New Device and Method for Soil Shrinkage Curve Measurement and Characterization
E. Braudeau,* J. M. Costantini, G. Bellier, and H. Colleuille
ABSTRACT
There is a need for a conceptual and analytical model to describe
and interpret the common shrinkage curves of structured soil samples.
In this work, we propose a new device for direct shrinkage measurement of unconfined structured soil samples and discuss the relevance
of parametric models of the shrinkage curve to fit the experimental
data. The experimental procedure consists of simultaneous and continuous measurements of the diameter, height, and weight of an initially
saturated soil sample as it dries. The shrinkage measurement can be
completed for the full moisture range in a short time (2-3 d), and all
shrinkage phases can be established easily and accurately identified.
The points of transition between the different shrinkage zones of the
shrinkage curve are considered as characteristics of the shrinkage
process. They are used as parameters to model the experimental
shrinkage curves by linear shrinkage zones separated by curvilinear
transition ones. The efficiency of two parametric models to fit the
experimental data and to provide the characteristic points of the curve
are compared and discussed. A new procedure for fitting the shrinkage
curves of structured soil samples and for determining the best position
of the characteristic points on the curves is proposed.
vance of existing models of parametric equations to fit
the experimental data for different kinds of soil structure is tested and discussed.
THEORY
Presentation of the Shrinkage Curve
Shrinkage data plotted as specific volume, v, against water
content is commonly termed a shrinkage curve (McGarry and
Malafant, 1987). On such a plot (Fig. 1), the location of experimental data is generally referred to the 1:1 saturation line,
also called the load line (Sposito and Giraldez, 1976), which
is characterized by an intercept equal to the specific volume
of the solids, vs, and by a slope equal to l/pw, pw being the
bulk density of water (1 kg m~3). This 1:1 line represents the
shrinkage curve of a theoretical two-component (solid and
water) saturated system where the water removal is not replaced by air entry throughout the full moisture range
(McGarry and Malafant, 1987). The point of intersection between the shrinkage curve and the 1:1 saturation line defines
the water content of the saturated soil sample 6S.
Shrinkage data may be also plotted as void ratio e against
moisture ratio •&, where e and •& relate the volume of voids
and water to the volume of solid, respectively (Groenvelt and
Bolt, 1972; Tariq and Durnford, 1993). They are related to
the variables used in this work by the following equations:
S
OIL STRUCTURE, the arrangement of particles and aggregates and the ensuing porosity, influences the
movement and storage of fluids, gases, and nutrients
in the soil, and hence most of the soil's hydrological,
physical, and agronomical processes. Soil structure decline is also regarded as one of the most important single
issues in soil degradation and declining crop production.
It is therefore essential that soil structure is accurately
represented in models to simulate its hydrostructural
behavior of soil. Shrinkage curve analysis is one of the
few methods allowing an accurate quantitative assessment of soil structure (Coughlan et al., 1991) and hydrostructural properties (Braudeau and Bruand, 1993). A
shrinkage curve represents the specific volume change
of a soil relative to its water content (Haines, 1923; Stirk,
1954), which may be interpreted as soil pore changes
on drying. Shrinkage curve analysis provides pertinent
indices of soil structure and soil behavior (McGarry and
Daniells, 1987; Braudeau, 1988b) and has been successfully applied to investigate soil structure degradation
under crops (Chan, 1982; Reeve and Halls, 1978;
McGarry, 1988; McGarry and Smith, 1988). However,
the use of shrinkage curves as a measure of soil structure
depends first on the development of an accurate method
of measurement and modeling of the shrinkage curve.
Thus, we present an improvement of the method for
continuous and automated measurement of soil shrinkage curves suggested by Braudeau (1987), and the rele-
= (e + 1) vs and
6 = -9 vs pw
[1]
Three distinct shrinkage zones are generally distinguished
in a typical shrinkage characteristic curve (Fig. 1), although
there exist different terminologies and more than three zones
may be noticed. These zones are referred to as: (i) structural,
(ii) normal, (iii) residual (Haines, 1923; Lauritzen and Stewart,
1941; Stirk, 1954; Reeve and Hall, 1978). The points of transition between these different shrinkage zones are generally
considered as characteristic parameters of the soil samples
(Coughlan et al., 1991). According to the literature, the structural and the residual shrinkage zones correspond with a volumetric change smaller than in the normal shrinkage zone
where it is maximum. A zero shrinkage zone, corresponding
with a loss of water without volumetric change, is sometimes
recognized (Bronswijk, 1988). In the normal zone, or basic
zone, as suggested by Mitchell (1992), the volumetric change
is often assumed to be equal to the water loss (Kbs = dv/d6 =
1) (McGarry and Malafant, 1987; Tariq and Durnford, 1993b).
However, this assumption is only valid for a structureless clay
paste (Sposito and Giraldez, 1976; Chan, 1982). Generally,
the slope of the shrinkage curve in this zone does not correspond with a unitary shrinkage (Lauritzen and Stewart, 1941;
Croney and Coleman, 1954; Bruand and Frost, 1987; MacGarry and Daniells, 1987) and may be very low, near 0.1
(Braudeau, 1987).
In fact there are actually two conceptions for modeling a
shrinkage curve (Fig. 2), depending on the number of measured points of the curve: either by three straight lines corresponding with the three structural, normal, and residual
shrinkage zones (McGarry and Malafant, 1987; Coughlan et
E. Braudeau, IRD, BP 434,1004 Tunis-el-Menzah, Tunisia; J.M. Costantini, Geodit, 29 Grande Rue, 13002 Marseille, France; G. Bellier,
IRD, Lab Hydrophysique, 32 Av. Henri Varagnat, 93143, Bondy,
France; H. Colleuille, NVE, Hydrology Dep., Middlelthunsgate 29,
P.O. Box 5091 Maj, Oslo, Sweden. Received 6 May 1997. "Corresponding author (erik.braudeau@ird.intl.tn).
Abbreviations: PL, polynomial model; XP, exponential model; SL,
straight lines model; TSL, three straight lines model.
Published in Soil Sci. Soc. Am. J. 63:525-535 (1999).
525
526
SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
al., 1991; Mitchell, 1992), or by a sigmoidal curve divided
alternatively by linear and curvilinear zones (Lauritzen, 1948;
Braudeau, 1988a; Tariq and Durnford, 1993b).
Our study addresses the second type of modeling, where
all the shrinkage zones are distinguished. These zones are
referred to as linear and curvilinear residual shrinkage, basic
shrinkage, and linear and curvilinear structural shrinkage
zones. Assuming that each zone corresponds with a different
stage of the shrinkage process, the endpoint of these zones
(O, A, B, C, D, S in Fig. 1) are points of transition between
two kinds of hydrostructural states of the soil (Braudeau,
1988b), such as the well-known air entry point (Sposito and
Giraldez, 1976). These transition points are thus considered
as characteristic parameters of the soil's hydrostructural behavior. They may be determined using fitting parametric models on measured shrinkage curves. By this way, Sposito and
Giraldez (1976) determined the air entry point AE, while
MacGarry and Daniells (1987) determined the air entry point,
the swelling limit MS, and the saturated point S; then
Braudeau (1988b) and Braudeau and Bruand (1993) identified
the shrinkage limit SL, the air entry point in primary aggregates AE, the friability point C, and the swelling limit of
primary aggregates MS (Fig. 2).
Measure of the Shrinkage Curve
Numerous methods for soil volumetric shrinkage measurement have been suggested. Two approaches may be distinguished. The first one is a direct assessment of a soil sample's
volume by measuring its displacement in fluid commonly using
the Archimedes' principle. Some authors have measured the
displacement of very small soil samples in petroleum (Monnier
et al., 1973), in water after coating the soil samples with paraffin (Lauritzen and Stewart, 1941; Lauritzen, 1948), or by saran
0.62
0.61 --
i
O-
0.6
R.JN.J
I
0.05
Nb
I
—i—
0.1
0.15
0.2
0.25
Water content 6 (kg/kg)
Fig. 1. An example of soil core shrinkage curve (a) obtained by continuous measurement (Braudeau 1987). Dotted lines (b) and (c) are
two possible adaptations of the three straight lines model of
McGarry and Malafant (1987) on the continuous-measured curve;
(b) is done in this paper and (c) was done by Tariq and Durnford
(1993b). Shrinkage zones are: Z = zero, R = residual, N = normal,
and S = structural, with subscript b or c corresponding with the
two interpretations.
resin (Brasher et al., 1966; Reeve and Hall, 1978; Bronswijk,
1988,1991). In this way, recently, Tariq and Durnford (1993a)
suggested an original method to determine soil bulk volume
of clods in water by means of a flexible rubber membrane. The
second approach consists of indirectly measuring volumetric
change by measuring the physical dimensions of soil samples
(Berndt and Coughlan, 1976; Towner, 1986; Braudeau, 1987;
Wires et al., 1987; Hallaire, 1987,1991), assuming or not assuming isotropic shrinkage.
Direct measurement does not allow the continuous monitoring of shrinkage of soil samples, the shrinkage curve commonly consisting of 10 to 20 points for the full moisture range.
However, one indirect method using displacement transducers
allows not only automation of the measurement but also continuous and accurate measurement of shrinkage as the samples
dry. Braudeau (1987) first proposed such an automatic measurement with an apparatus called the Retractometre modified
later by Braudeau and Boivin (1995). The method consists of
monitoring the diameter of a cylindrical soil sample in the
vertical direction during its drying. The shrinkage of lowswelling material could be measured by this method, but the
issue of isotropic shrinkage was neglected (Hallaire, 1991).
The new method we present takes into account this problem
by using several laser captors to measure the diameter and
height of soil samples (Costantini, 1997).
Parametric Models of the Shrinkage Curve
Each of the five shrinkage zones in Fig. 1 may be modeled by
either linear or a nonlinear parametric equations parameters
using the coordinates of the endpoints (0, A, B, C, D, S).
Four parametric models are presently available in the literature (Giraldez et al., 1983; McGarry and Malafant, 1987;
Braudeau, 1988a; Tariq and Durnford, 1993b). They are presented in Fig. 2. These models do not describe exactly the
same shrinkage zones but they are developed on the same
principle: the linear zones are modeled by straight lines equations, and the curvilinear zones by exponential (Braudeau,
1988a) or polynomial (Giraldez et al., 1983; Tariq and Durnford, 1993b) parametric equations, the parameters being the
coordinates of the endpoints of the zone.
In the two last models in Fig. 2, referred to as exponential
(XP) and polynomial (PL) models, the curvilinear functions
are chosen such that their derivatives at the endpoints are
continuous and equal to the slopes of the adjacent linear
zones (Appendix I, II). However, there are some noticeable
differences between the two models. Tariq and Durnford
(1993b) have taken some working hypotheses derived from
the three straight lines (TSL) model of McGarry and Malafant
(1987): (i) the normal shrinkage zone was assumed to have a
slope equal to unity (K^ - 1 dm3 kg"1), which restricted
the application range of their model to clayey soils; (ii) the
structural shrinkage was considered as a single zone from the
saturation point to the onset of the basic shrinkage zone,
corresponding with the swelling limit as defined by McGarry
and Malafant (1987). Figures 1 and 2 show how the PL and
the XP models are related to the TSL model of McGarry and
Malafant (1987).
In the model suggested by Braudeau (1988a), the slope of
the basic zone did not have a fixed value and the structural
shrinkage range was divided into two zones, a curvilinear
[D-C] and a linear [S-D] zone, only the latter corresponding
with the structural shrinkage of the TSL model of McGarry
and Malafant (1987) as shown in Fig. 1.
In order to compare the parametric curvilinear equations
of the two models suggested by Braudeau (1988a) and Tariq
and Durnford (1993b), the latter has been slightly modified
527
BRAUDEAU ET AL.: SOIL SHRINKAGE MEASUREMENT AND CHARACTERIZATION
Experimental curve (Braudeau 1987)
5 shrinkage phases recognized
A, B, C, D, transition points of
the shrinkage zones
Model of Giraldez and al. (1983)
2 shrinkage zones modeled by one
linear and one polynomial equation.
Model of MacGarry and Malafant (1987)
3 shrinkage zones modeled by
3 straight lines. (TSL model)
MS= maximum swelling limit
Model of Braudeau (19880)
5 shrinkage zones modeled by 3 linear
and 2 exponential equations.
_______(XP model) ______
Model of Tariq and Durnford (1993)
4 shrinkage zones modeled by 2
linear and 2 polynomial equations
(PL model)
Fig. 2. Presentation of a typical continuously measured shrinkage curve and the four parametric models of soil shrinkage curves published in
soil literature.
to take into account the same five shrinkage zones and their
endpoints (Fig. 3). The slopes of the three linear zones (residual, basic, and structural) are Kte, K^, and Kst, respectively,
and the corresponding end-points are (6A, VA); (0B, VB); (6C,
v
c); (OD, VD); (6S, vs). The equations of the two models are
given in Table 1, where variables are expressed in their reduced form, 9n and vn, defined for each curvilinear zone. Explanations are reported in Appendixes I and II.
We have to observe, in Table 1, that the number of parameters needed to describe the normalized curvilinear zones,
[A-B] or [C-D], is not the same for the two XP and PL
models: two and three parameters, respectively. The reason
why only two parameters (the slopes at the two endpoints)
are necessary for the exponential equation of Braudeau to
describe each normalized curvilinear zone, is that it expresses
an equation of state, v(6), that obeys the Law of Corresponding States (Sposito and Giraldez, 1976). Thus, v(6) can be
expressed in a form that is the same for all macroscopic analogous systems, soil organizations in our case. If variables are
taken in their normalized form, parameters will be even in
number and they will depend on only the conditions at the
endpoints. The parametric polynomial equation Giraldez et
al. (1983) proposed for expressing the residual shrinkage zone
also obeys this law, so the two parametric equations could be
called General Soil Volume Change equations. Appendix III
shows the close similarities between the coefficients of these
two general equations, the exponential equation being expanded to its polynomial form.
This property of the exponential model is expressed by the
following relations between the parameters (cf. Appendix I,
Eq. 3] and [5]) that do not exist with the PL model:
PCD = (vc - VD)/(0C - 6D)
- 2] +
[2]
PAB = (VA - vB)/(6A - 6B)
/
= {Kbs[exp(l) - 2] + JU/[exp(l) - 1]
[3]
These two equations allow us to relate the XP model to the
TSL model by adding only two other parameters (one by
curvilinear zone) to those of the three straight lines fitting the
three classical linear zones, in order to have the same number
of parameters in the two cases. The new chosen parameters
were the y-axis distances between the intersection points of
the straight lines and the experimental curve, MM' and NN'
E 0.74
tj
3
S
U
1 0.73
0.72
0.1
0.12
0.14
0.16
0.18
0.2
Water content (kg/kg)
Fig. 3. Representation of the straight lines model fitting the linear
zones of the classical sigmoidal shrinkage curve. The v coordinates
of N and M are the two needed parameters in addition to those
of the three lines to calculate the parameters of the exponential
model (the transition points A, B, C, D).
528
SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
(Fig. 3). The correspondence between the parameters of the
two models is given by the following equations (calculations
in Appendix IV).
6D = OM- + 4.8 MM'/(Kbs 6c
= eM- - 3.46
ral curvilinear shrinkage
(normalized form)
[C-D]
6A = 8N- - 4.8
6B = ON- + 3.46 NN'/(^bs - KK)
[4]
[5]
The straight lines parameters system can easily be obtained
by linear regression on the linear zones of the shrinkage curve.
Thus, it may be a good method for calculating the coordinates
of the characteristic transition points defined by the XP model.
i <*,
MATERIALS AND METHODS
Soils
a.
V
.
-.5050
afi
u n"
•S .*
Different types of soil widespread in the intertropical zone
were used in this study. The samples were collected in Senegal.
The name of the soils in the French taxonomy (Duchaufour,
1977) is used, and their corresponding names according to
U.S. soil taxonomy (Soil Survey Staff, 1975) are given in parentheses. We used a strongly desaturated ferrallitic soil (Plinthustox) from Kolda in Haute-Casamance (Maignien, 1961;
Chauvel, 1977); a leached ferruginous tropical soil (Plinthustalf) from Thysee Kaymor near Nioro du Rip (Bertrand, 1972);
a Vertisol (Pellustert) and a weakly developed soil on alluvial
deposits (Torrifluvents) from, respectively, the lower and upper part of the alluvial basin of Nianga-Podor (Valley of the
Senegal River) (Colleuille, 1993; Coquet et al., 1996). They
are referred to as Silty alluvial, Ferruginous, Ferrallitic soil,
and Vertisol in the text and figures. Their main chemical and
physical characteristics are given in Table 2.
I
5,
e
Q.
curvilinear shri
rmalized form)
[A-B]
Sampling and Samples Preparation
2 eq
U
A
§
I
X
u
I.5
3o
_c
iill
"3
B
*o "S
o
V I cS
*\
a.:
I Jfi
1&1
In order to avoid problems related to the spatial variability
of the soil structure in situ, particularly the macroporosity
variability, reconstituted soil samples were used. Soil samples
were gently broken into small aggregates by passing through
a 2-mm wire screen. Polyvinyl chloride rings (diameter 55 mm;
height 35 mm) were then filled in a homogeneous way with
aggregates smaller than 2 mm. The inside wall of the rings
was covered with a polyethylene film for easier removal of
the soil core. A weight of 0.5 kg was put on the soil samples
during their capillary wetting on a porous ceramic plate (12
h). They were then equilibrated (48 h) at a matrix potential
of -100 kPa using a pressure membrane apparatus before
being smoothly compacted under a given pressure in order to
reach the specific volume of the undisturbed soil (Table 2).
This matrix potential has been related to a particular hydrostructural status of the soil, namely the soil friability point
(Colleuille and Braudeau, 1996), at which it is possible to
decrease soil interaggregate porosity without damage to the
soil aggregates. Afterwards, they were turned out and then
preserved in airtight plastic boxes in a cool room (seven replications by type of soil). Before measurement of their shrinkage, the soil samples were slowly saturated by capillarity with
deionized water (2-3 d). This operation was carried out under
atmospheric pressure and without load, which allowed the
complete saturation and swelling without disturbing the structure of the sample (Dickson et al., 1991).
o .|S—
III!
s
o
2 PM
Shrinkage Curve Measurement
The experimental procedure consisted of the simultaneous
and continuous measurements of the height H, diameter D,
529
BRAUDEAU ET AL.: SOIL SHRINKAGE MEASUREMENT AND CHARACTERIZATION
Table 2. Main chemical and physical characteristics of the soil samples.
Soil name
French classification
(U.S. taxonomy)
Locality
Horizon
Depth
Clay
Silt
(<50|un)
Dry bulk
Sand
CEC
S/T
density
0.37
21.76
100
1.79
0.59
5.6
100
1.57
Alluvial soils (Senegal River, Podor)
Vertisol (Pellusterts)
Weakly develop, soil
(Aquic torrifluvent)
Strongly desaturated
ferrallitic soil
(Plinthustox)
Leached ferruginous soil
(Plinthustalf)
Lower part of
basin
Upper part of
basin
B
0.8
0.43
0.14
B
0.4
0.13
0.27
Ferrallitic-ferruginous tropical soils (Senegal)
Plateau
B
0.6
0.43
0.14
0.43
5.94
24
1.52
Plateau
BA
0.3
0.19
0.21
0.60
3.91
36
1.42
and weight w, of an initially saturated soil core sample during
water removal by evaporation. The measurements were determined in isothermal conditions (30 ± 0.5°C), using a new
version of the Retractometre (Costantini, 1997). The principle
of this apparatus is given in Fig. 4. The vertical sensor is a
laser spot that measures the height of the soil sample by
triangulation (resolution 10 u,m), and the two horizontal sensors give the diameter of the sample by measuring the parts not
intercepted by the laser beams. The swing-plate was equipped
with eight porous removable sample racks on which saturated
soil samples were carefully put flat. Every 10 min the plate
did a one-eighth turn and lowered to put down the sample
rack on the pan of the digital balance (nominal precision 0.01
g). The experiment was stopped when the sample diameter
remained constant during a minimum of 10 h. For the studied
soil samples, the complete shrinkage lasted =2 to 3 d.
After the experiment, some measurements were done on
each core in order to convert the data, into volume and water
content, Ms and vf, where Ms is the mass of the oven-dried
soil sample (105°C), and vf is the specific volume of the soil
sample at the end of the experiment (dry bulk volume of soil
sample per unit mass of solids, expressed in dm3 kg"1). vf =
Vf/A/s where the final volume of the sample Vt was assessed
by measuring the water displacement of the soil sample coated
with paraffin wax using the Archimedes' principle.
In order to calculate the 1:1 line, the specific volume of dry
solids vs, (volume of solids per unit mass of dry solids, expressed in dm3 kg"1) has been determined for each soil type
using a water pycnometer.
height measurement may be due to the movement (as shrinkage occurred) of the little reflector put on top of the soil
samples. Therefore, the shrinkage curves have been determined from the diameter change data, using Eq. [8], with an
exponent P that has been smoothed after direct calculation
from D and H data (Eq. [11] and [13]).
According to Towner (1986):
v = vf (D/Df)P
[8]
Aviv = a dH/H = p dD/D = 7 dD'/D'
[9]
coming from
where a, P, -y are the coefficients of variation of H, D, D'.
P depends on the anisotropy of the shrinkage between the
height and the diameter of the soil core, and is not constant
along the shrinkage. It is calculated as follows.
Assuming an axial isotropy of the soil cores (P = -y), and
from the small strain theory (Towner, 1986):
dv/v = dH/H + 2dD/D
and combining Eq. [9] and [10]
[10]
P = 2 + 3/a = 2 + lid
[11]
where C, is defined as the coefficient of isotropy of the soil
sample: C, = a/p.
C, may be easily calculated from experimental data by Eq.
[13] considering the following approximations:
Shrinkage Curve Calculation Taking the Anisotropy
of Shrinkage into Account
Assuming an axial isotropy for the volume change process,
the shrinkage curve V(9) was determined by the following
equations:
v = Vf (D/Dsy
[6]
e = (w - MS)/MS
where 6 is the gravimetric water content in (kg kg"1), and Df
and Hf are the last data of diameter and height at the end of
the shrinkage process.
Despite that the resolution of the sensors given by the
manufacturer is the same for the vertical and horizontal sensors (10 (Jim), the measurement of the height of the soil samples
was not as precise as the horizontal diameter of the sample
because the core stopped at slightly a different location each
time and the impact point of the laser beam was not exactly
the same at each measure. Moreover, the variability of the
Moveable
perforated
support
Fig. 4. Retractometre for continuous shrinkage measurement of soil
core samples.
530
SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
v/v, = (HIHfY = (D1D,)P
[12]
C, = a/3 = [(D - Df)/£>d/[(H - //f)///f] [13]
C; is calculated at each point of the curve and then fitted
by a three-order polynomial equation. It supplies a smooth
variation of the exponent p calculated by Eq. [11].
The two fitting methods were tested with four soil types
and their efficiency to determine the characteristic transition
points of the shrinkage curve compared. Practically, seven
samples by soil type were analyzed.
Shrinkage Curve Adjustment and Characteristic
Points Determination
The equations of the four soil shrinkage models given in
Fig. 2 have as parameters the coordinates of the endpoints of
the shrinkage zones these models consider. A fitting of these
equations on the measured data leads to a determination of
the parameters. As we considered all the linear and curvilinear
shrinkage zones and their endpoints (or transition points), the
only existing models it was possible to use for their determination were the XP and the PL (modified) models. Two ways
are chosen to maximize the adjustment.
One consists of using the classic multidimensional simplex
method (Nedler and Mead, 1965) that converges by progressive fitting to the best combination of parameters. The equations of the two models used are given in Table 1. Note that
10 parameters are needed with the polynomial model, such
as Kre, Kte, Kst, 6A, VA, 6B, VB, 9C, 8E, and VE, but only eight
parameters with the exponential model (Kre, Kbs, Kst, 8A, VA,
6B, 9C, and 6E) because of the two relations, Eq. [2] and [3].
The second way consists of using the relationships between
the straight lines (SL) model, which fits the linear zones of
the measured shrinkage curve, and the XP model. Therefore
the six parameters (KK, Kst, 6M-, VM., 6 N ', and V N -) of the system
of three lines, which may be obtained by a linear regression
on the linear zones of the curve, plus the two parameters VM
and V N read on the curve at 0M- and 0N', allow us to calculate
the corresponding eight parameters of Braudeau's model following Eq. [4] and [5].
RESULTS AND DISCUSSION
Silty alluvial soil
Shrinkage Curve Calculation
It was observed that the measured shrinkage data of
unconfined reconstituted soil samples covered practically the complete moisture content range, i.e., from
near the fully saturated and swollen state to the air-dry
and fully shrunken state. The measurement was realized
in a relatively short time (2 d for the kaolinitic materials,
3 d for Vertisols). The experimental data gave smooth
shrinkage characteristic curves (600 measured points for
the kaolinitic materials and 800 for Vertisols), without
sharp change but exhibiting distinct linear and curvilinear zones of shrinkage that corresponded with different
moisture ranges.
Figure 5 illustrates the steps of calculation of the
shrinkage curve. For each soil type, a sample is taken
as an example.
Untreated shrinkage data plotted as diameter D and
height H against time are presented in Fig. 5a. The total
change in D and //was very small, 5 mm for the Vertisol
and 2 mm for the other samples. In all the cases, change
in D was more marked and more regular than change
in H. For the ferruginous soil, for example, the residual
standard deviation limited to only the linear structural
shrinkage zone (70 data points) was 6 X 10~6 m for the
Ferrallitic soil
Ferruginous soil
2000
4000
Time (min)
Fig. 5. The different calculation steps of the shrinkage curve for four selected soil samples: (a) untreated shrinkage data from the retractometre,
plotted as diameter D and height H against time; (b) plot of the coefficient of isotropy, C, calculated and modeled over time by means of a
third-degree polynomial regression, and plot of the deducted exponent ((5), against time; (c) plot of the relative specific volume vlv, against time.
BRAUDEAU ET AL.: SOIL SHRINKAGE MEASUREMENT AND CHARACTERIZATION
diameter and 25 x 10~6 m for the height. Note that in
Fig. 5a the relatively important shrinkage that occurred
in a first shrinkage phase, before the structural one, for
soils with low clay content (Ferruginous, Ferrallitic, and
weakly developed soils). During this shrinkage phase
process, the lower shrinkage of the height compared
with that of the diameter may be due to the influence
of gravity during the wetting procedure. Cores bear their
own weight when they remain saturated for many days
before being put into the Retractometre, so they do
not swell upward as much as horizontally in diameter.
Thereafter, shrinkage reflects this anistropy of swelling.
Towner (1986) reported an analogous observation
about shrinkage of clay cores that had been previously
stressed in one direction.
For each type of soil Fig. 5b gives an example of the
coefficient of isotropy C, and the corresponding exponent. C, has been calculated at each measure point by
Eq. [13] and modeled over time by means of a thirddegree polynomial regression. (3 has then been directly
calculated from C, by Eq. [11]. Except for Ferrallitic
soil, (3 is smaller than three all along the shrinkage
ranges, which confirms the anisotropy of shrinking and
the more marked change in D than H.
Figure 5c shows shrinkage curves as relative specific
volume v/vf plotted against time. This relative specific
volume was calculated from Eq. [8], with two different
values of the exponent for comparison, (3 calculated
from data and equal to three, corresponding with an
isotropic shrinkage. From these curves, it is possible to
assess the error of not taking into account the anisotropy
of the shrinkage by taking (3 = 3 as a hypothesis, as in
the classical measurement. This error A(v p - v3)/vf is
-5% at high moisture content for the ferruginous soil
sample and decreased as v neared vf. That error is maximal for C, = oo (i.e., H = constant during shrinkage) so
(3 = 2: Amax/v{ = (D,/Df)3 - (D,/Df)2 » (DjDf - 1),
subscript i referring to the initial value. For cases in Fig.
5c, Amax/Vf = 11% for Vertisols and <5% for the others.
531
The Continuous Shrinkage Curve
The corresponding soil shrinkage curves, v = /(6), are
shown in Fig. 6. In accordance with the data published in
the literature, the basic shape of these experimental
shrinkage curves is sigmoidal ("S"-shaped curve). The
curvilinear zone [C-D] of the structural range, is often
more important than the linear normal zone [B-C] for
the weakly swelling soils (Fig. 6). Moreover, for this
kind of soil, a new shrinkage zone is observed at the
higher moisture content, with shrinkage two to three
times greater than shrinkage in the normal zone. Its
slope nears 1 dm3 kg'1, showing a volumetric change
approximately equal to the water loss. This behavior
was attributed to a weak cohesion of the reconstituted
soil structure and to the removal of the water spacing
the fabric members (skeleton grains, soil aggregates) at
high moisture range (Colleuille and Braudeau, 1996).
Note that this shrinkage zone parallel to the saturated
line, is >5 dm3 kg"1 apart from it. Therefore, a great
amount of air (20-30%) remained in the soil sample
though it behaved as if it were fully saturated. This fact
is well known for this kind of soil for which a complete
saturation is never reached in the field conditions. So
we attributed the term pseudo-saturated to that shrinkage zone (Fig. 6).
With these results, it is obvious that the slope Kbs of
the basic shrinkage (6B < 6 < 0C) may differ greatly
from the unity and may be very small, as was observed
for the weakly swelling soils. This property was interpreted as the influence of structure by Bruand and Prost
(1987) and Braudeau (1988a, 1988b), who assumed that
Kbs = dv/dv^ with v^ being the massic volume of the
clayey matrix contained within the soil structure.
Determination of the Characteristic
Transition Points
The XP model of Braudeau (1988a) and PL model
of Tariq and Durnford (1993b) were first tested on their
Vertisol
OS
0.77
Ferrallitic soil
Ferruginous
0
0.1
0.2
0.3
"'" 0
0.1
0.2
0.3
Water content (kg/kg)
Fig. 6. Experimental and modeled shrinkage curves of the four selected soil samples of Fig. 5. Localization of the characteristic transition points
A, B, C, D, E, and F have been performed by the straight lines fitting method including the new pseudo-saturated zone [E-GJ.
532
SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
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CONCLUSION
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i
ability to fit the data using the simplex method. Table
3 shows that there was no difference between the results
derived from the PL and XP models; the fitting of the
two models led to the same average value of the parameters and the residual variance concerning the whole
curve was not higher than 10~8 at each test. These results
confirm that only two parameters instead of three are
sufficient to determine the curvilinear zones in their
normalized form, which consequently justifies Eq. [2]
and [3] of the XP model and then, the relations Eq. [4]
and [5] setting the correspondence between the shrinkage transition points and the straight lines system parameters.
The SL model has been then used as a simple method
for determining all the transition points of the shrinkage
curve, not only those of the residual and structural curvilinear zones, but also the new points E and F of the
pseudo-saturated shrinkage zone (Fig. 6). The intersection point of the two lines fitting [D-E] and [F-G] zones
was named L' and the corresponding point of the curve
L. Equations relating LL' to the E and F points were
analogous to Eq. [5] concerning the residual zone [A-B].
The fits were excellent with a standard deviation always
near 10~8; the results given in Table 3 show that, except
the determination of the new transition points, there is
no marked difference with the previous results obtained
by the simplex method. Figure 6 shows the adjustment
of the XP model obtained by this straight lines method
on the four selected soil samples of the Fig. 5. Experimental and modeled curves are superposed and are not
easily distinguished. Localization of the characteristic
transition points A, B, C, D, E, and F in Fig. 6 have
also been performed by the straight lines fitting method
including the new pseudo-saturated zone [E-G]. The
fact that a same exponential equation described well
the three curvilinear zones of the shrinkage curve leads
us to think that it was a similar hydrostructural physical
process which presided over these three shrinkage
phases.
od
Sll
odd odd
odd
odd
-1
>o©
ooo
J EL. O.
B.X X
j o, a.
-1
a. XX
J
j a. a,
a. XX
In this study we have shown that in order to determine
accurately the different shrinkage phases and characteristic transition points of a soil sample, the shrinkage
curve sample has to be measured continuously by displacement transducers. New forms of shrinkage curves
have been described with this method and some concepts of the shrinkage of soil organizations should be
discussed in the future. In particular, the fact that soils
with low clay content, which in the past were considered
as nonswelling soils, have a clear shrinkage curve with
seven easily identified linear and curvilinear zones.
Then, we showed how parametric models of the shrinkage curve adjusted to the measured data by the simplex
method allowed us to maximize the localization of the
phase transition points. A study of models of soil shrinkage curves available in the literature led us to suggest
a fitting method based on the exponential model given
by Braudeau (1988a, 1988b) and its relation with the
straight lines system fitted on the linear zones of the
533
BRAUDEAU ET AL.: SOIL SHRINKAGE MEASUREMENT AND CHARACTERIZATION
curve. This suggested method more easily determines
the characteristic transition points of all of the different
phases of the soil shrinkage behavior with good reproducibility. These results allow us to consider the shrinkage curve, like the soil water potential curve, as a characteristic of the hydro structural functioning of soil.
and ^re, the slope of the residual range (9A £ 9), instead of
K* in Eq. [Al]:
K. (g - *•)
r\ K,f
VA ~
_ ^bs^c
e - B
KK
0.718 JCb
1.718
A
APPENDIX I
The Exponential Model
The linear ranges of the shrinkage curve (Fig. 1) are modeled by straight lines:
where
_ 6 - 9D
~ —— ~
,
a
Vn
_ v —
~
in between 9S and 6D (structural range)
9 - 6B
eC ~
f\
,
and
«B
v - VA
VB - VA
v - VB
" = ————
v
APPENDIX II
between 9C and 9B (basic range)
. _ 8 - 9A
and vn = v - VA
V0 - V A
6A
between 6A and 6 = 0 (if v0 ^ VA)
The curvilinear ranges [C-D] and [A-B] are described by
the following equations given by Braudeau (1988a).
For [C-D]
en _ i \
/een _ i
+ KJl - -——e-l
d6
and
(2.718 6n [A6]
Kn
0.718
The total number of parameters used in these equations are
14: (9S, vs), (9D, VD), (9C, vc), (9B, VB), (9A, VA), (v0), K*, K*,
Kn. The five relations between the slopes Kst, K^, KK, PAB, PCE
and the other parameters decrease the number of parameters
necessary to describe the whole curve to 9 (including point S).
vn =
n
[A5]
[Al]
where
The Polynomial Model
Using the reasoning of Tariq and Durnford (1993b) for the
structural zone [C-D], we calculate the coefficients of the
third-degree polynomial equation of the fitted curve that
passed by the points C and D, of which the derivatives are
determined in these points.
Written in their normalized form between C and D, the
variables are:
6n =
vn =
6 ~ Or
e - eD
"o
vc - V D
T Ul On T U2
n
;i "n
[A7]
Dundaries conditions are:
e. = 0;
e. = i;
By integration of Eq. [Al] between C and D:
v - VD
Kbs(e»° - 9. - 1) + Kst(eQn
e - 1
vn = 0;
dv/d9 = K«
vn = 1;
dv/d9
= ^bs
[A8]
hence
[A2]
that leads to the following relations when v = vc
p
/•ViTj
fli = Kst(Qc ~ 6D)/(e - 6D) = KJPCD
_
vc - VD _ Kbs(e - 2) + Ksl
— —————————— —
6C - 9 D
e- 1
=
0.718 Kbs + Kst
1.718
[A3]
a, + 2a2 + 3a3 = KJPCD
giving:
K* 9n + (3PCD - ^bs - 2KA) 62n +
(K,s + Kst - 2PCD) 63n
then, combining Eq. [A2] and [A3],
Vn
=
[A9]
[A10]
v - VD
Vn =
vc
VD
e
en -
- - en -
_L L[A4]J
0.718 Kbs + Ka
For the range [A-B], similar equations are obtained with taking
_ 9-9A
"n ~ ^————7T~
where PCo = (VG - vD)/(6c ~ 60) is not determined
In the same way for [A-B] range:
6 -
re
9n
(3PAB - Kbs - 2KK)
„ + KK - 2PAB) Ql
[All]
534
SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
In this polynomial model, the two parameters PAB and PCD
are not related to the slopes Ka, K^ or KK, so there are two
parameters more than in the previous model.
Hence, after simplification and using Eq. [A3]:
6n(M') = (a' - 8D)/(8C - 6D) = ll(e - 1) = 0.582
[A17]
APPENDIX III
Comparison between the Equations Suggested by
Giraldez et al. (1983), Braudeau (198Sa), Tariq and
Durnford (1993b), for the Shrinkage Range AB
Otherwise, from Eq. [A16], by dividing each term by PCD =
(vc - vD)/(9c - 6D), v n (M') = (P' - VD)/(VC - VD) = 6»(M')
KJPco that gives by using Eq. [3]
v.(M') =
Expanding the exponential Eq. [A4] gives:
^bs ~ ^re
Vn =
I 1-718/Lre
0.718tfbs + KK \tfbs - Kn
"
+ 0.4366? + 0.2826?)
•^-^-^yr^
[A12]
By substituting PAB in the polynomial Eq. [All] by its value
in the exponential model (Eq. [A5]) PAB = 0.418A:bs + 0.582KK,
we obtain:
A:bs - KK
11.718KK
0.718ATbs + KK \KU + KK
The Eq. [A3], [A17], [A18], and [A19] represent a system of
four equations with four unknown variables. Using [18] and
[19] yields
vc - VD = (3 - P')/[v n (M) - vn(M')]
"
c
[A13]
(Braudeau, 1988a)
•^bs ~ ^st
and
VD =
(Tariq and Durnford, 1993b)
(Giraldez et al., 1983)
[A14]
APPENDIX IV
As an example, we consider the case of the structural shrinkage zone [C-D]. Each fo the curvilinear zones [A-B] and
[C-D] were calculated with five parameters instead of six
owing to Eq. [A3] and [A5] which relate the parameters among
them. Therefore, knowing the five following parameters (cf.
Fig. 3): the slopes (Kbs, Kst), the intersection point M' (a', P')
of the two straight lines that fit to the linear zones [B-C] and
[D-S], and the corresponding point M of the shrinkage curve
(a. = a', p), it may be calculated the C and D points as follow:
The straight line equations are:
(y - V)l(x - «') = K«
and
(y - p')/(* - a') = ATbs
[A15]
The C, D, and M' points are related by the following expressionsion:
(VD - P')/(eD - «') = Kst
and
(vc - PW - «') = *bs
[A16]
p>
+
M^(P' - P)
[A21]
^bs ~ ^st
Combining [A3], [A20], and [A17] yields 6D and Oc:
_ 8.26(P - P').
oc ~ OD — —r^————,
_
4.8(P' - p)
OD — « + ———————
^bs ~ ^st
and
Relation between the Three Straight Line Model
(McGarry and Malafant 1987) and the
Exponential Model (Braudeau 1988a)
[A20]
3.46JCbs(E' - p)
Vc — p — ——————————
vn = 0.6086?, + 0.3926?
vn = 0.8956? + 0.1046?
_ v _ (P ~ P')/0.718£bs + KA
D
0.208 \ Kbs-Ks{
thus the values of vc and VD:
It is then possible to compare these expressions with the polynomial expression given by Giraldez et al. (1983) by taking
KK = 0 and ATbs = 1
vn = 0.6956S + 0.2366? + ...
[A18]
0.718^bs + Kst
The value of vn(M) is determined using 9n = 6n (M') = 0.582
in Eq. [A4]:
en + o.502n
+ 0.1666? + ... + ?s 9 + ..
Ktt
_ ,,
6r = «' -
3.46Q' - P)
^MJS ~ ^st
^bs
KX
[A22]
ACKNOWLEDGMENTS
We gratefully acknowledge program "Jachere et Biodiversite," contract TS3-CT93-0220 (DG12HSMU), and its Coordinator in Senegal, Dr. C. Floret, who initiated this work and
supported the first essays in Senegal. We thank the Soil Directorate (Ministry for Agriculture, Tunis) and the Director, Dr.
A. Mtimet, for contributing to the installation of this new
methodology in his laboratory, and thus allowing the completion of this work.
BRAUDEAU ET AL.: SOIL SHRINKAGE MEASUREMENT AND CHARACTERIZATION
535