The complex variation of hydraulic conductivity in natural aquifer materials is represented in a continuum sense as a spatial stochastic process which is characterized by a covariance function. Assuming statistical homogeneity, the theory of spectral analysis is used to solve perturbed forms of the stochastic differential equation describing flow through porous media with randomly varying hydraulic conductivity. From analyses of unidirectional mean flows which are perturbed by one-and three-dimensional variations of the logarithm of the hydraulic conductivity, local relationships between the head variance and the log conductivity variance are obtained. The results show that the head variance produced by three-dimensional statistical isotropic conductivity perturbations is only 5% of that in the corresponding one-dimensional case. The head variance is also strongly dependent on the correlation distance of the log conductivity covariance function. These results emphasize the importance of including spatial correlation structure and multidimensional effects in stochastic simulation of groundwater flow. HYDRAULIC CONDUCTIVITY AS A STOCHASTIC PROCESSCasual observation of roadcuts, gravel pits, and other outcrops of sedimentary deposits which are potential water-transmitting units demonstrates that properties which affect hydraulic conductivity, such as grain size, are highly variable even within a given geologic deposit. One would also notice that the variation of the properties is not completely disordered in space; rather, one may observe a structured arrangement of bodies of different sediment types which may exhibit typical dimensions but are not completely regular. These two characteristics would also be seen in quantitative observations of flow properties such as geophysical well logs or laboratory tests of core samples. The properties are highly variable; hydraulic conductivity may vary by 3 orders of magnitude and porosity by tens of percent within a single sedimentary deposit. Such data also show some spatial structure which might be described as layers of clay, sand, or gravel with recognizable but variable thickness.Aware of this complex structure and extreme variability of flow properties, groundwater hydrologists and others have undertaken the formidable task of trying to observe and predict the quantity and quality of waters moving through these materials. This has been accomplished to some degree by ignoring the complexity or more appropriately by some implicit averaging of the flow equation to introduce an average flow property. Transmissivity is an example of such an averaged property which results from integration of the flow equation over depth. Major advances in computer-based methods during the last decade have made it possible to solve very complicated flow equations with complex boundary conditions and parameter configurations. However, most hydrologists now recognize that the predictive capabilities of such models are limited because the parameters of the models are difficult to determine. Much...

Steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques. The hydraulic conductivity K is related to the capillary pressure head • by K = K s exp (-•),where Ks is the saturated conductivity, and • is a soil parameter. A general formulation is presented for the case with Ks and • represented as statistically homogeneous spatial random fields. In part 1, solutions are developed assuming • is constant and representing K s variability by one-dimensional and three-dimensional isotropic random fields. Results are obtained for head variances and covariance functions, effective hydraulic conductivities, variances of the unsaturated hydraulic conductivity, flux variances, and variance of pressure gradient. When the parameter • is relatively large, corresponding to coarse textured soils, the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one-dimensional vertical flow. The effective conductivity is dependent on the correlation scale of In K s and the mean hydraulic gradient.

The longitudinal dispersion produced as a result of vertical variations of hydraulic conductivity in a stratified aquifer is analyzed by treating the variability of conductivity and concentration as homogeneous stochastic processes. The mass transport process is described using a first‐order approximation which is analogous to that of G. I. Taylor for flow in tubes. The resulting stochastic differential equation describing the concentration field is solved using spectral representations. The results of the analysis demonstrate that for large time the longitudinal dispersivity approaches a constant value which is dependent on statistical properties of the medium. The analysis also describes the transient development of the dispersive process and some non‐Fickian effects which occur early in the displacement process.

Steady unsaturated flow in heterogeneous soil with an arbitrarily oriented mean hydraulic gradient is analyzed using spectral solutions of the stochastic perturbation equation which describes the capillary pressure head ½. The unsaturated hydraulic conductivity is related to ½t by K = Ks exp (-•½), where Ks is the saturated hydraulic conductivity and • is a soil parameter, and both K s and • are treated as three-dimensional statistically homogeneous, anisotropic random fields. Analytical results are obtained for the capillary pressure head variance and the effective (mean) unsaturated hydraulic conductivity. The head variance depends upon the degree of anisotropy of the In K s covariance; when • is random, the head variance increases with mean capillary pressure head. The effective hydraulic conductivity for arbitrary orientation of the mean hydraulic gradient J is shown to have tensorial properties, but its components depend on the magnitude and direction of J and the orientation of the stratification in the soil. When • is random, the degree of anisotropy of the effective conductivity depends strongly on mean capillary pressure.

Abstract. This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical (LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the "true" transmissivity field, or the presence of "anomalies" such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the 1og•0 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method. •2Stanford University, Stanford, California.•3Duke Engineering and Services, Inc., Austin, Texas.•4University of Arizona, Tucson.•Slnstitut Franqais du Pftrole, Rueil-Malmaison, France.•6University of California, Berkeley.Copyright 1998 by the American Geophysical Union. Paper number 98WR00003.0043-1397/98/98WR-00003509.00 tion, or performance assessment of planned waste disposal projects, it is no longer enough to determine the "best estimate" of the distribution in space of the aquifer parameters. A measure of the uncertainty associated with this estimation is also needed. Geostatistical techniques are ideally suited to filling this role. Basically, geostatistics fits a "structural model" to the data, reflecting their spatial variability. Then, both "best estim...