A three‐dimensional stochastic analysis of the contaminant transport problem is developed in the spirit of Naff (1990). The new derivation is more general and simpler than previous analysis. The fast Fourier transformation is used extensively to obtain numerical estimates of the mean concentration and various spatial moments. Data from both the Borden and Cape Cod experiments are used to test the methodology. Results are comparable to results obtained by other methods, and to the experiments themselves.

The origin of nonlocality in "macroscale" models for subsurface chemical transport is illustrated. It is argued that media that are either nonperiodic (e.g., media with evolving heterogeneity) or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean. A metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window, constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal. The intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations. Deng et al. (1993) present a first-order, nonlocal, Eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions. Hu et al. (this issue) extended these results to account for nonequilibrium linear sorption with random partition coefficient K a but deterministic constant reaction rate Kr. These theories are localized herein, and comparisons are made between the fully nonlocal (FNL), nonlocal in time (NLT), and fully localized (FL) theories. For conservative transport, there is little difference between the first-order FL and FNL models for spatial moments up to and including the third. However, for conservative transport the first-order NLT model differs significantly from the FNL model in the third spatial moments. For reactive transport, all spatial moments differ between the FNL and FL models. The second transverse-horizontal and third longitudinal-horizontal moments for the NLT model differ from the FNL model. These results suggest that localized firstorder transport models for conservative tracers are reasonable if only lower-order moments are desired. However, when the chemical reacts with its environment, the localization approximation can lead to significant error in all moments, and a FNL model will in general be required for accurate simulation.

A nonlocal, first-order, Eulerian stochastic theory was developed by Deng et al. (1993) for the mean concentration of a conservative tracer. Here that result is extended to account for linear nonequilibrium sorption with random partition coefficient. The resultant theory is nonlocal in space and time. An important observation is that unlike Deng et al. (1993), nonlocality is manifest not just in the dispersive flux, but in an effective convective flux and in sources and sinks as well. The fully nonlocal theory is solved exactly in Fourier-Laplace space and converted to a real-space solution via fast Fourier transform in the spirit of Deng et al. (1993). Where possible, comparisons are made with Bellin et al. (1993) and Dagan and Cvetkovic (1993). Positive, negative, and uncorrelated models relating the fluctuating partition coefficient to the fluctuating log conductivity are used to examine the evolution of mean concentration via contours and spatial moments up to the third. The initial sorbed concentration and the deterministic reaction rate can have a significant effect on the moments, especially the second longitudinal moment. The first moment is relatively insensitive to the various correlation structures, but the second and third may exhibit a sensitivity. In the long time asymptotic limit the first two moments are consistent with Fickian theory; however, in the preasymptotic regime the process is nonlocal and non-Fickian. for the concentration C in solution holds: OC OS oD-+ 77 = v. (d. VC) -V. (VC) (•) where S is the sotbed concentration defined as sotbed solute mass per formation solid volume, d is the local-scale dispersion tensor, and V is the locally homogeneous but macroscopically stochastic Darcy velocity. It is further assumed that the mean flow is constant in the x• direction so that V = (V, 0, 0) and that the dispersion tensor is diagonal with entries d• = az• V, d2 = arnV, and d3 = ar•V, where az•, am, and ar• are the so-called Darcy scale longitudinal, transverse-horizontal, and transverse-vertical dispersivities, respectively. It is important to note that in this Eulerian framework we do not neglect local-scale dispersivities as is commonly done in the Lagrangian framework [cf. Dagan, 1989]. As pointed out by Kapoor and Gelhar [1994], the inclusion of local-scale dispersivity may have significant physical importance at long time. We further assume that concentrations C and S are related by a rate equation of the form aS Ot = Kr(KaC -S)where Ka is the partition coefficient which controls local-scale chemical exchange, and Kr is a reaction rate parameter between solution and sorbed phases. In subsequent analysis it is first assumed In K (K is conductivity) is random with Ka a 2239 2240 HU ET AL.: NONLOCAL REACTIVE TRANSPORT--NONEQUILIBRIUM SORPTION deterministic constant, and then later we assume both are

Second‐order log fluctuating conductivity variance (σƒ2) corrections to the head and velocity covariance functions are derived for a lognormal, stationary hydraulic conductivity field. The Fourier transform method proposed by Deng et al. (1993) is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance. It is shown that the velocity covariance is insensitive to second‐order corrections in the head field. The velocity covariance, on the other hand, is highly sensitive to second‐order corrections in the velocity when the log fluctuating conductivity variance approaches unity. A closed expression is derived for a second‐order correction to the velocity variance when there is no second‐order correction to the head field. The longitudinal second‐order correction to the velocity variance is 0.4σƒ2 different from the first‐order approximation in isotropic media, 1.5σƒ2 different in a highly stratified formation, and no different when the ratio of vertical to horizontal integral scales approaches infinity. The second‐order corrections to the horizontal and vertical transverse velocity variances are 2σƒ2 different from the first‐order approximations for both isotropic and anisotropic systems.

Transport moments up to fourth order for convolution-, quasi-, and classical-Fickian dispersion of a conservative tracer are presented and compared. Convolution-Fickian yields a non-Gaussian distribution while quasi-and classical-Fickian result in Gaussian type distributions regardless of the distribution of a stationary velocity field. The first and second central spatial moments are the same for convolution-and quasi-Fickian dispersion, while the third and fourth central moments differ. Introduction For steady mean flow velocity, Deng et al. [1993] have derived a nonlocal transport equation where the dispersive flux of a conservative tracer is a convolution of a time-space dependent dispersion tensor and the gradient of mean concentration. This type of dispersion will be referred as convolution-Fickian. Several researchers [e.g., Dagan, 1984; Sposito and Barry, 1987;Zhang and Neuman, 1995a, b, c, d] have in effect adopted the assumption that the mean concentration gradient varies in space and time much slower than the dispersion tensor, resulting in a dispersive flux which is the product of a time dependent dispersion tensor and the gradient of mean concentration. This type of dispersion will be referred as quasi-Fickian, and it is generally believed to hold at later times. When the dispersive flux is the product of a constant dispersion tensor with the gradient of mean concentration, it will be referred to as classical-Fickian and it is an asymptotic result. Such is the case for Gelhat and Axness [1983], Winter et al. [1984], and Kapoor and Gelhar [1994]. The objective of this note is to illustrate analytically for a conservative tracer the effect on transport spatial moments of the various types of dispersion flux just mentioned.In passing, we should point out that far more general constitutive equations obtain when constraints (e.g., stationarity, smallness in perturbations, etc.) resulting in the above fluxes are relaxed [Neuman, 1993;Cushman et al., 1994;Cushman and Ginn, 1993; Deng et al., 1993, equation (13)]. These more general results do not admit analytic solutions for the various moments and hence are not studied in this note.