资源描述
Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Diffusion, Electrical Conduction, and Flow in Porous Media,Continuum Percolation Theory for Truncated Random Fractal Media,1,Two Basic Applications of Percolation Theory to Flow,Near Perc. Threshold,Gives scaling of hydraulic, electrical conductivity, air permeability with distance from threshold moisture 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)!,2,And One to Accessibility (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.,3,Apply to Random (Truncated) Fractal Model of Porous Media,What is optimal form of percolation theory?,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.,4,Basic Results for Saturation Dependence of Properties,K(S) pore-size distribution dominates at high sat., connectivity/tortuosity at low sat.,Electrical Conductivity similar,Air permeability connectivity/tortuosity dominates throughout,Solute and gas diffusion, connectivity/tortuosity dominate throughout,5,Consider First Unsaturated K on Lattice of Tubes,6,7,Appropriate Analogy in Continuum Percolation Representation,8,9,Result for K(,),Result from bottleneck,pore.,Result from average,resistance to flow over,all larger pores.,Note argument is,not,-,t,(unless,=1)!,10,Effects of Tortuosity, Connectivity,Hydraulic Conductivity,Electrical Conductivity,t=1.88 (3D),=1.28 (2D),Require both K and,dK/d,to be continuous,at unknown,x,; this,generates prefactor,and ,x,.,In vicinity of,t,conductivity,must scale as,11,Hanford site data (Rockhold et al., 1988),Connectivity/tortuosity,Pore-sizes,12,Comparisons with Other Results,Balberg, Phil. Mag. 1987,if,W,(,g,) continues to,g,=0,But if distribution continues to,g,=0, this means that smallest pore has,zero radius. That means that,=1 in Rieu and Sposito model and,yields Balberg result since,at,=1 (also for,),13,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,Experiment finds,m=1.86 Thompson et al., 1987,Kuentz et al., 2000, find (in 2-D simulations) m=1.28,14,What happens if Archies law is not quite valid (cross-over occurs,before saturation)?,Electrical conductivity at,saturation is enhanced by,15,R,2,increases to over 0.3 if single point is eliminated,Data from Katz and Thompson, Advances in Physics, 1987,16,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.,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.,17,Theory has consequences for saturation dependence of electrical conductivity,Data from Tusheng Ren for silt loams (two adjustable parameters,for entire family of curves).,18,Data from Tusheng Ren for,sands. Power ca. 2.5, not 1.88;,Balberg non-universality?,19,Additional data from Andrew Binley,Both cases one adjustable parameter,other parameters from Cassiani,20,Data from Jeffrey Roberts,21,What About Air Permeability?,22,23,Result observed by Moldrup et al., (2003) with power equal to,1.84,0.54. 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 al. (1999) compared,with power of 1.28; one adjustable,parameter.,24,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,=0.88,25,Final form substituting moisture,content for porosity,26,27,28,29,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 parameters).,30,Data compiled by Werner et al., 2004 (VZJ),“The Moldrup relationship,D,pm,/,D,a,=,2.5,/, originally proposed for sieved,and repacked soils, gave the best predictions of several porosity-based,relationships” (Werner et al., 2004),31,Repacked soils (presumably without structure),32,Rieu and Sposito WaterRetention Curves,Continuous truncated,random fractal; analogy,to Rieu and Sposito model,Water retention curve,Porosity,33,34,35,Hydraulic Conductivity Limited Equilibration,Cross-over to regime,of rapidly diminishing,K will have consequences,for equilibration on,ceramic plates,36,Hypothesis: At Moisture Contents Approaching Threshold Actual Water Removed is reduced by ratio of,to,This is effectively a zero-dimension model with correct constitutive relations,rather than a column model with inaccurate constitutive relations.,37,38,The deviations in the fractal scaling of the water retention curves are predicted by the hydraulic conductivity derived from the fractal model using percolation theory for the hydraulic conductivity.,This means that the fractal model actually predicts the deviations from the fractal model.,Note that the deviations occurred at a moisture content related to the moisture content at which solute diffusion vanished in soils from another contintent (Moldrup et al., 2001).,39,Hysteresis has Two Components,Water removed from a pore must pass through a pore throat; water imbibed must fill the entire pore body. Pressure ratio is ratio of two radii (typically about 2).,All pores that allow water will contain that water during drainage, but only those connected to the infinite percolation cluster will contain water during imbibition.,40,Percolation Component,Rieu and Sposito random fractal,water retention curve.,Same accessibility factor as for,gas diffusion.,Product of two gives imbibition,curve.,41,Data from,Bauters et al.,(1998),Air entry pressure,=11cm,Characteristic pressure,=5.5cm,Ratio of pressures,consistent with other,experiments.,42,Summary,K,S,Competition,K(S)Mainly CPA,(S)Mainly Scaling,k,a,(=0)Competition,k,a,()Entirely Scaling,DEntirely Scaling,43,Conclusions,Application of continuum percolation theory to fractal models yields results for saturation scaling of,Hydraulic conductivity,Air permeability,Electrical conductivity,Pressure-saturation curves including hysteresis and lack of equilibration,Solute diffusion,Gas diffusion (worst comparison with expt., still best theory),in agreement or accord with experiment.,Theory yields consistent interpretations, parameters, analysis of relevance of pore size distributions compared with, e.g., connectivity/tortuosity issues.,Theory reinterprets fundamental soil physics.,44,Relevant References,Hunt, A. G., and Gee, G. W., 2002, Application of Critical Path Analysis to Fractal Porous Media: Comparison with Examples from the Hanford Site,Advances in Water Resources,25, 129-146.,Hunt, A. G., and Gee, G. W., 2002, Water Retention of Fractal Soil Models Using Continuum Percolation Theory: Tests of Hanford Site Soils,Vadose Zone Journal,1, 252-260.,Hunt, A. G., 2003, Percolative Transport and Fractal Porous Media,Chaos, Solitons, and Fractals,19, 309-325.,Hunt, A. G., and Ewing, R. P., 2003, On The Vanishing of Solute Diffusion in Porous Media at a Threshold Moisture Content,Soil Science Society of America Journal,67, 1701-1702, 2003.,Hunt, A. G., and Gee, G. W., 2003, Wet-End Deviations from Scaling of the Water Retention Characteristics of Fractal Porous Media,Vadose Zone Journal,2, 759-765.,Hunt, A. G., 2004, Continuum Percolation Theory for Water Retention and Hydraulic Conductivity of Fractal Soils: 1. Estimation of the Critical Volume Fraction for Percolation,Advances in Water Resources,27, 175-183.,Hunt, A. G., 2004, Continuum Percolation Theory for Water Retention and Hydraulic Conductivity of Fractal Soils: 2. Extension to Non-Equilibrium,Advances in Water Resources,27, 245-257.,Hunt, A. G., 2004, A note comparing van Genuchten and percolation theoretical formulations of the hydraulic properties of unsaturated media, accepted, June., 2004 to,Vadose Zone Journal,.,Hunt, A. G., 2005, Continuum Percolation Theory for Saturation Dependence of Air Permeability, February issue of,Vadose Zone Journal,.,Hunt, A. G., 2004, Continuum Percolation Theory and Archies Law Oct., 2004 issue of,Geophysical Research Letters,.,Hunt, A. G., and T. E Skinner, 2005, Hydraulic Conductivity Limited Equilibration: Effect on Water-Retention Characteristics, February issue of,Vadose Zone Journal,.,Hunt, A. G., 2005, Comment on Ultra-low Frequency Magnetic Precursors to the Loma Prieta earthquake, by S. Klemperer and Merzer, Accepted to Pure and Applied Geophysics, January, 2005.,45,
展开阅读全文