Stochastic Analysis of Flux and Head Moments in a Heterogeneous Aquifer System
Evan K. Paleologos and Theofilos S. Sarris Dept. of Environmental Engineering, Technical University of Crete, Chania, Crete, 73100 Greece
Contact Author: Evan K. Paleologos Dept. of Environmental Engineering Technical University of Crete Chania, Crete, 73100 Greece Tel: 302-8210-37771 E-mail:
[email protected]
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Abstract This study investigates the behavior of flux and head in a strongly heterogeneous threedimensional aquifer system. The analyses relied on data from 520 slug tests together with 38,000 one-foot core intervals lithological data from the site of the General Separations Area in central Savannah River Site, South Carolina, USA. The skewness in the hydraulic conductivity histograms supported the geologic information for the top two aquifers, but revealed stronger clay content, than was reported for the bottom aquifer. The log-normal distribution model described adequately the hydraulic conductivity measurements for all three aquifers although, other distributions described equally well the bottom aquifer measurements. No apparent anisotropy on the horizontal plane was found for the three aquifers, but ratios of horizontal to vertical correlation lengths between 33 and 75 indicated a strong stratification at the site.
Three-dimensional Monte Carlo stochastic simulations utilized a grid with larger elements than the support volume of measurements, but of sub-REV (representative elementary volume) dimensions. This necessitated, on one hand, the use of upscaled hydraulic conductivity expressions, but on the other hand did not allow for the use of anisotropic effective hydraulic conductivity expressions (Sarris and Paleologos, 2004). Flux mean and standard deviations components were evaluated on three vertical crosssections. The mean and variance of the horizontal flux component normal to a no-flow boundary tended to zero at approximately two to three integral scales from that boundary. Close to a prescribed head boundary both the mean and variance of the horizontal flux component normal to the boundary increased from a stable value attained at a distance of about five integral scales from that boundary. The velocity field was found to be mildly anisotropic in the top two aquifers, becoming highly anisotropic in the bottom aquifer; was anisotropic in all three aquifers with directions of high continuity normal to those of the field; finally, was highly anisotropic in all three aquifers, with higher continuity along the east-west direction. The mean head field was found to be continuous, despite the high heterogeneity of the underlying hydraulic
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conductivity field. Directions of high continuity were in alignment with field boundaries and mean flow direction.
Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of the horizontal flux components by as much as fifty percent in the bottom aquifer. Variability in the head cross-sections was affected only marginally, with an average 10% reduction in the respective standard deviation. Finally, the location of the conditioning data did not appear to have a significant effect on the surrounding area, with uniform reduction in standard deviations.
Keywords: Stochastic modeling, Flux moments, Upscaling, Boundary effects
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Introduction One of the most challenging issues in predicting the flow of water and the movement of contaminants in the subsurface environment is accounting for the spatial variability of the parameters that characterize the physical properties of natural porous media. The situation is accentuated by the few field measurements that are usually available, their spatial variability, and the reliability and variable support volume of different measurement techniques which introduce stochastic and scaling issues in the mathematical and numerical description of the problem. This hydrogeologic properties‟ variation creates significant fluctuations in the velocity field with which contaminants are transported making the prediction of a plume‟s location and movement challenging. Interest in the velocity field is motivated by its dominant role in the hydrodynamic dispersion of contaminants in the subsurface environment (Gelhar and Axness, 1983).
In this paper the issue that is addressed initially is the statistical and geostatistical characterization, and of the upscaling of hydraulic conductivity data of a multi-layered aquifer system located in the central Savannah River Site, in South Carolina. Hydraulic conductivity was considered a random scalar quantity and through the use of Monte Carlo stochastic simulations the geostatistical characteristics of the mean flux and head fields were investigated. Of interest to our study was the effect of the no-flow and prescribed head boundaries on the mean and standard deviation of the flux field. Previous studies had shown the existence of a transition zone until a limiting behavior at the boundary was reached. Naff et al. (1998) had shown that under uniform mean gradient conditions the variance of the flux component normal to a no-flow boundary departed from zero within a distance of two to three integral scales from the boundary. Despite the conditioning effect of constant head boundaries the high underlying variability of the hydraulic conductivity field was shown to result in great variability of the corresponding flux term (Rubin and Dagan, 1988). Desbarats and Srivastava (1991) had shown that the spatial distribution of head fluctuations in a single, two-dimensional, heterogeneous field was not greatly affected by the underlying hydraulic conductivity field variations. Naff and Vecchia (1986) had developed analytical expressions for the
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head covariance in three-dimensional, anisotropic media, bounded by constant head and no-flow boundary conditions.
Finally, stochastic Monte Carlo conditional simulations investigated the extent to which the use of conditional data can reduce the uncertainty in the flux and head fields, through a reduction of their standard deviations. Gómez-Hernández and Gorelick (1989) had found that in a two-dimensional simulated field the standard deviation of the head can be reduced by as much as one order of magnitude.
2.
Description of Study Area and Numerical Model
The Savannah River Site (SRS) is located in the state of South Carolina covering an area of 350 mi2 on the Aiken Plateau of the United States Atlantic Coastal Plain, with an average elevation of 300 feet above mean sea level. The site was constructed during the early 1950s as a controlled area for nuclear material production, primarily of tritium and plutonium-239. Five reactors were constructed at the site together with support facilities, which included two chemical separation plants, a heavy water extraction plant, nuclear fuel and target fabrication units, and waste management facilities. Parts of SRS were contaminated by, in general low-level, radioactive waste that included 35 million gallons of radioactive material buried in waste tanks, transuranic waste, mixed waste with hazardous and radioactive components, and sanitary waste. Ongoing environmental restoration efforts include closure of one third of the waste units, remediation of 3 billion gallons of groundwater, and removal of 500,000 pounds of solvents. The General Separations Area (GSA) covers approximately 15 mi2 and is situated in the central part of SRS. Operations taking place in the area are chemical separations, tritium processing, and receipt of offsite fuel for processing. Contamination plumes originated from two waste tank farms at GSA containing about fifty large underground storage tanks with high-level radioactive waste that leaked into the aquifer system. Plumes ended up discharging radionuclides, metals, and nitrates into a nearby creek, the Fourmile creek. The GSA has low to moderate topographic relief and is drained by several rivers. It is
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partly bounded by three river branches: the Upper Three Runs (UTR), located at the northern boundary, the Fourmile Branch, which forms the southern boundary, and the McQueen Branch on the northeastern boundary. The aquifers that are present in the GSA (Figure 1) are the upper and lower zones of the Upper Three Runs Aquifer (UTRA), separated by the Tan Clay confining unit, and the Gordon aquifer, which is confined at its top by the Gordon confining unit and at its base by the Meyers Branch confining unit (Aadland et al., 1995). Monitoring wells provide information about the hydraulic head in all three aquifers.
The upper zone of the unconfined UTRA is characterized by very dense clayey sediment with some gravelly sands (Smits et al., 1997). This zone consists, primarily, of sand and clayey sand with minor intercalated clay layers and it is geologically similar to the lower aquifer zone, which is also composed of sand and clayey sand. The composition of the intermediate „tan clay‟ confining zone is light yellowish to orange clay and sandy clay, inter-bedded with clayey sand and sand. This layer is dispersed both vertically and horizontally and it is not continuous over large areas. The Gordon confining unit usually referred to as „green clay,‟ separates the lower zone of the UTRA from the Gordon aquifer and consists of layers of inter-bedded silty and clayey sand, sandy clay, and clay. The lowermost unit, the Gordon aquifer, constitutes the basal unit of the Floridan aquifer system. It contains loose sand and clayey sand. The sand within the aquifer is yellowish to grayish orange, sub-to-well-rounded, moderately to poorly sorted, and medium to coarse grained. Thin clay layers have been found close to the base of the Gordon aquifer. The study area was discretized in a 129×101×19 grid. Grid spacing was constant in the horizontal directions and equaled 200ft. Vertical spacing varied with location, with 19 elements used always in the vertical direction, irrespective of depth. The entire top surface of the mesh was assigned a recharge and/or drain boundary condition. Based on field studies in the central SRS (Hubbard, 1984, 1985; Parizek and Root, 1986) a constant infiltration rate of 15.5 inches/year (based on the 1979-84 precipitation record) was assigned in our model. A more recent study (Paleologos et al., 2005) that analyzed the precipitation record of the GSA area for the last 112 years has concluded that deviations
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in the infiltration rate (such as those from drought conditions in the south-eastern US during the years of 1999-2001) has not influenced the hydraulic head in the area. Additionally, at the GSA there exist a number of artificial surface facilities, which contribute to the recharge of the aquifer system (Flach and Harris, 1999). Their location and corresponding infiltration rates can be found in Sarris (2003). For the bottom boundary, which corresponds to the bottom of the Gordon aquifer, no flow conditions were assigned.
Vertical surfaces between the top and bottom boundaries of the model were assigned either no-flow, or prescribed head conditions, based on field studies and previous modeling efforts (Flach and Harris, 1999). No flow was assumed beneath rivers and streams, as well as along the northern boundary of the domain. The north boundary of the UTRA formation is outcropped before the Upper Three Runs river branch, and no flow conditions were assigned at the outcrop location. No flow conditions were also applied on the eastern and southern boundaries in UTRA. Prescribed head boundaries were assigned along the south and east boundaries of the Gordon aquifer, as well as along the whole west boundary (Figure 2).
3.
Geostatistical Analysis of Hydraulic Conductivity
Field measurements of the hydraulic conductivity had been obtained in various locations of the study area using different methods. Approximately 520 slug tests, 120 pumping tests (single- and multiple-well), and 260 laboratory permeability tests were available for all aquifer units. From those only the number of slug test measurements was adequate to provide reliable geostatistical parameters, and these were used in this study. Other reasons for not analyzing the remaining data were: scaling issues, errors in pumping test results, and the documented bias of the laboratory permeability data, which were, mainly, collected in low permeability areas (Flach and Harris, 1999).
Nearly 38,000 data from one-foot core intervals (Smits et al., 1997) related to lithological characteristics of the units were utilized to supplement the slug test data. This was done
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because only coordinates and screening depths had been collected for the slug tests and in order to perform the geostatistical analyses the slug test measurements had to be matched with the corresponding hydrostratigraphic units. Slug test measurements where no correlation with lithological information could be established were discarded, as they could not be used for meaningful geostatistical analyses.
Some slug tests at the GSA had been conducted at water quality monitoring wells that had been drilled with mud rotary techniques resulting in the development of a low permeability “mud skin” on the wall of the borehole (Sadler, 1995). This “skin effect” of slug tests, where the permeability of a formation near a borehole decreases as a result of poor drilling and completion practices is well documented (Faust and Mercer, 1984). On the other hand, Peres et al. (1989) found that the skin effect for a number of wells was negative and the measured hydraulic conductivities appeared to be higher than those of the real medium. Based on these findings and the lack of information regarding the skin effect for the GSA slug tests our analysis proceeded with the assumption that errors in these tests were random.
The experimental histograms for the hydraulic conductivity of each aquifer were tested against several probability density function models, which consisted of the normal, lognormal, exponential, Weibull, gamma, extreme value, logistic, Pareto, and beta models. Three statistical tests tested the fit of each probability model to the experimental histograms; these consisted of the Kolmogorov-Smirnov, the Anderson-Darling, and the Chi-Square tests. The Kolmogorov-Smirnov and the Anderson-Darling tests use a maximum vertical difference criterion to determine the best fit between empirical and modeled distributions, with the former emphasizing the mid-range and the latter the tails of a distribution. The Chi-Square uses the sum of squared differences between empirical and model distributions to determine the best fit model.
Figure 3 plots the experimental histograms of the slug test data for each aquifer, with the solid line indicating the best-fit probability density function. All experimental histograms were highly skewed to the left indicating a preference to lower values. For the upper and
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lower UTRA aquifer zones, this behavior is in accordance with the geologic information (which indicated a strong presence of clay), but, it is, perhaps, surprising for the Gordon aquifer, indicating either a stronger clay content than is usually reported, or an influence of the “skin effect” in this aquifer that resulted to a bias of the conductivity values.
In the upper and lower UTRA zones the stochastic literature stipulation of lognormal distribution of hydraulic conductivity (for example, Desbarats, 1988; Dagan, 1989; Gomez-Hernandez and Gorelick, 1989; Paleologos et al., 2000; Sarris and Paleologos, 2004) was found to fit the field data the best. In the Gordon aquifer the KolmogorovSmirnov and the Anderson-Darling tests favored both the beta and lognormal distributions, whereas the Chi-Square test favored the gamma or Weibull models over the lognormal distribution. Although, previous studies have also found other distributions, than the lognormal, that fit hydraulic conductivity data (Smith, 1981; Woodbury and Sudicky, 1991) the inadequate documentation of the tests, and the poorly controlled testing practices at the GSA site does not give us enough confidence to depart from the log-normality assumption for this aquifer.
Figure 4 shows the experimental variograms of Y(x) = ln K(x) for the UTRA upper zone, the UTRA lower zone, and the Gordon aquifer in three directions (east-west; north-south; and vertical), as well as the fitted exponential models. Experimental variograms γ(h) were calculated based on the expression: N(h)
γ(h) =
1 2 N(h)
(Y - Y i
i+h
)2
(1)
i=1
where N(h) is the number of pairs of data separated by a distance h. Other horizontal directions were also analyzed but no significant anisotropy in the horizontal plane was observed. For the horizontal variograms slicing widths and heights of 500ft and 70ft, respectively, were used together with a 45 degree tolerance angle. For the vertical variograms the corresponding length was 50ft with the tolerance angle remaining the same. For each aquifer exponential anisotropic variogram models were calculated with λi the integral scale in the ith direction, and the variogram sill cY obtained iteratively as the quantity that minimizes the sum of squared differences between experimental and model
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values. The models did not account for nugget effect since no information was available concerning repeated testing at the same location. Table 1 lists the parameters of each exponential model for each aquifer, and the large difference in horizontal and vertical integral scales indicates a strong stratification at the site.
4.
Upscaling of Hydraulic Conductivity
Numerical simulations were performed on a grid with larger elements than the support volume of measurements, necessitating the use of upscaled hydraulic conductivity expressions (Paleologos et al., 1996; Renard and de Marsily, 1997; Harter, 2000; Zhang, 2002). The numerical grid has dimensions of 200×200 ft in the North-South and EastWest directions in all layers and an average depth of 9.1ft for the two zones of UTRA and 27.8ft in the Gordon aquifer. Several authors (for example, Rubin and Dagan, 1988; Paleologos et. al, 1996; Paleologos and Sarris, 2000) have found that numerical elements need to have dimensions of, at least, four integral scales if the hydraulic conductivity is to be treated as a deterministic parameter at the numerical element level. This characteristic length has been argued as necessary in order to define an REV (Representative Elementary Volume) with respect to the hydraulic conductivity (in which case the hydraulic conductivity can equal an effective value in a grid cell). The fact that our numerical elements are of sub-REV scale - horizontal dimensions of 200ft versus integral scales of 630ft, 755ft, and 873ft for the UTRA upper and lower zones, and the Gordon aquifer, respectively; and similarly for the vertical direction - makes inapplicable the use of anisotropic aquifer effective hydraulic conductivity expressions (Sarris and Paleologos, 2004), and requires the treatment of the hydraulic conductivity at this discretization level as a stochastic process and the extraction of its statistical structure at this level from that of the measurement scale.
The statistical structure of the upscaled hydraulic conductivity of a typical numerical element of our grid was determined via the Sequential Gaussian Simulation (Journel and Gomez-Hernandez, 1989). Generation of synthetic hydraulic conductivity point fields required the use of variogram sills (given in Table 1), as well as the statistical anisotropy
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λH /λV – horizontal to vertical integral scale for each aquifer - and the dimensionless length L/λ – ratio of element dimension to correlation length – for both horizontal and vertical directions for the three aquifers, all of which were calculated from the entries of Table 1 and the element dimensions. For each aquifer element through 500 realizations the equivalent horizontal and vertical hydraulic conductivities were calculated via the expression Keq = qL/H, where Keq was the (calculated) equivalent hydraulic conductivity of each element‟s simulated heterogeneous hydraulic conductivity field, q was the total flux through the element, and H the head difference of two boundaries separated by a distance L. Because of the large difference between horizontal and vertical length of an element our model simulated flow through an aquitard with no-flow conditions at infinity and Dirichlet conditions at the top boundaries. In order to provide for both the horizontal and vertical components of the equivalent hydraulic conductivity the flow direction was alternated in our simulations. Table 2 summarizes the results of these simulations for the three aquifers by providing the mean of the equivalent hydraulic conductivities, , in the horizontal and vertical directions, as well as the ratio of these two values, the physical anisotropy. This ratio was later used in our model in order to retrieve the anisotropic behavior of the aquifer system.
The hydraulic conductivity variograms were upscaled following Journel and Huijbregts (1978). Under conditions of stationarity the upscaled variogram is given by these authors as: γ V (h) = γ(v,vh ) - γ(v,v)
(2)
where vh denotes the support volume translated from v by a vector h, γ(v,vh ) is the mean value of the point variograms γ(u) when u varies randomly between v and vh. When h>>v equation 2 can be approximated by: γ V (h) γ(h) - γ(v,v)
(3)
where γ(h) is the point variogram and γ(v,v) can be evaluated numerically for various geometries and point variograms models. The calculated upscaled sills cY are shown in the third column of Table 3.
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Several authors (for example, Fenton and Griffiths, 1993; Sarris and Paleologos, 2004) have shown that block conductivities of log-normally distributed point measurements are also log-normally distributed. Designating as μK the mean (given in Table 2), and σK2 the variance of the horizontal (block) equivalent hydraulic conductivities KH, the mean μY, and variance σY2 (given in the third column of Table 3), of the transformed variable YV=ln (KH(x)) can be calculated through expressions (Law and Kelton, 1982):
(μ 2 +σ 2 ) μ 2K and σ Y2 =ln K 2 K μ Y =ln (μ 2K +σ K2 ) μK
(4)
In the system above the values of μ K and σ 2Y are known and they allow the calculation of the mean value of the logarithm of the horizontal hydraulic conductivity that corresponds to the grid element (second column of Table 3).
5.
Unconditional Simulations
The components of the flux vector were calculated using Darcy‟s law and with the hydraulic conductivity of a node equaling the harmonic mean of the hydraulic conductivities of the two adjacent blocks in the direction considered. According to our seven-point, block-centered finite difference scheme flux terms were calculated at the boundary between the nodes and not at the nodes where the head solution was obtained.
Figure 2 shows a plan view of the study area, the boundary conditions, and the locations of three sections, AA, BB, and CC, where the mean components and standard deviations of the flux were evaluated. For each vertical cross-section flux calculations were performed along the row of elements that form the midline of each aquifer and were based on 350 simulations. The number of aquifer units present in AA, BB, and CC varied. Thus, in all sections the upper UTRA was absent, the lower UTRA was highly discontinuous along AA, and the Gordon aquifer existed in all three sections. In the subsequent figures the right-hand side of each graph corresponds to the eastern and the left-hand side to the western boundary, respectively. On the vertical axis index x corresponds to the East–West, and y corresponds to the North–South direction, respectively. Our results for the vertical component of flux are omitted since it was found 12
to be negligible (of the order of 10-5 ft/day or 1mm/year) both for the UTRA and the Gordon aquifer. The absence of vertical flow makes is reasonable considering the existence of confining units at the top of these two aquifers and the fact that flux calculations took place at a significant distance from the confining units, and hence where horizontal flow is expected to have been established.
Figure 5 plots the x- and y-flux terms for the lower UTRA and sections BB and CC. The x-direction is normal to the eastern and western aquifer boundaries. For the lower UTRA and for section BB the eastern boundary is a no-flow boundary and the western is a constant head boundary. For section CC both eastern and western boundaries are no-flow boundaries. For both sections and σqx tend to zero as they approach the eastern noflow boundary, and in addition for CC and σqx tend to zero as they approach the western no-flow boundary. The same behavior appears to hold for the y-direction along section BB, but along cross-section CC appears to be stable with σqy increasing toward the eastern boundary. Naff et al. (1998) had shown that under uniform mean gradient conditions the variance of the flux component normal to a no-flow boundary departed from zero within a distance of two to three integral scales. For the lower UTRA this translates to developing a non-zero variance at a distance of about 1500ft to 2200 ft from the eastern no-flow boundary, which agrees with the results shown in Figure 5.
The effect of prescribed head boundaries is shown in Figure 6 for the Gordon aquifer. The eastern and western boundaries here are prescribed head boundaries. Despite the conditioning effect of the constant head boundaries (Rubin and Dagan, 1988) the high underlying variability of the hydraulic conductivity field, results in great variability of the corresponding flux term. For all three sections both and σqx increase towards the eastern boundary, while maintaining a stable value in the mid-section. The transition distance is about 4 to 6 integral scales, which compares well with the theoretical results by Rubin and Dagan (1988). In the direction parallel to the boundary there is no apparent trend, since σqy maintains a stable value. It should be noted that in this study the prescribed head boundaries were not set at constant value, as theoretical studies consider.
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The linearly-varying boundary values can thus contribute to flow in directions parallel to the boundary.
With respect to the conditioning effect of the boundaries on the variability of our numerical experiments showed that there appears a transition zone extending approximately 2-3 integral scales of the log-K field over which the standard deviation of decreases to zero towards the prescribed head boundaries (Rubin and Dagan, 1988). The spatial structure of the mean head field is shown in the covariance maps of Figure 7. The covariance of was calculated for east-west and north-south lags at 400 ft intervals. The value σ(0,0) is shown at the center of the plot, with the value σ(hx,hy) shown at offset (hx,hy) from the center. This representation provides a very concise visualization of spatial structure, where directions of anisotropy become easily evident.
The covariance maps are normalized by the variance of on the considered plane. The field is remarkably smooth considering the rather large degree of heterogeneity of the hydraulic conductivity field for all three aquifers. Figure 7 shows the field of the upper zone of UTRA to be almost isotropic, with greater spatial continuity along the north-south direction. For the other two formations the field becomes anisotropic with maximum spatial continuity along directions N68oW for the lower zone of UTRA, and N52oW for the Gordon aquifer, respectively. These two directions coincide approximately with the direction of the north and south boundaries of the GSA area, and with the mean direction of the equipotentials. Desbarats and Srivastava (1991) reached a similar conclusion considering the spatial distribution of head fluctuations of a single, two-dimensional, heterogeneous field. Naff and Vecchia (1986) developed analytical expressions for the head covariance in three-dimensional, anisotropic media, bounded by constant head and no-flow boundary conditions. The head covariance in the direction of mean flow, compares extremely well with the covariance in the direction of minimum continuity in Figure 7 for all three aquifers.
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The distributions of , and are in most cases rather symmetric with the exception of , which is bounded by zero, indicating an almost exclusively downwards vertical flow, forced by the high infiltration and recharge rates from the top boundary. The spatial continuity of and was also examined with the corresponding covariance maps available in Sarris (2003). was mildly anisotropic in both zones of UTRA, becoming highly anisotropic in the Gordon aquifer where it exhibited greater continuity along the north-south direction. was anisotropic in all three aquifers with directions of higher continuity approximately normal to those of the field. Further investigation of the , cross-covariance did not reveal any spatial relation other than that the two fields were positively correlated. Finally, appeared to be highly anisotropic in all three aquifers, with higher continuity displayed along the east-west direction. Additionally, it was highly discontinuous with its covariance decreasing to zero in very short lags. The result was not surprising considering that continuity was examined over the horizontal plane, while qz corresponded to vertical flow.
6.
Conditional Simulations
350 conditional simulations were performed, with conditioning initially on the point scale -or measurement scale –, followed by upscaling on the conditioned fields. Our procedure followed that by Gómez-Hernández and Gorelick (1989), and Gómez-Hernández (1991) for conditioning block-simulated fields to point measurements in two-dimensional fields. These authors had generated random conditional fields at the location of conditioning data; assigned the geometric mean of the conditional random field to the block, and then used this block value as conditioning data for the simulation of the block field.
The main flow patterns remained largely invariant with consideration of the conditioning data. Figure 8 plots the mean and standard deviation of the flux vector components along the three cross-sections for the Gordon aquifer. Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect 15
on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of flux (both for qx and qy) by approximately 30% in the lower zone of UTRA and by approximately 50% in the Gordon aquifer. Variability in the head crosssections was affected only marginally, with an average 10% reduction in the respective standard deviation.
Gómez-Hernández and Gorelick (1989) found that in a two-
dimensional simulated field the standard deviation of head can be reduced by as much as one order of magnitude, a result which was not supported by our study. An interesting observation from our study was that the location of the conditioning data did not appear to influence the level of reduction in the flux standard deviation. Even though the conditioning data were clustered towards the center of the field the reduction of the conditional standard deviation there was not noticeably greater than that of areas further away.
The statistical structure of the mean head field remained largely insensitive to the conditioning procedure. The covariance maps showed identical directions of continuity, with slightly higher covariances than in the unconditional case (Sarris, 2003). This greater spatial continuity of the conditional case can be attributed to the lower variability of the average solution obtained by conditional simulations.
7.
Conclusions
This study investigates the behavior of flux and head in a strongly heterogeneous threedimensional aquifer system. Data were obtained from the General Separations Area, situated in the central Savannah River Site, South Carolina. Uncertainty was associated with hydraulic conductivity, with boundary conditions described deterministically based on field studies, previous modeling efforts and field measurements. Approximately 520 hydraulic conductivity measurements were used to describe the statistical and spatial structure of the three aquifers that are found in the area. The skewness in the hydraulic conductivity histograms supported the geologic information for the top two aquifers, but revealed stronger clay content, than was reported for the bottom aquifer. The log-normal 16
distribution model described adequately the hydraulic conductivity measurements for all three aquifers although, other distributions described equally well the bottom aquifer measurements (Smith, 1981; Woodberry and Sudicky, 1991). A geostatistical analysis indicated that an anisotropic exponential model could describe the spatial continuity of the three hydraulic conductivity fields. No apparent anisotropy on the horizontal plane was found for the three aquifers, but ratios of horizontal to vertical correlation lengths between 33 and 75 indicated a strong stratification at the site.
Monte Carlo simulations investigated the geostatistical characteristics of the mean flux and head fields and standard deviations. The numerical grid consisted of larger elements than the support volume of measurements, but of sub-REV (representative elementary volume) dimensions. This necessitated, on one hand, the use of upscaled hydraulic conductivity expressions, but on the other hand did not allow for the use of anisotropic effective hydraulic conductivity expressions (Sarris and Paleologos, 2004). Flux mean and standard deviations components were evaluated on three vertical cross-sections. The effect of prescribed head and no-flow boundaries on the variability of the mean flux field was analyzed. The mean and variance of the horizontal flux component normal to a noflow boundary tended to zero at approximately two to three integral scales from that boundary. Close to a prescribed head boundary both the mean and variance of the horizontal flux component normal to the boundary increased from a stable value attained at a distance of about five integral scales from that boundary. The velocity field was found to be mildly anisotropic in the top two aquifers, becoming highly anisotropic in the bottom aquifer; was anisotropic in all three aquifers with directions of high continuity normal to those of the field; finally, was highly anisotropic in all three aquifers, with higher continuity along the east-west direction. The mean head field was found to be very continuous, despite the relatively high heterogeneity of the underlying hydraulic conductivity field.
The directions of higher continuity were
approximately in alignment with the direction of the field boundaries and as a consequence with the mean direction of flow.
Analytical expressions for the head
covariance of three-dimensional anisotropic media were generally supported by our numerical results.
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Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of the horizontal flux components by as much as fifty percent in the bottom aquifer. Variability in the head cross-sections was affected only marginally, with an average 10% reduction in the respective standard deviation. Finally, the location of the conditioning data did not appear to have a significant effect on the surrounding area, with uniform reduction in standard deviations. Even though the conditioning data were clustered towards the center of the field the reduction of the conditional standard deviation there was not noticeably greater than that of areas further away.
Acknowledgements The authors would like to acknowledge Dr. Mary Harris and Dr. Greg Flach, Savannah River Site National Lab, for their helpful suggestions during the modeling phase and for providing us with the data for this work.
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References Aadland, R. K., Gellici, J. A., and Thayer, P. A.: 1995, Hydrogeologic Framework of West-Central South Carolina, South Carolina Department of Natural Resources, Water Resources Division Report 5, pp. 200 + 47 plates. Dagan, G.: 1989, Flow and Transport in Porous Formations, Springer-Verlag, New York, pp. 465. Desbarats, A. J.: 1988, Estimation of effective permeabilities in the lower Stevens formation of the Paloma field, San Juaquin Valley, California, SPE Reservoir Eng., Nov., 1301-1307. Desbarats, A.J. and Srivastava, R. M.: 1991, Geostatistical characterization of groundwater flow parameters in a simulated aquifer, Water Resour. Res, 27(5), 687-698. Faust, C.R., and Mercer, J.W.: 1984, Evaluation of slug tests in wells containing a finite thickness skin, Water Resour. Res, 20(4), 504-506. Flach, G. P., and Harris, M. K.: 1999, Integrated Hydrogeological Modeling of the General Separations Area, Volume 2: Groundwater Flow Model, Westinghouse Savannah River Site Report: WSRC-TR-96-0399. Fenton, G. A., and Griffiths, D. V.: 1993, Statistics of block conductivity through a simple bounded stochastic medium, Water Resour. Res, 29(6), 1825-1830. Gelhar, L. W., and Axness, C. L.: 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res., 19(l), 161-180. Gómez-Hernández, J. J., and Gorelick, S. M.: 1989, Effective groundwater model parameter values: Influence of spatial variability of hydraulic conductivity, leakance, and recharge, Water Resour. Res., 25(3), 405-419. Gómez-Hernández, J. J.: 1991, A stochastic approach to the simulation of block conductivity fields conditioned upon data measured at a smaller scale, Ph.D. dissertation, Dep. Appl. Earth Sci., Stanford University, Stanford, CA. Harter, T.: 2000, Application of stochastic theory in groundwater contamination risk analysis: Suggestions for the consulting geologist and engineer, in D. Zhang and L. Winter (eds.), "Theory, Modeling and Field Investigation in Hydrogeology: A
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Special Volume in Honor of Shlomo P. Neuman's 60th Birthday," GSA Special Volume, pp. 43-52. Hubbard, J.E.: 1984, Water Budget for the SRP Burial Ground Area, DPST-83-742 Report. Hubbard, J.E.: 1985, An Update on the SRP Burial Ground Area Water Balance and Hydrology, DPST-85-958 Report. Journel, A. G., and Huijbregts, Ch. J.: 1978, Mining Geostatistics, Academic Press, London, UK. Journel, A.G., and Gomez-Hernandez, J.J.: 1989, Stochastic imaging of the Wilmington clastic sequence, 64th Annual Technical Conf. and Exhibition of the Soc. of Petroleum Engineers, San Antonio, TX, October 8-11, SPE 19857, p. 591-606. Law, A. M., and Kelton, W. D.: 1982, Simulation Modeling and Analysis, McGraw-Hill, New York. Naff, R. L., and Vecchia, A. V.: 1986, Stochastic analysis of three-dimensional flow in a bounded domain, Water Resour. Res., 22(5), 695-704. Naff, R. L., Haley, D. F., and Sudicky, E. A.: 1998, High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media, 1. Methodology and flow results, Water Resour. Res., 34(4), 663-677. Paleologos, E. K., Sarris, T., and Tolika, M.: 2005, Integration of an analytic element model in a stochastic analysis of infiltration into a complex unconfined aquifer system, Jour. of Hydroinformatics, 7, 53-59. Paleologos, E. K., Sarris T., and Desbarats, A.: 2000, Numerical estimation of effective hydraulic conductivity in leaky heterogeneous aquitards, in D. Zhang and L. Winter (eds.), Theory, Modeling and Field Investigation in Hydrogeology: A Special Volume in Honor of Shlomo P. Neuman's 60th Birthday, GSA Special Volume, pp. 119-127. Paleologos, E.K., and Sarris, T.: 2000, Upscaling of hydraulic conductivity in threedimensional, heterogeneous, bounded formations, in Bentley et al. (eds.), XIII Computational Methods in Water Resources, Balkema Pub., Rotterdam, 2, pp. 743-747.
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Paleologos, E.K., Neuman, S. P., and Tartakovsky, D.: 1996, Effective hydraulic conductivity of bounded, strongly heterogeneous porous media, Water Resour. Res., 32(5), 1333-1341. Parizek, R.R. and Root, R.W. Jr.: 1986, Development of a ground-water velocity model fro the Radioactive Waste Management Facility, Savannah River Plant, South Carolina, DPST-86-658 Report. Peres, A.M.M., Onur, M., and Reynolds, A.C.: 1989, A new analysis procedure for determining aquifer properties from slug test data, Water Resour. Res., 25(7), 1591-1602. Renard, Ph., and Marsily, G. de: 1997: Calculating equivalent permeability: A review, Advances in Water Resour., 20, Nos 5-6, 253-278. Rubin, Y. and Dagan, G.: 1988, Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers, 1, Constant head boundary, Water Resour. Res., 24(10), 1689-1697. Sadler, W. R.: 1995, Groundwater Model Calibration and Remediation Well Network Design at the F-area Seepage Basins, Westinghouse Savannah River Site Report: WSRC-RP-95-237, Rev. 0. Sarris T. S.: 2003, Stochastic analysis of flow in three-dimensional, heterogeneous, anisotropic aquifers: Synthetic simulations and case study, Ph.D. Dissertation, Dept. of Geol. Sc. Univ. of South Carolina, Columbia, SC, pp. 173. Sarris, T. S., and Paleologos, E. K.: 2004, Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media, Jour. of Stoch. Environ. Res. & Risk Assess., 18,188-197. Smith, L.: 1981, Spatial variability of flow parameters in a stratified sand, Math. Geology, 13(1), 1-21. Smits, A. D., Harris, M. K., Hawkins, K. L., and Flach, G. P.: 1997, Integrated Hydrogeological Modeling of the General Separations Area, Volume 1, Hydrogeologic Framework (U), Westinghouse Savannah River Site Report: WSRC-TR-96-0399. Woodbury, A. D., and Sudicky, E. A.: 1991, The geostatistical characteristics of the Borden aquifer, Water Resour. Res, 27(4), 533-546.
21
Zhang, D.: 2002, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, California, pp. 349.
22
Figures
Figure 1. Idealized cross-section of the GSA (Flach and Harris, 1999). Figure 2. Location of river branches, boundary conditions, and the three cross- sections analyzed. Figure 3. Histograms and best-fit models for the hydraulic conductivity (in ft/day). Figure 4. Experimental variograms and models for the log-hydraulic conductivity. Figure 5. Cross-section of mean q and σq, in UTRA lower zone: Unconditional simulations. Figure 6. Cross section of mean q and σq, Gordon aquifer: Unconditional simulations Figure 7. Normalized mean head spatial covariance σh(hx,hy)/σ2h. Figure 8. East-West cross-section of mean q and σq, Gordon aquifer: Conditional simulations.
23
Figure 1. Idealized cross-section of the GSA (Flach and Harris, 1999).
24
A
B
B McQueen river
UTRA: prescribed head Gordon: prescribed head
A
Fourmile river
C
C
UTRA: no flow Gordon: prescribed head
UTRA: no flow Gordon: no flow
Upper Three Runs river
UTRA: no flow Gordon: prescribed head
Coordinates in feet
Figure 2. Location of river branches, boundary conditions, and the three cross-sections analyzed.
25
UTRA “upper”
zone
UTRA “lower” zone
Gordon
aquifer
Figure 3. Histograms and best-fit models for the hydraulic conductivity (in ft/day).
26
North-South direction
Vertical direction
Gordon Aquifer
UTRA lower zone
UTRA upper zone
East-West direction
Figure 4. Experimental variograms and models for the log-hydraulic conductivity.
27
0,04
CC
0,01
BB
0,00
CC
0,03
(ft/d)
(ft/d)
0,02
BB
-0,01 -0,02
CC
0,02 BB
0,01
CC
-0,03 -0,04 45000
50000
55000
60000
65000
0,00 45000
70000
50000
55000
60000
65000
70000
Easting (ft)
Easting (ft) 0,06
0,05 BB
0,04
0,05
CC
σqy (ft/d)
σqx (ft/d)
BB
0,03
BB
0,02 0,01
0,04 0,03 0,02 0,01
0,00 45000
CC 50000
55000
60000
65000
0,00 45000
70000
Easting (ft)
CC
BB
BB
CC 50000
55000
60000
Easting (ft)
Figure 5. Cross-section of mean q and σq, in UTRA lower zone: Unconditional simulations.
28
65000
70000
0,15
0,08
BB AA BB
0,05
CC
0,04
(ft/d)
(ft/d)
0,10
CC
AA
0,00
BB
-0,05
AA
0,00
AA
BB CC
-0,04
CC
-0,10 45000
50000
55000
60000
65000
-0,08 45000
70000
50000
55000
0,25
70000
0.12 AA
AA
AA
BB
0.10
BB
CC
0,15 0,10
σ qy (ft/d)
σqx (ft/d)
65000
Easting (ft)
Easting (ft)
0,20
60000
CC
0,05 0,00 45000
0.08
AA AA
CC
BB CC
0.06 0.04
BB
0.02 0.00 50000
55000
60000
65000
70000
45000
50000
55000
60000
65000
Easting (ft)
Easting (ft)
Figure 6. Cross section of mean q and σq, Gordon aquifer: Unconditional simulations.
29
70000
UTRA upper zone -2000
0
2000
UTRA lower
4000
6000
8000
-6000
-4000
-2000
0
2000
4000
6000
8000 .11
0.33
18 0.
6000
-0
-0.11
6000
-8000
6000
-4000
6000
zone -6000
-8000
0.1 0.4
0.3
-6000
-6000
0.48
0
2000
4000
6000
8000
0.33
-8000
-6000
-4000
East-West Lag (ft)
2000
-6000
-4000
-2000
0
2000
4000
6000
8000
0.18
6000
6000
3
0.48
4000
33 0.
GORDON Aquifer
-0 .1 1
18 0.
2000
0.6
2000
03 0.
48 0.
63 0.
0. 48
3
0.3
3
33
0
0.
78 0.
0.9
8
78 0.
-2000
3
48 0.
0.1
0. 48
63 0.
-4000
-4000
33 0.
03 0.
-6000
-0
-6000
0.48
1 .1 0.33
18 0.
-8000
4000
East-West Lag (ft)
0.3
North-South Lag (ft)
0
4000
-8000
-2000
0
-2000
-2000
-4000
-6000
-4000
-2000
2000
8
3
1
-6000
-2000
0
0.93
3
3
1 -0.
-8000
3
0.63
0.6
0.0
0
2000
4000
6000
8000
East-West Lag (ft)
Figure 7. Normalized mean head spatial covariance σh(hx,hy)/σ2h.
30
4000
4000 2000 0 0. 33
18 0.
-6000
0.3
8
-4000
-0.11
-4000
-2000
0.03
0.33
3
-0.11
0.18
0.4
3
0. 78 0. 78
0.33
-6000
3
-2000
0
0.63 0.48
0. 93
0.63
-2000
0.6
8
0.18
0. 48
0.3
-4000
8
0.0
0.4
0
0.7
0.18
North-South Lag (ft)
2000
0.3
3
0.03
2000
.48
0.48
-4000
0.18
4000
0
0.03
North-South Lag (ft)
0.03
4000
8
0.33
6000
8000
0,15
0.08
0,05
AA
0.06
BB
0.04
(ft/d)
(ft/d)
0,10
CC
0,00 -0,05
BB AA
-0,10 45000
AA
BB CC
0.02 AA
0.00
BB
-0.02
CC
CC
-0.04
50000
55000
60000
65000
45000
70000
50000
65000
70000
0,12
0,25 AA
0,20
0,10
0,15
σqy (ft/d)
BB
σqx (ft/d)
60000
Easting (ft)
Easting (ft)
CC
0,10
AA 0,05
55000
CC
0,00 45000
50000
55000
60000
0,06
65000
BB
CC
AA
BB
0,00 45000 BB 50000
70000
CC
0,04 0,02
BB
AA
0,08
Easting (ft)
55000
60000
65000
Easting (ft)
Figure 8. East-West cross-section of mean q and σq, Gordon aquifer: Conditional simulations.
31
70000
Tables
Table 1. Parameters of exponential variogram models. Table 2. Horizontal and vertical mean equivalent hydraulic conductivity. Table 3. Statistical properties of YV.
32
λi (ft)
cY
Aquifer
Horizontal
Vertical
UTRA upper zone
2.4568
630.84
13.35
UTRA lower zone
2.7458
755.67
10.01
Gordon Aquifer
2.7259
873.16
26.70
Table 1. Parameters of exponential variogram models.
(ft/d)
Aquifer
(ft/d)
KG (ft/d)
Physical Anisotropy
Horizontal
Vertical
UTRA upper zone
4.81
1.55
1.34
4.63
3.45
UTRA lower zone
2.99
0.88
1.68
3.05
1.82
Gordon Aquifer
8.27
1.32
1.85
5.97
3.23
Table 2. Horizontal and vertical mean equivalent hydraulic conductivity.
33
Aquifer
μY
cY
UTRA upper zone
0.602
UTRA lower zone Gordon Aquifer
λi (ft) Horizontal
Vertical
1.86
630.84
13.35
0.126
1.98
755.67
10.01
0.818
1.94
873.16
26.70
Table 3. Statistical properties of YV.
34
