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[1] Flow of uniform mean velocity U takes place in a heterogeneous medium made up from a matrix of conductivity K-0 and inclusions of a different conductivity K. The inclusions of given shape are implanted at random and independently in the medium, without overlapping. The aim of the study is to derive simple, approximate solutions of advective transport of solutes in such heterogeneous formations for arbitrary permeability ratio kappa = K/K-0 and inclusions volume fraction n. Transport is characterized by the spatial moments, which in turn are equal to the one particle trajectory statistical moments for ergodic plumes. The flow and transport problems are solved for isotropic media, for circular (2D) and spherical (3D) inclusions by using the model of composite inclusions of Hashin and Shtrikman [1962]. The results tend to the dilute limit analyzed in the past by Eames and Bush [1999] for n = o(1). Asymptotic, analytical results are derived for weak heterogeneity (kappa = 1); they coincide with those of Rubin [1995] for a similar structure. Similarly, simple asymptotic expressions of the macrodispersivity are derived for low-permeability inclusions, kappa = o(1). The theoretical developments are applied to effectively computing trajectories moments in part 2 [Fiori and Dagan, 2003].

The theoretical results of part 1 [Dagan and Fiori, 2003] for modeling time-dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast kappa, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by alpha(L)(t; n, kappa)=(2U)(-1) dX(11)/dt, where X-11 is the trajectories second moment in the mean flow direction. The general behavior of the time-dependent longitudinal dispersivity alpha(L)(t; n, kappa) and, in particular, its constant, large time limit are examined. The tendency of alpha(L) to the "Fickian'' limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (kappamuch less than1) it may be extremely slow. It is shown that the first-order approximation in the conductivity contrast kappa is of limited validity (0.3

We analyze the influence of conditioning to hydraulic transmissivity measurements on transport of a conservative tracer in a two-dimensional evolving-scale formation with a large-scale cutoff of the hydraulic transmissivity. First-order analytical solutions of ensemble and effective plume moments, conditional to a single measurement of hydraulic transmissivity, are provided for an instantaneous release of solute within a linear source normal to the mean flow direction. The proposed solutions show that both ensemble and effective moments are significantly affected by the additional information brought into flow and transport models through conditioning to available data. We conclude that the effect of conditioning to available measurements is more pronounced on the ensemble moments than on the effective moments, and uncertainty is not reduced in the same proportion for longitudinal and transverse effective second moments, since the former shows the largest reduction of uncertainty for a given measurement location. This result confirms the experimental findings showing larger uncertainty in the estimation of the transverse moments. Furthermore, for all moments the impact of the measurement reduces with the distance of the measurement point from the mean plume trajectory. (C) 2003 Elsevier Science Ltd. All rights reserved.

[1] We consider steady groundwater flow of uniform mean in aquifers of random, spatially variable, hydraulic conductivity. Analytical expressions for the statistical moments of mass fluxes of sorbing solutes in presence of pore-scale dispersion are derived, where the reactive solutes undergo first-order sorption kinetics. The developments which lead to the analytical formulation of the solute flux are rigorous in the first-order analysis framework, and results obtained are valid for weakly heterogeneous formations. The methodology is exemplified for a two-dimensional aquifer, assuming that the source is of small transverse extent compared to the heterogeneity length scales. The examples show that pore-scale dispersion has a relatively small effect on the mean point flux, whereas the point flux variance shows much larger sensitivity to pore-scale dispersion. The variance first decreases as the reaction rate departs from the nonreactive limit, but for equilibrium reactions it is of the same order as for nonreactive solutes. The effect of averaging the solute flux over a finite sampling area is also investigated. It is found that for the expected area-averaged flux the mixing effect induced by sampling tends to supersede that caused by pore-scale dispersion. On the contrary, pore-scale dispersion may have a strong effect on the flux variance also when sampling effects are taken into account.

The spreading of a contaminant in a heterogeneous aquifer depends on the scales of variability effectively explored by the plume. In particular, we observe two major contributions of the fluctuating velocity field in the contaminant movement: (i) the spreading caused by velocity variations of scales lesser than that of the plume size, which we will call "relative' spreading, and (ii) the meander-like movement of the plume as a whole caused by velocity variations of scale larger than that of the plume size. The aim of this work is to consider the effects of the finite size of the contaminant plume on the local concentration moments and sigma (C). In particular a "relative' concentration, which depends on the scales of variability effectively explored by the plume, is defined. First, the mathematical formulation of the problem is developed along the Lagrangian framework. In particular, the expressions for the relative mean concentration and its variance are presented. Then, the methodology is applied to the regional transport problem, where the influence of the size of the plume and the pore-scale dispersion are quantitatively assessed.

The aim of the present study is to determine the longitudinal dispersion coefficient D-L for transport in formations of long-range permeability fields by considering both large-scale advection and local-scale dispersion; the nonergodicity of the plume will be considered throughout the work. The scope is twofold: (1) to analyze the mutual role played by both the "macroscale" and the local scales of heterogeneity in determining the overall transport properties and (2) to check the validity of the results obtained in the past, in particular concerning the occurrence of anomalous transport. The results are obtained through the Lagrangean formulation of transport, by the means of a few simplifying assumptions. Two models of permeability K variations are considered: (1) a stationary Y = In K with unbounded integral scale and (2) a formation of Y of stationary increments. For both cases, the longitudinal macrodispersion coefficient D-L always grows with time when local-scale dispersion is present, indicating that transport is always anomalous for the random fields examined. The results are in variance with those obtained in the past by considering nonergodic transport but neglecting the local-scale dispersion [e.g., Dagan, 1994; Bellin et al., 1996] and in qualitative agreement to those obtained by adopting the ergodic assumption [e.g., Neuman, 1990; Glimm and Sharp, 1991], which, however, predicted higher rates of growth of D, with time. We conclude that the interplay between large-scale, advective displacement and local-scale dispersion has a fundamental impact on the occurrence of anomalous transport in long-range correlated permeability random fields.

Transport of nonreactive solutes in aquifers of random permeability is considered. The local concentration C, which is a random function of space and time, can be modeled through the ensemble mean, and sigma-(2)(c), the variance. Because of the irregular spatial distribution of the concentration, which is a consequence of the spreading and dilution of the solute plume, the uncertainty in the local concentration prediction can be very large, and the mean concentration may not accurately predict the concentration distribution in the field, in particular, the peak concentration. The aim of the present paper is to quantify dilution through the peak concentration by introducing a quantity denoted as Lagrangian concentration (C-L). According to the proposed methodology the combined effects of dispersion, which is ruled by large-scale advection, and dilution, which is a consequence of pore-scale dispersion, are analyzed by following each Darcy-scale solute particle constituting the initial plume by focusing on the mass exchanges between the examined particle and the surrounding ones. After definition and derivation of C-L and its statistical moments, illustrative examples are shown for a small solute injection. Then the comparison of the theoretical results with the Borden and Cape Cod experimental findings shows that the Lagrangian concentration gives a reasonable prediction of the peak concentration: conversely, the often employed mean concentration underestimates the maximum concentration. Furthermore, the fluctuations of C-L are also smaller than those for C, as indicated by the analysis of the variance. The results are of some importance for applications in which the environmental regulations are formulated in terms of maximal admissible local concentrations.

Flow and transport take place in a formation of spatially variable conductivity K(x). The latter is modeled as a lognormal stationary random space function. With Y = ln K, the structure is characterized by the mean [Y], the variance sigma(Y)(2), the horizontal and vertical integral scales I-h and I-v. The fluid velocity field V(x), driven by a constant mean head gradient, has a constant mean U and a stationary two-point covariance. Transport of a conservative solute takes place by advection and by pore-scale dispersion (PSD), that is assumed to be characterized by the constant longitudinal and transverse dispersivities alpha(dL) and alpha(dT). The local solute concentration C(x, t), a random function of space and time, is characterized by its statistical moments. While the mean concentration [C] was investigated extensively in the past, the aim here is to determine the variance sigma(C)(2), a measure of concentration fluctuations. This is achieved in a Lagrangean framework, continuous limit of the particle-tracking procedure, by adopting a few approximations. The present study is a continuation of a previous one (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595-1606) and extends it as follows: (i) it is shown that the indepence of the advective component of a solute particle trajectory from the trajectory component associated with PSD, is a rigorous first-order approximation in sigma(Y)(2). This independence, that was conjectured in the work of Dagan and Fiori (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595-1606), simplifies considerably the solution; (ii) the covariance of two-particle trajectories, needed in order to evaluate sigma(C)(2), is rederived, correcting for an error in the previous work. The general results are applied to determining CVC = sigma(C)/[C] at the center of a small solute body, of initial size much smaller than I-h = I-v, as function of sigma(gamma)(2), t' = tU/I and Pe = UI/D-dT = I/alpha(dT). Though PSD reduces considerably CVC as compared with advective transport (Pe = infinity), its value is still quite large for time intervals of interest in applications. This finding is in agreement with the analysis of field data by Fitts (Fitts, C.R., 1996. Uncertainty in deterministic groundwater transport models due to the assumption of macrodispersive mixing: evidence from the Cape Cod (Massachussets, USA) and Borden (Ontario, Canada) tracer tests. J. Contam. Hydrol., 23, 69-84). (C) 2000 Elsevier Science B.V. All rights reserved.

The fate of contaminants in groundwater is controlled by many factors; the most relevant are the formation heterogeneities, the geochemical properties of the aquifer and the source size which is a limiting factor for the spreading of the contaminant. When the source size is much lager than the characteristic scale of the formation heterogeneities, the transport develops under ergodic conditions and the spatial moments of the plume are not influenced by the source size. In this case, the plume samples all the relevant scales of heterogeneity and the spreading of the solute reaches its maximum. To date, a number of solutions are available for transport under ergodic conditions and only few studies address the non-ergodic case, which develops when the source size is of the same order of magnitude of the formation integral scale, In this paper, we study how the source size affects the transport of linear kinetically sorbing solutes, characterized by constant reaction parameters. First, the general expressions for the effective plume moments in natural formations are obtained by using the Lagrangian approach. Then, closed form solutions are provided for instantaneous injection over a plane normal to the mean flow direction and in the limiting case of large travel times, in addition, we analyze the plume moments at small and moderate times in an isotropic two-dimensional formation characterized by the Gaussian log-conductivity covariance function. We found that the source size is the key factor controlling the relative impact on transport of the formation heterogeneities and the geochemical properties of the aquifer. In addition, we found that the relative importance of the contribution to dispersion stemming from the formation heterogeneities increases with the source size. Conversely, when the source size is comparable with the Darcy's scale, the formation heterogeneities do not contribute to solute spreading, which is controlled by the reaction only. The results show also that the ergodic conditions for the longitudinal second order moment are obtained with a progressively smaller source size as the reaction rates increases. Hence, the presence of reaction facilitates the occurrence of ergodic transport, which is obtained for plumes significantly smaller than for nonreactive solutes transport. (C) 1999 Elsevier Science B.V. All rights reserved.

In a recent work [Dagan and Fiori, 1997] we developed a Lagrangian theoretical framework to compute the concentration variance in transport of a conservative solute in a heterogeneous formation of random stationary structure. This variance, a measure of concentration fluctuations in the plume, depends on the spreading effect of advection and on the mixing effect of pore-scale dispersion. Comparison between theory and the numerical simulations of Graham and McLaughlin [1989] showed very good agreement. Here we compare the same theoretical formulae with the field measurements at Borden Site and Cape God. We rely on the analysis of these experiments carried out by Fitts [1996], who characterized the concentration fluctuations by a global measure. The latter is the standard deviation of the logarithm of the ratio between measured and modeled concentrations over the entire plume and at different times. The theoretical relationships require computing the two particles trajectories' covariances. We simplify the computations considerably by taking advantage of the anisotropy of the heterogeneous structures of the two aquifers. Thus the major effect of advection is in the shearing of the plume in the longitudinal direction, whereas pore-scale dispersion acts mainly through its transverse effect in the vertical direction. We found a good agreement between theory and experiments by using parameter values determined independently in the past, provided that the values of transverse pore-scale dispersivities were 0.5 mm for Cape Cod and 0.44 mm for Borden Site. These values are somewhat smaller than the vertical macrodispersivities inferred from measurements.

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D-11; we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Peclet number, defined as Pe = U lambda/D, where U, lambda, D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Peclet approximation is in most cases quite adequate, at least in the range of Peclet number usually encountered in practice (Pe > 10(2)). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D-11,especially when the solute plume initial dimensions an not much larger than the heterogeneities' lengthscale. In all cases, D-11 appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.

The study, a continuation of that of Indelman et at. [1996], aims at deriving the second-order moments of flow variables such as hydraulic head, its gradient,and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer. The lag conductivity Y = In K is modeled as a three-dimensional stationary function of Gaussian correlation of anisotropy ratio e. By using first-order approximations in sigma(Y)(2) and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradient E-r and of the radial component of the specific discharge q(r). Owing to the nonuniformity of the average flow, these quantities are functions of the distance from the well. It is shown that the variances of the head sigma(H)(2), and of its gradient sigma(E)(2), as well as the crossvariance sigma(E),(Y) between E-r and Y, vanish at the well, whereas the discharge variance sigma(qr)(2) tends to the product between the log conductivity variance sigma(Y)(2) and the squared mean discharge < q(r) >(2). This behavior pertains to a stratified formation surrounding the well. Far from the well (approximate to 75 horizontal Y integral scales I) the head variance approaches a constant value. For r greater than or equal to 10I the moments sigma(E)(2), sigma(qr)(2) and sigma(E,Y) tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one. The head vertical integral scale grows indefinitely with r, whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow. This latter property is explained by the presence of the source line, which increases the correlations in the vertical direction. The present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells.

Transport of an inert solute in a heterogeneous aquifer is governed by two mechanisms: advection by the random velocity field V(x) and pore-scale dispersion of coefficients D-dij. The velocity field is assumed to be stationary and of constant mean U and of correlation scale I much larger than the pore-scale d. It is assumed that D-dij = alpha(dij)U are constant. The relative effect of the two mechanisms is quantified by the Peclet numbers Pe(ij) = UI/D-dij = I/alpha(dij), which as a rule are much larger than unity. The main aim of the study is to determine the impact of finite, though high, Pe on [C] and sigma(C)(2), the concentration mean and variance, respectively. The solution, derived in the past, for Pe = infinity is reconsidered first. By assuming a normal X probability density function (p.d.f.), closed form solutions are obtained for [C] and sigma(C)(2). Recasting the problem in an Eulerian framework leads to the same results if certain closure conditions are adopted. The concentration moments for a finite Pe are derived subsequently in a Lagrangean framework. The pore-scale dispersion is viewed as a Brownian motion type of displacement X-d of solute subparticles, of scale smaller than d, added to the advective displacements X. By adopting again a normal p.d.f. for the latter, explicit expressions for [C] and sigma(C)(2) are obtained in terms of quadratures over the joint p.d.f. of advective two particles trajectories. While the influence of high Pe on [C] is generally small, it has a significant impact on sigma(C)(2). Simple results are obtained for a small V-0, for which trajectories are fully correlated. In particular, the concentration coefficient of variation at the center tends to a constant value for large time. Comparison of the present solution, obtained in terms of a quadrature, with the Monte Carlo simulations of Graham. and McLaughlin [1989] shows a very good agreement.

The temporal moments associated with the breakthrough of a sorbing solute in a heterogeneous aquifer are investigated assuming an anisotropic distribution of hydraulic conductivity and deterministic linear sorption kinetics. The expressions for the temporal moments are based on a solution to the advection-dispersion equation with constant coefficients which implies that they are valid in the large-time regime. For the evaluation of the ''macrodispersion'' coefficient we use the asymptotic results of Fiori [1996], who quantified the effect of pore-scale dispersion on nonreactive solute and showed that porescale dispersion affects field-scale transport primarily through transverse mixing between Bow paths. The present study shows that linear sorption kinetics decrease the effect of transverse mixing on the temporal moments as compared to the nonreactive case. It is concluded that in the considered large-time regime, the pure-advection approach to quantify field-scale spreading is appropriate under the conditions encountered in most natural formations.

Dagan's [1984] landmark solution for transport in heterogeneous porous formations is extended to the case of finite Peclet numbers. It is suggested that pore-scale dispersion matters only with reference to transverse spreading, and that Dagan's solution, valid for Pe = infinity, is an adequate approximation in a wide range of finite Peclet numbers.

Solute transport in natural formations at the regional scale is influenced by several scales of heterogeneity which correspond to the presence of several geological units called facies. As customarily assumed in stochastic theories, inside the facies the transport can be characterized by a single scale of heterogeneity. At the regional scale Several geological units are present such that a hierarchy of relevant scales needs to be defined. A possible model for this spatial variability assumes the log conductivity as a random space function of stationary increments characterized by a power law semivariogram. With this hypothesis the ergodic dispersion coefficient grows unbounded as time increases, leading to the phenomenon called anomalous dispersion, An alternative approach considers the plume in nonergodic conditions and assumes the effective dispersion coefficient, which is defined through differentiation in time of the expected value of the spatial second-order plume moment, as representative of macrodispersion. Large differences have been observed in the resulting plume spreading while approaching the problem using the above alternative definitions. In this paper we provide first-order analytical solutions for the longitudinal effective dispersion coefficient, D-L, as well as for the expected value of the longitudinal spatial plume moment, [S-11] that complement semianalytical expressions recently proposed in literature. Furthermore, we provide a semianalytical expression for the standard deviation of the longitudinal second-order moment which is important in assessing the interval of confidence of the estimation provided by [S-11]. Suitable numerical simulations are performed to validate analytical and semianalytical expressions as well as to assess the impact of the cutoff in the log conductivity power spectrum imposed by choosing a finite domain dimension, We conclude that according to recently published results, the dispersion is anomalous when the Hurst coefficients, H, is larger than 0.5 while it is Fickian for H < 0.5. This is in contrast with the ergodic analysis which concludes that the dispersion is anomalous irrespective of the Hurst coefficient, Hence the effective dispersion coefficient is more effective than the ergodic dispersion. coefficient to represent the plume spreading. However, the standard deviation of the longitudinal spatial second-order moment is of the same order of magnitude as the expected value leading to the conclusion that the estimations provided by D-L and [S-11] are affected by large uncertainties. Numerical results are in good agreement with the analytical solutions, and under some hypotheses they are not influenced by the cutoff. This is not the case for the ergodic second-order longitudinal moment, which strongly depends on the imposed cutoff.

We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius r(w) (Figure 1). The hydraulic conductivity K is modeled as a three-dimensional stationary random space function. The two-point covariance of Y = In (K/K-G) is of axisymmetric anisotropy, with I and I-v, the horizontal and vertical integral scales, respectively, and K-G, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head [H] and the mean specific discharge [q] as functions of the radial coordinate v and of the parameters sigma(Y)(2), e = I-v/I and r(w)/l. An approximate solution is obtained at first-order in sigma(Y)(2), by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between (H) over bar, (q) over bar, space averages over the vertical, and [H], [q], ensemble means. An equivalent conductivity K-eq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head H-w in the well and a given head H in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures H-w, (H) over bar, and Q. The main result of the study is the relationship (19) K-eq = K-A(1 - lambda) + K(efu)lambda, where K-A is the conductivity arithmetic mean and K-efu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient lambda < 1 is derived explicitly in terms of two quadratures and is a function of e, r(w)/I, and r/l. Hence K-eq, unlike K-efu, is not a property of the medium solely. For r(w)/I < 0.2 and for r/I > 10, lambda has the simple approximate expression lambda* = in (r/I) ln (r/r(w)). Near the well, lambda congruent to 0 and K-eq congruent to K-A, which is easily understood, since for r(w)/I << 1 the formation. behaves locally like a stratified one. In contrast, far from the well lambda congruent to 1 and K-eq congruent to K-efu, the flow being slowly varying there. Since K-A > K-efu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between K-A and K-efu is small for highly anisotropic formations for which e << 1. A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].