S. C. Lessoff

Detailed information on the spatial structure of hydraulic conductivity ( K ) is important for understanding and predicting groundwater flow and transport. Direct‐push injection logging (DPIL) is a promising technology for rapid measurement of K in unconsolidated formations. This technology was used to gain information on the highly heterogeneous aquifer at the Lauswiesen test site in Germany. Using a large body of DPIL and direct‐push slug testing measurements, we characterize the structure of K on scales not previously possible. Two new applications of DPIL are put forward: (1) use of raw DPIL measurements of relative conductivity K r to characterize the spatial distribution of K and (2) transformation of K r measurements to K values based on their statistical moments. The DPIL results are compatible to those obtained using more conventional methodologies. The main achievement of the methodology is the possibility to delineate deterministic aquifer subunits as well as the identification of the statistical parameters of the log conductivity for each subunit. In particular, the horizontal integral scale I , a parameter affecting solute transport, is difficult and costly to identify using other approaches. Nevertheless, further studies are needed to clarify questions on low K r measurements and the nature of the relationship between K r and K .

Transport of a conservative solute takes place in a formation made up from a matrix of conductivity K 0 and porosity ϑ 0 and inclusions of properties K , ϑ. For given inclusions shape, the system is characterized by the two parameters κ = K / K 0 and the inclusions volume fraction n . In the past, approximate solutions of the flow and transport problems were obtained under the limit of low variability, i.e., κ − 1 ≪ 1, and arbitrary n [ Rubin , 1995]. The present study aims at solving the problem under the opposite limit of a dilute system, i.e., n ≪ 1 and arbitrary κ. We are particularly interested in elongated inclusions (high length/thickness ratio) of high‐permeability contrast to the matrix. Such configurations are related to applications in which lenses or cracks are present in a medium of highly different conductivity (Figure 1). The basic procedure was developed by Eames and Bush [1999] for cylindrical or spherical inclusions, with no porosity contrast. They compute the macrodispersion coefficient, for advective transport past a large number of inclusions located at random. It is based on the solution for the distortion of a material surface of marked particles, moving past an individual inclusion in an unbounded domain and with uniform flow at infinity. In the present study we extend the approach to inclusions of arbitrary porosity and elliptical shape, characterized by the parameter e , the ratio between the small and large axes, with emphasis on e ≪ 1. We present the analytical solution of the flow problem and the procedure, requiring two quadratures, to calculate the macrodispersivity. Analytical solutions are obtained for two particular limits: κ ≪1 and κ ≃ 1. The latter is compared with the limit n ≪ 1 of the solution of Rubin [1995]. The theoretical results are applied to a few cases of hydrological interest [ Lessoff and Dagan , this issue].

The theoretical results of Dagan and Lessoff [this issue] are applied to three types of media (Figure 1): horizontal lenses submerged in a homogeneous matrix, sparse cracks of random orientation in a matrix of contrasting permeability, and channels of high permeability at the surface of a homogeneous medium. These discrete features are modeled as sparse elliptical inclusions of arbitrary conductivity. The longitudinal macrodispersivity is determined by the methodology of part 1 as function of the parameters characterizing the medium: the conductivity ratio κ, the anisotropy ratio of the ellipsis e , the porosity ratio ϑ/ϑ 0 , and the volume fraction n ≪ 1 or the fracture number per unit volume. Unlike existing stochastic continuum solutions that are first order in the logconductivity variance, the model developed here applies for an arbitrary permeability variance. This is of great advantage in media with high conductivity contrasts between the matrix and the inclusions. Simple results are obtained for inclusions of low conductivity that lead to high macrodispersivity values that are underpredicted by the first‐order continuum approach. In contrast, the presence of thin and highly conductive cracks leads to a finite longitudinal macrodispersivity that depends mainly on their length and the number density. An attempt is made to compare the present approach with the numerical simulations of Desbarats [1990].

Numerical solution of regional‐scale aquifer flow requires discretizing the transmissivity T . Typically, the numerical element scale R and the transmissivity integral scale I are both of the order of hundreds of meters. Then the upscaled block transmissivity is random, and its statistical moments depend on those of T , on flow conditions, and on R / I . Modeling Y = ln T as a two‐dimensional normally distributed stationary random field, we derive unconditional statistics of the upscaled = ln accurate to the first order in σ Y 2 for strongly nonuniform flow in a circular block of radius R centered on a well of radius r w in an unbounded domain. After a proper definition, it was found that the ensemble mean 〈 〉 is approximately but not exactly equal to ln T G (the geometric mean) and that the variance ratio 2 / σ Y 2 , which depends on r w / I and the shape of the correlation ρ , drops slowly from unity for R / I = 0 to approximately 0.6 for R / I = 10. Hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow. Additionally, defining the equivalent T eq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge, we find that in radial flow T eq ≅ T H the harmonic mean and grows slowly with increasing R / I , whereas for mean uniform flow T eq = T G . Application of the procedure is illustrated for an example of aquifer with selected values of parameters.

Abstract Single‐well injection/recovery tracer tests are considered for use in characterizing and quantifying matrix diffusion in dual‐porosity aquifers. Numerical modeling indicates that neither regional drift in homogeneous aquifers, nor heterogeneity in aquifers having no regional drift, nor hydrodynamic dispersion significantly affects these tests. However, when drift is coupled simultaneously with heterogeneity, they can have significant confounding effects on tracer return. This synergistic effect of drift and heterogeneity may help explain irreversible flow and inconsistent results sometimes encountered in previous single‐well injection/recovery tracer tests. Numerical results indicate that in a hypothetical single‐well injection/recovery tracer test designed to demonstrate and measure dual‐porosity characteristics in a fractured dolomite, the simultaneous effects of drift and heterogeneity sometimes yields responses similar to those anticipated in a homogeneous dual‐porosity formation. In these cases, tracer recovery could provide a false indication of the occurrence of matrix diffusion. Shortening the shut‐in period between injection and recovery periods may make the test less sensitive to drift. Using multiple tracers having different diffusion characteristics, multiple tests having different pumping schedules, and testing the formation at more than one location would decrease the ambiguity in the interpretation of test data.