A stochastic analysis of groundwater flow leads to probability distributions on the predicted hydraulic head values. This variability reflects our uncertainty in the system being modeled due to the spatial heterogeneity of hydraulic conductivity. Monte Carlo techniques can be used to estimate the head distributions. This approach relies on the repetitive generation of discrete-block conductivity
realizations. In this study, steady state flow through one and two-dimensional flow domains is investigated. A space law based on a first order, nearest neighbour stochastic process model is used to generate the multilateral spatial dependence in the hydraulic conductivity values within the block structure. This allows consideration of both statistically isotropic and anisotropic autocorrelation functions.
It is shown that the probability distribution of hydraulic head and the head gradient or the flux across the boundaries of the flow domain, must be interpreted in terms of:
1) The spatial variation of expected head gradients.
2) The standard deviation in the conductivity distribution.
3) The ratio of the integral scale of the autocorrelation function for conductivity to the distance between boundaries on the flow domain.
4) The arrangement of stationary units within the flow domain.
The standard deviations in hydraulic head increase with an increase in either the conductivity standard deviation or the strength of the correlation between neighbouring conductivity values. Provided the integral scales of the medium are preserved, the standard deviations in head show only a minor dependence on the discretization interval. The head standard deviations are approximately halved in a two-dimensional model from those in a one-dimensional model with an equivalent space law. Spatial trends in the mean conductivity can considerably alter the magnitude and spatial variation in the hydraulic head standard deviations.
The geometric mean has been suggested by others as a suitable effective conductivity in a heterogeneous two-dimensional flow domain. This study shows that only in the case of uniform flow through a single stationary unit is this concept valid. If the mean gradient field is nonuniform, or if the mean conductivity has a spatial trend, predictions based on the geometric mean do not satisfy the necessary equivalence criteria.
Direct comparisons cannot be made, but the Monte Carlo and spectral approaches to the solution of the stochastic flow equations predict a similar behavior.
A first order, nearest neighbour model is matched to a data
set collected from a relatively uniform but stratified, unconsolidated sand deposit. The data show statistically anisotropic autocorrelation functions, both in the integral scale and in the functional form of the correlation. A broader class of spatial models may need to be considered to describe the cyclic behavior of sedimentary sequences. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/21288 |
| Date | January 1978 |
| Creators | Smith, James Leslie |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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