Stochastic analysis of spatial variability in subsurface flows: 1. Comparison of one‐ and three‐dimensional flows

Abstract

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...

Stochastic modelling of groundwater flow and solute transport in aquifers

“…Gelhar, 1976;Bakr et al, 1978;Gutjahr et al, 1978;Mizell et al, 1982;Gelhar et al, 1979;and Gelhar and Axness, 1983;Yeh et al, 1985a, b, c;Mantoglou and Gelhar, 1987a, b, c). To illustrate the approach, let us consider steady-state flow in a heterogeneous but locally isotropic aquifer (i.e., the hydraulic conductivity at the scale of core samples is isotropic) with infinite lateral extent.…”

“…Using the perturbation-spectrum technique and assuming statistical isotropy in In K , Bakr et al (1978) solved Equation 12 to analyse the effects of spatial variability on steady-state groundwater flow. Closed forms of head variance expressions (a statistic measure of the deviation of the head value at a point from the mean head) were derived for one-and three-dimensional flow fields in which hydraulic conductivity is a spatially-varying stochastic process.…”

Spectral approach to seawater intrusion in heterogeneous coastal aquifers

“…(22), the solution of Eq. (20) can now be developed using the principle of superposition (e.g., Farlow, 1993) and the spectral representation (e.g., Bakr et al, 1978;McLaughlin, 1991, 1995) based on Fourier-Stieltjes representation for the perturbed quantities in wave number domain. Superposition is a useful tool in analyzing linear groundwater problems (e.g., Townley, 1995;Trefry, 1999).…”

A brief review of bacterial transport in natural porous media

“…Upscaling the dispersion (of a nonreactive tracer) in physically heterogeneous media has received significant attention in the literature (e.g., Dagan 1986). The upscaled dispersive process is known as "macrodispersion" and is represented either through modification of regular (small-scale) dispersion coefficients in time pagan 1984) or space (Bakr et al 1978), or through introduction of memory into the spreading process (Cushman and Ginn 1993). Exploration of stochastic theories for representing dispersive mixing in the heterogeneous subsurface has brought substantial progress in characterizing macrodispersion in certain aquifer types (cf.…”