Soil Science Society of America Journal 66:1784-1787 (2002)
© 2002 Soil Science Society of America
DIVISION S-1—NOTES
Modeling soil compaction under uniaxial compression
S. Assouline*
Institute of Soil, Water and Environmental Sciences, A.R.O., the Volcani Center, P.O.B. 6, Bet Dagan 50250, Israel
* Corresponding author (vwshmuel{at}agri.gov.il)
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ABSTRACT
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Knowledge about soil compaction is increasingly important within agriculture and for environmental protection. The objective of this study is to modify a previous two-parameter model to generalize it to account for preconsolidation effect. A three-parameter model for the relationship between soil bulk density and applied stresses during compaction is presented. It relies on the physical approach of the previous two-parameter model and satisfies the same boundary conditions. The proposed model is applied to compaction data of four silt loam and loam soils. A good fit to data is obtained for a wide range of applied stresses, including stresses less than the preconsolidation stress. The proposed model is compared with the three-parameter version of a recent model. The performances of the two models are similar. The root mean square errors for the four soils ranged between 0.004 to 0.011 for the recent equation and 0.005 to 0.018 for the proposed model. The advantage of the proposed model is that it releases the physically unrealistic constraint in the recent model that the maximal bulk density of a compacted soil is equal to its particle density.
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INTRODUCTION
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SOIL COMPACTION occurs in unsaturated soils when subjected to mechanical forces (Gill and Vanden Berg, 1967). In agricultural systems, the risk of soil compaction increases with the growth in farm size, increased mechanization and equipment weight, and the drive for greater productivity. Soil compaction has negative effects for agriculture as well as the environment: it adversely affects soil structure, reduces crop production, increases runoff and erosion, and accelerates potential pollution of surface water by organic wastes and applied agrochemicals. Therefore, knowledge on soil compaction is increasingly important within agriculture and for environmental protection.
The basic mathematical description of soil compaction is the relationship between soil bulk density, 
, and applied stress, 
. A logarithmic model was generally used to represent this relationship (Bailey and Vanden Berg, 1967; Larson et al., 1980; Gupta and Larson, 1982). In the logarithmic model, the bulk density is undefined at a stress of zero. To remove this limit, Bailey et al. (1986) developed a three-parameter multiplicative model:
 | [1] |
where o is the bulk density at zero stress and a, b, and c are empirical parameters determined by fitting Eq. [1] to compression data. The models based on the logarithmic function, including Eq. [1], predict that bulk density continues to increase as the applied stress increases (Assouline et al., 1997). This is in disagreement with experimental observations, which indicate that soils can be compacted to a finite maximal bulk density, max, which depends on soil properties and initial conditions (Amir et al., 1976; Faure, 1981). To satisfy this important boundary condition, Assouline et al. (1997) have proposed the following model:
 | [2] |
where max and 
are fitting parameters, and the other symbols retain their previous definition. Eq. [2] has two additional advantages over Eq. [1]: (i) it requires only two fitting parameters instead of three and (ii) the parameter max has a physical meaning and its value in the field can be predicted by means of compression tests under laboratory conditions (Hartge, 1986; O'Sullivan, 1992). The model in Eq. [2] performs as well as the one in Eq. [1] with one less fitting parameter (Assouline et al., 1997). Recently, Fritton (2001) has stressed that Eq. [1] and [2] are not flexible enough to represent the variability of the compression curve shapes, which range from nearly linear to partially S-shaped curves when expressed on a logarithmic scale. He introduced the following power expression:
 | [3] |
where m is soil particle density, 
, n, and m are empirical fitting parameters, and the other symbols retain their previous definition. This expression was shown to perform better that Eq. [1] or [2] for three of the four soils studied (Fritton, 2001). The model in Eq. [3] requires at least three parameters, and has also been used as a five-parameter model, considering m and o as additional parameters (Fritton, 2001). It is not clear why the bulk density in Eq. [3] is expressed as a function of the applied stress plus one. In terms of the physics of the process, the bulk density should be expressed as a function of the applied stress, and there is no mathematical reason to deviate from this. The expression in Eq. [3] is well defined for the whole range of applied stresses, from 0 to 
, without the need to add any constant to the independent variable . Moreover, in Eq. [3], = m - (m - o)(1 + n)-m for = 0, while by definition, should be exactly equal to o for = 0. Therefore, Eq. [3] should be written
 | [4] |
In its three-parameter version, the model of Fritton (2001) assumes that a soil can be compacted up to its particle density, m. This is physically unrealistic and is incompatible with experimental observations (Amir et al., 1976; Faure, 1981). This constraint is theoretically released in the five-parameter version, but in fact, the fitted values for m are greater than the particle density for two of the four soils, which is also physically unrealistic (Fritton, 2001; Table 2). The objective of this study is to modify the previous model of Assouline et al. (1997) to generalize it and make it more flexible for cases where the two-parameter model in Eq. [2] does not provide satisfactory results, especially when the applied stress is below the preconsolidation stress.
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Model Description
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The concept at the basis of Eq. [2] is that as the soil bulk density increases because of increases in applied stress, 
, the increment of increase, , becomes smaller. This may not be valid when the applied stresses are less than the preconsolidation stress. In such case, increasing the applied stress has by definition no effect on the bulk density, and the () function should include an inflection point. It is thus suggested to add a third parameter, 
, to Eq. [2] to allow the inflection point in () when and if required:
 | [5] |
where max and 
are fitting parameters, as defined for Eq. [2]. The proposed model in Eq. [5] fulfills the boundary conditions:
 | [6] |
 | [7] |
For a given soil, all the parameters in the model are dependent on the initial soil water content.
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Materials and Methods
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Compression data obtained for three silt loam soils (Bucks, Glenelg, Rayne; mesic Typic Hapludult; Fritton, 2001), and for a loam soil (Nahal Oz; Calcic Haploxeralf; Hadas, 1967, unpublished data), were used. The compression procedure of the silt loam soils is described in detail in Fritton (2001). It consists of uniaxial drained compression tests carried out on small undisturbed and disturbed soil samples submitted to stresses between 0 to 2971 kPa. The compression of the loam soil consists of uniaxial drained compression tests of air-dried disturbed samples submitted to stresses between 0 to 10 077 kPa. The initial bulk density, o, of the Bucks, Glenelg, Rayne, and Nahal Oz soils were 1.06, 1.43, 1.21, and 1.19 Mg m-3, respectively. In all the cases, m in Eq. [3] was set equal to 2.65 Mg m-3 (Fritton, 2001). The different models (Eq. [2], [3], and [5]) were fitted to the data by an iterative nonlinear regression procedure by the Levenberg-Marquardt method to minimize the error sum of squares between observed and computed values of the dependent variable (Glantz and Slinker, 1990). The evaluation of the performances of the respective models is based on the root mean square error (RMSE).
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Results
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The two-parameter model (Eq. [2]) provides the lower performance (Table 1). Adding the third fitting parameter, , to Eq. [2] increases its flexibility and improves significantly the fit to the data, especially for the Bucks soil (Table 1) and the Nahal Oz soil (Fig. 1)
. For these two soils, the proposed model performs somewhat better than the three-parameter equation suggested by Fritton (2001) (Table 1). For the Glenelg and the Rayne soils, Eq. [3] fits the data better than Eq. [5] (Table 1; Fig. 2)
. However, for practical purposes, the performances of these two models can be considered as similar. The main difference between the two expressions remains in the max values obtained for Eq. [5], which are much more realistic values of maximal soil bulk density than the particle density m = 2.65 Mg m-3 value assumed in Eq. [3].
For the Rayne soil, the improvement gained by the addition of the third parameter to Eq. [2] is slight, and the max value also remains practically unchanged (Table 1). Therefore, in some cases, the two-parameter version of the proposed model (Eq. [2]) might be sufficient. This is another advantage of the proposed model, which offers the possibility of reaching a good fit to data for the least number of fitting parameters.
Data for the Glenelg and the Rayne soils present a strong preconsolidation effect while data for the Bucks and the Nahal Oz soils do not. The different behavior of the soils during compaction is expressed by the relationship between the rate of increase of the soil bulk density and the applied stress, d/d(). The addition of to Eq. [2] affects d/d() significantly. While for Eq. [2], d/d() is:
 | [8] |
for Eq. [5], it becomes:
 | [9] |
The respective functions d/d() for the four soils on the basis of Eq. [9] are depicted in Fig. 3
on a semi-logarithmic plot. For the soils with the low preconsolidation effect, d/d() decreases monotonically with an exponential-like trend, because < 1.0. For the soils with the strong preconsolidation effect, d/d() increases drastically at low applied stresses to a maximum [indicating the inflection point in ()] and then decreases almost linearly because > 1.0. Note that in a semilogarithmic plot, the trend of d/d() according to Eq. [8], where = 1.0, is linear. Therefore, for soils with a low preconsolidation effect, the expected value of the third parameter is 0 < 
1. In such condition, Eq. [2], with = 1.0, is a particular case that can be appropriate to some soils. For the soils with a strong preconsolidation effect, the value of the third parameter should be > 1.0.
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Discussion and Conclusion
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The models of Bailey et al. (1986) and Assouline et al. (1997) were developed to represent the relationship between soil bulk density and applied stress during the compression of disturbed soil samples. In such a case, there is no concern about the preconsolidation stress, this concept being relevant only when undisturbed soil samples are compacted. Fritton (2001) has proposed a model that takes into account also applied stresses less than the preconsolidation stress. The independent variable of the model was set arbitrarily equal to the applied stress plus one. This transformation is not necessary as the power function chosen is well defined for the whole range of applied stresses, from 0 to . Moreover, it introduces a physical inaccuracy, as it does not satisfy the boundary condition = o for a zero stress. Also, the three-parameter version of the model assumes that the maximal bulk density of a compacted soil sample tends to the soil particle density, in disagreement with experimental observations. In this study, the two-parameter model of Assouline et al. (1997) is improved to account also for the range of applied stress that are lower than the preconsolidation stress. The increased flexibility is gained through the addition of a third parameter. The model satisfies the two boundary conditions for = 0 and 
(Eq. [6] and [7]). The model conserves the advantage that one of the parameters, max, has a physical meaning, and values under field conditions can be predicted from compaction tests under laboratory conditions (Hartge, 1986; O'Sullivan, 1992).
Bailey et al. (1986) have stated that the main disadvantage of their multiplicative model (Eq. [1]) was that it has three empirical parameters. In that sense, the model of Assouline et al. (1997) presents an important advantage, as it requires only two parameters, one of them being physically based and predictable. Taking into account the sources of error related to bulk density estimation in compaction experiments (Fritton, 2001), one can conclude that in most of the cases it was applied, the two-parameter model has provided a good representation of the relationship between bulk density and applied stress, leading to RMSE values below 0.04 for all cases but the Bucks soil (Assouline et al., 1997; Fritton, 2001; Table 1). The chances of developing predictive ability in empirical models, through the definition of relationships between the parameters and soil properties, increase as the number of parameters is lowered. Therefore, the two-parameter model should be preferred whenever it provides accurate results. For the specific cases where additional flexibility is needed, such as when preconsolidation effects are observed, the three-parameter model (Eq. [5]) can be used, while conserving the same physical approach and satisfying the same boundary conditions.
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NOTES
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Contribution no. 612/02 from the Agricultural Research Organization.
Received for publication August 28, 2001.
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REFERENCES
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TOP
ABSTRACT
INTRODUCTION
