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DIVISION S-1—SOIL PHYSICS

Characterizing Nonrigid Aggregated Soil–Water Medium Using its Shrinkage Curve

^a Institut de Recherche pour le Développement (IRD), Centre de Montpellier, 911 av. Agropolis, BP64501, 34094 Montpellier, France
^b Laboratoire Environnement et Développement (LED), Université Paris VII, 2 place Jussieu, BP7071, 75251 Paris Cedex 5, France
^c Agricultural and Biological Engineering Dep., Purdue Univ., West Lafayette, IN 47906, USA

^* Corresponding author (erik.braudeau{at}ird.fr).

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
The properties of the soil–water medium are presented in the literature independent of its internal organization and operation. The objective of this study is to develop and test a conceptual model that used a continuously measured shrinkage curve (SC) to describe the functional organization of the soil–water medium. In this model, two functional porosities (micro and macro) are delineated and quantified by the SC. In addition, the equilibrium for four functional water pools is represented and parameterized by the SC. A set of eleven parameters was found necessary to model the seven phases of the SC and to describe the corresponding soil hydrostructural changes. A method to accurately obtain the parameters of this model by a specific analysis of the continuously measured SC is demonstrated. Examples of continuously measured and modeled SCs according to the pedostructure model (PS) are presented and discussed.

Abbreviations: CARHYS, software for soil hydrostructural characterization • PS, pedostructure (model of shrinkage curve) • SC, shrinkage curve • SL, straight lines (method of shrinkage curve parameters determination) • XP, exponential (model of shrinkage curve)

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
MODELING THE SC can be classified in three categories: 1. Those considering certain parts of the curve only such as the model of McGarry and Malafant (1987) and that of Giraldez et al. (1983), which amounts to neglecting or fixing a certain number of parameters; 2. Those modeling its mathematical shape as in the case of Groenevelt and Grant (2001), where they used the sigmoid curve. A limited number of parameters are then necessary (two or three), but the selected parameters do not have any relevance to the processes being modeled; and 3. Those considering the entire SC but making assumptions on the shape of the phases, as is the case of the models of Tariq and Durnford (1993) and Braudeau et al. (1999): a straight line for the quasi-linear parts, and polynomial or exponential function for the curvilinear parts. The parameters of the models are, therefore, representative points of the SC and, contrary to the preceding case, are in connection with the process being analyzed. These points represent particular hydrostructural (see Appendix 1 for definitions) states of the soil and constitute the boundary conditions (V, W) for each shrinkage phase. However, these models do not provide any explanation for the corresponding configuration of the distribution of air, water, and solids in soil, which remains to be explained. The PS proposed in this paper addresses this issue by modeling the entire SC without any embodied assumptions related to its shape and functionality.

Braudeau et al. (1999) developed a method for continuously measured SCs providing accurate estimate of the soil water content at each transition point of the shrinkage phases. They recognized seven shrinkage phases on a typical curve (Fig. 1) , successively linear and curvilinear, and divided into four stages: interpedal, structural, basic, and residual. They suggested that the transition points (A, B, C, D, E, and F of Fig. 1) are characteristic of the hydrostructural behavior of the soil and they supplied a method to determine these points from soil volume change data using optimal fitting techniques. Appendix 2 describes the equations used by their model. However, these equations were empirically established and thus do not supply any information about the hydrostructural processes of the observed shrinkage phases of the curve; neither do they allow a better understanding of the mechanical, agronomic, and hydraulic properties associated with these shrinkage phases. To quantitatively characterize these properties, we need a physically based equation of the soil SC that is an equation linked to a conceptual model of the soil structure and water interaction.

View larger version (22K): [in this window] [in a new window]   Fig. 1. A typical continuously measured shrinkage curve (SC) of a reconstructed soil sample using three hundred points of measurement. Points A, B, C, D, E, and F are the transition points of the shrinkage phases as determined by parametric modeling of the SC according to the exponential model (XP) of Braudeau et al. (1999). LN, CV refers to as linear and curvilinear phases, respectively.

Two major approaches exist for modeling soil water physical properties in soil matrix. The first views soil structure as an assembly of capillary tubes between and within rigid aggregates, forming only one or two levels of structure. Soil structure is then characterized by a mono- or bimodal distribution of pore volume by class size of equivalent pore diameter (Coppola, 2000). The second approach considers soil structure as an arrangement of particles and swelling aggregates at various scales: clayey plasma, primary peds and assembly of primary peds that constitutes the soil fabric at the horizon level (Braudeau, 1988a, 1988b; Colleuille and Braudeau, 1996). Figure 2 schematizes this approach and defines the pedostructure as the soil–water fabric of a soil horizon.

View larger version (83K): [in this window] [in a new window]   Fig. 2. The pedostructure concept is shown taking into consideration the hierarchical functional levels of the soil medium.

The objective of this study is to develop and test a physically based model of the pedostructure fitting the continuously measured SC during drying. The goal is to characterize and parameterize the soil–water medium using its measured SC to model its internal hydrostructural configuration.

We divide the article into two parts; (i) description of the shrinkage processes and their relationships caused by the removal of the water pools held by each of the two pore systems, inter and intra primary peds, and (ii) application of the PS to the characterization of four soil types.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
The Different Water Pools in the Pedostructure
For interpreting the SC we define two pools of water in the two pore systems, inside and outside primary peds: swelling water, w_sw, and condensed water or nonswelling, w_cn. Swelling water occupies a pore space acquired by the spacing of particles or aggregates under the effect of osmotic pressure. Its removal from the sample causes shrinkage of the concerned pore system. Condensed water, on the other hand, occupies an interstitial pore space and is replaced by air (or water vapor at saturation pressure) when it leaves the pore; its loss causes little or no shrinkage. During drying, each linear phase of the SC is caused by the predominant departure of only one pool of water, w_sw or w_cn, from either the micro- or the macropore system (Fig. 3) . The curvilinear components of the SC, which constitute a transition between the two adjacent linear phases, are caused by the simultaneous departure of the two water pools (w_sw and w_cn) corresponding to these two linear phases. Table 1 shows the different water pools acting in each of the shrinkage phases. The basic relationship between the pedostructure volume change dV and water pools can be written as follows:

[1]

where K_re, K_bs, K_st, and K_ip are slopes of the linear phases (residual, basic, structural, and interpedal) of the SC that are considered constant for the entire range of the water content. The water pools within the pedostructure are defined as follows:

View larger version (63K): [in this window] [in a new window]   Fig. 3. Various configurations of air and water partitioning into the two pore systems, inter and intra primary peds, related to the shrinkage phases of a standard shrinkage curve (SC). wcn represents the condensed water lodged in interstitial pore site Scn, and wsw represents the swelling water lodged in interstitial pore site Ssw that can be interped macropores (m) or matrix micropores (µ). Asterisk means that the site is active and the corresponding numbers of the active sites are nsw and ncn. The various pools of water, wre, wbs, wst, wip, are represented with their domain of variation. The inflection points Ibs and Ist, and by extension Ire and Iip, are placed on the graph for representing the quasi-linear shrinkage phases delimited by the transition points (A, B, C, D, E, and F). Points N', M', and L' are the intersection points of the tangents at those linear phases and Points N, M, and L, the corresponding points of the SC.

w_bs such that dw_bs dW in the basic phase (I_bs);

w_st such that dw_st dW in the linear structural phase (I_st); and

w_ip such that dw_ip dW in the interpedal phase W > W_F.

All the water pools (w_re, w_bs, w_st, w_ip) and W, total water content of the sample, are expressed as mass of water divided by mass of dry soil (kg kg^–1); thus the relationship between the different water pools of the pedostructure is:

[2]

The constants K_re and K_st are near zero and much smaller than K_bs and K_ip. The above definitions and assumption, and definition of the specific volumes in Fig. 2, makes it possible to quantitatively represent the volume change of the different elements of the pedostructure in terms of the main SC variables, V and W (Fig. 4) .

View larger version (19K): [in this window] [in a new window]   Fig. 4. Graphical representation of the specific volumes (Vµ and V), the specific pore volumes (Vpµ and Vpm), the air contents and the water pools (W, wre, wbs, wst, and wip) of the pedostructure starting from a measured SC. Vµ is the specific volume of primary peds, equal to (Vpµ + Vs) = [max (wre)w + wbs w + Vs], and w is the water bulk density.

Let us consider the curvilinear parts of the SC, J (J = L, M, N), located between the inflection points of two adjacent linear phases: L for (I_ip and I_st), M for (I_st and I_bs) and N for (I_bs and I_re) (Fig. 3). According to Table 1, each Segment J is associated with a swelling (of micro or macropore) and condensed (of micro or macropore) water pair such as ^L, ^M, and ^N for J = L, M and N, respectively. Let ^J_sw be the probability of water pool pair w_sw (or w_cn) that is lost from the sample during drying at a given water content of Region J, then:

[3]

where w_sw can be replaced by w_bs or w_ip, and w_cn by w_re or w_st depending on the stage of J (N, M, or L; Table 1). In fact, ^J_cn and ^J_sw represent the respective fractions of the two water pools of the SC Region J, available for evaporation at the water content W. These two fractions can be considered at the same energy level depending on W assuming that evaporation is sufficiently slow so that the SC can be regarded as a succession of equilibrium states after each change of water content dW.

Each water pool is in interstitial sites S_sw or S_cn (of swelling or nonswelling) that can be active or not as defined in Fig. 3. Let n_sw and n_cn be the numbers of active sites of swelling and condensed at J. It is assumed that, for a variation of water content dW, the number dn_sw (or dn_cn) of active sites which appear for one water pool and disappear for the other (Fig. 3), is proportional to dW and to the number of active sites n_sw (or n_cn) present such that:

[4]

This amounts to assuming an exponential progression of the number of active sites with water lost during drying, increasing for one water pool and decreasing for the other:

[5]

Since, for the shrinkage Region J, the loss of a water pool is proportional to the number of its active sites, the probability ^J_sw can be expressed as:

which can be written as:

[6]

where a, b, A, B are constants, k_J is a constant characteristic of segment J (= N, M, or L) of the SC, and W_J is the water content at Point J (= N, M, or L) of this segment such as ^J_sw = ^J_cn = 1/2. Integration of Eq. [6] between 0 and W gives:

[7]

In Region L of the SC, dW dw_ip + dw_st, and

[8]

with k_L > 0 since for W >>W_L, _ip 1 and _st 0; and for W << W_L, _ip 0 and _st 1.

In Region M, dW dw_bs + dw_st and:

[9]

In Region N, dW = dw_bs + dw_re and

[10]

Each probability ^J is a logistic equation that varies asymptotically between 0 and 1, as shown in Table 2.

View this table: [in this window] [in a new window]   Table 2. Values of the probabilities Jsw and Jcn = 1 – Jsw in the different shrinkage phases during drying for each water pool. The subscript sw correspond to the swelling water pool (wip or wbs) and cn to the condensed water pool or nonswelling (wst or wre) of the same pair associated with Region J of the shrinkage curve (SC).

[11]

(b) For the case of w_bs, the variation of w_bs occurs in Segments N and M, dw_ip can be ignored rather than dw_re, dw_bs, dw_st, therefore, we can write:

[12]

The product (dw_re dw_st) is negligible and therefore ignored in the following equation:

[13]

Therefore:

[14]

Considering the values of ^J given in Table 2, Eq. [14] can be expressed as:

[15]

where W_Ibs is the water content at the inflection point of the basic shrinkage phase.

According to Eq. [7], integration of [15] gives:

[16]

One can verify that since k_N > 0 and k_M < 0, max (w_bs) (W_M – W_N) for W >> W_M. (c) Similar to the previous case, the case of w_st, which varies in Segments M and L of the SC, gives:

[17]

[18]

and since k_L > 0, max(w_st) (W_L – W_M) for W >> W_L. d) For the case of w_ip, that varies only in Segment L of the SC, according to Eq. [7] where ^J_sw = ^L_ip and ignoring exp(–k_LW_L) below 1:

[19]

And

[20]

Significance and Determination of Water Pools Parameters
According to Eq. [1], the parametric equation of the SC is:

[21]

where V_o is the specific volume of the dry sample. Equation [21] can be considered as the pedostructure variation law in which K_re, ..., K_ip, are the (geometrical) parameters, and w_re, ..., w_bs, the parametric functions of water pools at equilibrium represented by Eq. [11], [16], [18], and [19]. The variation law (Eq. [1]) with its parameters and functions represent the PS in which all of its variables shown in Fig. 4 are at equilibrium defined by the SC.

The physical significance of the six parameters (k_N, k_M, k_L, W_N, W_M, W_L) used in the water pool functions is demonstrated in Appendix 3. The points of the SC for water contents W_{J (J = N, M, L)} have the same x-coordinates as the intersection points (N', M', and L') of the tangents to the linear parts of the SC (Fig. 3); and the constants k_J represent the reverse of the vertical (y axis) distance (NN', MM', and LL') between those intersection points and the SC segments (k_N,L > 0 and k_M < 0):

[22]

Using relationships in Eq. [22] the six parameters of the water pools are easily calculated from the continuously measured SC after having drawn the four tangents of the SC at the inflection points.

Comparison with the Exponential Model
The foregoing conceptual PS uses a total of eleven parameters, six for the water pools and five for the structure:

[23]

The independent parameters of the SC exponential model (XP) proposed by Braudeau et al. (1999) also uses a total of 11 parameters, namely:

[24]

The method of determining the parameters of the PS using Eq. [22] and the tangents of the SC, is similar to the Straight Lines (SL) method for determining the phase transition Points A, B, ... and F proposed by Braudeau et al. (1999) for the XP model. This method draws the four tangents to the SC and uses the vertical distances between the points N', M', L', and their homologs N, M, L on the SC (Fig. 3) to determine the position of the transition phases using the relationships shown in Table 3 such as:

[25]

and

[26]

Using the Shrinkage Curve for Soil Characterization
As shown above, parameter lists [23] and [24], can be easily estimated from the measured SC using the SL method (Braudeau et al., 1999). An optimization algorithm was implemented to fit the data to the above equations using the simplex method. Accordingly, W_N, W_M, W_L characterizes the pore systems using the water pools definition, such as:

[27]

where _w is the water bulk density.

A special case occurs where the PS and XP act differently. This is the case of very swelling soils where the macroporosity of the pedostructure is not significant and the polyhedral aggregates are assembled by their microsurfaces. The structural shrinkage phase (E-D) is reduced and may even not exist so the I_ip and I_bs shrinkage phases (Table 1) are overlapping. In addition, the primary peds begin to retract before all the aggregates are in complete contact (at Point E) giving W_D > W_E. In this case and for the PS model, K_st must be taken as equal to zero and should not be confused with the slope of the shrinkage Phase E–D as it was done with the XP (Braudeau et al., 1999).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
The soil samples used in this study came from two sources:

1. A pedological cartography and characterization of an irrigated area in the lower valley of Majerda, 20 km north of Tunis (North Tunisia), on recent deltaic alluvia near the Mediterranean Sea. This study (Braudeau et al., 2001) was undertaken within the framework of a development project with emphasis on pedological information system where the physical soil characterization was to be based on the retractometric analysis. Shrinkage curves were measured and analyzed on recast soil cores. The SC measurements were made by means of a "retractometer" (Braudeau et al., 1999). Soil samples were gently disaggregated and 2-mm sieved, then reconstructed in consistent cylinders by piling up the aggregates in a 5.5-cm diameter and 4.0-cm long PVC cylinder and packing down the cylinders by one cycle of moistening–drying under free air and stress conditions (Zidi and Braudeau, 1998). The SC measurements constitute 400 to 800 points, equivalent to one data point per minute. Three of the SCs from their study will be used in this paper: one for a soil located in an old raised bank with loam texture and the other in the decantation plain with clay-loam texture. Their great group name in Soil Taxonomy Survey (Soil Survey staff, 1996) is Xerofluvents. They are referred to as Alluvial 1, 2, and 3 in Table 4 and Fig. 5 .
   
2. Four soils, all in Senegal, studied by Braudeau et al. (1999). Their great group names are Pellustert, Torrifluvents, Plinthustox, and Plinthustalf. Soil samples were reconstituted with <2-mm sieved aggregates in consistent cylinders (diameter 55 mm; height 35 mm). After their wetting on a porous ceramic plate (12 h) they were equilibrated (48 h) at a matrix potential of –100 kPa before they were smoothly compacted under a given pressure to reach the specific volume of the undisturbed soil. Unlike the precedent study above, the SC data files are not available and only the XP parameters of the last five shrinkage phases of the SC (eight parameters) are provided. Parameters were determined by minimizing the difference between the XP and the measured SC. Two clayey soils were taken as examples in this work for comparison between the PS model and the XP: the Pellustert and the Plinthustox referred to in tables and figures as Vertisol and ferrallitic soil. Their SC was redrawn according to the equations and parameters of the XP (Braudeau et al., 1999) and was then fitted according to the PS.
   

View larger version (35K): [in this window] [in a new window]   Fig. 5. Continuously measured and modeled shrinkage curves (SCs) for three alluvial soils in the Lower Majerda Valley, Tunisia, using the retractometry procedure. The graph on the right hand side shows the fit for the whole data range. One out of three points is represented. The parameters determined by software for soil hydrostructural characterization (CARHYS) are shown in Table 4. The cumulative variation curves of the water pools, wre, wbs, wst, and wip, added to Vs are represented referring to as (1), (2), (3), and (4), respectively. Points A, B, C, D, E, and F are the transition points of the shrinkage phases defined by the exponential model (XP) and represented on the corresponding water pools curves.

View larger version (21K): [in this window] [in a new window]   Fig. 6. Shrinkage curves (SCs) and the corresponding micro and macropore volume (Vpµ and Vpm) of two clayey soils in Senegal: (a) vertisol, and (b) ferralitic soil. For the swelling soil case (Fig. 6a) Curve 1 is the modeled wbs water pool contribution (Kbswbs + VA) to the SC when Kst = 0.4 and Curve 2 when Kst = 0. (Points A, B, C, and D are the initial parameters determined by the exponential model (XP) with taking Kst = 0.4; Points C'', D'', E'', and F'' were obtained by CARHYS using the pedostructure (model for SCs) model with Kst = 0).

Step 1: Manually estimating the initial location of Points B, C, and D on the measured SC. Points B and C are located inside the linear basic shrinkage phase to determine the parameters of the tangent (of slope K_bs) passing through the basic shrinkage phase, which remains constant during the optimization. Point D is placed inside the linear structural shrinkage phase to limit the interval subjected to adjustment in the second step between the Point D and the last measured point of the curve {V_D – V_final}. V_final is the specific volume corresponding to the last reading.

Step 2: Adjusting segment {V_D – V_final} of the measured SC using the following relationship V1 = V_N' + K_bs w_bs and considering (i) K_re = 0 and V₀ = V_N', (ii) the tangent to the basic shrinkage phase, of slope K_bs, is given (determined using linear regression) and that (iii) W_N, V_N', W_M, k_N, k_M are the variable parameters to be determined (V_N' is the y-axis coordinate corresponding to W_N on the given tangent).

Step 3: Adjusting the curve (V_measured – V1) versus W for the whole range of the water content using V2 = K_st w_st + w_ip while keeping K_st, W_L, and k_L variables with the constraint that the maximum value of K_st is fixed at 0.07. Beyond that, K_st is taken as zero, considering that the linear phase of this structural shrinkage is missing and that an overlap exists between the two curvilinear Phases M and N caused by w_bs and w_ip water pools.

Step 4: Adjusting the whole SC using V = V₀ + K_bs w_bs + K_st w_st + w_ip considering only W_M, k_M, W_L, and k_L as variable parameters.

The above procedure is repeated for various initial conditions to ensure global optimal solution is reached.

The optimization procedure used to determine these PS parameters at each step is based on the simplex algorithm. The final adjustment evaluated by the mean square deviation between the measured and the PS-modeled SC is around 10^–6, 10^–7. The output of CARHYS is a file containing the resulting parameters of the PS, and also those of the XP that are deducted from the PS parameters using the relationships shown in Table 3.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
Figure 5 shows the measured and PS-based modeled SCs of the three alluvial soils of Tunisia along with the changes in water pools and in specific micro- and macropore volumes calculated using the PS. In the three examples, the measured and modeled SCs cannot be distinguished, the mean deviation being 3.6 x 10^–4, 3.8 x 10^–4, and 7.0 x 10^–4 dm³ kg^–1 for the three soils with a coefficient of correlation >0.9995. The two sets of parameters (PS and XP) determined by CARHYS for these soils are shown in Table 4. Results are presented as means and standard deviations of the parameters using CARHYS analysis for twelve different initial locations of the transition points. It can be seen that the robustness of the CARHYS procedure based on the PS model equations for determining the pedohydral parameters is very good.

Figure 6a and 6b shows a comparison between the XP and PS. The data given by Braudeau et al. (1999) of the two clayey soils from Senegal, vertisol, and ferrallitic soil were used. Equations in Table 3 were used to calculate the PS parameters from the XP parameters (Table 4, Columns 5 and 6) to draw PS SCs. The XP SC were also drawn from the XP parameters. The XP and PS curves represented in Fig. 6a and 6b for the two soils cannot be distinguished. The mean deviation between the PS and XP modeled SC, calculated for 200 points for the interval {W_A–W_E}, was found to be <1.5 x 10^–4 and 0.9 x 10^–4 for the vertisol and ferrallitic soil, respectively. However, when CARHYS was used for these SC for determining the pedohydral parameters, they were found to be the same as the values given for the ferrallitic soil and were different for the vertisol (Columns 5 and 6 in Table 4). This is explained by the fact that CARHYS fits the SC using the pedostructure conceptual model that does not accept a value of 0.4 for K_st, the slope of the structural shrinkage phase, as determined by the XP. As mentioned above, K_st values are often near zero and cannot exceed 0.07. If a higher value of K_st is found by the XP procedure (SL method, Braudeau et al., 1999) as in the case of the vertisol (Fig. 6a), the interpedal swelling water pool, w_ip, is not completely vanished when the shrinkage of the primary peds starts to be effective. In this case, there is no structural shrinkage phase and K_st must be taken as zero.

Figure 6a shows how the particular case of the vertisol was considered by CARHYS and the XP. For this soil the contribution of the swelling water, w_bs, calculated using K_st = 0.4, is lower (Curve 1) than that calculated by using K_st = 0 (PS, Curve 2). This correction (taking K_st = 0) shows that the volume change of primary peds was underestimated by the XP, but above all, W_E can be smaller than W_D, signifying that the beginning of the clayey plasma shrinkage takes place before the interped shrinkage limit at Point E.

Figure 6b shows the case for ferrallitic soil, clayey but weakly swelling soil. The structural linear shrinkage phase (E–D) is well discernible with a slope of K_st = 0.034 dm³ kg^–1. CARHYS determined the same values of the parameters as those given by the XP.

The decrease in macropore volume shown in the figure in the range {W_F–W_E} is directly related to the decrease in w_ip. In the range {W_D–W_A}, this change is related to the micropore volume decrease, depending on K_bs being smaller or greater than unity, according to the following relationships: dVp^m = (K_bs – 1) dVp^µ.

The above results show the entire compatibility of both models. Both fit well the continuous measured SC even though their basic hypotheses and the resulting implications are different (Fig. 7) . The PS, more realistically, is based on a conceptual model of the soil fabric in equilibrium with its water pools. It models all the state variables of the pedostructure. On the other hand, the XP is based on partitioning of the SC into linear and exponential segments. It determines an empirical delineation of the shrinkage phases and thus defines phase transition points that are taken as parameters.

View larger version (29K): [in this window] [in a new window]   Fig. 7. An example of how the exponential functions of the exponential model (Eq. [A1]) simulates the curvilinear part of the curve wbs = f(W) of the vertisol (Eq. [16]), and how it can thus determine the effective beginning and end of variation of the curve. The mean deviations between the two fitted curves in the ranges {WD–WC} and {WB–WA} are 3.5 x 10–4 and 1.4 x 10–4 dm3 kg–1, respectively.

The pedostructure concept is then composed of the XP model, which defines the change in macroscopic soil physical properties, and the PS model that simulates the internal soil–water medium equilibrium for each water content. Together they describe, internally and externally, the description space of the pedostructure, of which the eigenvalues are the set of parameters [23] or [24] (Appendix 3).

	CONCLUSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
This article shows how the SC can be considered and modeled as a result of soil structural variation law and logistic probability functions for coupled water pools (swelling and nonswelling) for micro- and macropore system equilibrium.

As a result, the continuously measured SC can be used as a unique characteristic of the soil–water medium that offers a quantitative description of its internal organization and operation.

A new pedostructure concept was described for characterizing the external and internal behavior of the soil horizon. This new paradigm will have significant implications on modeling and characterizing soil–water dynamics.

	APPENDIX 1

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
List of definitions
Functional scales: levels of scales in the soil medium defined by their soil–water interaction and operation contrary to their classification by size.

Hydrostructural: interaction between soil water and soil structure.

Pedohydral parameters: parameters of the shrinkage curve.

Pedostructure: the soil-water medium of the soil horizon composed of the assembly of peds in several organizational levels, the last of which is that of the primary peds. It refers to the combination of the soil fabric and its dynamic operation with water.

Nomenclature
V, Pedostructure specific volume, dm³ per kg of dry soil horizon

V^µ, Primary peds specific volume, dm³ per kg of dry soil horizon

V_s, Solids (primary particles) specific volume, dm³ per kg of dry soil horizon

Vp, Pedostructure pore specific volume, dm³ per kg of dry soil horizon

Vp^m, Macropore specific volume of pedostructure, dm³ per kg of dry soil horizon

Vp^µ, Micropore specific volume of pedostructure, dm³ per kg of dry soil horizon

W, Pedostructure water content (soil moisture), kg per kg of dry soil horizon

w_re, Pedostructure residual water pool, kg per kg of dry soil horizon

w_bs, Pedostructure basic water pool, kg per kg of dry soil horizon

w_st, Pedostructure structural water pool, kg per kg of dry soil horizon

w_ip, Pedostructure interpedal water pool, kg per kg of dry soil horizon

A, B, C, D, E, F, Shrinkage transition points of the SC defined by the XP model

N', M', L', Intersection points of the tangents to the SC at the linear phases

N, M, L, Characteristic points of the SC at the vertical (y axis) of N', M', L'

I_st, I_bs, Inflection points of structural and basic shrinkage phases

W_A, W_B, ... W_F, Pedostructure water content (kg kg^–1) at Points A, B, ..., F

W_N, W_M, W_L, Pedostructure water content (kg kg^–1) at Points N, M, L

V_A, V_B, ..., V_F, Pedostructure specific volume (dm³ kg^–1) at Points A, B, ..., F

V_N, V_M, V_L, Pedostructure specific volume (dm³ kg^–1) at Points N, M, L

V_N', V_M', V_L', y-axis values (dm³ kg^–1) of Points N', M', L' in the SC graph

K_re, K_bs, K_st, K_ip, Slopes of the SC linear phases (dm³ kg^–1)

k_N, k_M, k_L, Shape parameters (kg kg^–1) of the SC equation

	APPENDIX 2

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
Equations of the Exponential Model
Points A, B, C, ..., and F were defined as the parameters of the exponential equation (XP model) that fits the curvilinear phases of the SC. For example, the expression for the curvilinear Phase D–C is:

[A1]

where W_n = (W – W_D)/(W_C – W_D) and e = exp(1) = 2.718.

This leads to the important relationships between parameters when W = W_C in Eq [A1]:

[A2]

This last equation is used to establish the relationships in Table 3.

	APPENDIX 3

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 
Significance of the Parameters k_J and W_J, (J = N, M, and L)
Equations for the tangents at the linear phases represented by Points I_re, I_bs, I_st, and I_ip in Fig. 3 are obtained by integrating Eq. [1] (see main text) into each Segment L, M, and N of the curve. For the Segment M of the SC, integration of Eq. [1] and [9] according to [7] gives:

[A3]

Equations for the tangents are:

[A4]

(in the linear structural shrinkage phase), and in the linear basic shrinkage phase:

[A5]

Thus, the coordinates of the intersection Point M' of the two lines are:

[A6]

The intersection Point M' and the Point M of the SC have the same x-coordinate and k_M represents the reverse of the distance MM' except for the factor (K_bs– K_st) Log2. The same reasoning applies for Parts L and N of the SC that gives:

[A7]

[A8]

These three last equations are listed in Table 3 and are used to transform the PS parameters into XP parameters.

The SL Method for the XP and PS Models, and the Required Parameters

Building the SC according the XP and using the SL method proposed by Braudeau et al. (1999) needs, as for the current PS model, the four tangent lines at the inflexion points of the SC (eight parameters), plus the y axis of the three points, N, M, and L, (three parameters) of the SC corresponding to the intersection points N', M', and L' of the tangents. This leads to 11 parameters. If the enclosed porosity (nonconnected) is neglected, then the interpedal shrinkage line (IP-line) is identified to as the saturation line of which the slope is unity and the necessary parameters number becomes 10. In that case, W_sat = W_L and V_s = V_L– W_L.

Thus, if there are no other pore subsystem in the pedostructure other than the primary peds and their assembly, the two slopes at the extremes of the curve reduce to zero for W = 0 and one for W >> W_sat. Therefore, the pedostructure description space is nine-dimensional if the enclosed porosity is neglected (IP-line and saturation line are merged).

In most cases, simplifications can be made: K_re = 0; K_st = 0; V_s = 0.4 (d_s = 2.5 kg dm^–3), which reduces the parameters number. Additionally, if the interpedal shrinkage is also neglected (for modeling soil in situ, for example) the five-phases SC is considered with only six parameters required, such as:

or

V_B and V_D being calculated using (Eq. [A2]).

	ACKNOWLEDGMENTS

 
This work was partially conducted while two of the authors were on leave at Cirad in the COTONS Decision Support team. The authors thank the support and the encouragement provided by IRD, CIHEAM-IAMM, Cirad, and Purdue University.

Received for publication November 23, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSION APPENDIX 1 APPENDIX 2 APPENDIX 3 REFERENCES

 

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