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The relationship between water content and water potential for a soil is termed its water retention curve. This basic hydraulic property is closely related to the soil pore size distribution, for which it serves as a conventional method of measurement. In this paper a general model of the water retention curve is derived for soils whose pore size distribution is fractal in the sense of the Mandelbrot number‐size distribution. This model, which contains two adjustable parameters (the fractal dimension and the upper limiting value of the fractal porosity) is shown to include other fractal approaches to the water retention curve as special cases. Application of the general model to a number of published data sets covering a broad range of soil texture indicated that unique, independent values of the two adjustable parameters may be difficult to obtain by statistical analysis of water retention data for a given soil. Discrimination among different fractal approaches thus will require water retention data of high density and precision.

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We use multifractal analysis (MFA) to investigate how the Rényi dimensions of the solid mass and the pore space in porous structures are related to each other. To our knowledge, there is no investigation about the relationship of Rényi or generalized dimensions of two phases of the same structure.Images of three different natural porous structures covering three orders of magnitude were investigated: a microscopic soil structure, a soil void system visible without magnification and a mineral dendrite. Image size was always 1024 × 1024 pixels and box sizes were chosen as powers of 2. MFA was carried out according to the method of moments, i.e., the probability distribution was estimated for moments ranging from − 10 b q b 10 and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. A meaningful interval of box sizes was determined by estimating the characteristic length of the pore space and taking the next higher power of 2 value as the smallest box size, whereas the greatest box size was determined by optimizing the coefficients of determination of the log/log fits for all q. The optimized box size range spans from 32 to 1024 pixels for all images. Good generalized dimension (Dq) spectra were obtained for this box size range, which are capable of characterizing heterogeneous spatial porous structure. They are alike for all images and phases which the exception of the solid mass of the soil void system, which shows a rather flat D q behavior. A closer examination reveals that similar patterns of structure gain similar spectra of generalized dimensions. The capacity dimension for q = 0 is close to the Euclidian dimension 2 for all investigated images and phases.

To study relationships between soil hydraulic properties and soil structural properties, a computer micro model of soil is constructed. We present first a general method of building a two-dimensional porous structure, including both pores and particles, with different levels of aggregation resulting from a fragmentation process. A fractal structure is obtained when self similarity is imposed over the successive scales of fragmentation. Emphasis is put upon the modeling of the retention curve. A classical capillary model and methods taken from percolation theory enable us to simulate qualitatively the primary and secondary loops of this hysteretic curve. In the fractal case, theoretical analytical expressions proposed for adjusting retention data are tested. The unsaturated hydraulic conductivity is also calculated on the same simulated soil by analogy with an electrical network. The soil structures are deformable and simulation proves to be a useful tool to investigate the behavior of swelling soils. IntroductionKnowledge of the specific hydraulic behavior of a given soil is needed to model water transport in the unsaturated zone. The basic hydraulic properties, mainly the retention and conductivity curves, are usually determined by controlled flow experiments either in the laboratory or in the field. It is an old dream among soil scientists to directly relate these hydraulic properties to structural properties which could be obtained more easily on dry soil samples. Numerous attempts have been made to find either statistical relations or deterministic links between structural data and hydraulic properties. This paper is concerned with a deterministic approach.Research has been done [e.g.,Arya and Paris, 1981; Haverkampf and Parlange, 1986; Tyler and Wheatcraft, 1989, 1992] to link hydraulic properties to the particle size distribution in a soil, which structural information is easily and widely obtained through mechanical sieving. The texture models whose definition of the soil is based on matrix properties such as particle diameter or particle shape (e.g., the spheres models) view the solid phase as a set of discrete particles and the void phase as a continuum. The complex geometry of the remaining voids makes the analysis of flow too difficult unless one works at the Navier-Stokes scale, dealing with only a few particles as in lattice gas simulations [di Pietro et al., 1994]. At the pore scale, the pore space must be divided into a set of simple geometric parts, such as cylinders or parallepipeds, so that integrated forms of the fluid movement equations can be used. A common approach is therefore to invent a pore space model with a simple geometry that can be associated with the particle distribution. For example, Arya and Paris [1981] associated a tube with each particle size class and treated the pore space as a bundle of capillary tubes. However, to obtain good agree-Paper number 95WR02214. 0043-1397/95/95WR-02214 $05.00 ment between calculated and observed retention data, they needed to add an empirical fi...

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