Channeling and Fickian Dispersion in fractal simulated porous media

Grindrod, Peter; Impey, M. D.

doi:10.1029/93wr01286

Cited by 39 publications

(25 citation statements)

References 2 publications

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“…However, Neuman 's [1990] estimate of H (his ω) based on using ln K fBm to explain the observed scale dependence of apparent dispersivity was 0.25. When the Molz and Boman [1993] data were reanalyzed assuming that the vertical variation of ln( K ) was represented by the nonstationary function fBm [ Molz and Boman , 1995], the resulting Hurst coefficient was around 0.3, implying negative correlation of the ln K increments, reasonably consistent with the Neuman [1990] ln K result and the porosity result ( H = 0.34) of Grindrod and Impey [1993]. Early work by Kemblowski and Chang [1993] using relatively small data sets resulted in fBm‐based H values in the vertical and horizontal directions of 0.18 and 0.53, respectively.…”

Section: Fractals and Multifractals In Subsurface Hydrologymentioning

confidence: 66%

Stochastic fractal‐based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions

Molz

Rajaram

2004

Reviews of Geophysics

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Modern measurement techniques have shown that property distributions in natural porous and fractured media appear highly irregular and nonstationary in a spatial statistical sense. This implies that direct statistical analyses of the property distributions are not appropriate, because the statistical measures developed will be dependent on position and therefore will be nonunique. An alternative, which has been explored to an increasing degree during the past 20 years, is to consider the class of functions known as nonstationary stochastic processes with spatially stationary increments. When such increment distributions are described by probability density functions (PDFs) of the Gaussian, Levy, or gamma class or PDFs that converge to one of these classes under additions, then one is also dealing with a so‐called stochastic fractal, the mathematical theory of which was developed during the first half of the last century. The scaling property associated with such fractals is called self‐affinity, which is more general that geometric self‐similarity. Herein we review the application of Gaussian and Levy stochastic fractals and multifractals in subsurface hydrology, mainly to porosity, hydraulic conductivity, and fracture roughness, along with the characteristics of flow and transport in such fields. Included are the development and application of fractal and multifractal concepts; a review of the measurement techniques, such as the borehole flowmeter and gas minipermeameter, that are motivating the use of fractal‐based theories; the idea of a spatial weighting function associated with a measuring instrument; how fractal fields are generated; and descriptions of the topography and aperture distributions of self‐affine fractures. In a somewhat different vein the last part of the review deals with fractal‐ and fragmentation‐based descriptions of fracture networks and the implications for transport in such networks. Broad conclusions include the implication that models based on increment distributions, while more realistic, are inherently less predictive than models based directly on stationary stochastic processes; that there is presently an unresolved ambiguity when a measurement is attempted in a medium that exhibits property variations on all scales; the strong possibility that log(property) increment distributions that appear to be described by the Levy PDF are actually superpositions of several PDFs of finite variance, one for each facies; that there are apparent similarities in the transport behavior of heterogeneous porous media and fractured rock at the field scale that appear to be related to the existence of a few preferential flow paths in both types of media; and finally, that additional carefully collected data sets are needed to clarify and advance the fractal‐based theories, particularly in the case of three‐dimensional fracture networks where few data are available. Further refinement is needed also in the understanding of instrument spatial weighting functions in heterogeneous media and how measuremen...

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Section: Fractals and Multifractals In Subsurface Hydrologymentioning

confidence: 66%

Stochastic fractal‐based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions

Molz

Rajaram

2004

Reviews of Geophysics

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“…The influence of aperture variability on flow and solute transport in two-dimensional fissures is evident in several numerical simulation studies [e.g., Brown, 1987;Moreno et al, 1988;Tsang and Tsang, 1989;Grindrod and Impey, 1993] and has been characterized theoretically [e.g., Gelhar, 1987Gelhar, , 1993. In particular, significant flow channeling is evident in these numerical simulation studies.…”

Section: Introductionmentioning

confidence: 99%

Influence of aperture variability on dissolutional growth of fissures in Karst Formations

Hanna¹,

Rajaram²

1998

Water Resources Research

149

171

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Abstract. The influence of aperture variability on dissolutional growth of fissures is investigated on the basis of two-dimensional numerical simulations. The logarithm of the fissure aperture before dissolution begins is modeled as a Gaussian stationary isotropic random field. The initial phase of dissolutional growth is studied up to the time when turbulent flow first occurs at a point within the fissure (the breakthrough time). The breakthrough time in variable aperture fissures is smaller than that in uniform fissures and decreases as the coefficient of variation of the aperture field (cr//x) increases. In comparing uniform and variable aperture fissures in limestone, the breakthrough time with cr//x = 0.1 is about a factor of 2 smaller than that in a uniform fissure. The breakthrough time is reduced by about an order of magnitude with cr//x = 2.0. The mechanism leading to reduced breakthrough times is the focusing of flow into preferential flow channels which are enlarged at a faster rate than the surrounding regions of slower flow. Dissolution channels are narrower and more tortuous as cr//x increases. Investigations of the influence of reaction rate reveal that the influence of aperture variability is more pronounced in rapidly dissolving rock. In uniform fissures in rapidly dissolving minerals, breakthrough times are very long since water becomes saturated with respect to the mineral within a short distance of the entrance to the flow path. However, in variable aperture fissures, breakthrough occurs rapidly because of selective growth along preferential flow channels, which progressively capture larger fractions of the total flow. These results partly explain why conduits develop rapidly in gypsum, although previous one-dimensional studies suggest that conduit growth will not occur. IntroductionQuantitative modeling of the evolution of hydraulic conduits in karst formations is an area of active research. Initiation of conduits during the early evolution of karst is of particular interest because it is believed to be the slowest phase of cave development. In previous modeling studies, Dreybrodt [1990Dreybrodt [ , 1996, Palmer [1991], and Groves and Howard [1994a, b] observed that once turbulent flow is initiated, further dissolutional growth occurs at a relatively rapid rate. In these works the duration of the initial phase of karst evolution is quantified using the concept of "breakthrough time," which refers to the time at which turbulent flow is first encountered. As breakthrough is approached, the flow rate through the system increases rapidly, leading eventually to turbulent flow. After breakthrough occurs, conduits are expected to grow at a relatively rapid rate of a few millimeters a year [Dreybrodt, 1990[Dreybrodt, , 1996 Groves and Howard, 1994a, b]. Thus the occurrence of breakthrough ensures further development of caverns. The time required for the onset of turbulent flow typically depends on the length of the fissure, the hydraulic gradient, and the initial aperture. Typical breakthrough time...

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“…The change of type of diffusion may not necessarily be related to the scale. Grinrod and Impey (1993) simulated solute transport in saturated two-dimensional fractal porous medium and found that at early stages an anomalous transport occurs, whereas the later breakthrough is predominantly Fickian. Changes of the diffusivity scaling as the water transport in soil progresses seem to be an interesting issue to explore.…”

Section: Resultsmentioning

confidence: 99%

Water transport in soils as in fractal media

Pachepsky

Timlin²

1998

Journal of Hydrology

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Fractal scaling laws of water transport were found for soils. A water transport model is needed to describe this type of transport in soils. We have developed a water transport equation using the physical model of percolation clusters, employing the mass conservation law, and assuming that hydraulic conductivity is a product of a local component dependent on water content and a scaling component depending on the distance traveled. The model predicts scaling of water contents with a variable x/t v(2dJ) wherefl deviates from the zero value characteristic for the Richards equation. A change in the apparent water diffusivity with the distance is predicted if the apparent diffusivity is calculated using the Richards equation. An equation for the time and space invariant soil water diffusivity is obtained. Published data sets of five authors were used to test the scaling properties predicted by the model. The value offl was significantly greater than zero in almost all data sets and typically was in the range from 0.05 to 0.5. This exponent was found from regression equations that had correlation coefficients from 0.97 to 0.995. In some cases a dependence offl on water content was found indicating changes in scaling as the water transport progressed.

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Channeling and Fickian Dispersion in fractal simulated porous media

Cited by 39 publications

References 2 publications

Stochastic fractal‐based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions

Stochastic fractal‐based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions

Influence of aperture variability on dissolutional growth of fissures in Karst Formations

Water transport in soils as in fractal media

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