soil water retention curves based on idealized models

ISSN 1068�3674, Russian Agricultural Sciences, 2013, Vol. 39, No. 5–6, pp. 522–525. © Allerton Press, Inc., 2013.Original Russian Text © V.A. Sysuev, I.I. Maksimov, V.V. Alekseev, V.I. Maksimov, 2013, published in Doklady Rossiiskoi Akademii Sel’skokhozyaistvennykh Nauk, 2013, No. 5,pp. 63–66.

522

The water retention curve (WRC) is useful in deter�mining how the hydro�physical properties and physi�cal status of soils are changed while they are used forcrop growing. Since the use of the WRC makes it pos�sible to model the moisture and nourishment trans�port, WRC building is of great interest for researchersfrom both Russia and other countries. The basic soilproperties are the most efficiently described by pedot�ransfer functions, which are divided in physically jus�tified, single�point, and parametric pedotransfer func�tions [1–4]. The use of idealized models in the aerody�namic method allows us to combine advantages andpeculiarities of all pedotransfer functions (PTF) listed.

METHODS

Soil moisture is limited by, on the one hand, thegaseous phase and, on the other hand, by the solidphase. This means that there is the surface energy ofinteraction with the soil solid phase (E ') and the soilair (E ''):

E = E ' + E '', (1)

where E is the total surface energy. The WRC repre�sents dependence of a free energy change from thecontent of soil moisture. Practically, the dependenceon the moisture potential Ψ = E/m (a ratio of energyto water mass) or the dependence on equivalent pres�sure p = ρΨ (ρ is the water density) is used.

A potential value resulting from the interaction ofmoisture and the soil solid phase can be found accord�ing to the formula [2]:

(2)

where ρ is water density, kg/m3; Ω0 is bulk specific sur�face, m2/m3; w is bulk moisture, m3/m3; A is constant,J; Π0 is porosity of the dry sample, m3/m3. Ψ '' poten�tial is specified by the interaction of moisture and thesoil gas and determined by the expression:

(3)

where Ωcf is bulk specific surface of the water�air inter�face, J/m2. A value of the bulk specific surface of thewater�air interface can be found in the experiment ofthe air leakage through the soil. A shape of the depen�dence of this surface from bulk moisture Ωcf(w) can beobtained from the idealized modeling.

The most informative and genetically justifiedcharacteristics of the pore space are the shape, thesquare area, and the pore orientation [6]. Let us con�sider idealized models of soil samples with cylindricalpores (Fig. 1). The volume of pores in a unit of thesample volume (porosity) is a function of the radius ofthe cylindrical pore in all three cases:

for the model A, (4a)

Π = for the model B, (4b)

ψ ' AΩ03/ρ 1/w3 1/Π0

3–( ),=

ψ '' Ωcfσlg/ρ,=

Π 3πr2l 2 4/3( )πr3–[ ]/l,≈

πr2l/l3

MODELING

Soil Water Retention Curves Based on Idealized ModelsV. A. Sysueva, I. I. Maksimovb, V. V. Alekseevc, and V. I. Maksimovd

aRudnitsky Research Institute of Agriculture of the Northeast, Kirov, 610007 Russia bChuvash State Agricultural Academy, Cheboksary, 428032 Russia

cCheboksary Cooperative Institute (Branch) of the Russian Cooperation Institute, Cheboksary, 428025 Russia dNoyabr’sk Institute of Oil and Gas (Branch), Tumen State Oil and Gas University, Noyabr’sk, 629810 Russia

e�mail: [email protected] July 1, 2013

Abstract—Energy consideration of soil moisture as a medium bounded by the soil air and the soil solid phaseallows us to obtain the water retention curve (WRC). A pedotransfer function (PTF) reflects approximationsand simplifications made in the soil modeling. The main parameters of the PTF are the experimentally mea�sured porosity and the volume of the specific surface area of the soil solid phase. This approach resulted in aPTF with the lowest disagreement with the experimental study.

Keywords: water retention curve, soil, idealized soil pore space model, the specific surface energy, bulk poroc�ity, particle size distribution

DOI: 10.3103/S1068367413060219

RUSSIAN AGRICULTURAL SCIENCES Vol. 39 No. 5–6 2013

SOIL WATER RETENTION CURVES BASED ON IDEALIZED MODELS 523

(4c)

where l is cube edge and r is pore radius.When a preassigned gradient of gas pressure is

applied to the sample ends along the selected directionthrough the tube of the radius r, a stationary gas flow isestablished. If a flow is stationary, the loss of thekinetic energy of the flow equals the work against theforce of friction on the surface of the condensed phase.Let us consider a cylindrical tube of the effective radiusRef of the same length as the sample. The loss of thekinetic energy in the tube, while the pressure drop isthe same, equals to the real loss of the kinetic energy inthe soil. Let us consider the pressure drop values Δpthat allow us to describe the laminar flow through thetube of the radius Ref by the Poiseuille’s equation:

(5)

where ΔV is the volume of gas flowing through thesample, m3; Δt is tme, s; and η is gas viscosity, Pa s. Thetubes of the flow of the two radiuses r and Ref are simi�lar if they are geometrically similar, the same continu�ous medium flows through them, and two flows meetthe condition of the equality of their similarity criteria,which are numbers of Reynolds (Re), Froude (F),Mach (M), and Strouhal (Sr). If we increase the lengthof the flow tube of the radius r, to a value of L, the tubewill be similar to the tube of the radius Ref and thelength l. In this case, the Poiseuille’s equation is

(6)

The air consumptions ΔV/Δt, if the pressure dropΔp is given, are equal in the formulae (5) and (6). Letus define the length of the tube L of the flow of theradius r. The gas flow that passes the tube has the sameenergy loss as it has in a soil sample:

Π πr2l 2 2 1+( ) 2 π+( ) 8/3( )r3–[ ]/l3,≈

for the model C,

ΔVΔt�� π

8η��Δp

l��Ref

4,=

ΔVΔt�� π

8η��Δp

L��r4

.=

(7)

The surface of the interface between the condensedphase of the sample and the air equals the lateral sur�face of the flow tube S = 2πrL. The specific surface ofthe condensed phase Ωcf is a ratio of the surface S tothe volume of the condensed phase Vcf = l3(1 – Π).

(8)

Since the airflow loses its energy when it flowsthrough only one of the three tubes, it is necessary totriple the value of the total surface (model A) or to

multiple this value by 1 + 2 (model C). If thedependence of the radius r of the porosity is expressednumerically as a power function from the formulae(4a), (4b), and (4c), then the specific surface of thecondensed phase can be found for the models A, B,and C, respectively:

(8a)

(8b)

(8c)

The first multipliers in these expressions are con�stants, the second multipliers can be determined afterthe porosity is found, and the third ones are deter�mined after the flow time Δt of the air, when it passesthrough a sample, is found (the air volume and thepressure drop are known and equal to ΔV and Δp,respectively).

When moisture is decreased, the porosity of the wetsample Π = Π0 – w transits to the porosity of the dry

L r4πΔpΔt8ηΔV��.=

Ωcf2πrl

l3 1 Π–( )��.=

2

Ωcf3 2× πrL

l3 1 Π–( )�� π2

ηl3��0.00735Π2.92

1 Π–( )��ΔpΔt

ΔV��,= =

Ωcf2πrL

l3 1 Π–( )�� π2

ηl3��0.01429Π2.50

1 Π–( )��ΔpΔt

ΔV��,= =

Ωcf1 2 2+( )πrL

l3 1 Π–( )�� π2

ηl3��0.00340Π3.14

1 Π–( )��ΔpΔt

ΔV�� .= =

r

l

l

l

r

r

l

l

l

r

l

l

l

r

rr

(a) (b) (c)

Fig. 1. Idealized models A, B, and C.

524


SYSUEV et al.

sample Π0 and the specific surface of the condensedphase Ωcf transits to the specific surface of the soil solidphase Ω0. Hence, the needed dependence Ωcf(w) is

(9)

where α is the parameter determined by the modelgeometry (α = 2.92, 2.50, and 3.14 for the models A,B, and C, respectively). The parameter α is increasedas the volume of pores, which are perpendicular to thedirection of the gas flow (those the gas does not flowthrough) is increased. This parameter makes it possi�ble to consider implicitly the particle size distributionbecause it specifies the average size, orientation, andthe number of pores that, altogether, determine theloss of the kinetic energy of the gas flow. Thus, sincethe loss change is known, it is possible to estimate thechange of the average pore size and, hence, the parti�cle size distribution.

The calculating formula for the WRC becomes

(10)

The dependence for the equivalent pressure is

(11)

Ωcf Ω0 1 w1 Π0– w+��–⎝ ⎠

⎛ ⎞ 1 wΠ0

��–⎝ ⎠⎛ ⎞ α

,=

ψ ψ' ψ ''+AΩ0

3

ρ�� 1

w3�� 1

Π03

��–⎝ ⎠⎛ ⎞= =

+Ω0σlg

ρ�� 1 w

1 Π0– w+��–⎝ ⎠

⎛ ⎞ 1 wΠ0

��–⎝ ⎠⎛ ⎞ α

.

p p ' p ''+ AΩ03 1

w3�� 1

Π03

��–⎝ ⎠⎛ ⎞= =

+ Ω0σlg 1 w1 Π0– w+��–⎝ ⎠

⎛ ⎞ 1 wΠ0

��–⎝ ⎠⎛ ⎞ α

.

With the use of packages for the analytical calcula�tions or electronic tables, it is possible, based on thesuggested formulae, to build WRC plots sufficientlyfast. The hydrophysical soil parameters have beenmeasured with the devices that were created in theChuvash Agricultural Academy (Fig. 2).

RESULTS AND DISCUSSION

To assess the PTF we have developed, we usedexperimental data obtained by centrifugation of sam�ples of the light�gray forest soil (Ω0 = 46 × 106 m2/m3;Π0 = 0.53 m3/m3). A range of moisture values wasstudied. This included a variety of soil states from thecomplete filling of pores by water to the maximummolecular water�retaining capacity consistent with the“mature” state of soil, which was optimal for particleaggregation and microaggregates, in particular. Con�ventionally, building the WRC plot suggests the use ofthe decimal logarithm of the pressure absolute valueexpressed as centimeters of water (pF). As Fig. 3shows, experimental pF points from 0 to 2 are veryclose to the plot built for model B. In other words,when the pressure is low, perpendiculars to the flowdirection are not essential. For the higher pressure(pF values are from 2 to 3) the best approximation, inthe beginning of the interval, has model A and thenmodel C, which means that pores perpendicular to theflow directions are important in the latter interval. Anassessment of fitting of the WRC to the experimentaldata with the use of the χ2 test showed that, in all cases,the observed values for the models A, B, and C (χ2 =0.669, 0.225, and 1.457, respectively) are less than thecritical value (3.2551), that is, the suggested PTFdescribe the experimental data adequately.

Fig. 2. A permeameter and a device used for porosity mea�surement.

5

0.20

4

3

2

1

0.1 0.3 0.4 0.5 W, m3/m3

pF

1A

BC

Fig. 3. WRC built according to models A, B, and Ctogether with experimental data (1).


SOIL WATER RETENTION CURVES BASED ON IDEALIZED MODELS 525

REFERENCES

1. Teorii i metody fiziki pochv. Kollektivnaya monografiya(Theories and Methods of Soil Physics: CollectiveMonograph), Moscow: Grif i K, 2007.

2. Development of pedotransfer function in soil hydrol�ogy, Developments in Soil Science, Pachepsky, Ya. andRawls, W.J., Ed., 2004, vol. 30.

3. Seki, K., SWRC fit – a nonlinear fitting program with awater retention curve for soils having unimodal andbimodal pore structure, Hydrol. Earth Syst. Sci. Dis�cuss., 2007, vol. 4(1), pp. 407–437.

4. Vogel, T., van Genuchten, M.T., and Cislerova, M.,Effect of the shape of the soil hydraulic functions nearsaturation on variably�saturated flow predictions, Adv.Water Res., 2000, vol. 24(2), pp. 133–144.

5. Alekseev, V.V., Mksimov, I.I., Maksimov, V.I., and Sya�kaev, I.V., Energetic assessment of mechanical influ�ence on soil of tillage machines and instruments, Agrar.Nauka Evro�Sev.�Vost., 2012, no. 3(28), pp. 7–72.

6. Skvortsova, E.B. and Utkaeva, V.F., Soil pore spacearrangement as a geometric indicator of soil structure,Eurasian Soil Sci., 2008, vol. 41, no. 11, pp. 1198–1204.

Translated by A. Boutanaev

soil water retention curves based on idealized models

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