No abstract

Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory.The variance of the space-averaged concentration over a volume V may be quite large unless the lengthscale of the initial solute body or of V is large compared with the conductivity integral scale. This condition is bound to be obeyed for transport at the local scale, in which case the concentration may be assumed to satisfy the ergodic hypothesis. This is not generally the case at the regional scale, and the solute concentration is subjected to large uncertainty. The usefulness of the prediction of the concentration expectation value is then quite limited and the dispersion coefficients become meaningless.In the second part of the study, the influence of knowledge of the conductivity and head at a set of points upon transport is examined. The statistical moments of the velocity and concentration fields are computed for a subensemble of formations and for conditional probability distribution functions of conductivity and head, with measured values kept fixed at the set of measurement points. For conditional statistics the velocity is not stationary, and its mean and variance vary throughout the space, even if its unconditional mean and variance are constant. The main aim of the analysis is to examine the reduction of concentration coefficient of variation, i.e. of its uncertainty, by conditioning. It is shown that measurements of transmissivity on a grid of points can be effective in reducing concentration variance, provided that the distance between the points is smaller than two conductivity integral scales. Head conditioning has a lesser effect upon variance reduction.

The problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two-and three-dimensional flows. Approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity. The solute concentration, regarded as a random variable, is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty. The uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale. In any case the concentration expectation does not obey a diffusion type equation in the case of two-dimensional flows, unless the solute body has traveled a distance larger than a few tens transmissivity integral scales. This distance may be exceedingly large in many conceivable applications. streamlines become tortuous even if the flow is uniform in the average. According to our basic approach of Part 1, heterogeneity is modeled as random because of uncertainty. The main 835 aim of the study is to determine the various possible paths of the solute body in probabilistic terms and subsequently the distribution of the expectation and variance of concentration in space and time. SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS GeneralThe last topic of the present study is the problem of transport of a solute carried by water which flows in a large, heterogeneous, porous formation. Only an inert solute, which does not change significantly the density and viscosity of water to affect its flow and which preserves its total mass (no decay or adsorption), is considered here.The problem has been studied intensively in the past for flows through homogeneous porous media at laboratory scale under the designation of hydrodynamic dispersion. It has been found that the macroscopic concentration satisfies a convection-diffusion type of equation in which the molecular diffusion coefficient is supplemented by the pore scale dispersion coefficients (for a review, see, for example, Fried [1975]). The investigation can be traced back to the chemical engineering literature, where such problems are encountered in industrial applications. Hydrologists were attracted to the area later, in the early sixties, in relation to problems of salt water intrusion and waste disposal in aquifers. Unfortunately, the approach which is appropriate to laboratory columns or sand boxes has been applied to problems of solute transport in large natural formations, while not realizing that the change in scale might alter the nature of the phenomenon (see, as an example of hydrological applications, Fried [1975]). Field tests, costly and complicated, have shown, however, that the apparent dispersion coefficients in natural formations are larger by a few orders of magnitude than those determined in laboratory. It has been suggested, consequently, that the same dispersion-convection equations can be used in aquifers but w...

A Lagrangian framework is used for analysing reactive solute transport by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The reaction considered is kinetically controlled sorption–desorption. Transport is quantified by the expected values of spatial and temporal moments that are derived as functions of the non-reactive moments and a distribution function which characterizes sorption kinetics. Thus the results of this study generalize the previously obtained results for transport of non-reactive solutes in heterogeneous formations (Dagan 1984; Dagan et al. 1992). The results are illustrated for first-order linear sorption reactions. The general effect of sorption is to retard the solute movement. For short time, the transport process coincides with a non-reactive case, whereas for large time sorption is in equilibrium and solute is simply retarded by a factor R = 1+Kd, where Kd is the partitioning coefficient. Within these limits, the interaction between the heterogeniety and kinetics yields characteristic nonlinearities in the first three spatial moments. Asymmetry in the spatial solute distribution is a typical kinetic effect. Critical parameters that control sorptive transport asymptotically are the ratio εr between a typical reaction length and the longitudinal effective (non-reactive) dispersivity, and Kd. The asymptotic effective dispersivity for equilibrium conditions is derived as a function of parameters εr and Kd. A qualitative agreement with field data is illustrated for the zero- and first-order spatial moments.

The expected values of the spatial second‐order moments of a solute body transported by groundwater are derived for flow through heterogeneous formations of a stationary random anisotropic structure. They are based on a general formulation, which reduces to most existing results in the literature as particular cases. Detailed results are given for the spatial variance as a function of time in the case of axisymmetric anisotropy, average flow parallel to the plane of isotropy, first‐order approximation in the log conductivity variance σY2, and high Peclet numbers. These results fill the gap existing between the studies of Dagan (1982, 1984), on one hand, and those of Gelhar and Axness (1983) and Neuman et al. (1987), on the other. A preliminary investigation of the higher‐order effects in σY2 suggests that the use of the first‐order approximations is warranted, at present, only for σY2 ≪ 1. The impact of the errors of estimation of the parameters on which transport depends is briefly analyzed.