The most realistic representation of a naturally occurring porous medium is a stochastic set of macroscopic elements in which the values of the three basic hydrogeologic parameters (hydraulic conductivity K, compressibility α, and porosity n) are defined by frequency distributions. A homogeneous formation under this representation is one in which the frequency distributions do not change through space. All soils and geologic formations, even the ones that are homogeneous, show random variations in the values of the hydrogeological parameters through space; that is, they are nonuniform, and a measure of the nonuniformity is provided by the standard deviation of the frequency distributions. If K and α are log normally distributed and n is normally distributed, and if we define Y = log K and C = log α, then the parameters Y, C, and n can be generated from a multivariate normal density function with means μy, μc, and μn, standard deviations σy, σc, and σn, and correlation coefficients ρyc, ρyn, and ρcn The analysis of groundwater flow in nonuniform media requires a stochastic-conceptual approach in which the effects of stochastic parameter distributions on predicted hydraulic heads are analyzed with the aid of a set of Monte Carlo solutions to the pertinent boundary value problems. In this study, two one-dimensional saturated flow problems are analyzed: steady state flow between two specified heads and transient consolidation of a clay layer. The primary output is the statistical distribution of hydraulic head ϕ, through space and time, as indicated by the mean values and their standard deviations Sϕ¯(x, t) Results show that the standard deviations of the input hydrogeologic parameters, particularly σy and σc, are important index properties; changes in their values lead to different responses for even when the means μy, μc, and μn are fixed. The degree of uncertainty associated with hydraulic head predictions increases as the degree of nonuniformity of the porous medium increases. For large values of σy and σc it becomes virtually impossible to obtain meaningful hydraulic head predictions. For transient flow the output distribution of hydraulic head values is almost never normal; in some cases it approaches a uniform distribution. The results of this study throw into question the validity of the hidden assumption that underlies all deterministic groundwater modeling; namely, that it is possible to select a single value for each flow parameter in a homogeneous but nonuniform medium that is somehow representative and hence define an ‘equivalent’ uniform porous medium. For transient flow there may be no way to define an equivalent medium. The fact that nine index parameters rather than three are required to describe a nonuniform geologic formation, the large uncertainties in predicted hydraulic heads for relatively simple flow problems in nonuniform soils, and the contention that there may be no simple way to define an equivalent uniform porous medium all have important implications in the development of groundwater flow theory and in its most fundamental applications.

A Stochastic-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media

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.

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

One of the most important limitations to the application of modeling techniques for the analysis of mass transfer in groundwater systems is the difficulty in characterizing the dispersive character of natural systems. Although large quantities of experimental data exist for laboratory scale experiments, the handful of measurements which have been obtained from regional systems suggest that dispersion produced by large-scale porous medium nonidealities is considerably more important. In this paper the effects of macroscopic dispersion are simulated for uniform heterogeneous porous media under conditions of one- dimensional flow. The idealized media considered for detailed analysis consist of low permeability inclusions within a higher-permeability medium. When the inclusions are not arranged rather homogeneously within the region, a unique dispersivity value for the medium cannot be defined, and dispersivity changes as a function of space. The magnitude of dispersion is controlled by the contrast in hydraulic conductivity between the inclusions and the remainder of the medium, the number of inclusions, and the mode of aggregation. Generally, dispersivity is found to decrease as the conductivity contrast decreases and the structure of the medium is regularized. It will be possible to estimate the dispersivity of a medium by using stochastic analysis. Hypothetical porous media with characteristics similar to those of some actual medium will yield a range of dispersivity values. The basic data for this technique will be detailed statistical analysis of the mode of porous medium aggregation and the conductivity contrasts within the medium.

Macroscopic dispersion in porous media: The controlling factors

Notes on stochastic shales; from outcrop to simulation model

Abstract Whilst it may be easy to describe and quantify a sedimentary succession from outcrop or core, using these data to predict sedimentary facies, diagentic features and fracture distribution in unsampled areas still requires considerable intuition. Frequently, at the current state of geological knowledge, reservoir datasets must be supplemented by consideration of analogues. The aim of this paper is to sound caution on the indiscriminate use of analogues in reservoir geology. Analogues used in reservoir geology fall into three main categories: comparisons with modern depositional processes, experimental simulations, and rock record analogues. All analogues used for reservoir geology suffer from limitations due to the physical and temporal scales at which the comparisons are made. Difficulty in selecting appropriate analogues results from the amount of interpretation of the nature of the reservoir required before an analogue may be selected, the large number of degrees of freedom in the processes of formation of reservoir rocks, and the limited numbers of well-documented analogues which are easily available.

A discussion on the use of analogues for reservoir geology

A method was developed for the analysis of seepage through nonhomogeneous soil deposits. The layers of pervious soils in the deposit were considered as inclusions within a medium of much lower permeability. The locations and dimensions of the inclusions were assumed to be random variables. Field studies of excavations provided the basis for the statistical representation of the inclusions. Data obtained from the Mississippi River alluvium and fluvial glacial deposits were interpreted by means of the statistical model. The results were used to compute the seepage at the Port Allen Lock excavation and the Granville Dam. The computed seepage is of the right order of magnitude when compared with the measured seepage.

Shyam K. Vyas

Papers

Probabilistic analysis of seepage