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DIVISION S-1-SOIL PHYSICS

On Diffusion in Fractal Soil Structures

^a Selby Biolab, 5/43-51 College St., Gladesville, NSW, 2111, Australia
^b Soil Plant Dynamics Unit, Scottish Crop Research Inst., Invergowrie, Dundee, DD2 5DA, Scotland, United Kingdom
^c Dep. of Agricultural Chemistry and Soil Sci., Univ. of Sydney, NSW, 2006, Australia

alisona{at}selbybiolab.com.au

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusions REFERENCES

 
Fractal models of soil structure can be used to predict the scaling properties of associated transport coefficients. For gas diffusion, the structure of the soil pore space is relevant, while the structure of the solid matrix is most implicated in heat conduction. In fractal soil structures, the magnitude of the relevant diffusivities can be written in the generic form , where D(r) is a length-dependent diffusion coefficient, A is the normalization coefficient, r is the Pythagorean length, and is a structure-dependent constant. The dependence of on structure has been described elsewhere; however, the influence of structure on the magnitude of A has not been previously elaborated. Here, we determine the functional dependence of A on the structural parameters of the soil. The heterogeneity and connectivity, as quantified by the mass fractal dimension (D_m) and spectral dimension (d), respectively, and porosity are estimated from sections of undisturbed soil cores. For these soil structures, we demonstrate that the magnitude of the thermal and gas diffusivities is more sensitive to the porosity than to the scale dependency inherent in fractal structures. A methodology is developed and applied to rank the predicted thermal and gas diffusivities for the soil structures studied.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusions REFERENCES

 
FRACTAL GEOMETRY(Mandelbrot, 1983) has been used in recent years to quantitatively describe soil structure (e.g., Bartoli et al., 1991; Hatano and Booltink, 1992; Crawford et al., 1993; Anderson et al., 1996). It provides a more accurate representation of the heterogeneous nature of soil structure than approximations that use the concept of a characteristic scale (e.g., homogeneous or dual-porosity models). In addition to the role that fractal geometry has in the quantification of soil structure, it can also be used to address the relation between structure and a range of physical processes occurring in the soil. This ability to directly relate structure to function is one of the most important and beneficial features of fractal soil models.

A number of studies have applied fractal theory to soil physical processes. Tyler and Wheatcraft (1989, 1990) applied fractal theory to estimating soil water retention. Anderson and McBratney (1995) discussed the implications that the mass fractal dimension has for estimating the pore-size distribution and the moisture characteristic. Hatano and Booltink (1992) and Hatano et al. (1992) empirically related fractal dimensions of flow paths to bypass flow and Brenner numbers of chloride breakthrough, respectively. In Crawford (1994), fractal theory forms the basis of a calculation of the relation between soil structure and the hydraulic conductivity of soil. Guerrini and Swartzendruber (1994) discussed the fractal characteristics of the horizontal movement of water in soil. Crawford et al. (1993) discussed the use of fractal dimensions in diffusion theory applied to pedal soil. Anderson et al. (1996) subsequently applied this theory to images of different soil types. Here we explore the possibility of using image analysis of soil thin sections to compare the relative diffusivities of soil samples. We identified key structural parameters that determine the magnitude of the diffusivity coefficient, and determined their relative importance.

	Theory

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Theoretical Derivation of the Magnitude of the Diffusion Coefficient
The motion of a particle moving through a fractal network is constrained by the heterogeneity and tortuosity of that network. The rate of movement of a particle moving randomly through a fractal network is related to the mass fractal dimension (D_m) and spectral dimension (d) of that network. The mass fractal dimension is associated with the heterogeneity and space-filling properties of an object, while the spectral dimension reflects the connectedness of a fractal network. For example, a pore network that is homogeneous and makes up a relatively high percentage of the total soil volume will have a large value of D_mp and if the pore space is also continuous it will have a high value of d_p as well. Methods that can be used to estimate these fractal dimensions are described in Anderson et al. (1996).

The classical method of describing gas diffusion in soil is by Fick's Law (Campbell, 1985):

(1)

where f_g is the flux density (g m^-2 s^-1), D is the diffusion coefficient (m² s^-1), c is the concentration (g m^-3), and x is the distance (m). Campbell (1985) calculates the diffusion coefficient using

(2)

where D₀ is the diffusion coefficient of gas in free air (m^-2 s^-1), is the ratio of soil to free air gas diffusivity, and _g is the air-filled porosity. It is generally accepted that tortuosity and blockage of the pore space (e.g., by water) increases the effective diffusion length by a value of about 2, so Eq. [2] becomes

(3)

However, the motion of a randomly moving particle confined to a fractal network cannot be described by classical (Fickian) diffusion. For fractal networks, the conventional diffusion coefficient, D, is replaced by a length-dependent diffusion coefficient, D(r), where r is the Pythagorean length (Orbach, 1986). It may be postulated that the soil pore network is a fractal network. A description of diffusion in a fractal soil pore space is necessary. The magnitude of the diffusion coefficient [D(r_c)], which corresponds to a particular fractal two-dimensional soil image, depends on the characteristic diameter of the image, r_c, the value of (Orbach, 1986; Crawford et al., 1993), and the normalization coefficient A, where

(4)

The physical meaning of is that the diffusion coefficient decreases with increasing if A is independent of and r_c is constant. Clearly, however, the value of A can be dependent on structure, and therefore the relation between the diffusion coefficient and soil structure, as derived in Crawford et al. (1993), is incomplete. One of the aims of this paper is to relate A to structural parameters.

In Crawford (1994), the relation between soil structure and scaling of hydraulic properties was derived. Since any solution to the heat equation can be mapped onto the diffusion problem, we develop that work here to obtain a full expression for the influence of structure on the magnitude and scaling of the diffusion coefficient, and the related thermal conductivity. The calculation is based on the construction of a random recursive lattice (Fig. 1 and 2) , as described in Crawford (1994). We describe the construction in two dimensions, but it can be extended to three. A square in space is divided into M x M equal square subareas with side r₀. With probability P, these M² cells are occupied by a particle of characteristic diameter r₀. The remaining cells constitute pores of diameter r₀. The cell of size represents the smallest soil volume that can be approximated by a fractal. At the next hierarchical level of clustering, a square area of side M²r₀ is divided into M² square subareas of side Mr₀, and PM² of these cells are filled with a random realization of the aggregate produced in the previous generation. The construction is continued to higher levels such that at the nth level, the aggregate has diameter Mⁿr₀ and is built from PM² random realizations of the aggregates of diameter M^n-1r₀ generated at the n-1th level of construction. Between the length scales of r₁ and r₀Mⁿ, the structure shares the scaling properties of a fractal with dimension given by (Crawford, 1994):

(5)

View larger version (11K): [in this window] [in a new window]   Fig. 1 First two stages in the construction of a random recursive lattice in two-dimensional Euclidean space with

View larger version (69K): [in this window] [in a new window]   Fig. 2 An example of a random recursive lattice with and , giving . The image size is 200 x 200 pixels

(6)

where r is a length scale, D₀ is the effective diffusion coefficient at scales below r₀ (i.e., below the fractal regime and where diffusion is Fickian). The porosity of the starting structure can be taken into account by noting that Eq. [6] can be rewritten as:

(7)

Equation [5] can be rewritten as

(8)

and the probability, P, can be written in terms of porosity, since it is equal to the porosity of the smallest structures that are fractal (i.e., the portions of size r₁). If we denote the porosity at that scale ₁, then Eq. [8] can be written as

(9)

Using Eq. [9] in Eq. [6] yields

(10)

that is

(11)

This is in a rather unsatisfactory form, since ₁ is not easily measured. However, the porosity is scale-dependent in a manner that can be calculated with knowledge of D_m,

(12)

which can be written

(13)

Finally then, the full expression for the diffusion coefficient can be written in the form

(14)

From Eq. [14], we see that both porosity (r_c) and sample size r_c influence the value for the diffusion coefficient. If the soil structure is fractal, porosity will be scale-dependent. Here we are referring to the apparent porosity for the sample size of interest.

We can explore the relative roles of porosity and scale-dependency by their influence on diffusion, by analyzing and ranking thin-section images prepared from undisturbed soil samples. Since all of the images discussed in this paper are the same size (constant r_c, and ), if scale dependency is of secondary importance to porosity in determining the diffusion coefficient, an adequate way to rank them is to compare the values of (r_c), taking D₀, r_c, and r_c/r₁ to be constant. The value of r₁ is the value of the length scale below which there is significant departure from linearity. In principle, the values of r₁ can be measured from the plots used to obtain the fractal dimensions. However, in the absence of any such knowledge, we proceed with our assumptions of constant r_c/r₁. Alternatively, if scale dependency dominates porosity in determining the diffusion coefficient, we can rank the images according to the values for .

	Materials and methods

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Vertical images of soil structure were digitized from black-and-white photographs of thin sections. Two images were made from each photograph. One represented the upper half of the thin section while the other represented the lower half of the thin section. All images were 1000 by 1000 pixels, representing 50 by 50 mm. The solid phase (the black areas of the photographs) was made to equal grey level 255, while the pore phase (the white areas) was made to equal grey level 0. All images had to be binary for analysis. The solid phase referred to here includes all material that appears solid at the scale of examination. This solid phase will include pores that will only become apparent at finer scales of examination. Scale-dependent porosity is an inherent property of fractal structures. However, the attempt to measure the fractal dimension in both phases necessarily misses structure below the resolution limit in both phases. There is no a priori reason for assuming that you do not get small-scale matrix structure, but you do get small-scale pore structure.

The location from which each thin section was taken, and the soil type and characteristics of each thin section are described in Anderson et al. (1996). The images produced from the thin sections are shown again here for ease of comparison (Fig. 3) . Soil 1 and Soil 5 are from the top 10 cm of a soil profile. Soil 3, Soil 4, and Soil 6 were taken from the 10- to 25-cm region, and Soil 2 is from a depth of 70 to 80 cm.

View larger version (178K): [in this window] [in a new window]   Fig. 3 Binary images of the vertical soil thin sections, (a) top of thin section and (b) bottom of thin section (from Anderson et al., 1996)

The pore space and solid phase cannot be simultaneously modeled as fractals over the same range in length scale. This is because the density of a mass fractal decreases as aggregate size increases. For example, if the solid phase is fractal and its density is decreasing with increasing aggregate size (as has been reported by Chepil [1950] and Eghball et al. [1993], for example), the same cannot be occurring for the pore space. In practice it can be difficult to determine which phase is more appropriately modeled by a fractal, especially if the fractal has a value of D_m close to 2 (or 3 for three dimensions) because artefactual scaling of the fractal complement will occur (Crawford and Matsui, 1996). For the images studied by Crawford and Matsui (1996), it was concluded that it was the solid phase that was fractal. Without all the data that Crawford and Matsui (1996) used to make their conclusions, the goodness of fit of the line used to estimate D_m was examined to determine validity of the fractal model. The value of D_m is estimated from a plot of the ln(number of boxes of size m that contain the phase of interest) vs. the ln(box size), where each box has sides equal to m pixels (Anderson et al., 1996). A line is fitted to the data by linear regression. If the line is a good fit to the data (as indicated by R²% values) then the object being examined is considered fractal. An example of a plot is shown in Fig. 4 .

View larger version (16K): [in this window] [in a new window]   Fig. 4 Plot of ln(number of boxes of size m) vs. ln(m). The gradient of the line is equal to -Dm. This plot is for the pore space of structure 4 (b) ( )

	Results and discussion

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusions REFERENCES

Values obtained for D_m, d, d/D_m, and of the pore space have been discussed with regard to these images in Anderson et al. (1996). Inasmuch as intuition is a guide, the values obtained for D_mp, d_p, D_ms, and d_s are what would be expected from a visual examination of the images (Fig. 3). We would expect, by looking at the images, that a pore network that appears to fill more of the total area relative to another would have a larger value of D_mp, and a pore network that is connected and less tortuous relative to another to have a larger value of d_p. This can also be applied to the solid phase. In most cases, the larger the solid fraction in a soil image (smaller porosity), the larger the estimate of D_ms. Values of D_mp range between 1.852 (structure 4 [b]) and 1.682 (structure 2 [a]). Values of D_ms range between 1.978 (structure 2 [a]) and 1.898 (structure 4 [a]). Values of d_p range between 1.668 (structure 4 [a]) and 1.236 (structure 2 [a]), while values of d_s range between 1.727 (structure 2 [a]) and 1.601 (structure 5 [a]).

An immediate observation for the soil images used in this study is that in the majority of cases the solid phase is disconnected, while the pore space is connected. Even though this is the case here, this is probably not so for most two-dimensional images of soil structure. It is more likely that the solid phase will be largely connected with some connected pores and some disconnected pores (in two dimensions at least) running through it. For a structure with low porosity we would expect the solid phase to be more connected than for a structure with a high porosity. As noted in Crawford and Sleeman (1998), and as is well known from percolation theory (Bunde and Havlin, 1991), the connectivity of a large class of random structures is associated with a critical value for the fractal dimension. In our samples, we find that the solid phase appears connected for D_ms > 1.96, while the pore phase is connected for D_mp > 1.78. It should be stressed that the calculation of the spectral dimension (d_s or d_p) reflects only the structure of the connected components of the images.

It was found previously that values calculated for d_p/D_mp for these samples (Table 1) did not always agree with what may be expected, given the values of D_mp and d_p (Anderson et al., 1996). The value of d/D_m will be equal to 1 in Euclidean space and less than 1 for diffusion through a fractal network (Orbach, 1986). The smaller the value of d/D_m, the smaller the mean square distance traveled in time t [r²(t)] by a particle will be. There were cases where structures had large values of D_mp and d_p but comparatively small values of d_p/D_mp, implying a lower value for the diffusion coefficient, provided all other factors are constant. However, the calculations derived above show that the normalization of the diffusion coefficient is a function of structure and so does not remain constant when D_mp and d_p are varied. Indeed, the normalization increases for more open structures. We proceed to determine the relative insensitivity of the diffusion coefficient to the scale-dependent and porosity-dependent terms.

For constant sample size, we find that the relative change in the magnitude of the diffusion coefficient [D(r)/D(r)] with respect to the relative change in the porosity [(r)/(r)] can be written as

(15)

Similarly, the sensitivity of the diffusion coefficient to sample length can be expressed as

(16)

From Tables 1 and 2, it is clear that the values of are typically an order of magnitude greater than the values of . Therefore, for these soil structures, the diffusion coefficient is more sensitive to the porosity (defined at the scale of examination) than the scale dependency inherent in a fractal soil structure. This suggests that the structures could be ranked in order of increasing magnitude of their associated diffusion (conductivity) coefficients according to the value of for the pore (solid) phase. Since the spectral dimension refers only to the connected phases, only those images where the relevant phases are connected should be ranked.

According to Table 1, if we remove the images with disconnected pore spaces from consideration, the ranking agrees with a visual comparison of the structures. That is, if we observe a structure with large connected pores, we would expect particles to diffuse more rapidly because they are relatively unobstructed compared with a structure observed to have small, discrete, or tortuous pores. However, it should be realized that qualitative judgment may not be a good guide in estimating transport coefficients in these heterogeneous domains. For the pore space, structure 4 (a) has a value of , which is much higher than for any of the other structures, while 5 (b) and 6 (b) have very low values of , equal to 0.061 and 0.070, respectively. This suggests that gas diffusion will be more rapid through the pore network of structure 4 (a) than through the others. For the solid phase, again considering only those images with connected solid phases, structure 1 (a) has the highest value of , (equal to 0.171), while the lowest is 1 (b) with . Structures 2 (b) and 2 (a) have intermediate values of . This suggests that heat diffusion will be most rapid in structure 1 (a). By comparison with the pore space, the values for the connected solid phases vary less between the images. This may be a consequence of the fact that the solid phases are more nearly homogeneous.

Finally, since our measurements are based on two-dimensional thin sections, we must be careful in extrapolating the results of image analysis to three dimensions. Provided the structure is isotropic and the flow approximately one-dimensional, then the rankings based on the two-dimensional sections should be valid for the intact soil samples. At present, we postulate that the rankings we derive are relevant to the corresponding undisturbed three-dimensional cores if isotropy is not satisfied, and the concentration or heat gradients are complex.

	Conclusions

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusions REFERENCES

 
In previous work (Crawford et al., 1993; Crawford, 1994), it was demonstrated that as a consequence of fractal structure, the transport coefficients in soil should depend on sample length up to scales of 5 cm or more. This scale dependency is a consequence of the effect of the observed structural variability across a broad range of scales below 5 cm. Heterogeneity at these scales affects transport processes implicated in phenomena that may be manifest at far larger scales, for example, denitrification and soil-borne pathogen movement. Here we further develop the theory of diffusion and heat transport in soil. In particular, we derive a full expression for the magnitude of the coefficient, which includes both a scale-dependent part and a normalization that depends on porosity. We have shown that for the soil structures studied here, the diffusion or conductivity coefficient is more sensitive to porosity than to sample volume.

A method for ranking soil samples according to gas or heat transport based on direct measurement of the structure in thin sections is demonstrated. The method is based on measurements of porosity. A correction in the form of a power-law exponent, depending on the mass fractal and spectral dimensions of the appropriate phases, is derived to account for heterogeneity and connectivity of the phase.

To test the model, it would be desirable to prepare thin sections from soil cores where measurements of diffusion, or heat conduction, have previously been made. This would also allow the ranking methodology to be tested.

	ACKNOWLEDGMENTS

 
The authors would like to thank Dr. E.A. FitzPatrick for providing the photographs of the thin sections and Robert Farrell for writing the computer program STRUCTURA. This research was supported by the Commonwealth of Australia through the Australian Research Council Small Grants Scheme.

Received for publication May 23, 1996.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Conclusions REFERENCES

 

• Anderson A.N., McBratney A.B. Soil aggregates as mass fractals. Austr. J. Soil Res. 1995;33:757-772.
• Anderson A.N., McBratney A.B., FitzPatrick E.A. Soil mass, surface, and spectral dimensions estimated from thin section photographs. Soil Sci. Soc. Am. J. 1996;60:962-969.[Abstract/Free Full Text]
• Bartoli F., Phillipy R., Doirisse M., Niquet S., Dubuit M. Structure and self-similarity in silty sandy soils: The fractal approach. J. Soil Sci. 1991;42:167-185.
• Bunde, A., and H. Havlin (ed.) 1991. Fractals and disordered systems. Springer-Verlag, Berlin.
• Campbell G.S. Soil physics with BASIC. Developments in soil science 14. Amsterdam: Elsevier, 1985.
• Chepil W.S. Methods for estimating apparent density of discrete soil grains and aggregates. Soil Sci. 1950;70:351-362.
• Crawford J.W. The relationship between structure and hydraulic conductivity of soil. Eur. J. Soil Sci. 1994;45:493-502.
• Crawford J.W., Matsui N. Heterogeneity of the pore and solid volume of soil: Distinguishing a fractal space from its non-fractal complement. Geoderma 1996;73:183-195.
• Crawford J.W., Ritz K., Young I.M. Quantification of fungal morphology, gaseous transport and microbial dynamics in soil: An integrated framework utilising fractal geometry. Geoderma 1993;56:157-172.
• Crawford J.W., Sleeman B.D. From particles to architecture: Fractals, aggregation and scaling in environmental soil science. In: Huang P.M., Senesi N., Buffle J., eds. Structure and surface reactions of soil particles. New York: John Wiley & Sons, 1998:41-45 IUPAC Environ. Anal. Phys. Chem. Ser. Vol. 4..
• Eghball B., Mielke L.N., Calvo G.A., Wilhelm W.W. Fractal description of soil fragmentation for various tillage methods and crop sequences. Soil Sci. Soc. Am. J. 1993;57:1337-1341.[Abstract/Free Full Text]
• Guerrini L.A., Swartzendruber D. Fractal characteristics of the horizontal movement of water in soils. Fractals 1994;2:465-468.
• Hatano R., Booltink H.W.G. Using fractal dimensions of stained flow patterns in a clay soil to predict bypass flow. J. Hydrol. 1992;135:121-131.
• Hatano R., Kawamura N., Ikeda J., Sakuma T. Evaluation of the effect of morphological features of flow paths on solute transport by using fractal dimensions of methylene blue staining patterns. Geoderma 1992;53:31-44.
• Mandelbrot B.B. The fractal geometry of nature. New York: W.H. Freeman, 1983.
• Orbach R. Dynamics of fractal networks. Science (Washington, DC) 1986;231:814-819.[Abstract/Free Full Text]
• Tyler S.W., Wheatcraft S.W. Applications of fractal mathematics to soil water estimation. Soil Sci. Soc. Am. J. 1989;53:987-996.[Abstract/Free Full Text]
• Tyler S.W., Wheatcraft S.W. Fractal processes in soil water retention. Water Resour. Res. 1990;26:1047-1054.

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