1. Sample Scale Liquid Retention and Liquid-Vapor Interfacial Area Dani Or & Markus Tuller Dept. of Plants, Soils and Biometeorology Utah State University, Logan, Utah

2. Outline for Section 2 How can we use simplified Y-L expressions in a unit cell to represent a population of pores (i.e., a soil sample)? Statistical representation of pore size distribution (normal, log- normal, and Gamma) Relationships between psd and soil water characteristic curve Measurable soil attributes that provide constraints for psd estimation A proposed estimation scheme Examples for various soils – capillary and film water content Liquid-vapor interfacial area – an important function for gas exchange processes (e.g., bioremediation)

3. Statistical representation of soil pore size distribution From Brutsaert (1966) to Assouline (2000) many have proposed to represent soil pore size distribution by various statistical PDF’s such as: normal, log-normal, Gamma, and Weibul PDF’s. For example, Kosugi (1994) proposed a log-normal expression: There are some subtle differences between f(r) and f(r 3 ) – psd of size usually pertains to pore radii and not volume distribution… Relationships between psd and SWC – water capacity (d  /d  ) with f(r) based on the capillary rise equation.

4. Statistical representation of soil pore size distribution Pore size (radii) distribution is calculated by taking the derivative of the SWC (d  /dh); and By employing the capillary rise equation (r=a/h) Matric Potential - h - [m] Water Content -  - [m 3 m -3 ]

5. Upscaling from Pore-to Sample-Scale WetDry L1L1 L2L2 L3L3 L4L4 L5L5 22 11 33 L6L6 f(L) Slits L1L1 L2L2 L3L3 L4L4 L5L5 L6L6 Gamma Distribution for L  A statistical approach using Gamma distributed cell size is employed to represent a sample of a porous medium.  Gamma distribution – facilitates analytical solutions and preserves the observed skewness in psd.  In developing “upscaled” equations for liquid retention, one must keep track of portions of pore population at various filling stages (due to differences in their pore size).

6. Limits of Integration for the Upscaling Scheme L max Full Cells Full Slits- Partially-Filled Pores Partially-Filled Slits & Pores L min L2=L2= L1=L1= Filling Stage Boundary Cell Size

7. The application of Limits of Integration in the Upscaling Scheme full cells corners pore full slits films pore films pore +slits

8. Measured Media Properties Provide Constraints for Geometry and PSD  Measured soil specific surface area and the “dry end” of the SWC curve provide constraints for  and  parameters.  The “bubbling pressure” defines the largest pore to be considered.  The smallest pore size is bounded by slit-spacing.

9. Measured and Modeled Water Retention Curve Millville Silt Loam films corners/ full pores

10. Measured and Modeled Water Retention Curve Salkum

11. Measured and Predicted Liquid-Vapor Interfacial Area Sand

12. An Overview of the Proposed Scheme