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At regional scale, it is common to model groundwater flow as 2-D in the x, y, horizontal plane, by integrating the full 3-D equations over the vertical. Furthermore, adopting the Dupuit assumption results in the local transmissivity T as a formation property, equal to the vertically integrated hydraulic conductivity K. In practice, the related block transmissivity T b, defined for a volume of area ω (square of side L) in the horizontal plane and height D, is the property of interest. However, most aquifers are of a heterogeneous 3-D structure, and Y = ln K is commonly modeled as a normal and stationary random function which is characterized by the variance σ Y2, the horizontal I, and vertical I v integral scales. The Dupuit assumption is generally not obeyed for formations of 3-D spatially variable Y, and transmissivity is no more a meaningful property, independent of flow conditions. Useful generalizations of local and block transmissivity are possible for steady mean uniform flow in the horizontal direction and formations of constant thickness. In that case T and T b become random stationary variables characterized by their mean, variance, and integral scales. These moments are determined for the first time in an analytical form or by a few quadratures, by adopting a first-order approximation in σ Y2, and they depend on the ratio D/ I v, e = I v/ I and L/ I. The block conductivity expected values are compared with the numerical solutions of Dykaar and Kitanidis (1993), and the agreement is very good for σ Y2 ≤ 1. The main conclusion of the study is that for this simple flow configuration and for common parameter values, T b is practically deterministic and equal to K eff( e) D, where K eff is the effective conductivity in uniform flow in an unbounded formation. At regional scale, T b may be regarded as a local property which changes slowly in the horizontal plane. Analysis of numerous field data shows that this variation is also random and characterized by integral scales I reg, of the order of kilometers. The separation of scales makes possible to regard the local T b, as determined along the lines of the present study in a support volume of extent of a few D, as a point value at the regional scale. Practical implications and topics for future investigations are outlined. [ABSTRACT FROM AUTHOR]

Abstract: We consider 3D steady flow of fresh water over a salt water body in a confined aquifer of constant thickness D, with application to a pumping well in a coastal aquifer. With neglect of mixing, a sharp interface separates the two fluid bodies and an existing analytical solution, based on the Dupuit assumption, is adopted. The aim is to solve for the mixing between the fresh and salt waters for α T/D ≪1 (α T transverse dispersivity), as field studies indicate that α T = O(10−3 −10−2 m). The mixing zone around the interface is narrow and solutions by existing codes experience numerical difficulties. The problem is solved by the boundary layer (BL) approximation, extending a method, applied previously to two-dimensional flows. The BL equations of variable-density flow are solved by using the Von Karman integral method, to determine the BL thickness and the rate of entrainment of salt water along the interface. Application to the pumping well problem yields the salinity of the pumped water, as function of the parameters of the problem (well discharge, seaward discharge, well distance from the coast and density difference). [Copyright &y& Elsevier]

Numerical solution of regional scale aquifer flow requires discretizing the transmissivity T. For spatially varying T and by the stochastic approach, T is modeled as a random field. Typically, both the numerical element scale R and the log transmissivity integral scale I are of order of hundreds to thousands meters. Consequently, the upscaled block transmissivity is random and its statistical moments depend on those of the fine scale T, on flow conditions, and on R/ I. Modeling Y = ln T as a two-dimensional normally distributed stationary random field, we have derived [ Dagan and Lessoff, Water Resources Research, 43, W05431, doi:10.1029/2006WR005235 2007] the unconditional statistics of the upscaled = ln , accurate to the first-order in σ Y2, the log transmissivity variance. Both cases of mean uniform flow [solved previously by Indelman and Dagan, Transport in Porous Media, 12, 161-183, 1993] and of strongly nonuniform flow in a circular block of radius R centered at a well of radius r w, were considered. Such upscaling applies either to elements in regions of natural gradient flow or those surrounding a projected well, respectively, which are sufficiently far from points of measurement of T. The present article extends the analysis to account for the presence of a measurement removed in the center of the element (e.g., result of a pumping test). The fine scale conditional log transmissivity Y c is modeled as multi-Gaussian, of nonstationary mean and of two point covariance which depend on the measured value and distance from the measurement location. Upscaling in mean uniform flow is related, for instance, to the case of a pumping well which is not operative, but served to determine T while natural gradient flow conditions prevail in the area. It was found that upscaling has a significant effect upon the mean upscaled log transmissivity that changes from the measured value for R/ I 1 to the effective one Y = ln T G for R/ I > 6. Similarly, the conditional log transmissivity variance is greatly reduced compared to the unconditional case. It tends to zero for R/ I → 0 or R/ I → ∞, and it reaches a maximal value of around 0.15 σ Y2 for R/ I 1. The second case, of upscaling in well flow, applies to an element with an operative well at its center, which was used to determine the transmissivity T w. Unlike mean uniform flow, it is found that upscaling has a very small effect for the realistic values of r w/ I = 0(10−3) and R/ I < 5. In this case the upscaled transmissivity is practically deterministic and equal to the given T w. A summary of the two articles, (Dagan and Lessoff, 2007) and the present one concludes the paper. [ABSTRACT FROM AUTHOR]

Flow of uniform mean velocity U takes place in a heterogeneous medium made up from a matrix of conductivity K0 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 κ = K/ K0 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 (κ ≃ 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, κ = o(1). The theoretical developments are applied to effectively computing trajectories moments in part 2 [ Fiori and Dagan, 2003]. [ABSTRACT FROM AUTHOR]

Abstract: Solute transport is investigated in a heterogeneous aquifer for combined natural-gradient and well flows. The heterogeneity is associated with the spatially varying hydraulic conductivity K(x, y, z), which is modelled as a log-normal stationary-random function. As such, the conductivity distribution is characterized by four parameters: the arithmetic mean K A, the variance σ Y 2 (Y =lnK), the horizontal integral scale I of the axisymmetric log-conductivity autocorrelation and the anisotropy ratio e = I v/I (I v is the vertical integral scale). The well fully penetrates an aquifer of constant thickness B and has given constant discharge QB, while the background aquifer flow is driven by an uniform mean head-gradient, − J. Therefore, for a medium of homogeneous conductivity K A, the steady-state capture zone has a width 2L = Q/(K A|J|) far from the well (herein the term capture zone is used to refer both to the zone from which water is captured by a pumping well and the zone that captures fluid from an injecting well). The main aim is to determine the mean concentration as a function of time in fluid recovered by a pumping well or in a control volume of the aquifer that captures fluid from an injecting well. Relatively simple solutions to these complex problems are achieved by adopting a few assumptions: a thick aquifer B ≫ I v of large horizontal extent (so that boundary effects may be neglected), weak heterogeneity σ Y 2 <1, a highly anisotropic formation e <0.2 and neglect of pore-scale dispersion. Transport is analyzed to the first-order in σ Y 2 in terms of the travel time of particles moving from or towards the well along the steady streamlines within the capture zone. Travel-time mean and variance to any point are computed by two quadratures for an exponential log-conductivity two-point covariance. Spreading is reflected by the variance value, which increases with σ Y 2 and I/L. For illustration, the procedure is applied to two particular cases. In the first one, a well continuously injects water at constant concentration. The mean concentration as function of time for different values of the controlling parameters σ Y 2 and I/L is determined within control volumes surrounding the well or in piezometers. In the second case, a solute plume, initially occupying a finite volume Ω 0, is drawn towards a pumping well. The expected solute recovery by the well as a function of time is determined in terms of the previous controlling parameters as well as the location and extent of Ω 0. The methodology is tested against a full three-dimensional simulation of a multi-well forced-gradient flow field test ([Lemke, L., W.B. II, Abriola, L., Goovaerts, P., 2004. Matching solute breakthrough with deterministic and stochastic aquifer models. Ground Water 42], SGS simulations). Although the flow and transport conditions are more complex than the ones pertinent to capture zones in uniform background flow, it was found that after proper adaptation the methodology led to results for the breakthrough curve in good agreement with a full three-dimensional simulation of flow and transport. [Copyright &y& Elsevier]