The magnitude of longitudinal dispersivity in a sandy stratified aquifer was investigated using laboratory column and field tracer tests. The field investigations included two‐single‐well injection‐withdrawal tracer tests using 131I and a two‐well recirculating withdrawal‐injection tracer test using 51Cr‐EDTA. The tracer movement within the aquifer was monitored in great detail with multilevel point‐sampling instrumentation. A constant value for dispersivity of 0.7 cm was found to be representative (and independent of travel distance) at the scale of an individual level within the aquifer. A dispersivity of 0.035 cm was determined from laboratory column tracer tests as a representative laboratory‐scale value for sand from the field site. The scale effect observed between the laboratory dispersivity and the dispersivity from individual levels in the aquifer is caused by the greater inhomogeneity of the aquifer (e.g., laminations within individual layers) and the averaging caused by the groundwater sampling system. Full‐aquifer dispersivities of 3 and 9 cm obtained from the single‐well tests indicate a scale effect with the value obtained being dependent mainly on the effect of transverse migration of tracer between the layers and the total injection volume. The full‐aquifer dispersivity of 50 cm from the two‐well test is scale‐dependent, controlled by the distance between the injection and withdrawal wells (8 m) and hydraulic conductivity distribution in the aquifer. Scale‐dependent full‐aquifer dispersivity expressions were derived relating dispersivity to the statistical properties of a stratified geologic system where the hydraulic conductivity distribution is normal, log normal or arbitrary. In the developed expressions, dispersivity is a linear function of the mean travel distance. Proportionality constants ranged from 0.041 to 0.256 for the hydraulic conductivity distributions obtained from the field tracer tests.

Solute transport through fractured media is described by numerically combining advective‐dispersive transport, which is dominant in the fractures, and diffusive transport, which is usually dominant in the unfractured matrix. Transport is considered in a manner conceptually similar to ‘double‐porosity’ or ‘intra‐aggregate’ transport models. A finite element model is developed for simulating nonreactive and reactive solute transport by advection, mechanical dispersion, and diffusion in a unidirectional flow field. The effect of the value of the solute diffusion coefficient in the matrix (termed the matrix diffusion coefficient) is illustrated by solute breakthrough curves and concentration profiles in the fracture as well as in the matrix. The illustrated conditions are similar to the laboratory tracer study on fractured till described in the accompanying paper (Grisak et al.). The effects on solute transport of fracture aperture size, water velocity in the fracture, matrix porosity, matrix distribution coefficient, and dispersivity in the fracture are illustrated with breakthrough curves and concentration profiles. The net effect of large matrix diffusion coefficients and/or large distribution coefficients in the matrix is to reduce significantly the effective solute velocity in the fracture. Reduction in the effective solute velocity can also be seen to be possible in materials with low matrix porosities such as crystalline rocks. The aperture size, matrix porosity, matrix diffusion coefficient, and distribution coefficient, all are important in determining the relative amounts of solute transported in the fracture and stored in the matrix. Implications with regard to aquifer recharge, groundwater chemistry, contaminant transport, tracer tests, and groundwater age dating in fractured media are discussed. The numerical model and the laboratory tracer test data provide considerable insight into the processes controlling solute transport in fractured media.

Pickens and Grisak [1981] present an interesting analysis of modeling of scale-dependent dispersion in hydrologic systems. They conclude that for systems that exhibit a constant dispersivity at large times or large mean travel distances, the importance of scale-dependent dispersion at early times or short travel distances is minimal in long-term predictions of solute transport. The purpose of this comment is to point out that this conclusion may not be appropriate.The effect of scale-dependent dispersivity on spreading of the contaminant plume is self-evident. A graph (Figure 1) of spatial concentration variances of contaminant plumes resulting from a constant dispersivity and a scale-dependent dispersivity versus mean travel distance or time illustrates the effect. The spatial concentration variance of a contaminant plume is a measure of spreading of the plume. Thus the value Arrx 2 in Figure 1 represents the difference between the sizes of the plumes resulting from a constant dispersivity and a scale-dependent dispersivity. This value depends on the mean displacement distance, the development distance or time of the asymptotic dispersivity, and the magnitude of the dispersion coefficient (D = a U) or the asymptotic macrodispersivity a if the velocity, U is constant. This figure shows that asymptotic macrodispersivities with large values of development time will produce much smaller variances (curve 1) at large values of distance than the constant dispersivity (curve 2). Also, the difference in the spatial concentration variances increases with the magnitude of the asymptotic dispersivity (Arrx 2' > Arrx2). However, at this moment there is no adequate information to quantify the relationship between spatial variances and mean travel distances and the magnitude of asymptotic macrodispersivity. It is premature to conclude that the scale-dependent dispersion at early times or short travel distances has little effect in long-term predictions of solute transport, although the relative error in such predictions may diminish with distances. It should also be pointed out that a well-defined scale-dependent dispersivity function does not warrant the use of the classical diffusion equation in the prediction of large-scale dispersion processes [Dagan, 1982]. (2) (1) distance x Fig. 1. Spatial concentration variance versus distance. Unless experimental data from various geological environments are available, the effect of scale-dependent dispersivity remains at large. REFERENCES Dagan, G., Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, The solute transport, Water Resour. Res., 18(4), 835-848, 1982. Pickens, J. F., and G. E. Grisak, Modeling of scale-dependent dispersion in hydrogeologic systems, Water Resour.

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A new technique is employed to calibrate a regional‐scale groundwater flow model to an extensive data base of undisturbed (i.e., assumed to be steady state) and transient heads. The methodology presented in this study is similar in concept to one presented by de Marsily et al. (1984) in that an adjoint sensitivity technique is coupled with a kriging algorithm to calibrate a flow model. The notable difference of the methodology presented in this paper is that it directly identifies the regions where modification of the model's kriged transmissivity or boundary pressure values will directly improve the overall fit between measured and model‐calculated heads at selected wells. At the locations identified as most sensitive to transmissivity changes, synthetic transmissivity values, referred to as pilot points, are added to the transmissivity data base and used as input for kriging the transmissivity field. An application of the methodology to data originating from approximately 10 years of regional hydrogeologic site characterization efforts that have been conducted in the Culebra dolomite at the Waste Isolation Pilot Plant in southeastern New Mexico is presented.