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1、Diffusion, Electrical Conduction, and Flow in Porous Media,Continuum Percolation Theory for Truncated Random Fractal Media,Two Basic Applications of Percolation Theory to Flow,Near Perc. Threshold Gives scaling of hydraulic, electrical conductivity, air permeability with distance from threshold mois
2、ture content. Connectivity/ tortuosity,Far from Perc. Threshold Gives dependence of same properties on moisture content through dependence of “bottleneck” pore size. Pore-size dependence,NOTE: These two applications are not independent (same critical percolation probability for both)!,And One to Acc
3、essibility (in Hysteresis),The probability that a given site (or volume) is connected to the infinite cluster above the percolation threshold is given by a universal function of percolation theory.,Apply to Random (Truncated) Fractal Model of Porous Media,What is optimal form of percolation theory?
4、Continuum. What are relevant variables? Critical volume fraction (moisture content, air-filled porosity). What about potential non-universal behavior? Much, though not all behavior is universal.,Basic Results for Saturation Dependence of Properties,K(S) pore-size distribution dominates at high sat.,
5、 connectivity/tortuosity at low sat. Electrical Conductivity similar Air permeability connectivity/tortuosity dominates throughout Solute and gas diffusion, connectivity/tortuosity dominate throughout,Consider First Unsaturated K on Lattice of Tubes,Appropriate Analogy in Continuum Percolation Repre
6、sentation,Result for K(),Result from bottleneck pore.,Result from average resistance to flow over all larger pores.,Note argument is not - t (unless =1)!,Effects of Tortuosity, Connectivity,Hydraulic Conductivity,Electrical Conductivity,t=1.88 (3D) =1.28 (2D),Require both K and dK/d to be continuous
7、 at unknown x; this generates prefactor and x.,In vicinity of t conductivity must scale as,Hanford site data (Rockhold et al., 1988),Connectivity/tortuosity,Pore-sizes,Comparisons with Other Results,Balberg, Phil. Mag. 1987 if W(g) continues to g=0,But if distribution continues to g=0, this means th
8、at smallest pore has zero radius. That means that =1 in Rieu and Sposito model and,yields Balberg result since, at =1 (also for ),To find limit of validity of Archies law set,Archies law,Evaluate at saturation,Use proportionality of critical volume fraction to porosity (Hunt, AdWR),Archies Law,Exper
9、iment finds,m=1.86 Thompson et al., 1987,Kuentz et al., 2000, find (in 2-D simulations) m=1.28,What happens if Archies law is not quite valid (cross-over occurs before saturation)?,Electrical conductivity at saturation is enhanced by,R2 increases to over 0.3 if single point is eliminated,Data from K
10、atz and Thompson, Advances in Physics, 1987,Application to Loma Prieta Earthquake Precursor Signal,Ultra-low frequency magnetic field effects were interpreted (Merzer and Klemperer) as due to increase in fault zone conductivity by factor 15. Authors suggested Archies law exponent changed from 2 to 1
11、. Check: A change from 1.88 to 1.28 is expected for a change in dimensionality from 3 to 2. This is consistent with development of 2-D network of interconnected micro-fractures in hours before earthquake.,Theory has consequences for saturation dependence of electrical conductivity,Data from Tusheng
12、Ren for silt loams (two adjustable parameters for entire family of curves).,Data from Tusheng Ren for sands. Power ca. 2.5, not 1.88; Balberg non-universality?,Additional data from Andrew Binley,Both cases one adjustable parameter, other parameters from Cassiani,Data from Jeffrey Roberts,What About
13、Air Permeability?,Result observed by Moldrup et al., (2003) with power equal to 1.840.54. Note that the expected power in 2-dimensions is 1.28 (Derrida and Vannimenus, 1982, J. Phys.) (cases with power near 1 observed in clay-rich media, otherwise near 2).,Two-dimensional configuration Steriotis et
14、al. (1999) compared with power of 1.28; one adjustable parameter.,Solute and Gas Diffusion,Typical definition,Results of numerical simulations of Ewing and Horton x=system length,Finite-size scaling (Fisher, 1970),Value of universal exponent of percolation, =0.88,Final form substituting moisture con
15、tent for porosity,How to Modify for Gas Diffusion?,Trivial modification,But not all air allowable pores have air, only those accessible to the infinite percolation cluster.,Extra factor is from percolation theory and represents the fractional volume attached to the infinite cluster (no adjustable pa
16、rameters).,Data compiled by Werner et al., 2004 (VZJ),“The Moldrup relationship, Dpm/Da=2.5/, originally proposed for sieved and repacked soils, gave the best predictions of several porosity-based relationships” (Werner et al., 2004),Repacked soils (presumably without structure),Rieu and Sposito WaterRetention Curves,Continuous truncated random fractal; analogy to Rieu and Sposito model,Water retention curve,Porosity,Hydraulic Conductivity Limited Equilibration,Cr
