Search

The study, a continuation of that of Indelman et al. [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 log conductivity Y=ln K is modeled as a three-dimensional stationary function of Gaussian correlation of anisotropy ratio e. By using first-order approximations in σ2Y and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradient Er and of the radial component of the specific discharge qr. 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 σ2H and of its gradient σ2Er, as well as the crossvariance σE,Y between Er and Y vanish at the well, whereas the discharge variance σ2qr tends to the product between the log conductivity variance σ2Y and the squared mean discharge 〈qr〉2. This behavior pertains to a stratified formation surrounding the well. Far from the well (≈75 horizontal Y integral scales I) the head variance approaches a constant value. For r ≥ 10I the moments σ2Er, σ2qr and σErY 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.

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere $\Omega $, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux $% U_{\infty }$ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere $\Omega ,$ under the limits $n\rightarrow 1$ and $R_{0}/I_{Y}\rightarrow \infty .$ Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit $% n\rightarrow 0.$ It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulation...

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 〈C〉, the ensemble mean, and σC2, 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 〈C〉 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 (CL). 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 CL 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 〈C〉 underestimates the maximum concentration. Furthermore, the fluctuations of CL 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.

\

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 larger 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.