VOL. 14, NO. 2 WATER RESOURCES RESEARCH APRIL 1978 Stochastic Analysis of SpatialVariability in Subsurface Flows 1. Comparison of One- and Three-Dimensional Flows ADEL A. BAKR, • LYNN W. GELHAR,ALLAN L. GUTJAHR, AND JOHN R. MACMILLAN New Mexico Institute of Mining and Technology, Socorro,New Mexico 87801 The complex variation of hydraulic conductivity in natural aquifer materialsis represented in a continuum sense as a spatialstochastic process which is characterized by a covariance function. Assuming statistical homogeneity, the theoryof spectral analysis is used to solve perturbed formsof the stochastic differential equation describing flow through porous media with randomly varying hydraulic con- ductivity.From analyses of unidirectional mean flows whichareperturbed by one-and three-dimensional variationsof the logarithm of the hydraulicconductivity, local relationships between the head variance and the log conductivity varianceare obtained.The results showthat the head variance produced by three-dimensional statistical isotropicconductivity perturbations is only 5% of that in the corresponding one-dimensional case. The head variance is also strongly dependent on the correlationdistance of the log conductivity covariance function. These results emphasize the importance of including spatial correlation structure and multidimensional effects in stochastic simulationof groundwaterflow. HYDRAULIC CONDUCTIVITY AS A STOCHASTIC PROCESS Casual observationof roadcuts, gravel pits, and other out- cropsof sedimentary deposits which are potentialwater-trans- mitting units demonstrates that propertieswhich affect hy- draulic conductivity, such as grain size, are highly variable even within a given geologicdeposit. One would also notice that the variation of the properties is not completely dis- ordered in space; rather, one may observe a structured ar- rangementof bodies of different sedimenttypes which may exhibit typical dimensions but are not completely regular. These two characteristics would also be seen in quantitative observations of flow propertiessuchas geophysical well logs or laboratory tests of core samples. The properties are highly variable; hydraulic conductivitymay vary by 3 ordersof mag- nitude and porosity by tens of percent within a single sedi- mentary deposit. Such data also show some spatial structure whichmight be described aslayers of clay, sand, or gravelwith recognizable but variable thickness. Aware of this complexstructure and extremevariability of flow properties, groundwater hydrologistsand others have undertakenthe formidable task of trying to observe and pre- dict the quantity and quality of watersmoving through these materials. This has been accomplished to some degree by ignoring the complexityor more appropriately by someim- plicit averagingof the flow equation to introduce an average flow property. Transmissivity is an exampleof suchan aver- aged property which results from integration of the flow equa- tion over depth. Major advances in computer-based methods during the last decade have made it possibleto solve very complicated flow equations with complex boundary condi- tions and parameter configurations. However, most hydrol- ogistsnow recognize that the predictivecapabilities of such models are limited because the parameters of the models are difficult to determine. Much of this difficulty may stem from the natural variability of the medium. The generalobjective of this study is to evaluatehow the inherent spatial variability of aquifer properties will influencetraditional flow observations and modeling.Using a stochastic approach,we will emphasize •On leave from the Middle EasternRegional Radioisotope Centre for the Arab Countries, Dokki, Cairo, Egypt. Copyright¸ 1978by the AmericanGeophysical Union. Paper number 7W1137. 0043-1397/78/027W-1137503.00 two interrelated features which we feel are basic to the under- standing of the problem; these are (1) the continuum nature of the medium and flow system description and (2) the spatial correlation structure of the medium properties. First we want to emphasize what is already implicit in the use of terms like hydraulic coriductivityand porosity, i.e., the flow system is treated as a continuum. Conceptually, the gov- erning field equations result from an averaging over a small 'representative elementary'volume [Bear, 1972] which encom- passes many poresor flow passages but is small in comparison with the scale of heterogeneities in the system. Thus in terms of mathematical description the hydraulic conductivity, for ex- ample, is a function K(x) defined at every point x = (x•, x•.,xa) in space. In the usual statisticalapproach to describing variability of flow properties, the observations of hydraulicconductivity,for example, are treated as being statisticallyindependentregard- less of their spatial location, and the data are used to estimate a probability density function for the property. However, it is clear that this approach neglects the spatial structureinherent in the original data, because of the structuredarrangement of different sedimenttypes in the natural medium. A more com- plete approach involves a statisticaldescriptionwhich incor- porates the spatial structureof the property. In developing a stochastic description of the aquifer proper- ties we recognizethat it is neither desirablenor practical to observe every detail of the hydraulic conductivity field. Rather, it is appropriate to characterizethis highly complex spatial structure in terms of statistical quantities. Thus we choose to represent the hydraulicconductivitycontinuum as a spatial stochastic process or a random field in three dimen- sions. In contrast to a single random variable which is de- scribed completelyby its probability densityfunction, a con- tinuum random field requires joint probability density functions between all points in spaceto describethe process completely. Data will practicallyneverbe available to estimate these joint densities; a more practical approach is to consider only certainmoments of thejoint densities. The generalmixed moments of the field are defined by E[K'•(x,)K'•(x•.)] -- ff K,'•K•.'•iS(K,, K•.; x,, x•.) dK, dK•. (1) where• is thejoint density of K(x) at positions x• and x•., K• = 263