is presented, followed by the simulation results for the case of a simple conservative solute. These results are compared, as appropriate, with previously published analytic expressions.

2. Monte Carlo Simulation

Monte Carlo simulations were performed for a heterogeneous vadose-saturated zone system subject to two types of surface boundary conditions: (1) spatially and temporally uniform surface boundary flux and (2) a spatially uniform, temporally random boundary flux representative of the rainfall patterns of Bradenton, Florida, and Olympia, Washington. The first case investigates effects of soil and aquifer property heterogeneity on flow and transport in a coupled vadose- saturated zone system subject to a uniform surface boundary flux. This case is similar to that studied by Destouni and Gra- ham [1995], who derived an approximate analytic expression for field-scale solute breakthrough in the coupled flow problem utilizing a joint probability density function (pdf) between the travel times in the two zones. The objective of the second and third cases is to investigate the additional effects of random surface boundary flux on this system.

The computer code VS2DT was used to solve Richards' equation for flow in a variably saturated vertical cross section and the advection-dispersion equation for transport of a single conservative, nonreactive solute. Neither bare soil evaporation nor plant root uptake of water or solute were simulated. De- tails of the governing equations and the numerical approach used to approximate them are given by Healy [1990]. The VS2DT code was modified to accept random surface boundary flux and random soil properties and to use the conjugate gradient solver [Hestenes and Stiefel, 1952] for the flow equation. The transport equation was solved using a strongly implicit procedure algorithm.

Figure 1 shows the domain and boundary conditions used to simulate the coupled system. The domain spans 120 m in the horizontal direction by 12.6 m in the vertical direction. The thickness of the vadose zone ranges from 2.6 to 4.4 m over the domain. Constant flux is specified on the upstream (left) boundary. Above the approximate height of the water table is specified as no horizontal flux, and below this height a constant water flux enters the domain. The right boundary is constant hydrostatic head below the specified water table and no horizontal flux above the water table. The bottom of the aquifer is an impervious no vertical flux boundary. Temporally variable but spatially uniform rainfall of intensity R (t) [L T- • ], enters the entire domain width of 0 -< x -< 120 m for all t _> 0.

Hydrostatic initial conditions were assumed in the aquifer. An equilibrium pressure profile above the water table was assumed as the initial conditions in the vadose zone.

A hydrostatic specified head boundary condition was not used on the upstream boundary because we found that fixing the water table height on both sides of the domain artificially constrained the water table fluctuations. The boundary conditions adopted here allow the height of the water table at the upstream boundary to fluctuate, in an approximate manner, in response to the random recharge. However, since all boundary conditions tend to produce undesirable effects, we applied the solute in a region 30 correlation scales downstream of the right boundary to minimize the influence of the specified boundary conditions on solute transport.

A solute pulse of 1 kg was introduced at the soil surface between 30 and 40 m. The solute was applied for one day, on day 26 of the simulation period. The concentration of the solute in the infiltrating water within the application zone was constant during the application period and was zero otherwise. The left and right boundaries were treated as no dispersive flux

FOUSSEREAU ET AL.: COUPLED VADOSE-SATURATED ZONE TRANSPORT 1579

Rainfall

x=0m z=0m

over

Solute application zone X---- X----

30 m 40 m

Fixed water flux

depth

z=12.6

Impermeable layer

x= 120m

.... ¾VVVt

I " 2.6m __ CPi__

.... .,: ....................................................................... • ..................... 4.4 m

_ CP 2 CP 3 CP 4

10.0m :

• 8.2 m

6m / f/ / // // / I / / I / / / l / I / // ,rtr t rrr t r r r r t

Fixed head

i ß X

Figure 1. Physical description of the coupled soil-groundwater experiment.

boundaries, and no advective or dispersive flux was allowed out of the bottom boundary. Initially, the water in the vadose- saturated zone system was assumed to be solute free.

One hundred fifty Monte Carlo runs were performed for a fine sandy soil subject to each of three surface boundary flux patterns. For each soil-climate combination, simulations were run for heterogeneous saturated hydraulic conductivity. The random saturated conductivity fields were generated using the turning bands algorithm, originally developed by Journel and Huijbregts [1978], and then modified by Akpoji [1993]. The simulations were run for a fine sandy soil [Brooks and Corey, 1964] for which the hydraulic properties are presented in Table 1. On the basis of the work by Gelhar [1993, p. 291] the variance of log saturated hydraulic conductivity was assumed to be 1.0, and the spatial correlation length was assumed to be 1 m in all cases. The soil parameters which describe pore size distribution were assumed to be deterministic constants. This

assumption is justified for the sandy soil, shallow water table,

Table 1. Hydraulic Properties of the Fine Sandy Soil

Properties Fine Sand

Saturated soil water content 0s Residual soil water content Or Grain size distribution to

Air bubbling pressure h b Geometric mean of •fs (•fG) Variance of In •fs (O'•n Xs) Correlation scale of In •fs (Xin Xs)

0.350 0.063

3.7

-0.82 rn

5.0md -• 1.0

1.0m

humid climate combinations considered here because the

mean soil conditions are guaranteed to be sufficiently moist so that the contribution of the variability of the pore-size distribution parameters to flow variability will be negligible [Yeh et al., 1985b; Foussereau et al., 2000a, 2000b]. In fact, Monte Carlo simulations with and without heterogeneous pore size distribution parameters conducted by Sassner [1995] for these soil-climate combinations showed this to be the case.

The boundary flux series were generated using WGEN, a weather generator developed by Richardson and Wright [1984]. This computer simulation code generates daily rainfall series which reproduce the statistics of the rainfall from a specified region. WGEN uses a first-order Markov process to represent the relative occurrences of wet (Pw) and dry days (PD) and a gamma distribution to represent the daily volume of the rainfall. Throughout the simulation the June rainfall statistics were used from Bradenton, Florida, and the January statistics were used from Olympia, Washington, to maintain stationarity of the rainfall series (see Table 2). These statistics, although non- seasonal, produce realistic rainfall patterns that exhibit an av- erage of 70% dry days for the Bradenton scenario, and 30% dry days for the Olympia scenario. The magnitude of the temporally uniform rainfall series was selected to be equal to the mean rainfall rate for the Bradenton weather pattern. For the soil-climate combinations studied here, all rainfall infiltrated into the soil profile since the ratio of mean rainfall to mean saturated hydraulic conductivity ranges from 6 x 10 -4 to 4 x 10 -3 , and the rainfall coefficient of variation ranges from 1.43 to 2.58.


Table 2. Ensemble Statistics of the Two Rainfall

Scenarios a

Uniform Bradenton Olympia

Pi[W/W] 1.00 0.453 0.816 Pi[W/D] NA 0.240 0.452 P_u- 1.00 0.30 0.71 R, m/d 0.004 0.004 0.003 o-2•, X10 -s m 2 d -2 0.00 9.81 1.76 CVR 0.00 2.58 1.43 XR, days +• 0.6 0.6

ap,[W/D ], P,[W/W], and Pw stand for the probabilities of a wet day on day i given a dry day on day i - 1, the probability of a wet day on day i given a wet day on day i - 1, and the unconditional probability of a wet day occurring, respectively./•, o-2•, CVR, and XR stand for the mean, the variance, the coefficient of variation, and the temporal correlation scale of the rainfall series, respectively.

The numerical grid was chosen to include four grid points per hydraulic conductivity correlation scale [Ababou et al., 1989]. The solute transport parameters were chosen to meet the Peclet, Courant, and Neumann stability and accuracy cri- teria for the numerical solution. The isotropic dispersivities and a r were set equal to one half of the uniform mesh size of 0.25 m in both directions for numerical convenience.

The first control plane (CP 1) was placed in the unsaturated zone 1.125 m below ground surface above the top of the capillary fringe in the application zone (Figure 1). This control plane was 11.5 m in horizontal extent, slightly wider than the application region of 10 m at the soil surface, to ensure that solute that might deviate from the mean flow direction due to the random soil properties would still be detected. Control planes 2-4 were placed vertically at distances of 40, 60, and 80 m from the upgradient boundary, so that the first saturated zone CP was at the edge of the solute application zone.

3. Results and Discussion

3.1. Pore Water Velocity Statistics

Figure 2 is an example vector plot of the mean pore water velocity for the Bradenton rainfall pattern at observation day 194. Similar behavior was observed at earlier and later simu-

lation times for this weather pattern as well as the uniform and Olympia weather patterns. As expected, the mean vadose zone pore water velocity is predominantly vertical and is small in magnitude. The mean saturated zone pore water velocity, on the other hand, is predominantly horizontal and larger in magnitude. The magnitude of the mean pore water velocity is equivalent for the uniform rainfall and the Bradenton rainfall scenarios in both the vadose and saturated zones because these

scenarios have the same mean rainfall rate and the same sat-

urated zone boundary conditions. However, the mean pore water velocity is slightly smaller in the vadose zone and increases more slowly with the horizontal direction in the saturated zone for the Olympia scenario owing to the smaller mean rainfall rate. In all cases the saturated pore water velocity increases approximately linearly with horizontal distance due to the effects of uniform mean recharge from the unsaturated zone. Furthermore, the mean velocity tends to be approximately horizontal and constant over depth in the saturated zone. These results show that despite the variable depth of the phreatic aquifer the pore water velocity statistics in the two- dimensional (2-D) vertical cross section should be well approximated by pore water velocity statistics previously derived for 2-D infinite domain aquifers with linearly trending mean velocity [e.g., Rubin and Bellin, 1994; Li and Graham, 1999].

Figures 3-5 show the Monte Carlo and approximate theoretical ensemble mean and standard deviation of the saturated

zone pore water velocities as a function of the distance in the horizontal direction at a depth of 5 m from the soil surface for the three rainfall patterns. The theoretical ensemble mean horizontal pore water velocity for linearly trending velocity resulting from uniform mean recharge in a 2-D infinite domain randomly heterogeneous aquifer is [Rubin and Bellin, 1994]

(x -x0)) /J(X) : U 0 1 -}- /• •-lnXs ' where Uo [L T -j] is the mean horizontal pore water velocity at the reference point x o,/3 is the dimensionless nonuniformity coefficient which governs the linear rate at which the pore water velocity increases with horizontal distance, and Xln Xs is the isotropic log hydraulic conductivity correlation length.

The standard deviation of the pore water velocity for a

0o0--

-10.0 -

0.0 20.0 40.0 60.0 80.0 100.0 120.0

Horizontal distance (m)

--• = 0.$0

Figure 2. Vector plot of the mean velocity (m d -1) 194 days after solute application for Bradenton weather.


0.40 a) b)

0.35 -

0.30

0.2-

0.:0 IM. v4 .... '

0.15- :.•, '•i• ;• I ,•i •' •,,•,&'q , • ß ,' '

0.]0-•,• .;.. ,,, , . 0.05

0.00 , • , • , • , • , • , • , • , 0 15 30 45 60 75 90 105 120

Distance from origin (m)

0.40

0.35

0.30 0.25

0.20

0.15

0.10 0.05

0.00

15 30 45 60 75 90 105 120


..... Theoretical standard deviation ..... Theoretical standard deviation

...... Measured standard deviation. - ..... Measured standard deviation Theoretical mean

• Measured mean.

Figure 3. (a) Comparison between the predicted and the measured mean and standard deviation of the horizontal velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the uniform weather (/3 = 0.00894; Uo = 0.1508 m d-•). (b) Comparison between the predicted and the measured standard deviation of the vertical velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the uniform weather (/3 = 0.00894; Uo = 0.1508 m d-•).

two-dimensional infinite domain heterogeneous aquifer with a linearly trending mean velocity can be approximated [Rubin and Bellin, 1994]

3 ( (x -xo)] 1 ( (x-xo)] u: = •fs ON' • 1 + X,n I +

(2)

where •n •q is the variance of log saturated conductivity. For the case of zero recharge (/3 = 0), (2) is equivalent to that derived by Dagan [1984].

Originally, Rubin and Bellin [1994] developed these pore water velocity statistics for a 2-D infinite domain aquifer assuming the mean flow was unidirectional, and the spatially and temporally uniform recharge • was introduced as a source term in the flow governing equation. In this case the expression for the nonuniformity coefficient is /3 = •3,1nXs/(bUoOs), where b is the thickness of the aquifer, Uo is the mean pore water velocity at reference point Xo, and Os is the saturated water content. Li and Graham [1999] showed that these equations also hold for spatially uniform temporally random recharge to a 2-D infinite domain aquifer. The example vector plot of mean pore water velocity shown in Figure 2 justifies the use of (1) and (2) for the 2-D vertical aquifer considered here because the mean flow in the saturated zone is approximately unidirectional (horizontal) and linearly increasing in the horizontal direction. Furthermore, although introduced at the top

of the water table, the impact of recharge is distributed approximately uniformly over the full depth of the aquifer.

For the 2-D vertical aquifer cross section the analogous expression for the nonuniformity coefficient is /3 = •/•-ln Xs/ [h(x)UoO]. The rate of increase of the mean horizontal pore water velocity (controlled by/3) is not strictly linear for this case owing to the variable depth of the aquifer within the simulation domain. Therefore/3 and Uo were fit to the approximately linear trend in pore water velocity observed by running one deterministic simulation with VS2DT for each rainfall

pattern using the mean values of the parameters and boundary conditions. The values of/3 obtained for the linear fit, when

combined with the theoretical values of •, •.lnXs , and Uo, imply an effective aquifer depth of 8.2-8.8 m, which is within the range of aquifer depths observed over the simulated domain.

Figures 3a, 4a, and 5a show that for both uniform and random rainfall cases, the horizontal mean pore water velocity is well-predicted by (1), particularly in the central part of the domain away from the boundary conditions. Discrepancies near the boundaries are expected since the theoretical mean pore water velocity was derived for an infinite domain and therefore does not account for the boundary effect encoun- tered in the numerical simulations. Note that, in all cases, there is still considerable fluctuation in the ensemble mean pore water velocities calculated using the Monte Carlo simulation results. This indicates that more replicates are needed to get smooth estimates of nodal pore water velocity statistics. How-

1582 FOUSSEREAU ET AL.' COUPLED VADOSE-SATURATED ZONE TRANSPORT

0.40

0.35 a) b)

0.30- • 0.25-

•: 0.20-

•z 0.15-

• 0.10- .,•

0.05 -

0.00

0.40

0.35 -

0 15 30 45 60 75 90 105 120

Distance from origin (m) Distance from origin (m)

0.30- ß • 0.25-

• 0.20-

• o.15- =z • ,;• • , ',,• , •.,•. ,,,. " - ..,, f,,,.,

0.00 / ' I 0 15 30 45 60 75 90 105 120



• Measured mean.

Figure 4. (a) Comparison between the predicted and the measured mean and standard deviation of the horizontal velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the Olympia weather (/9 = 0.00658; Uo = 0.1557 m d-•). (b) Comparison between the predicted and the measured standard deviation of the vertical velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the Olympia weather (/9 = 0.00658; Uo = 0.1557 m d-•).

ever, while additional runs will eventually dampen the fluctuations of the estimated statistics, the general behavior of the statistics will not change significantly.

Figures 3a, 4a, and 5a also show that for all cases the standard deviation of the horizontal pore water velocity is also well predicted by (2) throughout most of the domain. However, Figures 3b, 4b, and 5b show that the vertical pore water velocity standard deviation appears to be slightly underestimated in all cases, perhaps due to increased variations in vertical pore water velocity over depth imposed by the top and bottom boundary conditions. A comparison of the uniform and random rainfall cases shows that (2) performs equally well under spatially uniform, temporally random recharge as predicted by Li and Graham [1999]. Thus the temporal randomness of the recharge does not affect the saturated zone pore water velocity variance as long as it is spatially uniform. Therefore the saturated zone pore water velocity statistics for the Bradenton rainfall case and the uniform rainfall case (which have equivalent mean rainfall rates) are approximately equivalent.

3.2. Spatial Moment Analysis

To further characterize the solute transport behavior, spatial moments of the ensemble mean plume were considered [Aris, 1956; Freyberg, 1986]. The first-normalized and second- centralized moments of the ensemble mean solute plume evolution were analyzed, providing a measure of the location of the center of mass and the dispersion of the solute around its center of mass, respectively. Figure 6 confirms that the plumes initially move vertically downward, then begin to move horizontally as they enter the capillary fringe and the saturated

zone. However, the center of mass trajectories also shows a vertical downward trend in the saturated zone due to the

downward sloping water table. As expected, the mean plumes for both the uniform and Bradenton rainfall cases show very similar mean center of mass trajectories because the mean recharge rate for the Bradenton rainfall case is equal to the uniform rainfall rate. The mean center of mass trajectory for the Olympia rainfall scenario is also very similar, indicating the small difference in mean rainfall rate does not affect the mean

plume trajectory significantly for this regional flow scenario. Figure 7 shows the evolution of the second-centralized lon-

gitudinal moments Xxx as a function of the dimensionless mean horizontal displacement for the three rainfall patterns. Also shown is the approximate analytical solution proposed by Destouni and Graham [1995] for second longitudinal moment in steady nonuniform (linearly trending) flow in an infinite 2-D aquifer. The theoretical longitudinal second moment was approximated using the following expression [Destouni and Gra- ham, 1995, equation 18]:

Xxx(•, r) = r + e -(3/s)* - 1 (3) 3/8 '

where f is the longitudinal starting position of the solute particle and r is the dimensionless mean solute displacement from the starting position, equal to

T = (X(•, t))/,•ln 7G = (e 13U(f>t/x'n•r' - 1)/• • if= 0

1- = (X(t))/Xlnx• = Sot/XlnX• • = O, (4)


a) b) 0.40 0.40

0.35 - 0.35 -

0.2 0.2

• 0.20 o.o

' I i I i, .•l i• i Iii • ii I • ß ! ', • , ,,•,• ,' ', •

• • • t , •', •,• ,'"•,' '•. •- •, . ,-, •'•

•.• •-•/ ' I ' I ' I ' I ' • ' I ' I '

Distance Dem eri•i• (m) Distance Dem eri•i• (m)



• Measured mean.

Figure 5. (a) Comparison between the predicted and the measured mean and standard deviation of the horizontal velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the Bradenton weather (/3 = 0.00857; Uo = 0.1516 m d-•). (b) Comparison between the predicted and the measured standard :deviation of the vertical velocity at a depth of 5 m along the domain length in a fine sandy soil subject to the Bradenton weather (/3 = 0.00857; Uo - 0.1516 m d-•).

120

and the values of/3, U({•), and Uo are as defined and determined above for the velocity moment expressions.

Figure 7 shows that for a given mean horizontal displacement the random recharge cases show larger longitudinal

spread than either the uniform case or the analytical prediction. This is due to the fact that initially, the longitudinal displacement variance is strongly controlled by the spatiotemporal distribution of ensemble mean solute plume entering the

-2

-4

-8

-10

-12

0 20 40 60 80 100


Uniform weather + Bradenton weather ß Olympia weather x Water table .....

Figure 6. Evolution of the center of mass trajectories for each weather pattern.

120


140

120

lOO

80

60 -

40 -

20 -

o

30

i i i i I i

40 50 60 70 80 90

Normalized mean horizontal displacement lOO

Uniform weather +

Olympia weather x Bradenton weather ß

Theoretical (Destouni and Graham, 1995, eq. 18) --

Figure 7. Comparison between the predicted and the measured horizontal displacement standard deviation along the domain length in a fine sandy soil for the three weather patterns.

saturated zone. Because the vertical movement and spread of the solute in the vadose zone is mainly controlled by the timing and variability of the rainfall pattern, the ensemble vertical dispersion in the unsaturated zone is greater under a temporally random flow regime than under an uniform flow regime [Foussereau et al., 2000a, 2000b]. This leads to the ensemble mean solute plume entering the saturated zone over a longer time period, causing greater apparent horizontal (longitudinal) dispersion in the saturated zone for the random rainfall case. The theoretical expression for particle displacement variance assumes the solute enters the saturated zone as a dirac pulse in space and time; thus it underestimates the initial longitudinal dispersion for the uniform and the random rainfall Monte Carlo simulations. However, within the saturated zone a solute plume moving under either steady state or temporally random recharge regimes (with the same mean pore water velocity) ex- periences identical longitudinal ensemble dispersion because the velocity variances are equivalent. Thus the asymptotic slope of X= for the random and uniform rainfall patterns are very similar to each other and to the slope of the analytical expression.

3.3. Temporal Moment Analysis

Figures 8a-8c show the temporal evolution of the Monte Carlo ensemble mean and standard deviation of the solute flux past the four subsurface control planes for the three weather patterns. The mean solute flux curves for the uniform rainfall case show higher peaks and less spreading than the random rainfall cases, particularly in the unsaturated zone. For all cases the mean BTCs become less skewed and more Gaussian as the solute moves into the saturated zone. Attenuation in peak and spread both increase slightly with travel distance from the source, but within the near-field saturated zone, solute plume translation is more dominant than attenuation, as predicted by Destouni and Graham [1995] for conservative solute transport in a nonuniform flow field.

For the uniform rainfall pattern the standard deviation of

the solute flux breakthrough curve is smallest at CP 1 in the unsaturated zone (coefficient of variation of-10%). The standard deviation of solute flux breakthrough increases abruptly as the solute enters the saturated zone (coefficient of variation of-30%) and then begins to attenuate slowly as the solute moves downstream in the saturated zone. Under the temporally random flow regimes, the opposite pattern occurs. The highest uncertainty occurs at CP 1 in the unsaturated zone (e.g., coefficient of variation of -1.0 for the Bradenton rainfall). The uncertainty abruptly decreases as the solute enters the saturated zone and then slowly attenuates as it moves downstream (coefficient of variation of-50%). Thus, in the vadose zone the solute flux uncertainty for the random rainfall case is approximately an order of magnitude larger than for the uniform rainfall case. In the saturated zone, however, the solute flux uncertainty is only slightly larger for the random rainfall case. These results suggest that for the high-rainfall, shallow- water table aquifer considered here, uncertain rainfall patterns dominate transport prediction uncertainty in the unsaturated zone, while uncertain solute breakthrough to the saturated zone and uncertain hydraulic conductMty dominate prediction uncertainty in the saturated zone. Foussereau et al. [2000a, 2000b] showed that the impact of uncertain rainfall patterns on water flow and solute transport predictions attenuate with depth in the vadose zone. Thus the shallow water table scenario presented here represents the case which would produce maximum impacts of uncertain rainfall patterns on solute transport predictions in the saturated zone. For cases with deeper water tables the impacts of rainfall uncertainty on the saturated zone prediction uncertainty would be even less significant.

The Monte Carlo simulation ensemble mean temporal breakthrough curves were compared to theoretical predictions based on the model developed by Destouni and Graham [1995]. This model quantifies the ensemble mean solute flux past an arbitrarily located CP in the saturated zone of an integrated, heterogeneous,

FOUSSEREAU ET AL.' COUPLED VADOSE-SATURATED ZONE TRANSPORT 1585

vadose-saturated zone subsurface system. The method assumes essentially vertical flow in the unsaturated zone, horizontal steady mean flow in the saturated zone, and that travel time pdf in the vadose and saturated zone systems are approximately indepen- dent. The equation for mean solute flux from a surface source of mass Mo and longitudinal extent h• past a vertical control plane located at x• is [Destouni and Graham, 1995, equation 12]

E[S(x, t)]

Mø fot •xø+hl " ( r" ' = • #•(t - T"; œ3) #• 'x•, •') d•' dT", •'X0

(5)

where #;(t - T"; L3) is the vadose travel time pdf for a control plane located at depth L 3 and #';(T"; x•, s •') is the saturated zone travel time pdf for a particle originating at s •' and travelling to a control plane located at x •.

To use (5), pdfs for travel time in the unsaturated and saturated zones must either be assumed, fitted to field or numerical data, or derived from soil and climatic properties. Destouni and Graham [1995] assumed a lognormal vadose zone travel time pdf and derived a nonstationary saturated zone travel time pdf, assuming that the particle displacement pdf was Gaussian. The resulting travel time pdf was nonstationary owing to the effects of flow nonuniformity in the saturated zone system caused by recharge from the unsaturated zone. The first and second moments needed for the pdf, (X) and Xxx, were estimated using (4) and (3), respectively.

In this study the unsaturated zone travel time pdf derived by Foussereau et al. [2000a, 2000b] was used for the random rainfall cases. This model utilizes the random rainfall statistics, soil property statistics, and control plane depth to derive analytical travel time pdfs for inert solutes transported in variably saturated, heterogeneous porous media subject to a random rainfall boundary condition. For the uniform rainfall case a lognormal unsaturated zone travel time pdf was used. In this case the mean travel time was estimated using the mean soil properties and control plane depth, and the travel time variance was estimated from local dispersion only, using the temporal moment equations derived by Goltz and Roberts [1987]. For the uniform rainfall case, use of only the mean soil properties in the unsaturated zone travel time pdf is justified by the very low uncertainty in unsaturated zone breakthrough shown in Figure 8a. In both the uniform and the random rainfall cases the

interface between the unsaturated and saturated zones was

located at the top of the capillary fringe, ---1.875 m below land surface. The nonstationary saturated zone travel time pdf developed by Destouni and Graham [1995] was used in all cases. The displacement statistics (X) and Xxx were estimated using (4) and (3), with the values of/3, U(s•), and Uo determined as described above for each rainfall scenario.

Figure 9a shows a comparison of the "predicted" analytic and the "measured" Monte Carlo saturated zone ensemble

mean solute flux breakthrough curves for the uniform rainfall case, and Figures 9b and 9c show similar results for the random rainfall cases. These plots show that the analytic predictions match the Monte Carlo simulation results very well for the uniform rainfall case and quite well for the random rainfall cases. However, the analytical predictions are less accurate for the Bradenton rainfall scenario, which has a higher rainfall coefficient of variation. The uniform rainfall and random Bra-

denton rainfall scenarios have identical saturated zone pdfs

0.06

a) Uniform weather

0.05-

0.04-

0.03-

0.02-

!. 0.00 , • , • , • , -' 0 100 200 300 400

Time (d)

0.06

0.05

0.04

0.03-

0.02-

0.01-

o.oo o

b) Olympia weather

.... Standard deviation at CP4 - - - Standard deviation at CP3

Standard deviation at CP2 Standard deviation at CP1

-- '- Mean at CP4 Mean at CP3

..... Mean at CP2 '-- - Mean at CP1

100 200 300 400

Time (d)

c) Bradenton weather

0.06 = 0.05 ,

•, o.04

'• ! ß • 0.03 - , ',,• •

• ,•1' • }'i•,.

',,, • 0.00 , , ,

0 100 200 300 400

Time (d)

Figure 8. Temporal evolution of the measured mean and standard deviation of the solute flux at the four control planes in a fine sandy soil subject to (a) the uniform weather, (b) the Olympia weather, and (c) the Bradenton weather.


0.0150

0.0125

0.0100

0.0075

o.oo5o

0.0025

0.0000

a) Uniform weather

' I ' I ' I '

100 200 300 400

Time (d)

0.012

O.OLO

0.008

0.006

0.0o4

0.002

0.000

b) Olympia weather

ß Measured mean at CP4 X Measured mean at CP3 O Measured mean at CP2

•c•• - - - Predicted mean at CP4 Predicted mean at CP3 -- Predicted mean at CP2

100 200 300 400

Time (d)

0.012

0.010-

0.008-

0.006

0.004

0.002

0.000

c) Bradenton weather

I\

x3 ,.

100 2• 300 400

Time (d)

Figure 9. Comparison between the predicted and the measured means of the solute flux over time at the three control

planes in a fine sandy soil subject to the (a) uniform weather, (b) Olympia weather, and (c) Bradenton weather.

since the pore water statistics are the same in both cases. The increased spread of the saturated zone solute flux breakthrough curve for the random rainfall case observed in the Monte Carlo simulations and predicted with the analytical solution is therefore solely due to the increased spread of the mean solute breakthrough from the unsaturated zone. Figures 9b and 9c indicate that this increased spreading is adequately captured using the vadose zone pdf predicted using the Fousse- reau et al. [2000a, 2000b] model. Similarly, the increase in saturated zone solute flux uncertainty observed in the Braden- ton rainfall scenario over the uniform rainfall scenario is due to

increased uncertainty in the solute breakthrough from the unsaturated zone in the random rainfall case since the saturated

zone pore water velocity uncertainty is the same for these cases.

3.4. Accuracy of the Monte Carlo Moments

Owing to the extreme computational demands associated with running large transient variably saturated flow and transport simulations, only 150 Monte Carlo replicates were used to estimate the ensemble moments. Using statistical sampling theory [Kendall and Stuart, 1977; Harter and Yeh, 1998] and the approximate moments predicted by the Monte Carlo simulations, the accuracy of the Monte Carlo moments can be estimated. For all weather scenarios, statistical sampling theory predicts that using 150 Monte Carlo replicates the mean velocity should be estimated within 10% of its true value, and the velocity standard deviations should be estimated within 11.5% of their true values, at the 95% confidence level. For the uniform rainfall case, at the 95% confidence level, the mean solute flux should be estimated within <1% of its true value at

control plane 1 and <3% of its true value at control planes 2-4 at the time of peak solute flux variance. For the Olympia rainfall, using 150 replicates should allow the mean solute flux to be estimated within <10% of its true value at control plane 1 and <4% of its true value at control planes 2-4 at the time of peak solute flux variance. For the most variable Bradenton rainfall case, using 150 replicates should allow the mean solute flux to be estimated within --•20% of its true value at control

plane 1 and <7% of its true value at control planes 2-4 at the time of peak solute flux variance. In all cases, using 150 replicates allows the standard deviation of the solute flux to be

estimated within 11.5 % of its true value at the 95% confidence

level. Statistical sampling theory predicts that doubling the number of Monte Carlo replicates would improve the estima- tion accuracy for the Bradenton case to --•7% for the mean velocity, 15% for the mean solute flux at control plane 1 at the time of peak solute flux variance, <5% for the mean solute flux an control planes 2-4 at the time of peak solute flux variance, and to 8% for all velocity and solute flux variances. However, this would also double the computation time for the Monte Carlo simulations from -2 to 4 months (on a Sun Ultra 1 workstation). For the purposes of this study it was decided that this relatively small increase in accuracy was not worth the significant increase in computer time. It should be noted that the accuracy estimates for the standard deviations assume that the underlying population distributions are normally distributed, which is not likely the case for the velocity or solute flux distributions. Thus these estimates can only be used as guide- lines.


4. Conclusions

Monte Carlo simulations presented in this study predict ensemble solute transport in a coupled vadose-saturated zone system when the soil-aquifer properties and rainfall are random processes. Velocity fields required for solution of the advection-dispersion transport equation were obtained by the approximate solution of Richards' equation. Mass balances for flow and solute transport were very good (•2% error). Of particular interest was the impact of random rainfall on the solute transport predictions, characterized by the unconditional ensemble moments of the pore water velocity, the plume spatial moments, and solute flux breakthrough at subsurface control planes.

Results showed that the mean saturated zone pore water velocity was approximately horizontal and uniform over depth for both the random and uniform rainfall scenarios studied

here. The mean pore water velocity increased approximately linearly with distance in the horizontal direction. Away from the boundary conditions the mean and variance of the horizontal pore water velocity were well predicted using the infinite domain analytical expressions developed by Rubin and Bellin [1994]. This confirms the first-order analytical work by Li and Graham [1999], which predicted that temporal randomness of recharge should not affect saturated zone pore water velocity statistics as long as it is spatially uniform. The standard deviation of the v•rtical pore water velocity obtained by the Monte Carlo predictions was slightly underpredicted using the infinite domain analytical development of Rubin and Bellin [1994], indicating some additional uncertainty in vertical pore water velocity may be introduced through the finite vertical boundaries. In all cases the fluctuations of the Monte Carlo velocity moment estimates around the analytic values were somewhat larger than predicted by statistical sampling theory.

The mean plume displacement, estimated by the trajectory of the mean plume center of mass, was very similar for the uniform and random rainfall cases studied here. In the tem-

porally random infiltration cases, the saturated zone mean plume experienced more spread in the direction of mean flow at early times due to the increased spread of the ensemble mean vadose zone plume as it enters the saturated zone. How- ever, the asymptotic rate of spatial spreading of the mean solute plume were found to be approximately equivalent for the uniform and random surface infiltration cases and well

predicted by the approximate expressions for longitudinal macrodispersivity in nonuniform flow proposed by Destouni and Graham [1995].

Random rainfall and random soil properties increased prediction uncertainty of the solute plume behavior in the vadose zone by an order of magnitude when compared to the uniform rain and random soil case. This effect was reduced consider-

ably when the solute entered the saturated zone where random rainfall produced only slightly larger prediction uncertainty than the uniform rainfall case. The analytic model developed by Destouni and Graham [1995] accurately predicted the temporal breakthrough of the mean solute plume at saturated zone control planes for all cases as long as transport through the unsaturated zone accounted for the effects of transient

rainfall using the methodology developed by Foussereau et al. [2000a, 2000hi. In all cases the fluctuations of the Monte Carlo solute flux moment estimates around the analytic values were well within those predicted by statistical sampling theory.

Results of this work indicate that while uncertain rainfall

patterns dominate transport prediction uncertainty in the shallow unsaturated zone, uncertain hydraulic conductivity domi- nates prediction uncertainty in the saturated zone. Thus the primary impact of rainfall uncertainty on saturated zone transport processes can be captured by appropriately accounting for the macroscopic spread and uncertainty of solute moving across the water table boundary. The shallow water table scenario considered here was selected to allow maximum impact of the transient random rainfall on the saturated zone flow and

transport processes [Foussereau et al., 2000a, 2000b]. For deeper water table aquifers the influence of uncertain solute breakthrough due to random rainfall should be even less significant than for the case studied here.

The first-order Lagrangian methodology tested here should be useful for predicting the movement of surface applied chemicals through the unsaturated zone and into the saturated zone. Quantifying mean mass flux past subsurface compliance boundaries is important for many environmental problems, including evaluating compliance with total maximum daily loads (TMDL) of nonpoint source contaminants to surface and groundwaters. The general methodology can be extended to apply to deepwater table conditions and/or arid climates by appropriately accounting for the important vadose zone transport process uncertainties in predicting solute movement across the water table boundary. For arid climates this should include the effects of spatial variability in the pore size distribution parameter in addition to the hydraulic conductivity variability considered here.

Acknowledgments. This work was supported in part by the U.S. Department of Agriculture (USDA) water quality grant 93-34214- 8807, the Florida Department of Agriculture and Consumer Services, and the Florida Agricultural Experiment Station. Florida Agricultural Station Journal Series paper R05917.

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G. A. Akpoji, South Florida Water Management District, 3301 Gun Club Road, West Palm Beach, FL 33416.

G. Destouni, Department of Civil and Environmental Engineering, Royal Institute of Technology, Stockholm S 10044, Sweden.

X. Foussereau, Camp Dresser and McKee, 2300 Maitland Center Parkway, Suite 300, Maitland, FL 32751.

W. D. Graham, Interdisciplinary Hydrologic Science Program, P.O. Box 110570, University of Florida, Gainesville, FL 32611. ([email protected])

P.S. C. Rao, School of Civil Engineering, Purdue University, West Lafayette, IN 47907.

(Received November 8, 1999; revised October 23, 2000; accepted November 30, 2000.)

Solute transport through a heterogeneous coupled vadose-saturated zone system with temporally random rainfall

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