Large-scale dispersion in a sandy aquifer: Simulation of subsurface transport of environmental tritium

Transcription

1 WATER RESOURCES RESEARCH, VOL. 32, NO. 11, PAGES , NOVEMBER 1996 Large-scale dispersion in a sandy aquifer: Simulation of subsurface transport of environmental tritium Peter Engesgaard and Karsten Høgh Jensen Department of Hydrodynamics and Water Resources, Groundwater Research Centre Technical University of Denmark, Lyngby John Molson and Emil O. Frind Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada Henrik Olsen 1 Department of Geology and Geotechnical Engineering, Groundwater Research Centre Technical University of Denmark, Lyngby Abstract. Large-scale dispersion in a sandy unconfined aquifer in Denmark was studied by simulating subsurface transport of environmental tritium. Subsurface transport included transport in a moderately deep unsaturated zone and in a relatively long cross section of the aquifer. The tritium data from the site enabled a four-step modeling analysis comprising (1) estimation of tritium content in the infiltration water, (2) transport in the unsaturated zone, (3) estimation of flux-averaged tritium concentration in the recharge water, and (4) transport in the groundwater zone. The groundwater model simulations were sensitive to the longitudinal and transverse dispersivity parameters, L and T,asa set of parameters, but a model sensitivity analysis showed that it was not possible to identify a unique set of parameter values. A likely range of variation for the two parameters could be identified: ( L, T ) [(1 m, m); (10 m, 0.0 m)] the two parameters being interdependent in that an increase in L results in a decrease in T and vice versa. The reported dispersivities represent a scale of 1000 m, the approximate travel distance from the water table to the observation wells. If the estimated L can be regarded as being of intermediate reliability following earlier defined criteria, the range or the representative set of values then represent the largest scale of earlier reported values. Including our range of L in the set of reported dispersivities suggests that L does not increase indefinitely with scale. Introduction Dispersion of solutes in the subsurface for both unsaturated and saturated conditions has been subject to both experimental and theoretical research for many years. Characterization of the dispersive behavior at a given site, as manifested in the dispersivity parameters, is a prerequisite for predicting the migration of contaminants at the site. Techniques to quantify these parameters are therefore of great practical significance. A number of tracer and modeling studies have been carried out to determine values of dispersivity in different geological strata. The results of the reported efforts have been compiled by Gelhar et al. [1985] and were later updated by Gelhar et al. [1992]. Dispersivity values are basically derived from two types of experiments: controlled events, where the quantity and duration of the tracer input are known, or uncontrolled events, where a contaminant is introduced in the aquifer unintentionally and the source input history is therefore not known in detail. Of the large number of controlled tracer experiments that have been reported, five are rather unique because the 1 Now at Rambøll, Virum, Denmark. Copyright 1996 by the American Geophysical Union. Paper number 96WR /96/96WR-02398$09.00 tracer clouds were monitored over relatively large distances ( m) in a dense network of sampling wells, and independent measurements of the hydraulic properties were carried out. These studies have provided new insight into the dispersion mechanism, and dispersivity values have been derived in longitudinal and transverse directions for increasing transport distances. Moreover, validation data for deterministic and stochastic models have been provided by these experiments which include (1) Borden Air Force Base [Mackay et al., 1986; Freyberg, 1986; Sudicky, 1986], (2) Cape Cod [Garabedian et al., 1988; LeBlanc et al., 1991; Garabedian et al., 1991], (3) Twin Lake [Killey and Moltyaner, 1988; Moltyaner and Killey, 1988a, b], (4) Columbus Air Force Base [Boggs et al., 1992], and (5) Vejen [Jensen et al., 1993]. These studies have provided high-reliability data for a scale less than a few hundreds of meters, and they suggest that the dispersivities are much lower than previously thought. To date, no controlled experiments have been undertaken for larger transport scales because of the very long time period and resources required. Dispersion data for larger scales are primarily obtained from transport studies of environmental tracers such as bomb tritium, helium, and chlorofluorocarbons. Tritium has been widely used for dating of groundwater, that is, to determine the length of time that the water has been isolated from the atmosphere [see, e.g., Solomon et al., 1992] and for estimating the 3253

2 3254 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER discuss our results, relate our main findings to other studies, and present conclusions on this investigation. Figure 1. Plan view of the Rabis Creek aquifer in Denmark (latitude: N; longitude: 9 11 E). groundwater recharge [see, e.g., Daniels et al., 1991; Solomon and Sudicky, 1991; Cook et al., 1994]. For these purposes the exact form of the input function may be less important in comparison to using the tritium data for determining the dispersive properties. Egboka et al. [1983] were among the first to determine the longitudinal dispersivity using tritium data. On the basis of a one-dimensional analysis along a stream tube of the aquifer at the Borden site they found values in the range m. This value is much larger than the value found in a controlled experiment at the same site [Mackay et al., 1986; Freyberg, 1986; Sudicky, 1986], indicating that the results may be affected by the ill-defined input function at the water table. Robertson and Cherry [1989] determined the transverse dispersivity in a vertical cross section of an aquifer in Ontario, neglecting the presence of the unsaturated zone. They were able to determine the longitudinal and transverse dispersivities in the part of the study area where the flow was near vertical, while they could not identify the longitudinal dispersivity in the area where the flow was predominantly horizontal. Later Solomon et al. [1993] determined the vertical longitudinal dispersivity for the same aquifer. In contrast to previous studies we have analyzed the coupled transport through the unsaturated and groundwater zones using tritium data from both zones. This allows not only for analyzing the dispersive properties of the unsaturated zone but the tritium input function to the groundwater zone can be determined more accurately, thus allowing for a more reliable analysis of dispersion in groundwater. Our analysis is based on tritium concentration profiles for the unsaturated zone that were measured in the late 1960s [Andersen and Sevel, 1974] and on tritium concentration profiles for the groundwater zone measured November 1988 and October 1989 [Kristiansen et al., 1990]. The field site is located in western Denmark. The paper is organized in the following way. First, the field site, the hydrogeology, and the tritium observations in the subsurface will be presented. Second, a four-step modeling approach for simulating transport in the unsaturated and saturated groundwater zones will be described. Third, the results from each of these steps will be presented. Fourth, we will Area Under Investigation The area under investigation is located in central Jutland, Denmark (Figure 1). The hydrogeological and tritium data originate from two localities within the greater Rabis Creek area. The unsaturated zone data were collected at Grønhøj, a location just outside the Rabis Creek catchment [Andersen and Sevel, 1974] and only 2 km away from the site in the Rabis Creek catchment where the data from the saturated zone were collected. The location of the study site is shown in Figure 1 [Olsen et al., 1993; Postma et al., 1991; Kristiansen et al., 1990]. The Rabis Creek area has been intensively studied during the last 8 years, mainly in connection with investigating nitrate contamination and denitrification processes in groundwater [Postma et al., 1991]. Originally, the area was vegetated by heath but is now mainly used for farming with only small areas of heath remaining. A small pine tree plantation is also present in the area. Hydrogeological Conditions An extensive geological and geophysical study has been carried out in the greater Rabis Creek area, including analysis of sediment samples from water supply drillings, borehole logs, and a reflection seismic survey [Olsen et al., 1993]. During the Quaternary age, Denmark experienced four glaciation periods [Sjørring, 1983]. During the latest glaciation in the Weichselian the northern and eastern parts of Denmark were covered by ice. The central and western parts of Jutland were situated at the front of the ice and were dominated by extensive outwash plains. The study area is therefore covered by a thick sequence of sandy outwash deposits, approximately m in thickness. These glacial deposits are underlain by shallow marine sand and clay deposits of Miocene age. A reflection seismic survey along part of the Rabis Creek (Figure 1) indicated the presence of three seismic sequences. By combining the seismic data and data from borehole logs and sediment samples, Olsen et al. [1993] were able to develop a detailed characterization of the geological structure (Figure 2). The uppermost sequence is composed of medium-grained sand ( mm), interpreted as an outwash deposit. A valley fill, up to 80 m in thickness, of alternating gravel/sand and silt/clay underlies the outwash deposit. The valley fill is interpreted as being subglacial. The valley fill is topped by a lacustrine silty clay layer, up to 5minthickness, which extends beyond the valley structure in a westward direction (Figure 2). The lowermost sequence is slightly tilted toward the west. It is dominated by two sand units, probably medium grained, separated by a silt/clay unit and a clay unit, each approximately 20 m in thickness. The lowermost sequence is interpreted as Miocene sediments of shallow marine origin. Three hydrogeological units may be distinguished within the cross section that forms the basis for the flow and transport analysis in the groundwater. The outwash sand and the uppermost shallow marine sand unit probably exhibit similar flow properties. A hydraulic conductivity in the range m/s is calculated based on pumping tests in the area [Miljøstyrelsen, 1983; Hansen, 1991]. The two sand units are in direct contact in the eastern half of the section shown in Figure 2 and accordingly act as one flow unit called deposit I. The subglacial valley fill was interpreted as a separate flow unit called deposit

3 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER 3255 Figure 2. Geological model of the Rabis Creek area. A lower-tilted sequence of sand, silt/clay, and clay units of Miocene shallow marine origin is truncated by a Quaternary subglacial valley fill. The valley fill is composed of alternating gravel/sand and silt/clay deposits and is topped by a lacustrine silty clay unit. The uppermost part is composed of Quaternary outwash sand. The model cross section A-A is also shown. II. No measurements of the hydraulic conductivity have been carried out for this deposit. On the basis of the very heterogeneous composition with alluvial channel gravel/sand and flood basin silt/clay, the hydraulic conductivity may span from 10 9 to 10 4 m/s. The lacustrine clay layer is also interpreted as a separate flow unit called deposit III. Within the model cross section (to be discussed later on) we assume that deposit III has a maximum thickness of 2 m at the west end and pinches out toward the east, not extending beyond the eastern boundary of the underlying valley fill (Figure 2). No information about the hydraulic conductivity is available, but typical values for this type of formation are in the range m/s, that is, much below the hydraulic conductivity of deposit I. The aquifer is unconfined, and monthly observations of groundwater levels in 25 wells (not shown in Figure 1) indicate horizontally dominated flow from close to the eastern water divide to near Karup Stream [Kristiansen et al., 1990; Engesgaard and Jensen, 1990]. Three-dimensional flow simulations by Engesgaard and Jensen [1990] confirmed this interpretation and furthermore showed that there is significant downward vertical flow close to the water divide and upward vertical flow close to Rabis Creek and Karup Stream. In fact, the low nitrate concentrations measured in Rabis Creek are partly caused by mixing of upper nitrate-rich groundwater and deeper nitratefree groundwater immediately below the creek. Groundwater discharge to Rabis Creek is generally constant over the season. The thickness of the unsaturated zone can be from 14 m in the upstream area of Rabis Creek to 22 m at Grønhøj. Seasonal water table fluctuations are in the range m [Engesgaard and Jensen, 1990; Andersen and Sevel, 1974]. Tritium The measured content of tritium in precipitation is shown in Figure 3. The values are monthly measurements taken at a station close to Ödum (N 56 18, E10 8 )[Andersen and Sevel, 1974] located approximately 50 km from the site. For 1961 and 1971, most of the data are from Plönninge, Germany [Andersen and Sevel, 1974]. Figure 3 shows that a peak in the input above 3500 tritium units (TU) occurred in Two sets of independently obtained data on tritium in the subsurface were available for this study. At Grønhøj, Andersen and Sevel [1974] installed four boreholes in the unsaturated zone numbered I (March 1966), IV (March 1968), V (November 1970), and VI (November 1972), all located within 15 m of each other. The lithology and grain-size distribution are similar at all wells. Well I was sampled every 0.2 m, and the rest of the wells were sampled every 0.5 m for measurements of tritium in the soil water (see Andersen and Sevel [1974] for details on the experimental procedure). Wells I, IV, and V show a tritium peak moving through the unsaturated zone, while this peak has left the unsaturated zone at well VI. Therefore only the data from wells I, IV, and V are used in the analysis. Postma et al. [1991] installed a series of multilevel sampling wells in the saturated zone near Rabis Creek along an approximate flow line trajectory (wells T1 T8 and T10) (Figure 1) to document the leaching of nitrate and the denitrification processes occurring in the aquifer. Wells T1, T2, and T10 were sampled at 1-m intervals over the entire depth in , capturing the 1963 peak. The tritium concentrations were measured only at a few levels in the rest of the wells. The sparse data from these wells are in general agreement with the other data and will not be included in this analysis. Modeling Approach A four-step approach will be used to simulate subsurface transport of environmental tritium at the site. This approach is schematically shown in Figure 4: (1) estimation of tritium in Figure 3. Monthly measurements of concentration of tritium in precipitation and simulated recharge in years after 1961.

4 3256 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER Figure 4. Illustration of the four-step modeling approach. the infiltration water, (2) one-dimensional transport analysis in the unsaturated zone, (3) prediction of the mass flux of tritium to groundwater, and (4) two-dimensional transport analysis in the saturated zone. In contrast to many other studies where transport through the unsaturated zone has been disregarded, the moderately deep unsaturated zone in the study area (14 22 m) necessitated that the transport through this zone be explicitly considered. Step 1 The first step was the construction of a representative tritium input function reflecting the concentration of tritium in that part of the precipitation that will infiltrate below a certain depth (here called upper recharge). This step can therefore be subdivided into the following two stages: the construction of a precipitation input function and the assessment of the effect of evapotranspiration and transient flow on the infiltration process. As in many other studies [e.g., Bradbury, 1991], it was necessary to construct an input function that reflects the actual conditions at the site as closely as possible. As mentioned above, it was possible to construct a fairly accurate input function for the Rabis Creek area because of the proximity of the Ödum station. The monthly measurements are shown in Figure 3. The tritium function of the infiltrating water needs to be transformed to a function representing the concentration of tritium in the upper recharge water in order to estimate the boundary conditions for the transport simulations in the unsaturated zone. Andersen and Sevel [1974] utilized a simple water and tritium balance for the upper 1mofthesoil, assuming complete mixing to derive the upper recharge function which we will revise as described below. Our objective has been to determine an effective steady state pore water velocity and a longitudinal dispersivity for the unsaturated zone that are representative for the period of observation, and that we will assume can be used to predict the subsequent transport of tritium in the unsaturated zone for the time period up to , the time period of observation of the tritium profiles in groundwater. The mass of tritium per unit area transported into the soil as t a function of time is M(t) 0 q z (t)c p (t)dt, where q z (t) is upper recharge flux (millimeters per year), c p (t) is concentration of tritium (TU), and t is time (year). Tritium as a tracer has the advantage over other tracers; it is not concentrated in the soil as a result of evapotranspiration [Allison and Hughes, 1978], and therefore c p can be approximated by the concentration measured in precipitation. M(t) is thus a function of the temporal changes in q z and c p. However, in this study we assume that q z is constant over the considered time period, and to incorporate the temporal changes of the input of mass M, we will introduce a scaled time t* defined as t* 0 t q z d q* z, (1) where q* z is a time-averaged recharge defined by q* z 1 t m q t z d, (2) m 0 where t m is the considered time period. By scaling time according to (1) the input M(t) becomes volume weighted. Although the tritium concentration in precipitation may be high during dry summer periods, the input of tritium to the unsaturated zone will be very low because it is assigned a very small pulse period. The upper recharge q z at a depth of 1mwas estimated using an unsaturated flow model developed by Jensen [1983]. The model was applied to the site using daily values for precipitation (corrected for wind effects) and potential evapotranspiration obtained from a nearby meteorological station. The model predicts unsaturated flow based on a finite difference solution to the one-dimensional Richards equation, and evapotranspiration is calculated and distributed over the

5 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER 3257 Figure 5. Model cross section A-A. The location of the cross section is shown in Figures 1 and 2. Deposit I is a medium-grained sand, deposit II is a fine-grained sand, and deposit III is a low-permeable clay. Boundary conditions for the flow modeling are also shown. Deposit II and III (paleovalley) is modeled with a set of rectangular blocks of finite elements. root zone according to semiempirical relations and subsequently introduced in the flow equation as a sink term. From the simulated flow at 1 m depth, q z (t), the scaled tritium input function can be found from (1) and (2). Step 2 In step 2, transport of tritium in the unsaturated zone at Grønhøj was investigated for the purposes of determining an effective steady state pore water velocity and longitudinal dispersivity. We assumed that flow in the unsaturated zone was one dimensional. The CXTFIT inverse model [Parker and van Genuchten, 1984] was used with the time-scaled tritium input function to fit an analytical solution to the observed tritium data for wells I, IV, and V. The analytical model solves the following transport equation: c/ t D 2 c/ z 2 v z c/ z c, (3) where c is the resident tritium concentration, D is the dispersion coefficient (m 2 /yr), v z is the vertical pore water velocity (m/yr), is the rate of decay (yr 1 ), t is time (yr), and z is vertical distance (m). Since decay is first order, is calculated as ln 2/t 1/2, where t 1/2 is the year half-life of tritium. Notice that concentrations are assumed to represent resident concentrations because they were measured as volumeweighted concentrations [see Andersen and Sevel, 1974]. The appropriate upper boundary condition for (3) is therefore of the flux type represented by the scaled tritium input function. The lower boundary condition is a zero gradient at infinity. Although the simulated region is finite, the semi-infinite solution nevertheless provides an accurate approximation [Parker and van Genuchten, 1984]. CXTFIT minimizes the sums of squares of the residuals between observed and calculated concentrations to find an optimum v z and/or D, which are the two unknowns in (3). The longitudinal dispersivity is then defined as L D/v z (meters). CXTFIT also calculates a correlation coefficient (r) and a 95% confidence interval for the fitted parameters. CXTFIT was used to fit analytical solutions of (3) for each set of well data and to all well data simultaneously. Step 3 CXTFIT can also be used to solve (3) in a forward mode, provided v z and D are known. In step 3, representative values for v z and D are selected, and the flux concentration of tritium at a depth of 14 m is predicted. The time-scaled input function was used as upper boundary condition. This depth corresponds to the thickness of the unsaturated zone at the trajectory of wells shown in Figure 1. The upper boundary condition for predicting the flux concentration is the tritium input function for the upper recharge derived in step 1. The assumption is therefore made that the transport parameters found at Grønhøj, located 2 km away, can be used to predict transport in the unsaturated zone in the cross section at Rabis Creek. This is a reasonable assumption because the origin of the sediments is the same, the sediments are relatively homogeneous, and the precipitation at the two locations is comparable. The only major difference between the two localities is the surface vegetation, primarily farmland at Rabis Creek and grass at Grønhøj. Step 4 In step 4, transport and decay of tritium in groundwater are simulated in the cross section A-A shown in Figure 1. We have assumed that transport is two dimensional. This is a reasonable assumption since the input of tritium occurred as an areal source, flow is approximately parallel to the direction of the section of wells, and the valley fill structure (perpendicular to the flow direction) is of a regional extent. Figure 5 shows schematically the cross section A-A. The location of the model cross section within the geological cross section is shown in Figure 2. The model cross section extends down to the clay-rich layer, which is assumed impermeable. The simulation

6 3258 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER period is from 1961 to 1988 (wells T1 and T2) and from 1961 to 1989 (well T10), the times of observation in the groundwater. It is assumed that flow is steady. Considering the approximately 28-year simulation time and the small seasonal water table fluctuations, a steady state flow approximation seems justified. Because of the sparse tritium data in wells T7 and T8 and because of the possible influence of Rabis Creek, where the flow field begins to deviate from the assumed twodimensional pattern, the cross section ends 700 m downstream from well T1. The groundwater flow and tritium transport simulations at Rabis Creek were made using a modified version of Flotrans, a finite element model capable of simulating steady state groundwater flow and advective-dispersive mass transport with linear decay [Guiguer et al., 1994]. Flotrans was modified for the Rabis Creek study to accommodate a larger grid size, to allow for the sloping aquifer base, and to permit a time-varying source concentration. The two-dimensional steady state groundwater flow system was simulated in Flotrans by solving the dual formulation [Frind and Matanga, 1985] expressed as x K xx x z K zz z 0 (4) x 1 K zz x z 1 0, (5) K xx z where x and z are the horizontal and vertical coordinate directions, respectively (m), K xx and K zz are the principal components of the hydraulic conductivity tensor (m/d), is the hydraulic head (m), and is the stream function (m 2 /d). This simulation approach assumes a saturated, steady state flow field, and a nondeforming porous medium. Flotrans uses the Galerkin finite element approach to solve (4) and (5) [see, e.g., Huyakorn and Pinder, 1983] and uses an efficient preconditioned conjugate gradient matrix solver [Schmid and Braess, 1988]. In the solution to (4) an iterative approach is used where the domain is allowed to deform vertically to conform to the equilibrium water table position. Boundary conditions for the Rabis Creek flow model consisted of variable recharge across the upper water table boundary, a no-flow condition at the upstream groundwater divide and along the bottom clay-rich layer, and a constant head at the downstream boundary (Figure 5). The governing equation for two-dimensional advectivedispersive mass transport of a dilute species undergoing firstorder decay within a porous medium can be written as [Bear, 1972] x i D ij R c x j x i v i R c c c t, (6) where x i are the spatial coordinates ( x, z) (m), v i is the average linear flow velocity (m/d), D ij is the hydrodynamic dispersion tensor (m 2 /d), R is the linear retardation factor, is the linear decay rate (d 1 ), t is time (d), and c is concentration (TU). The form of the dispersion tensor in (6) is given by Frind and Hokkanen [1987] and is dependent on the average linear flow velocities, the longitudinal ( L ) and transverse ( T ) dispersivities (m), and the effective molecular diffusion coefficient D* (m 2 /d) according to where D xx L v x 2 /v T v z 2 /v D* (7) D zz T v x 2 /v L v z 2 /v D* (8) D xz D zx L T v x v z /v, (9) v v x 2 v z 2 (10) From (4) (5) and (7) (9) it is clear that we allow the porous medium to be anisotropic with respect to flow but only isotropic with respect to dispersion. We will discuss this assumption below. The velocity v i in (6) is obtained using the steady state stream function solution of the dual formulation. In (6) the decay constant is given by ln 2 /t 1/ 2, (11) where t 1/2 is the tritium half-life (t 1/2 for tritium years). A retardation factor of R 1 was assumed for tritium. The tritium transport simulation used a Cauchy-type mass flux boundary condition across the water table. Tritium source concentrations were allowed to vary in time and were derived from the results of the one-dimensional tritium transport simulation in the unsaturated zone. The third type, or Cauchy boundary, represents continuity of mass flux across an external boundary and is written as q 0 C 0 / v n c D ij c/ n, (12) where q 0 C 0 / is a known mass flux term at the boundary, and n is the unit normal vector at the boundary [Frind, 1982]. Below we will discuss the compatibility of the lower boundary condition in the unsaturated simulations with the upper boundary condition in the groundwater simulations. At all remaining transport boundaries, including the left outflow boundary, a zero-concentration gradient condition was applied. The initial (pre-1961) background tritium concentration was assumed to be zero. The transport equation (6) is solved in Flotrans using the Galerkin finite element method with Leismann time weighting to increase efficiency and limit memory requirements [Leismann and Frind, 1989]. The Leismann scheme generates a symmetric matrix and produces a solution which is effectively second-order accurate in time. The transport solution is time marching and uses the same conjugate gradient matrix solver as in the flow solution. Results The results from each modeling step are presented separately. Step 1 Three observation time periods were considered, all starting at the beginning of 1961 before significant amounts of tritium were found in precipitation (Figure 3). The three periods correspond to the times tritium profiles were measured at the three wells at Grønhøj, that is, 1961 to mid-march 1966 (well I), 1961 to mid-march 1968 (well IV), and 1961 to mid- November 1970 (well V). For each of these periods the upper recharge (q z ) at a depth of 1 m was simulated, and the timeaveraged infiltration (q* z ) was calculated according to (2). The calculated time-averaged infiltrations were 457, 476, and 434

7 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER 3259 Table 1. Fitted v z and D Parameters Using Measured Tritium Data in Wells I, IV, V, and I IV V Together Well v z, m/yr* D, m 2 /yr* L, m r I IV V Average I IV V *Fitted parameters with 95% confidence interval. Here L is the longitudinal dispersivity; r is the correlation coefficient. Figure 6. Timescaled (volume-weighted) and measured input function for 1963 predicted in step 1. The estimated monthly infiltration at z 1 m is also included. mm/yr for wells I, IV, and V, respectively. The time-scaled tritium input function was derived from (1) and shown in Figure 6 for Also included in Figure 6 is a plot of the measured concentration in precipitation and the estimated upper recharge. The precipitation input function shows high concentrations in the summer period, where the upper recharge is relatively low, and consequently a direct use of this input function would overpredict the mass flux to the groundwater. For 1963 the time-scaled tritium input function reduces the total mass flux by approximately 30%. The time-scaled input function was only determined for From 1970 to 1989 a constant concentration equal to the measurement in the most upper sampling port in the wells was assumed. Step 2 CXTFIT was used to determine the transport parameters v z and D by fitting an analytical solution to the observed tritium profiles in the unsaturated zone. Both v z and D were assumed unknown, and data from each well were used both individually and together. Table 1 shows the results of this analysis. The results for the individual wells show a very consistent prediction of v z, with an average of 3.54 m/yr and a very narrow 95% confidence interval ( 10% of the fitted estimate). There is greater uncertainty in the fitted estimate of D with a factor of 1.7 in difference for the parameter values of well I and well IV. The average D is 3.17 m 2 /yr. The 95% confidence intervals are much wider, approximately 74% of the fitted value for well I, which also has the lowest correlation coefficient. Some of the uncertainty in D is caused by the estimate of v z, because D is a linear function of v z. The transport parameters were also determined by fitting to data from all three wells simultaneously. In this case, the fitted parameters for v z and D are 3.40 m/yr and 3.00 m 2 /yr, respectively, very close to the averages deduced from averaging between all wells. The calculated longitudinal dispersivity values L are given in Table 1. The average between all wells is 0.93 m, which is close to the value of 0.88 m obtained when using all data. The observed and individual fitted profiles are shown in Figure 7. For well IV and V, CXTFIT provides an excellent fit to the data, which is also indicated by the high correlation coefficients (Table 1). For well I the model compares less favorably to the data. This discrepancy may be a consequence Figure 7. Observed and fitted tritium profiles in the unsaturated zone.

8 3260 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER Figure 8. Simulated steady state water table and four observations of water table elevation (April 1988). of the large sensitivity of the inverse model to the input function at early times. As discussed above, a number of assumptions are invoked when estimating the tritium concentration of the recharge water, and uncertainties related to this procedure may influence the model predictions. At later times the accuracy with which the input function is determined becomes less critical because decay and dispersion now to a greater extent determine the shape of the tritium profiles. Data for well I also show a more fluctuating pattern than for the two other wells, which corroborates our hypothesis. Our predictions of tritium transport in the unsaturated zone are more consistent than those of Andersen and Sevel [1974], who used only the 1966 profile to manually fit v z and D and obtained values of 4.5 m/yr and 3.2 m 2 /yr, respectively. These can be compared to our estimates of 3.9 m/yr and 2.3 m 2 /yr or in terms of L, values of 0.7 and 0.6 m. The differences between the two sets of parameter values are rather small. Andersen and Sevel [1974], however, could not produce the same good fit for the two other profiles with their set of values. We conclude on this basis that because of the time-scaled tritium input function, we are able to obtain a more consistent prediction of the profiles in all three wells. Step 3 CXTFIT was used in a forward mode to predict the flux concentration of tritium for at a depth of 14 m, corresponding to the thickness of the unsaturated zone at the Rabis Creek trajectory of wells. Application of the model in a prediction mode requires that representative values for v z and D are specified. The time-scaled input function was used as upper boundary condition. We have chosen to use v z 3.4 m/yr and D 3.0 m 2 /yr, corresponding to the values obtained when fitting to all data (Table 1). Figure 3 shows the calculated tritium input recharge function. The peak is now much lower as a result of decay and dispersion and arrives at the groundwater table in 1967, which then suggests a residence time of approximately 4 years in the unsaturated zone. Step 4 A flow solution for the cross section of the groundwater zone is required before transport and decay of tritium for can be simulated. Figure 5 shows the boundary conditions that have been used to simulate flow, assuming steady state conditions. The time-averaged recharge of 434 mm/yr for the well V period was used as upper flux boundary condition so that the amount of recharge used to calculate the tritium recharge function is consistent with the generated flow field. The timeaveraged recharge is close to an estimated value of 482 mm/yr determined by Storm et al. [1990] for a larger regional catchment in the same part of Jutland and for a longer period Recharge at the pine tree plantation was reduced by 15% to account for higher evapotranspiration in the plantation. A constant head of 42.5 m was applied to the downstream end. An inappropriate choice of spatial and temporal model discretizations can result in numerical dispersion and oscillations. These undesired effects will mask the true mechanical dispersive effects and thus make it difficult to assess the dispersive transport mechanisms in the aquifer. This is a well-known problem when using numerical models for dispersion analysis [Gelhar et al., 1992]. A sensitivity analysis on the effects of different discretizations on predicted tritium profiles was therefore carried out in order to find an acceptable discretization that would minimize these effects without having excessive computing times. The result of this analysis suggested that an accurate simulation could be obtained using a nodal mesh of ( 171,114 nodes) in the horizontal and vertical directions, respectively. We will comment on this below. The grid elements are evenly distributed in each dimension resulting in a horizontal element length of 2.3 m and a vertical element length ranging from 0.3 near the upstream boundary to 0.5 m near the outflow boundary. Although the mesh adapts to the domain geometry (i.e., to the bottom slope of the aquifer and to the sloping water table), the conductivity distribution does not entirely adapt to the valley fill structure in the interior of the solution domain. Specifically, deposits II and III of this structure are discretized as two sets of rectangular blocks (Figure 5). The structure of the valley fill is therefore somewhat smeared at the bottom of the aquifer, but it was judged that further improvements to reproduce the valley fill structure would not influence the simulated results in the upper part of the aquifer. The hydraulic conductivities of deposits II and III were fixed at and m/s, respectively, and both were assumed to be isotropic. The horizontal hydraulic conductivity (K x ) of deposit I was calibrated to m/s. An anisotropy ratio of K x /K z 2 for deposit I was used based on measurements in a similar aquifer (Vejen aquifer [see, e.g., Jensen et al., 1993; S. Hvilshøj, personal communication, 1996]). The sensitivity of the simulated tritium distributions to this ratio was investigated and will be discussed below. Figure 8 shows that the simulated water table and point observations in April 1988 of the water table in well T1, T2, T10, and 28 compare reasonably well. The horizontal hydraulic conductivity for deposit I is also within the range suggested by others [Olsen et al., 1993; Engesgaard and Jensen, 1990]. Figure 9 shows the simulated streamlines. The impact of the lower-permeable paleovalley structure on flow is clearly seen Figure 9. Simulated streamlines in cross section. Contour interval is 0.1q, where q is the steady state recharge of 434 mm/yr.

9 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER 3261 in Figure 9. Very little flow traverses this geological structure, and all streamlines converge in the upper half of the aquifer right below wells T1 and T2. With a porosity of 0.35, the maximum horizontal pore water flow velocity in the vicinity of wells T1, T2, and T10 are approximately 96, 174, and 47 m/yr, respectively. The maximum vertical pore water flow velocity at the same wells are 1.3 m/yr (downward), 19 m/yr (upward), and 1.3 m/yr (downward), respectively. The maximum flow velocities all occur within the upper m of the cross section. To gain more insight into the dispersive transport mechanisms in this aquifer, we carried out a sensitivity analysis by comparing the simulated tritium concentrations for different dispersion parameters and the observed tritium profiles. In the analysis the simulated steady state flow field and the calculated tritium recharge function remained the same. The two remaining parameters are thus the longitudinal ( L ) and vertical transverse ( T ) dispersivities. The Peclet numbers in the horizontal and vertical directions are defined by [Kinzelbach, 1986] Pe x v x x/d xx and Pe z v z z/d zz, respectively, and they should be lower than 2, although Peclet numbers up to 10 sometimes can be accepted. From hand calculations we have verified that both Peclet numbers are not violated. However, at large ratios L / T 10 the discretization must be based on the order of the magnitude of T in order to obtain an adequate resolution of the concentration profile in the vertical direction. For this purpose it is useful to define a vertical Peclet number as Pe z z/ T. As a rule of thumb, z 10 T [Voss, 1984], a criteria that has been impossible to meet with the low transverse dispersivities found in this study (see below) and the available computer resources. To verify that the chosen nodal mesh does not introduce appreciable numerical dispersion, especially with respect to transverse dispersion, we compared simulation results using this nodal mesh with simulation results using a mesh where the number of nodes in the vertical direction was increased by a factor of 4. A time step of 25 days was adopted. There was virtually no difference in the simulated amount of spreading using these two nodal mesh configurations. The maximum Courant number in both directions C x V x t/ x and C z V z t/ z are 5.2 and 3.3, respectively, for the maximum flow velocities listed above. The maximum values occur locally at well T2. The C x value is more than a factor of 5 higher than the normal criteria. The results from a simulation using a time step of 5 days were, however, nearly identical to the simulation results using a time step of 25 days. Consequently, we assume that the chosen nodal mesh ( ) and time step (25 days) will not produce unacceptable numerical errors for the subsequent dispersion analysis. The effects of changing the anisotropy ratio K x /K z on the simulated profiles in the three wells were investigated in the Table 2. Simulation List of Dispersion Simulations L, m T, m Figure , 12, Figure 10. Simulated tritium distributions at two points in time using L 1 m and T m. Concentration is in tritium units (TU). Contour lines are 50 and 100 TU at 28.2 years and 50, 100, 200, and 400 TU at 11 years. T1, T2, and T10 are the three observation wells. A simulation time of 28.2 years is approximately the time of sampling (November 1988 for wells T1 and T2 and October 1989 for well T10). The sampling ports have been indicated on each of the wells. following way. The anisotropy factor was fixed at 1, 2, or 10, and K x was calibrated for these anisotropy factors to approximately obtain the same simulated position of the water table. The K x was in this way found to vary less than 10%, with a higher K x for the high anisotropy ratio and vice versa. The simulated tritium profiles were very similar (not shown here), which shows that since the horizontal gradient was the same in all three cases, the vertical gradients in head adjust to the specified anisotropy ratio, producing very similar flow fields. Therefore, in the following analysis we will adopt an anisotropy ratio of 2. Since we are simulating flow in an anisotropic medium, ideally, anisotropic dispersion should be considered as well. However, we have inferred that the number of tritium measurements available and the scale of the problem in combination with the low anisotropy ratio do not justify the inclusion of anisotropic dispersion. Table 2 lists the simulations for various values of the dispersivity parameters that will be presented in the following. The simulated tritium distribution at 2 points in time are shown in Figure 10. Assumed values for L and T are 1.0 and m, respectively (simulation 4, Table 2). After 11 years of simulation (1971), tritium only resides in the upper m of the cross section. This is 4 5 years after the 1963 peak entered the aquifer. Maximum tritium concentrations are above 400 TU. The distribution after 28 years (1989) represents approximately the conditions at the times of sampling. The 1963 peak can still be seen in the distribution; however, dispersion and decay have lowered the concentrations to slightly above 100 TU. The impact of the paleovalley structure on the simulated distribution at wells T1 and T2 is clearly seen in Figure 10. Figure 11 shows observed tritium profiles in wells T1, T2,

10 3262 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER Figure 11. Observed and simulated tritium profiles in wells T1, T2 (November 1988) and T10 (October 1989). Three simulations with three sets of ( L, T ) are shown: simulation 1 with (1 m, 0.0 m), simulation 2 with (1 m, m), and simulation 3 with (1 m, 0.01 m). and T10 together with simulations 1 3 (Table 2) for different values of T (0.0, 0.001, and 0.01 m) and L constant at 1.0 m. The simulation with T 0.01 m gives the best overall agreement with the observations, the two others overpredicting the peak concentrations. Values of T higher than 0.01 m tend to excessively smear out the simulated profiles. The location of the peak in T1 is simulated very well. For T2 and T10 the simulated location of the peak tritium concentration is offset by approximately 3 5 m in comparison with the observed tritium peak location. This may be due to an inaccurate location of the low-permeable deposit III layer or variations in local recharge rates. The observed T10 profile has a very dispersed peak which is not captured by the simulation. By inspecting the T1, T2, and T10 profiles it is notable that the tritium peak in well T10 is located deeper than in well T2, where the peak again is located deeper than in well T1. This can not be explained by the later time of sampling in well T10. Downstream from well T10 upward flow seems to take place. If the aquifer was homogeneous and the recharge was uniform, one would expect the peaks to be located deeper and deeper with increasing distance away from the groundwater divide. This is not the case, and one explanation can be the presence of the thin clay layer, which confines flow to the upper part of the groundwater in that part of the cross section. Incorporating this clay layer in the model, we are indeed able to simulate tritium concentrations at higher elevations in wells T1 and T2 than in well T10. Figure 12 shows the observed tritium profiles together with simulations 4 6 for different values of L (1, 10, and 20 m) and T constant at m. Clearly, the simulation results are very sensitive to L. The simulation with L 1 m gives a good agreement with the observed peak concentration. Figure 13 shows three possible sets of L and T that all describe the profiles equally well (simulation 4, 7, and 8). The three sets of parameter values for ( L, T ) are the following: (1.0 m, m), (5.0 m, m), and (10 m, 0.0 m). The model is sensitive to L because of the presence of significant vertical flow components in the cross section not only at the water table but also deep in the aquifer close to the paleovalley structure. With a high value for L, initial mixing at the water table will transport tritium deeper, and the concentrations at the water table will be lower. Likewise, upward flow and high mixing at the paleovalley structure will mix tritium-free and tritium-rich groundwater. To compensate for higher dispersion in the direction of flow, a smaller T must be used in order to match the observed profiles. Notice that the sets of parameter values have different effects on the simulated profiles in the three wells. For example, for ( L, T ) (1 m, m) the simulated peak concentration in T1 is the lowest of the peaks in three simulated profiles, while in T10 it is the second highest simulated peak concentration. The opposite is the case for ( L, T ) (10 m, 0 m). An explanation for this behavior is that since flow is more horizontally close to T1 than at T10, and because tritium is an areal source, vertical dispersion will be more important at T1 than at T10. From the dispersion analysis it is not possible to identify a unique set of parameter values for the investigated transport problem. However, it is important to realize that L and T are closely interdependent. Other sets of L and T values within the range suggested by simulations 4, 7, and 8 may result in simulations that compare just as well to the observations. Discussion and Conclusions The main objective of this study was to investigate field-scale dispersion in the Rabis Creek aquifer, resulting in estimates for longitudinal and vertical transverse dispersivity parameters. This has been done by tracing the movement of environmental tritium through the subsurface, consisting of a moderately

11 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER 3263 Figure 12. Same as Figure 11, except for simulation 4 with (1 m, m), simulation 5 with (10 m, m), and simulation 6 with (20 m, m). deep unsaturated zone and a relatively long cross section of the aquifer. The analysis did not result in a unique set of longitudinal and transverse vertical dispersivity values ( L, T ) and a number of parameter combinations such as (10 m, 0 m), (5 m, m), and (1 m, m) can describe the observed profiles equally well. From Figure 13 it can be seen that most of the tritium profiles are found in a vertical interval between 25 and 35 m of elevation. By inspecting the simulated streamlines in Figure 9 the approximate travel distances to T1, T2, and T10 can be estimated to m, m, and m, respectively. In the following we will use 1000 m as a mean travel distance from the water table to any of the observation wells. The sets of dispersivities thus represent the dispersion mechanisms on a scale of 1000 m. In view of the long transport distances and the rather homogeneous nature of deposit I, the values represent asymptotic values. Figure 13. Same as Figure 11, except for simulation 4 with (1 m, m), simulation 7 with (10 m, 0.0 m), and simulation 8 with (5 m, m).

12 3264 ENGESGAARD ET AL.: LARGE-SCALE DISPERSION IN A SANDY AQUIFER Figure 14. Longitudinal dispersivity values versus scale. Large and small circles identify data of high and intermediate reliability, respectively. All these data are from Gelhar et al. [1992]. Vertical solid line connecting triangles shows range of values reported in this study. To put this work into a broader context, it is of value to relate it to the study by Gelhar et al. [1992], who compiled dispersivity observations from 59 field sites. The studies involved a suite of different test conditions ranging from controlled tracer test to uncontrolled contamination events. Because of the nature of these studies, the reliability of the reported dispersivities were classified as being of high, intermediate, or low reliability. On the basis of all three classifications of data, Gelhar et al. [1992] argued that the longitudinal dispersivity could increase indefinitely with scale in contradiction to the stochastic dispersion theory. However, if the reliability of the data was taken into account, there was not supportive evidence of such a behavior, primarily because of the lack of values at scales larger than 300 m. A plot of the data of intermediate and high reliability (for unfractured media only) is shown in Figure 14. We have also plotted the range of longitudinal dispersivity values determined in this study (1 10 m) at a scale of 1000 m. If our analysis has produced a range of values that can be considered as being of intermediate reliability, then there may be a stronger argument for saying that the longitudinal dispersivity does not increase indefinitely with scale, but that it seems to reach an asymptotic value. Our study has confirmed that the vertical transverse dispersion is very low, of the order of m in terms of transverse vertical dispersivity. If it is assumed that the vertical transverse dispersivity is greater than zero, then the range of m is representative for the scale of 1000 m at Rabis Creek. Gelhar et al. [1992] give 2 values of the transverse dispersivity which are of high reliability, approximately T at a scale of m and 2 values of intermediate reliability, approximately T 0.06 at a scale of m. A plot of the transverse vertical dispersivity values found by Gelhar et al. [1992] and the range of values determined here are shown in Figure 15. Our values are close to those of high reliability indicating that even at a scale an order of magnitude larger, the vertical transverse dispersivity values are still low, reinforcing the concept of limited vertical mixing at very large scales. A low transverse dispersivity value, m at a scale of 200 m, was also found at another Danish location [Jensen et al., 1993]. We do not believe our parameter estimates can be considered as being of high reliability. The reasons for this are (1) the input of tritium to the groundwater is not known accurately, that is, it has not been measured, (2) at the scale of the cross section (3400 m) the hydrogeology is not known in sufficient detail, (3) we have only a small data set available for the analysis, and (4) we are using numerical models that inherently will carry some numerical errors in the simulations. Gelhar et al. [1992] defined sets of criteria for the high-, intermediate-, and low-reliability values. We will not review these criteria but just mention that some of our listed concerns are included in their definitions. Gelhar et al. [1992] classified the longitudinal dispersivities obtained through studies of environmental tritium in groundwater as being of low reliability, mainly because the tracer input was not clearly defined. The input of tracer was not measured and this uncertainty will produce uncertainty about the estimates of dispersion parameters. In our study we also do not have a measured tracer input to the groundwater. However, the tritium input recharge function that we have used is the result of an analysis of the movement of tritium through the unsaturated zone and we have been able to favorably compare our simulations with actual measurements of the content of tritium. The input of tritium to the groundwater zone is therefore known with more certainty. Another concern to be raised about our analysis is how well the model of the hydrogeology describes the actual aquifer characteristics in order to be able to quantify the dispersion parameters. Certainly, in this study we have only considered the macroscale geology, for example, the paleovalley structure shown in Figure 2, despite the many investigations that have been carried out at the field site. In general, however, the greater the scale of observation, the more difficult it will be to map the hydrogeology in detail, and the less feasible it will be to conduct controlled tracer tests [Gelhar et al., 1992]. At the Rabis Creek site the macroscale geology is not known in all details, and we have not been able to identify deposit-specific local heterogeneity, but the scale of tritium observation is so large that all heterogeneity has been sampled by the tritium plume so that our results should represent asymptotic values. The sensitivity of the simulation results to the longitudinal Figure 15. Transverse vertical dispersivity values versus scale. Large and small circles identify data of high and intermediate reliability, respectively. All these data are from Gelhar et al. [1992]. Vertical solid line connecting triangles shows range of values reported in this study.