scaling in soil physics

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This presentation provides an introduction to scaling in soil physics following Miller-Miller similar media theory. Scaling soil hydraulic functions and Richards' equation is emphasized.


  • 1. Scaling in Soil Physics Morteza Sadeghi Department of Plants, Soils, and Climate, Utah State University

2. Similar media TheoryScaling in soil physics is based on Miller and Miller (1956) Similar media concept 3. Two similar media - are similar in their microscopic geometry and differ only in scale - have identical porosities3 4. Two non-similar mediaIdentical particle size distribution Different pore size distribution 5. Two dissimilar mediaDifferent particle size distribution Different pore size distribution 6. similar media in similar state identical water content (%)6 7. Similar media are scalable into each other by a scaling factor, a ratio of two corresponding physical lengths. 2/1 can scale the first media into the second.7 8. Scaling soil-water suction, h Capillary equation:1 pore radiushSimilar media in similar state:h1 1h2 nh*Scaled suction head, h*, is the same for all similar media in similar state 8 9. Scaling hydraulic conductivity, K Poiseuille equation: Kpore radius2Similar media in similar state:K1K22 12 2...Kn 2K*nScaled hydraulic conductivity, K*, is the same for all similar media in similar state 9 10. Klute and Wilkinson (1958) tested Similar-Media Concept 11. - Five similar media were made by sand particles - Similarity was defined based on shape of the particle size distributions - Mean particle size was used as the physicals length scale (scaling factor) of each soil - Millers scaled h and K were calculated.104 125 2Identical porosity 12. Scaled particle size distribution 13. *hUnscaled retention curveh Scaled retention curve 14. K*Unscaled conductivity curveK 2Scaled conductivity curve 15. Some conclusions found 1. Klute and Wilkinson (1958): Disagreement was apparent, particularly when the volumetric water content was greater than 0.3. 2. Elrick et al. (1959): Scaling theory worked well when the medium was clean sand, but much less well when the amount of colloid increased in the media. 3. Tillotson and Nielsen (1984): Application of scaling theory is restricted to use in sand or sandy soils. 16. Warrick et al. (1977) modifications 17. First Owing to the fact that soils do not have identical porosity, Warrick et al. (1977) used degree of saturation (S = /s) rather than volumetric water content.Art WarrickDon Nielsen 18. By this modification: - Media do not need having identical porosities for Scaling . - Having identical degree of saturation isenough for having media in similar state. 19. Second Thereisnoneedtosearchforgeometric similarity. Scaling factor can be obtained by a least-square fitting to an average curve. 20. Assumersoils (locations) each havingidatapoints of retention curve, hr,i. At a given degree of saturation, minimizing following SS gives scaling factors (r) of eachsoil (location).h* rhIndividual curves hr ,iSS2hr r ,ir ,iAverage curveScaling factors 21. Functional form of average curve: h Sa0 S1 Sa1 1 S2This form was assumed for ease of mathematical derivations... an11 Sn 22. UnscaledScaled 23. A similar procedure was followed for scaling hydraulic conductivity curves.K*Kln K *2ln K2 lnrr ln K r ,iSS2r2ln K r ,ir ,iIndividual curves Average curve Average curve: ln K SScaling factorsb0 b1S b2 S2... bn Sn 24. UnscaledScaled 25. Distribution of scaling factors was found to be. 26. - Scaling provides a tool for describing soil heterogeneity.The soil heterogeneity is approximated by a single stochastic parameter of scaling factor having a log-normal distribution. Average soil hydraulic properties are described by the invariant scaled curves (the average curves). 27. - Scaling factors from K(s) were not the same as those calculated from h(s). But they were highly correlated. - Scaling factors from h(s) showed less dispersion. - Technology to measure K is not developed to the samedegree as that for h. 28. Sadeghi and Ghahram (2010) found a similar result. 29. SadeghiandGhahraman(2010)introducedaBetaparameter as:Ks 2Saturated hydraulic conductivity Scaling factor from retention curveThey theoretically indicated that must be the same for all similar soils when simultaneous scaling (equality ofscaling factors from K and h data) is expected. 30. Therefore, - Similarity is necessity for validity of Millers theory, but is not sufficient. - Equality of values gives the sufficiency for this validity.- This equality is related to the validity of capillary and Poiseuille equations in real soils. 31. Simmons et al. (1977) further developed a scaling method. They defined the similarity based on shapesimilarity of hydraulic functions. This definition helped Millers scaling theory to be applied in reality. Simmons, C.S., D.R. Nielsen and J.W. Biggar. 1979. Scaling of field-measured soil-water properties. I. Methodology. II. Hydraulic conductivity and flux. Hilgardia 47, 74-173. 32. The shape similarity can be easily investigated by shape parameters in hydraulic models.For example, in van Genuchen model,nandmareshape parameters in this model. Soils having identical n and m would be called as similar according to Simmons et al (1979). srr1hnm 33. For the purpose of scaling unsaturated flow, Simmons et al. (1977) considered different scaling factors for h and K: *hh hKK * K2For scaling unsaturated flow (e.g., scaling Richards equation), equality ofhandK isnot necessary.But, for describing soil variability, the differenceit is not desirable. 34. A similar idea of linear variability concept was described by Vogel et al (1991). 35. Linear variability deals with variability only in scale parameters (e.g.,,s, and r in van Genuchtenmodel). srr1hnmSoils are scalable when their variability is linear.Soils with nonlinear variability (e.g. different n and m) are considered as dissimilar soils. 36. To scale unsaturated flow (Richards equation), Vogel et al. (1977) considered different scaling factors for h, K, and as:hh , * hK h KK * h*,h *h*r * r 37. Scaling Richards Equation Different methods have been proposed for scaling Richards equation: -Miller and Miller (1956) Reichardt et al (1972) Youngs and Price (1981) Warrick and Amoozergar-Fard (1979) Warrick et al (1985) Kutilek et al (1991) Vogel et al (1991) Warrick and Hussen (1993) Sadeghi et al (2011) Sadeghi et al (2012a) Sadeghi et al (2012b) 38. Four of these methods are introduced here, as representatives of different generations -Warrick et al (1985) Kutilek et al (1991) Warrick and Hussen (1993) Sadeghi et al (2011) Sadeghi et al (2012a) 39. Warrick et al. (1985) 40. Richards equation:Scaled Scaled time Scaled depthScaled conductivity Scaled pressure head Scaled diffusivity 41. Scaled Richards equation:Scaled Hydraulic functions: 42. - Only n remains in the scaled RE and all other soil-dependent parameters (r, s, and Ks) go out. - Solution does not change by changing r, s, and Ks.- Soils having identical n may correspond similar soils of Millers. 43. Consider and infiltration process (the following IC and BC):Philips Solution to the scaled form of RE: 44. - A, B, and C are functions of n and Wi (scaled initial water content). A, B, and C were numerically calculated using the procedure of Philip (1968). 45. A, B, and C for van Genuchten functions. 46. A, B, and C for Brooks-Corey functions. 47. Comparing the solutions (points) with numerical solutions of Richards equation (line) 48. - Scaling provided a simple method for solving Richards equation. - The solutions of Warrick et al. (1985) needs identical scaled initial and boundary conditions. - To capture this limitations other methods were proposed. 49. Kutilek et al (1991) 50. Richards equation:Initial and boundary conditions for a constant flux infiltration: 51. Proposed scaled variables:q0: constant flux of infiltration , , and : scaling constants 52. Soil hydraulic functions:Scaled soil hydraulic functions: 53. Resulting scaled Richards equation:For the following conditions, q0 goes out of the scaled RE (solutions get invariant with respect to infiltration flux): 54. Scaled Richards equation:Invariant IC and BC: 55. Scaled solutions for three different q0 are the same. 56. Warrick and Hussen (1993)Warrick and Hussen (1993) developed a more general method for constant-head and constant-flux infiltration and drainage from a wet soil column. 57. Richards equation:Brooks-Corey soil hydraulic functions 58. 0 was defined: - to be soil water content (upper BC) in constanthead infiltration - to be initial water content in drainage - to give K(0) = q0, in the constant-flux infiltration (q0 is the constant flux). 59. Scaled variables:where: 60. Scaled Richards equation:Scaled soil hydraulic functions:- Scaled BC and IC are invariant. - Scaled RE depends only on v andm. 61. Scaled RE was solved for two different soils and different IC and BC.m and v are identical (soils are similar) 62. Scaled results for constanthead infiltration 63. Scaled results for drainage 64. Scaled results for constantflux infiltration 65. Sadeghi et al. (2011) - Methods of Kutilek et al (1991) and Warrick and Hussen (1993) are limited to special form of Hydraulic functions.- Sadeghi et al. (2011) developed a method in which all forms of hydraluc functions can be used. 66. Redistribution process was assumed. Boundary conditions:Initial conditions: 67. Scaled variables were defined based on initial conditions:vfi is the initial velocity of the scaled wetting front movement: 68. An invariant scaled initial condition was obtained: 69. Richards equation was numerically solved considering van Genuchten functions.Van Genuchten functions: 70. Twelve soils were considered. 71. Different initial conditions were assumed. 72. Scaled solutions were the same for medium- and finetextured soils and different initial conditions. 73. - All the previous methods were proposed for similar soils. This limits application of these methods to real (dissimilar) soils


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