Analytical Models for Soil Pore-Size Distribution After Tillage

for all properties whereas tillage practices contributed the most to the variation of infiltrability. Tillage causes soil fragmentation thereby increasing the proportion Quantifying the temporal dynamics of the PSD may of interaggregate (structural) pore space. The resulting tilled layer have important applications, but to do so in a determintends to be structurally unstable as manifested by a gradual decrease istic manner would be a difficult undertaking because of in interaggregate porosity until a new equilibrium has been reached between external loads and internal capillary forces at a rate governed the complexity of the pore geometry and our incomplete by the soil rheological properties. The soil pore-size distribution (PSD) understanding of relevant processes. Or et al. (2000) will change accordingly with time. We have previously applied the therefore proposed to describe the evolution of the PSD Fokker-Planck equation (FPE) to describe the evolution of the PSD with the Fokker-Planck Equation (FPE) (Risken, 1989). as the result of drift, dispersion, and degradation processes that affect The FPE is frequently used to quantify natural processes the pore space in unstable soils. In this study, we provide closed-form where the underlying processes or conditions cannot be solutions for PSD evolution, which can be used to predict temporal precisely captivated. The coefficients of the FPE encombehavior of unsaturated soil hydraulic properties. Solutions and mopass our understanding of the mathematical behavior of ments of the PSD were obtained in case: (i) drift and degradation the PSD in response to physical processes. A somewhat coefficients depend on time and the dispersivity is constant and (ii) similar approach was followed by Ozkan and Ortoleva drift and dispersivity are also linearly related to pore size. Both solu(2000), who used a Markov model for the evolution of tions can model the reduction in pore size during the growing season the particle-size distribution because of breaking and while the second solution can account for a reduction in the dispersion shearing in fault zones of porous media. of the PSD. The solutions for PSD were plotted for a mathematically Or et al. (2000) developed a conceptual framework convenient expression for the drift and degradation coefficients and for an expression derived from a model for soil aggregate coalescence. for the evolution of the PSD after soil tillage by postulatExperimental data on the settlement of a Millville (coarse-silty, caring that the PSD can be partitioned into a time-invariant bonatic, mesic Typic Haploxeroll) silt loam during wetting and drying textural component and a time-dependent structural comcycles were used to determine time-dependent drift and degradation ponent (Nimmo, 1997). Such partitioning is supported coefficients according to this coalescence model. The solution for the by experimental evidence that the tillage effect maniPSD was used to independently predict the water retention curve, fests itself only in the wet range for suctions 300 hPa which exhibited a satisfactory agreement with experimental retention (Ahuja et al., 1998). Following the work by Hara (1984), data at the end of two drying cycles. the drift coefficient in the FPE was defined by Or et al. (2000) from the displacement of the mean pore size and the dispersion coefficient as the product of drift coeffiT soil PSD greatly affects the movement of fluids cient and dispersivity. A pore-loss term accounts for the and dissolved substances and impacts the thermal collapse (disappearance) of pores. The evolution of the and mechanical properties of soils and other porous PSD was assessed by numerically solving the FPE for media. Soil tillage is intended to create conditions for a Millville silt loam (Or et al., 2000). The water retention gas, water, chemical, and heat movement in agricultural curve or soil water characteristic (SWC) was determined soils that provide an optimal habitat for crop growth. by calculating the soil water pressure head from the pore The upper part of the soil will generally be unstable after size with the Laplace-Young equation while the satutillage. The porosity and median pore size will decrease rated and unsaturated conductivities were estimated over time because of soil settlement and filling of pore based on the Kozeny-Carman equation (Hillel, 1980) and the approach by Mualem (1976), respectively. space instigated by mechanical compaction, wetting and The current work is intended to augment the concepdrying, and biological activity. Consequently, associated tual approach by Or et al. (2000) in two ways. First, soil hydraulic and transport properties will also vary the numerical solution procedure is quite adequate to over time (e.g., Mapa et al., 1986). Ahuja et al. (1998) predict the PSD but analytical solutions are more suitreviewed the literature and reported that tillage tempoable to elucidate the effect of different coefficients on rarily increases water retention in the wet range. These the evolution of the PSD and they can be used for authors proposed two approaches to predict the change moment analysis to determine transport parameters and in the retention curve according to the Brooks and predict trends. For the solution, we can utilize results Corey (1964) equation. Van Es et al. (1999) reported obtained for the advection-dispersion equation (ADE), on the sources of variability of infiltration and retention which is closely related to the FPE. For example, Su parameters determined in 3-yr field studies. Temporal (1995) obtained explicit solutions for transport in hetvariability was more significant than spatial variability erogeneous media by making simplifying assumptions regarding the FPE. Second, we want to apply our soluF.J. Leij, USDA-ARS, George E. Brown Jr. Salinity Lab. and Dep. of Environmental Sciences, Univ. of California, Riverside, CA 92507tions for the PSD to cases where the coefficients can 4851; T.A. Ghezzehei and D. Or, Dep. of Plants, Soils, and Biometeorology, Utah State Univ., Logan, UT 84322. Received 18 May 2001. Abbreviations: ADE, Advection-Dispersion equation; FPE, Fokker*Corresponding author ([email protected]). Planck equation; PSD, pore-size distribution; SWC, soil water characteristic. Published in Soil Sci. Soc. Am. J. 66:1104–1114 (2002).