Soil Science Society of America Journal 64:827-835 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Percolation Theory and Hydrodynamics of Soil-Peat Mixtures
Laure Vidal-Beaudet and
Sylvain Charpentier
UMR SAGAH, INH, 2, rue Le Nôtre, F49045 Angers cedex, France
laure.beaudet{at}angers.inra.fr
 |
ABSTRACT
|
|---|
Great quantities of organic matter are added reconstituting soils in urban conditions. To evaluate the efficiency of organic matter on physical properties of reconstituted soils, we studied the effects of loading on intrinsic permeability of soils mixed with peat. Percolation theory associated with a statistical porosity approach was used to explain variation of permeability related to porosity. After compression, sample porosity was measured and the pore-space morphology described by image analysis. An ellipse with major axis (a) and minor axis (b) was inscribed within each pore. All the pores appeared as lenses of different sizes which could be assumed to represent disk-shaped cracks characterized by a diameter d and an aperture e. The crack model was calculated by means of a percolation threshold,
, along with values d and e as well as the measured porosity. This allowed a determination of the crack interconnection factor f. During compression, the number of pores per unit area decreased. Increasing the loading closed off pores and modified flow pathways. Connected sites facilitating percolation became disconnected from each other and the flow was reduced. The effective porosity, which actually took part in flow, was determined for all the samples and was dependent on peat content.
Abbreviations: SLC, silty clay loam
|
INTRODUCTION
|
|---|
URBAN SOILS are very different from natural soils (Craul, 1992; Beyer et al., 1995). The intensive use and the lack of agricultural working induce soil compaction which inhibits plant development (Craul, 1992). During landscape planning, soils are reconstituted or strongly reworked with organic matter for the maintenance or the improvement of their physical properties.
The addition of peat material, which is characterized by high porosity and hydraulic conductivity, is used to create preferential pathways for water transfer in order to increase permeability. This improvement is rapidly reduced by vertical stress, the effectiveness of which depends on peat content.
Percolation theory was invented by English mathematicians (Broadbent and Hammersley, 1957) for describing transmission properties in disordered systems. It has been developed in the field of physics over the past 20 yr (Stauffer and Aharony, 1994), and has also been associated with statistical approaches (Guéguen and Dienes, 1989) to predict the permeability of porous media. Percolation theory has been applied in the geometrical characterization of porous media and in the determination of the physical properties of such systems (Berkowitz and Balberg, 1993).
In the present study, we used statistical distribution of porosity and percolation theory to interpret the efficiency of organic matter and porosity on physical properties of reconstituted soils. Results obtained on soil–peat mixtures compressed during cycles of repeated loading are correlated with porosity as estimated from image analysis and are then compared with estimates of permeability. In this work we determined to analyze peat mixtures and develop a model which can be applied to other organic additives like bark.
|
Theory
|
|---|
It is very difficult to describe water movement in soil. The fluid circulates in channels and cavities having different shapes and sizes, which may or may not be interconnected. Flow through a cylinder having radius R is obtained by integration of the Navier-Stokes equation, which gives Poiseuille's Law as follows:
 | (1) |
The mean fluid velocity 
(m s-1) in the cylinder is given by
 | (1') |
where 
(Pa s) is the fluid viscosity, 
(m3 m-3) is the density of the percolating fluid, g (m s-2) is the acceleration due to gravity, and h (m) is the total head.
In the same way, flow through a unit length of a crack in one direction (thickness between plane faces is D) is given by
 | (2) |
and the mean fluid velocity in the crack is equal to
 | (2') |
Darcy's Law describes the saturated macroscopic velocity as follows:
 | (3) |
where 
is the intrinsic permeability of the medium (m2), which is independent of the fluid and governed solely by the 3D geometrical configuration. 
is equivalent to flow through a unit area of porous medium and can be related to porosity 
and with
 | (4) |
Combining Eq. [1'] and [3] and [4] gives the determination of permeability
 | (5) |
for straight, uniform tubes parallel to the macroscopic water flow, and combining Eq. [2'] and [3] and [4] gives the determination of permeability,
 | (6) |
for cracks with unidirectional flow.
A bulk approximation to the permeability can be performed by means of a microscopic description of porosity starting from the simplest approach (one straight pipe or crack) and extending up to a more complex approach on the basis of a quantitative and qualitative analysis of the pore size distribution.
Permeability and Pore Models
Since pore orientations are not always parallel to macroscopic flow in a soil, it is necessary to introduce the concept of tortuosity. The mean tortuosity 
is defined as the length of the path actually followed between two points divided by the apparent path between these two points. The permeability is then given by the relation (Guéguen and Palciauskas, 1992)
 | (7) |
The Kozeny-Carman relation introduces the concept of an equivalent porous medium with the determination of an average hydraulic radius for the pores (Dunn and Phillips, 1991; Gimenez et al., 1997). The average hydraulic radius, Rm, is defined as the ratio between the pore volume and the pore surface area. It is directly related to the porosity as well as the geometry of the medium, i.e., tortuosity of the pores, and is written
 | (8) |
where n is a dimensionless constant. The hydraulic radius Rm can be estimated through image analysis (2D). It is equal to the surface of a pore divided by the perimeter of the pore (Gimenez et al., 1997).
Percolation Theory and Connected Crack Model
Previous models assume that porosity is made up of a set of individual cylindrical pores that are unlimited, unconnected, and possibly tortuous. In reality, soil pores are limited in length and are connected.
The determination of the percolation threshold pc of the probability p for two objects to be connected is very important for studying percolation on permeable object lattices. It has been observed that there exists a quasi-invariant Bc, a combination of pc with the coordination z of each object of the lattice (Berkowitz and Balberg, 1993; Charlaix et al., 1984). It is expressed as follows:
 | (9) |
where Bc is the average number of bonded objects per given object at the percolation threshold
It has been shown (Balberg et al., 1984) that Bc is also the average number of center objects which enter the excluded volume of a given object. That is,
 | (10) |
where N is the density of center objects and Ve the average excluded volume of an object. The excluded volume is the average volume around this object within which the center of a second object must be located in order for them to intersect (Fig. 1)
. Choosing a value for z and determining Ve, it is possible to approach p near the percolation threshold. Formulation of Ve is easy for spheres and was determined for simple convex shape volumes like pipes, sticks (Balberg et al., 1984), or thin disks (De Gennes, 1976). The choice of z is generally arbitrary and varies from 4 to 8.
An original approach using percolation theory associated with a statistical description of the porosity was proposed by Guéguen and Dienes (1989) to determine in concert with a crack model the intrinsic permeability of rocks. This model uses a simple lattice, which gives an exact solution to the percolation problem (Stauffer and Aharony, 1994). In the Bethe lattice, each site has z nearest neighbors and there are no closed loops. The Bethe lattice starts with a central point having z bonds ending in another site. There are (z-1) new bonds emanating from every new site apart from the direction towards the central starting point. Stauffer and Aharony (1994) show that the percolation threshold in the Bethe lattice is
 | (11) |
Guéguen and Dienes used the assumption of the Bethe lattice with
neighbors leads to
.
The crack model corresponds to a distribution of cracks resembling disks characterized by a diameter d and an aperture e with d >> e (Fig. 1). The excluded volume Ve for two disks is given by
 | (12) |
The volume of a crack is e
(d/2)2. If l is the average spacing between cracks, crack number density N (number of cracks per unit volume) is then 1/l3 and the porosity can be written:
 | (13) |
Combining porosity with Eq. [6] gives the following description of permeability:
 | (14) |
For a set of cracks of variable aperture and variable diameter isotropically distributed, Guéguen and Dienes (1989) have shown that the permeability is given by
 | (15) |
where 
, 
, 
are respectively the average crack diameter, average crack aperture, and average crack spacing if the crack distribution is assumed to be narrow. Geological or pedological media cover several orders of magnitude for pore effective radius. Pore size distribution of reconstituted soils is narrower than that of natural soils because biological porosity does not exist.
Only connected cracks need to be considered in the calculation of , which introduces the connectivity factor f (0 
f 1) representing the fraction of connected cracks that are part of an "infinite path". The permeability of connected cracks is given by the following equation:
 | (16) |
Percolation occurs in a network when the boundaries between sites create a continuous path to allow a property to be transmitted from one end of the system to another. Percolation begins when the probability p that two objects are linked attains a threshold probability pc. When p < pc, clusters (groups of connected sites, in the sense of Stauffer and Aharony, 1994) are independent from one another and continuous paths do not exist. Thus, the application of percolation theory to pore systems is linked to the analysis of the circulation lattices. In the crack model of Guéguen and Dienes (1989), p is the probability of interconnections between the cracks. Near the threshold probability, f approaches (p - pc)2 (De Gennes and Guyon, 1978). Factor f can be expressed as follows:
 | (17) |
When f 
0, lattice connections are poorly developed and 0. On the contrary, f 1 and 
0 when there are many connections. Combining Eq. [9], [10], [12], and [13] near the percolation threshold leads to the following:
 | (18) |
and, with Eq. [17],
 | (19) |
|
Materials and methods
|
|---|
Soils and Organic Materials
The silty clay loam (SCL) used in this study is an Alfisol sampled from near-surface horizons (0–30 cm). The particle density was 2.55 Mg m-3, and the sample was sieved through a screen of 20-mm mesh size before mixing with organic matter.
Organic material consisting of fibrous moss peat containing sphagnum originating from the Baltic region was added to the SCL. Organic material yielded a particle density of 1.58 Mg m-3 determined by pycnometer using petroleum ether (Blake and Hartge, 1986). In contrast to many other sources of organic matter, this material is widely used as a horticultural substrate and has the advantage of being relatively homogeneous, of constant quality, and widely available.
The different mixes were prepared by amending the SCL with 20 and 40% peat volume basis, which corresponds to 2.8 and 5.6% of the SCL dried at 105°C, respectively, on a weight basis. The mixtures were designated by the labels l20t and l40t, which referred to their percentage content of peat moss. Mixing was performed during 15 min in a concrete mixer using slightly moist materials (water content of the order of 0.15 m3 m-3) to obtain optimal homogenization and the least aggregate destruction of SCL. The pure soil was not placed in the mixer.
Sample Compression
The sample was placed in a methyl methacrylate cell (height 200 mm, internal diameter 190 mm) and moistened to different water contents. The amounts of water thus introduced into the samples were determined on the basis of the water-retention properties of the SLC and peat, which corresponded approximately to water potentials of -1000, -50, -3 and -1 kPa for the different mixtures. We assumed that the curves of the water content as a function of the water potential of the SLC and the peat could be added together, which allowed us to calculate the water contents in the mixture. The volumetric water contents were the same as in the mineral phase (with water contents of 0.15, 0.25, 0.35, and 0.45 m3 m-3) whatever the proportions of SLC and peat in the mixture. Initial sample thickness was 190 mm.
Sample compaction was carried out with a universal testing machine (Model DY 38 of MTS Systems S.A. Ivry/Seine, France) equipped with a 50 kN pressure transducer (resolution of 10 N, 0.1% precision at full-scale deflection) as well as an extensometer for measuring displacements in the range 6 to 600 mm with a resolution of 0.01 mm and a precision of 0.1%. The testing machine was servo-driven by a computerized system (Autotrac 6.30 software developed by MTS Systems S.A. Ivry/Seine, France) which automatically controlled the test to be carried out and automatically recorded the required data. The tested material underwent a mechanical stress of constant intensity by means of a piston that came up against the compression plate. This plate was directly linked to the displacement sensor of the extensometer and remained continuously in contact with the sample during performance of the test.
The SCL– and SCL–peat mixtures were subjected to compression-release cycles to reproduce the effect of repeated loads of short duration passing over urban soils (Bullock and Gregory, 1991). The test cycle comprised a brief loading for 5 s (Koolen, 1987; Paute et al., 1994) followed by a release to 0 kPa for 180 s. The compression-release cycle was repeated 100 times on each tested material and at seven different levels of compressive stress, i.e., at values of 30, 60, 100, 200, 300, 400, and 500 kPa. These load intensities covered a range varying from the tread of a pedestrian to the passage of a truck wheel. Vidal-Beaudet and Charpentier (1998) have shown that this cycle number is sufficient to approach a 90% stable value of consolidation. At each stage of loading or unloading in the test cycle, it was possible to estimate the void ratio of the material from the measured thickness of the sample and the quantities of material initially introduced into the cell. For each experimental condition, we tested one sample.
Intrinsic Permeability
After compression, each sample was water saturated for 48 h. The determination of hydraulic conductivity was based on a direct application of Darcy's law. A hydraulic constant head difference was imposed on the sample (0.25 m) for 24 h and the resulting flow rate measured four times (Klute and Dirksen, 1986). For some highly compacted samples, we had to wait 24 h to obtain sufficient volume to allow measurement and therefore the number of determinations was reduced to two. For some SLC samples, no measurable value for percolation could be obtained.
The intrinsic permeability (m2) was calculated from the following:
 | (20) |
where V (m3) is the volume of water that flows in time t (s) through a sample of cross-sectional area A, (m3 m-3) is the density of the percolating fluid, g (m s-2) is the acceleration due to gravity, 
H (m) is the hydraulic head difference imposed across the sample of length L (m), and (Pa s) is the fluid viscosity.
Image Analysis of Pore Space
After flow rate measurements, we allowed each sample to drain. An undisturbed core, 50 mm thick and 70 mm in diameter, was excavated at the base of the drained sample where it was less disturbed by the permeability measurement. The core was impregnated with a polyester resin containing Uvitex OB fluorescent dye (Ciba-Geigy, New York) (Moran et al., 1989). Each core was cut first parallel to the direction of compression and flow (vertical face) and secondly perpendicular to the first direction (horizontal face). Two images with low and high magnification were captured on each face under ultra-violet light with a JVC 3CCD KY-F30B camera (JVC, Yokohama, Japan) with a Micro-NIKKOR-P (Nikon Inc., Melville, NY) magnifying lens. Each image was digitized with a spectral resolution of 256 grey levels using a Matrox IM640 card, with a size of 640 by 480 pixels. The size of the image for the low and high magnification was respectively: 52.46 by 39.34 mm (pixel size: 0.08 mm) and 13.17 by 9.87 mm (pixel size: 0.02). All greylevel images were segmented and binarized into pore space and solid by threshold level. The threshold level was determined by comparing the size and occurrence of pore space by visual comparison between the digitized image and the camera original image. Binarized images allow the description of pores larger than 30 µm in diameter that are characteristic of soil macroporosity (Chen et al., 1993). Identification of the pores was performed by Optimas v5.2 software (Optimas Corporation, Bothell, WA) which allowed the determination of total pore area (mm2) and pore number. These specific parameters were used to characterize pore structure and arrangement (Murphy et al., 1977; Ringrose-Voase, 1996). The macropores were characterized by individual area, shape and orientation (Hallaire and Curmi, 1994). For each pore, the Optimas v5.2 software inserts the largest ellipse that can be inscribed in the pore and defines its major axis (a) and minor axis (b).
|
Results and discussion
|
|---|
Pore Space Characterization
For each binarized face, we defined 12 pore classes of surface porosity from 0.001 to 2.580 mm2 (the value is the median of the surface porosity of the class in mm2: Class 1 = 0.001; Class 2 = 0.003; Class 3 = 0.005; Class 4 = 0.010; Class 5 = 0.020; Class 6 = 0.041; Class 7 =0.081; Class 8 = 0.161; Class 9 = 0.323; Class 10 = 0.645; Class 11 = 1.290; Class 12 = 2.580).
Figure 2
compares, on l20t samples compressed under different loadings, relative porosity of all pore classes for vertical face and horizontal face (pore porosity/image surface). Differences between surface porosity distribution of the two faces were sometimes important. If the mixtures were considered as isotropic when just prepared, the uniaxial compression probably generated anisotropy of the pore orientation. Nevertheless, at the compression values used, this anisotropy is not great enough to de detected by image analysis and we considered that it did not invalidate the model. Figure 3
shows for l20t samples under different loadings mean minor axis (b) as a function of the mean major axis (a) for each pore class. The axis values were independent of loading for the first seven classes (0.001–0.08 mm2) and these results were observed for all the samples. For the biggest pore sizes (0.16–1.29 mm2) and compression at 400 kPa, the mean major axis increased and simultaneously the mean minor axis decreased. As illustrated in Fig. 4
, loading modified major and minor axis values for the biggest pore sizes and retained constant major and minor axis values for the smallest pore sizes. But all the pores could be considered as lenticular with a ratio of the mean major axis to the mean minor axis always being greater than 2. Each pore was defined by its major and minor axes and we calculated in Table 1
an overall geometric mean of axis values for cores compressed at the same vertical stress and with the same initial water content.
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 2 Comparison of the relative surface porosity (pore surface/image surface expressed in %) of all pore classes between vertical face and horizontal face for l20t under different loadings and for an initial water content of 0.25 m3 m-3
|
|
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 3 The mean minor axis (mm) as a function of the mean major axis (mm) of all pore classes for l20t samples under different loadings and for initial water content of 0.25 m3 m-3
|
|
View larger version (120K):
[in this window]
[in a new window]
|
Fig. 4 Pore density of l20t samples as a function of the applied load for initial water content of 0.25 m3 m-3
|
|
View this table:
[in this window]
[in a new window]
|
Table 1 An overall mean of major axis (a) and minor axis (b) (in mm) of inscribed ellipse in pores, independently of applied loading
|
|
Permeability Calculation
We applied the crack model to the SCL and the SCL–peat mixtures assuming that the lenticular pores were disk-shaped cracks characterized by a diameter (= mean major axis a) and an aperture (= mean minor axis b) (Table 1). If we calculated permeability by Eq. [16], assuming f = 1, it provided a good fit only for the uncompressed samples.
For the crack model, Eq. [16] and [19] give
 | (21) |
To be sure that we were near the percolation threshold, we calculated p - pc using values of , (Table 1), the measured porosity and pc = 0.33. Figure 5
shows measured permeability values as a function of p - pc. For all the samples compressed by more than 30 kPa, p - pc was smaller than 0.1 and we considered this to be near the percolation threshold. A plot of measured permeability as a function of p - pc produced curves by the equation = 
[p - pc]
. For l20t samples, the exponents depended on the initial water content. At initial water content of 0.25 and 0.35, the exponents were, respectively, 2.16 and 2.33, not very different from the value 2 proposed in the crack model.
Equation 21 was solved by values of , (Table 1), the measured porosity , and pc = 0.33 or 0.20 or 0.15, respectively, for values of (z = 4, 6, and 8). The best fit for the samples compressed by more than 30 kPa was obtained with z = 4. Only the fit for the samples compressed by 30 kPa was improved at z = 6. Given this, we therefore assumed that z = 4 was the best value for the percolation threshold. Figure 6
shows the fit obtained with the crack model using z = 4 for different calculated permeability values as a function of measured permeability values over the range of different experimental conditions. A plot of calculated permeability vs. measured permeability lead to linear regression equations and the resulting slopes of the regression lines were then compared with a slope of 1:1. Equation 21 showed a good fit with measured values for samples compressed by more than 30 kPa.
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 6 Comparison between calculated permeability obtained from crack model and measured permeability for silty clay loam, l20t and l40t, as a function of initial volumetric water content, for all values of vertical stress
|
|
Connection Factor and Effective Porosity
Crack connectivity is characterized by the factor f calculated by Eq. [19]. Figure 7
shows f variations as a function of the applied load for different initial water contents. Values of f decreased with vertical stress and tended to zero at stresses of more than 30 kPa for SCL at initial water contents of 0.25 and 0.35. The l20t sample showed f tending to zero at a water content of 0.35 for stresses greater than 200 kPa, while this occured in the SCL sample at water contents of 0.15 and 0.25 compared with 0.35 in the l20t sample. For sample l40t, f 0 only after compression at 500 kPa. Variations of f appeared to be directly linked to vertical stress and so to pore size and number.
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 7 Factor f as a function of vertical stress for silty clay loam, l20t and l40t, at different values of water content
|
|
Percolation can begin if f > 0 or if 
. We can call c the critical porosity, here defined as
 | (22) |
If the porosity is smaller than c, there is no percolation. If the porosity is greater than c, it is possible to determine the fraction of porosity that really takes part in flow transfer, i.e., the effective porosity eff. Effective porosity eff is calculated by
 | (23) |
Figure 8
represents the effective porosity eff for all the samples as a function of the applied load for different initial water contents. Effective porosity decreased in a way similar to factor, f, while the amount of decrease depended on the quantity of peat added and the initial water content. Effective porosity values were always positive for the l40t samples because f was always positive. The results obtained for effective porosity confirmed that the pathways used by water flow made up a small fraction of total porosity.
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 8 Effective porosity as a function of initial volumetric water content for silty clay loam, l20t and l40t, and for all values of vertical stress
|
|
The results show that increased loading produces pore disappearance, the closing up of pore connections, and the modification of flow pathways. During compression, the density number and size of pores is seen to decrease. Connected sites that are effective for percolation become disconnected from each other and the flow transfer is decreased. For the same loading value, addition of peat reduces the drastic decrease of permeability. The origin of the phenomenon can be attributed to the protection of the total porosity by the peat, or the contribution of its own porosity.
|
Conclusion
|
|---|
Permeability of soil–peat mixtures calculated by image-analysis information and the crack model of Guéguen and Dienes (1989) gave good agreement with measured permeability. Differences in hydrodynamic behavior between the silty clay loam and the silty clay loam–peat mixtures depend essentially on the pore density and not on the pore size. During increased loading, pore density decreases, interconnectivity is reduced, and the medium becomes less permeable. The value chosen for percolation threshold (pc = 0.33) and the variables obtained from image analysis allow calculation of a factor f for each sample, as well as a determination of the effective porosity which actually takes part in water flow within the sample. Increased loading induces a closing up of the links between the pores. Clustered sites that facilitate percolation become disconnected, and this occurs even more rapidly if the number of pores per unit surface area is initially small. The movement of water through the soil was dependent on the continuity of the macropores and their effectiveness in conducting water. In practice, the addition of 40% in volume of peat in silty clay loam improved effective porosity and connectivity between pores. For wet mixtures (0.25, 0.35), the addition of peat preserved a good permeability up to 400 kPa. A qualitative and quantitative two-dimensional description of pore geometry by image analysis can be extrapolated to water movement measured in a three-dimensional medium.
Received for publication December 6, 1998.
|
REFERENCES
|
|---|
- Balberg I., Anderson C.H., Alexander S., Wagner N. Excluded volume and its relation to the onset of percolation. Physical Rev. B 1984;30:3933-3943.
- Berkowitz B., Balberg I. Percolation theory and its application to groundwater hydrology. Water Resour. Res. 1993;29:775-794.
- Beyer L., Blume H.P., Elsner D.C., Willnow A. Soil organic matter composition and microbial activity in urban soils. Sci. Total Environ. 1995;168:267-278.
- Blake G.R., Hartge K.H. Particle density. In: Klute A., ed. Methods of soil analysis. Part 1, 2nd ed Madison, WI: ASA and SSSA, 1986:381-382 Agron. Monogr. 9..
- Broadbent S.E., Hammersley J.M. Percolation processes I, Crystals and mazes: Proc. Cambridge Philos. Soc. 1957;63:629-641.
- Bullock P., Gregory P.J. Soils: a neglected resource in urban areas. In: Bullock P., Gregory P.J., eds. Soils in the urban environment. Oxford, UK: Blackwell Sci. Publ, 1991:76-86.
- Charlaix E., Guyon E., Rivier N. A criterion for percolation threshold in a random array of plates. Solid State Comm. 1984;50:999-1002.
- Chen C., Thomas D.M., Green R.E., Wagenet R.J. Two-domain estimation of hydraulic properties in macropore soils. Soil Sci. Soc. Am. J. 1993;57:680-686.[Abstract/Free Full Text]
- Craul P. Urban soil in landscape design. New York: John Wiley & Sons, Inc., 1992.
- De Gennes P.G. The physics of liquid crystals. Oxford, UK: Oxford Univ. Press, 1976.
- De Gennes P.G., Guyon E. Lois générales pour l'injection d'un fluide dans un milieu poreux aléatoire. J. Mécanique 1978;77:403-432.
- Dunn G.H., Phillips R.E. Equivalent diameter of simulated macropore systems during saturated flow. Soil Sci. Soc. Am. J. 1991;55:1244-1248.[Abstract/Free Full Text]
- Gimenez D., Allmaras R.R., Huggins D.R., Nater E.A. Prediction of the saturated hydraulic conductivity-porosity dependence using fractals. Soil Sci. Soc. Am. J. 1997;61:1285-1292.[Abstract/Free Full Text]
- Guéguen Y., Dienes J. Transport properties of rocks from statistics and percolation. Math. Geol. 1989;21:1-13.
- Guéguen, Y., and V. Palciauskas. 1992. Introduction à la physique des roches. Hermann, ed. des sciences et des arts, Paris.
- Hallaire V., Curmi P. Image analysis of pore space morphology in soil sections, in relation to water movement. In: Ringrose-Voase A.J., Humphreys G.S., eds. Soil micromorphology: Studies in mangement and genesis. Dev. in Soil Sci. 22. Amsterdam: Elsevier, 1994:559-567.
- Klute, A., and C. Dirksen. 1986. Hydraulic conductivity and diffusivity: laboratory methods p. 687–734. In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed.Agron. Monog. 9. ASA and SSSA, Madison, WI.
- Koolen A.J. Deformation and compaction of elemental soil volumes and effects on mechanical soil properties. Soil Tillage Res. 1987;10:5-19.
- Moran C.J., McBratney A.B., Koppi A.J. A rapid method for analysis of soil macropore structure. I. Specimen preparation and digital binary image production. Soil Sci. Soc. Am. J. 1989;53:921-928.[Abstract/Free Full Text]
- Murphy C.P., Bullock P., Turner R.H. The measurement and characterisation of voids in soil thin sections by image analysis, Part. I. Principles and techniques. J. Soil Sci. 1977;28:498-508.
- Paute J.L., Hornych P., Benaben J.P. Comportement mécanique des graves non traitées. Bull. liaison Labo. P. et Ch. 1994;133:27-38.
- Ringrose-Voase A.J. Measurement of soil macropore geometry by image analysis of sections through impregnated soil. Plant Soil 1996;183:27-47.
- Stauffer D., Aharony A. Introduction to the percolation theory. Revised, 2nd ed DC: Taylor and Francis. Washington, 1994.
- Vidal-Beaudet L., Charpentier S. Mechanical behaviour of silty clay loam-peat mixtures: Cyclic compression-release tests and effects of initial water content. Agronomie 1998;18:261-274.
TOP
ABSTRACT
INTRODUCTION
