The spatial variability of hydraulic conductivity at the site of a long-term tracer test performed in the Borden aquifer was examined in great detail by conducting permeability measurements on a series of cores taken along two cross sections, one along and the other transverse to the mean flow direction. Along the two cross sections, a regular-spaced grid of hydraulic conductivity data with 0.05 m vertical and 1.0 m horizontal spatial discretization revealed that the aquifer is comprised of numerous thin, discontinuous lenses of contrasting hydraulic conductivity. Estimation of the three-dimensional covariance structure of the aquifer from the log-transformed data indicates that an exponential covariance model with a variance equal to 0.29, an isotropic horizontal correlation length equal to about 2.8 m, and a vertical correlation length equal to 0.12 m is representative. A value for the longitudinal macrodispersivity calculated from these statistical parameters using three-dimensional stochastic transport theory developed by L. W. Gelhar and C. L. Axness (1983) is about 0.6 m. For the vertically averaged case, the two-dimensional theory developed by G. Dagan (1982Dagan ( , 1984 yields a longitudinal di. spersivity equal to 0.45 m. Use of the estimated statistical parameters describing the In (K) variability in Dagan's transient equations closely predicted the observed longitudinal and horizontal transverse spread of the tracer with time. Weak vertical and horizontal dispersion that is controlled essentially by local-scale dispersion was obtained from the analysis. Because the dispersion predicted independently from the statistical description of the Borden aquifer is consistent with the spread of the injected tracer, it is felt that the theory holds promise for providing meaningful estimates of effective transport l•arameters in other complex-structured aquifers. terns. Because sufficient concentration measurements are sometimes unavailable in practical situations and because source boundary conditions are often unknown, a dispersivity value obtained by model calibration can become highly uncertain. The dispersivity in these cases can only be regarded as a curve-fitting parameter. Recent developments in the theory of contaminant transport recognize the complexity of groundwater systems by regarding the fundamental physical and chemical properties of earth materials that affect local solute transport as stochastic processes or as spatial random fields. Then by solving a stochastic form of the governing groundwater flow and mass transport equations in which random hydraulic parameters are represented statistically, average macroscale transport parameters are derived for the purpose of making large-scale transport predictions [Gelhar, 1985]. These macroscale parameters are intrinsically related to the statistical parameters describing complex three-dimensional heterogeneity of a medium's hydraulic properties, with the relationship being determined from the solution of the stochastic equations. Examples of the appro...
A general analytical solution is developed for the problem of contaminant transport along a discrete fracture in a porous rock matrix. The solution takes into account advective transport in the fracture, longitudinal mechanical dispersion in the fracture, molecular diffusion in the fracture fluid along the fracture axis, molecular diffusion from the fracture into the matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. Certain assumptions are made which allow the problem to be formulated as two coupled, one-dimensional partial differential equations: one for the fracture and one for the porous matrix in a direction perpendicular to the fracture. The solution takes the form of an integral which is evaluated by Gaussian quadrature for each point in space and time. The general solution is compared to a simpler solution which assumes negligible longitudinal dispersion in the fracture. The comparison shows that in the lower ranges of groundwater velocities this assumption may lead to considerable error. Another comparison between the general solution and a numerical solution shows excellent agreement under conditions of large diffusive loss. Since these are also the conditions under which the formulation of the general solution in two orthogonal directions is most subject to question, the results are strongly supportive of the validity of the formulation. tant attenuation mechanism exists in the form of molecular diffusion into the solid matrix [Golubev and Garibyants, 1971]. This mechanism acts to reduce contaminant concentrations in the fracture and thereby delays the migration of the contaminant.
Sudicky and Frind [1982] considered the physically important process of contaminant transport in a fractured porous medium. They developed analytic expressions for contaminant concentrations within equally spaced parallel fractures as well as within the matrix of the porous rock material. However, their expressions for the contaminant concentration in the porous matrix are incorrect, both for the general case with dispersion in the fracture and the simplified instance where dispersion is ignored. (Also note the error in their (35b); the term should be exp (-R2z/v) instead of exp (R2z/v)). We present the correct solutions in this paper. The steady state expressions given by Sudicky and Frind for concentrations in the porous matrix are correct. It must be emphasised that the main results illustrated and discussed by Sudicky and Frind are not affected by these errors, since they did not evaluate the expressions obtained for the porous matrix. The notation and numbering of equations used by Sudicky and Frind is retained in the working here. Where any differences occur, a definition will be given. Equations developed here are identified with numbers of the form (1'). Two errors can be identified in the expressions given by Sudicky and Frind for the concentration in the porous matrix. The first, an omission, relates to the concentration at the fracture wall, while the second stems from an error in integration. Any solution to a boundary value problem must, by definition , satisfy the imposed boundary conditions. The condition imposed at the interface between the fracture and the porous matrix is stated as c'(b, z, t) = c(z, t) (1') However, the solution given for the porous matrix (equation (31)) satisfies c'(b, z, t)= 0 (2') for all times. This indicates an omission in the solution given by Sudicky and Frind. The Laplace inversion in (29) is incomplete , since for x = b the result incorrectly gives L-•{1} =0 (Y) where L-• is the inverse Laplace transform operator. Equation (29) is therefore only correct for x > b. For x-b the inversion yields a delta function •5(t) and must be included in the overall solution (31). Sudicky and Frind make no covering statement, claiming (29) or (31) to be valid only in the region x > b. This immediately resolves the discrepancy between (1') and (2'), since the integrand in (31) is discontinuous at x = b. However, while both equations (29) and (30) are correct for x > b, a second error is apparent in expressions (31) and (36b). The time integration, in proceeding from (30) to (31), was performed incorrectly. Although they noted it earlier, Sudicky and Frind ignored the presence of u[T(0] in the kernel of the time integration of (30) by retaining the lower integral limit as zero. Here we have used the notation u[T0:)] = 0 •: < A r (4') u[T0:)] = 1 z > A Y where TOO = •:-A Y. Clearly the inclusion of (4') necessitates integration from A Y to t, not zero to t. This error by Sudicky and Frind required the introduction of the parameter fl'. In the correct solution it does not appear. Upon c...
Distinct plumes of septic system‐impacted ground water at two single‐family homes located on shallow unconfined sand aquifers in Ontario showed elevated levels of Cl−, NO3−, Na+, Ca2+, K+, alkalinity, and dissolved organic carbon and depressed levels of pH and dissolved oxygen. At the Cambridge site, in use 12 years, the plume had sharp lateral and vertical boundaries and was more than 130 m in length with a uniform width of about 10 m. As a result of low transverse dispersion in the aquifer, mobile plume solutes such as NO3− and Na+ occurred at more than 50 percent of the source concentrations 130 m downgradient from the septic system. At the Muskoka site, in use three years, the plume also had discrete boundaries reflecting low transverse dispersion. After 1.5 years of system operation, the Muskoka plume began discharging to a river located 20 m from the tile field. Almost complete NOs attenuation was observed within the last 2 m of the plume flowpath before discharge to the river. This was attributed to denitrification occurring within organic matter‐enriched riverbed sediments. The very weakly dispersive nature of the two aquifers was consistent with the results of recently reported natural‐gradient tracer tests in sands. Therefore, for many unconfined sand aquifers, the minimum distance‐to‐well regulations for permitting septic systems in most parts of North America should not be expected to be adequately protective of well‐water quality in situations where mobile contaminants such as NOs are not attenuated by chemical or microbiological processes.
[1] Pleistocene glaciations and their associated dramatic climatic conditions are suspected to have had a large impact on the groundwater flow system over the entire North American continent. Because of the myriad of complex flow-related processes involved during a glaciation period, numerical models have become powerful tools for examining groundwater flow system evolution in this context. In this study, a series of key processes pertaining to coupled groundwater flow and glaciation modeling, such as densitydependent (i.e., brine) flow, hydromechanical loading, subglacial infiltration, isostasy, and permafrost development, are included in the numerical model HydroGeoSphere to simulate groundwater flow over the Canadian landscape during the Wisconsinian glaciation ($À120 ka to present). The primary objective is to demonstrate the immense impact of glacial advances and retreats during the Wisconsinian glaciation on the dynamical evolution of groundwater flow systems over the Canadian landscape, including surface-subsurface water exchanges (i.e., recharge and discharge fluxes) in both the subglacial and the periglacial environments. It is shown that much of the infiltration of subglacial meltwater occurs during ice sheet progression and that during ice sheet regression, groundwater mainly exfiltrates on the surface, in both the subglacial and periglacial environments. The average infiltration/exfiltration fluxes range between 0 and 12 mm/a. Using mixed, ice sheet thickness-dependent boundary conditions for the subglacial environment, it was estimated that 15-70% of the meltwater infiltrated into the subsurface as recharge, with an average of 43%. Considering the volume of meltwater that was generated subsequent to the last glacial maximum, these recharge rates, which are related to the bedrock type and elastic properties, are historically significant and therefore played an immense role in the evolution of groundwater flow system evolution over the Canadian landmass over the last 120 ka. Finally, it is shown that the permafrost extent plays a key role in the distribution of surface-subsurface interaction because the presence of permafrost acts as a barrier for groundwater flow.
A complete reexamination of Sudicky's (1986) field experiment for the geostatistical characterization of hydraulic conductivity at the Borden aquifer in Ontario, Canada is performed. The sampled data reveal that a number of outliers {low In (K) valuest are present in the data base. These low values cause difficulties in both variogram estimation and determining population statistics. The analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate. The classical, Cressie/Hawkins and squared median of the absolute deviations {SMAD) estimators are used to compute experimental variograms. None of these estimators provides completely satisfactory variograms for the Borden data with the exception of the classical estimator with outliers removed from the data set. Theoretical exponential variogram parameters are determined from nonlinear (NL) estimation. Differences are obtained between NL fits and those of Sudicky (1986). For the classical-screened estimated variogram, NL fits produce an In (K) variance of 0.24, nugget of 0.07, and integral scales of 5.1 m horizontal and 0.21 m vertical along A-A'. For B-B' these values are 0.37, 0.11, 8.3 and 0.34. The fitted parameter set for B-B' data {horizontal and vertical) is statistically different than the parameter set determined for A-A'. We 'also evaluate a probabilistic form of Dagan's (1982, 1987) equations relating geostatistical parameters to a tracer cloud's spreading moments. The equations are evaluated using the parameter estimates and covariances determined from line A-A' as input, with a velocity equal to 9.0 cm/day. The results are compared with actual values determined from the field test, but evaluated by both Freyberg (1986) and Rajaram and Gelhar (1988). The geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with Dagan's stochastic theory for predicting the spread of a tracer cloud. INTRODUCTION Sudick2y [ 1986] described the results of a sampling program in which a large number of hydraulic conductivity measurements were taken along two transects at the site of an daborate tracer test performed in the Borden aquifer in Ontario, Canada. These measurements, combined with a derailed evaluation of the dispersion characteristics of the injected tracer cloud [Freyberg, 1986], provided a unique opportunity to examine the validity of modern stochastic theories of contaminant transport that have emerged over the past decade. Based on the field data and subsequent geostatistica! inferences, Sudiclo, [1986] computed mean values, variances and integral scales for the underlying log conductivity distribution of the Borden aquifer. Then, by using these quantities as input to stochastic transport theories by Dagan [1982, 1987] and Gelhar and Axness [1983], the predicted field-scale flow and dispersion parameters were shown to be consistent with the observed evolution of the tracer plume as interpreted by Freyberg [1986]. The geostati...
[1] In the recent literature, it has been shown that Pleistocene glaciations had a large impact on North American regional groundwater flow systems. Because of the myriad of complex processes and large spatial scales involved during periods of glaciation, numerical models have become powerful tools to examine how ice sheets control subsurface flow systems. In this paper, the key processes that must be represented in a continental-scale 3-D numerical model of groundwater flow during a glaciation are reviewed, including subglacial infiltration, density-dependent (i.e., high-salinity) groundwater flow, permafrost evolution, isostasy, sea level changes, and ice sheet loading. One-dimensional hydromechanical coupling associated with ice loading and brine generation were included in the numerical model HydroGeoSphere and tested against newly developed exact analytical solutions to verify their implementation. Other processes such as subglacial infiltration, permafrost evolution, and isostasy were explicitly added to HydroGeoSphere. A specified flux constrained by the ice sheet thickness was found to be the most appropriate boundary condition in the subglacial environment. For the permafrost, frozen and unfrozen elements can be selected at every time step with specified hydraulic conductivities. For the isostatic adjustment, the elevations of all the grid nodes in each vertical grid column below the ice sheet are adjusted uniformly to account for the Earth's crust depression and rebound. In a companion paper, the model is applied to the Wisconsinian glaciation over the Canadian landscape in order to illustrate the concepts developed in this paper and to better understand the impact of glaciation on 3-D continental groundwater flow systems.
This paper presents a computer algorithm that is capable of cogenerating pairs of three‐dimensional, cross‐correlated random fields. The algorithm produces random fields of real variables by the inverse Fourier transform of a randomized, discrete three‐dimensional spectral representations of the variables. The randomization is done in the spectral domain in a way that preserves the direct power and cross‐spectral density structure. Two types of cross spectra were examined. One type specifies a linear relationship between the two fields, which produces the same correlation scales for both variables but different variances. The second cross spectrum is obtained from a specified transfer function and the two power spectra, and it produces fields with different correlation scales. For both models the degree of correlation is specified by the coherency. A delay vector can also be specified to produce an out‐of‐phase correlation between the two fields. The algorithm is very efficient computationally, is relatively easy to use, and does not produce the lineation problems that can be encountered with the turning bands method. Perhaps most important, this random field generator is capable of co‐generating cross‐correlated random fields.
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