Stress-induced volume reduction of isolated pores in wet soil

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 3, 1067, oi: /2001wr001137, 2003 Stress-inuce volume reuction of isolate pores in wet soil Teamrat A. Ghezzehei 1 an Dani Or 2 Department of Plants, Soils an Biometeorology, Utah State University, Logan, Utah, USA Receive 18 December 2001; revise 8 November 2002; accepte 8 November 2002; publishe 21 March [1] This stuy eals with eformation of small pores in wet soils of relatively high bulk ensity such as in the final settlement phase of tille or isturbe soils. Pore eformation was moele by volume reuction of spherical vois embee in a homogenous soil matrix. External constant stress an overburen were consiere as steay stresses because the change in interaggregate contact stress uner overburen is slow compare to the associate strain rate. In contrast, stress ue to passage of farm implements was consiere as transient because the rate of change of interaggregate stress is comparable with the strain rate. Rheological behavior of the soil matrix uner steay an transient stresses was obtaine from inepenent rheological measurements. Experimental ata from the literature were use to illustrate the moel. Moel preictions of relative ensity compare favorably with experimental ata for constant stress application as well as for constant strain rate experiments. Results showe that the rate of ensification ecrease as the relative ensity approache unity (complete pore closure) an the relative stress require for riving ensification increase exponentially with increasing relative ensity. INDEX TERMS: 1866 Hyrology: Soil moisture; 1875 Hyrology: Unsaturate zone; 1899 Hyrology: General or miscellaneous; KEYWORDS: rheology, stress, strain, pore size, ensification, water content Citation: Ghezzehei, T. A., an D. Or, Stress-inuce volume reuction of isolate pores in wet soil, Water Resour. Res., 39(3), 1067, oi: /2001wr001137, Introuction [2] The structural properties of plow-layer of agricultural soil are in a constant state of change. The loose an unstable structural state that results from tillage graually evolves to a enser an more stable structure. These changes are accompanie by concurrent changes in soil hyraulic properties, such as soil water retention an hyraulic conuctivity. In previous stuies, we propose alternative moeling approaches for posttillage ynamic processes [Ghezzehei an Or, 2000, 2001; Or, 1996; Or et al., 2000]. These moels focuse on pore-scale mechanics an aresse soil structural changes inuce by internal capillary forces an external steay an transient forces. The unerlying concept behin these moels was the coalescence of spherical soil aggregates, resulting in growth of the interaggregate contact area an graual weling. Discrete representation of soil aggregates as iniviual units applies to the initial loose state of soil aggregate bes, while ajacent interaggregate contacts are not touching each other (see Figure 1). After attainment of a critical strain value e c, ajacent contact regions begin to overlap. Consequently, further eformation involves flow of soil material in a complex pattern towar the center of an enclose pore that cannot be represente aequately by the raial flow assume in the contact coalescence moel. The purpose of this note is to exten 1 Now at Lawrence Berkeley National Laboratory, Berkeley, California, USA. 2 Now at University of Connecticut, Storrs, Connecticut, USA. Copyright 2003 by the American Geophysical Union /03/2001WR TNN 1-1 the applications of the contact coalescence moel by introucing a complementary geometry that consiers volume reuction of pores enclose in soil. Thus unlike our previous analyses that focuse on loose surface soils, the current moel is irecte towar compaction of ense soils an unconsoliate seiments containing isolate pores. 2. Theoretical Consierations [3] The basic geometry subsequent to coalescence of iniviual aggregate contacts beyon the critical strain is that of spherical pores embee insie a homogeneous matrix of soil. The problem is mathematically similar to eformation of an elastic sphere embee insie a homogeneous viscous flui, for which Fröhlich an Sack [1946] evelope a simple self-consistent moel that uses perturbation calculations. The approach was later aapte by Mackenzie an Shuttleworth [1949] to moeling closure of isolate pores in the context of free-sintering of ceramic powers. In this note, we use these methos to erive equations of volume reuction for spherical isolate pores embee in wet soil matrix subjecte to omniirectional stress Geometrical Consierations [4] We consier a spherical pore of raius r 1 surroune by a concentric spherical shell of homogeneous (wet) soil with raius r 2, as shown in Figure 2. We consier a uniform stress acting normal to the pore surface efine by P ¼ P 1 þ P 2 þ P 3 : ð1þ 3 where P is the aheral stress; an P 1, P 2, an P 3 are stresses acting in the principal irections. From the simple

2 TNN 1-2 GHEZZEHEI AND OR: TECHNICAL NOTE Figure 1. Transition from open pore geometry to close pore geometry in aggregate soil subjecte to axial stress [Braunack et al., 1979]. After the aggregates have been eforme, ajacent interaggregate contacts begin to touch each other. The analogs using the conceptual moel of this stuy are shown. geometry, we efine the relative ensity of the unit cell relative to the ensity of the soil material forming the shell as r ¼ 1 ðr 1 =r 2 Þ 3 : ð2þ [5] The problem at han is to etermine the rate of ecrease in pore raius r 1, as etermine by the stress, P, an the rheology of the soil matrix. The problem is solve by equating the rate of energy issipation uring viscous ensification to the rate of work one by reuction of the pore raius [Frenkel, 1945]. [6] If the raial velocity of the pore surface is u 1 = r 1 /t, then the raial velocity of the material at any istance r from the origin is u = u 1 (r 1 /r) 2, an the raial strain at r is given as _e r ¼ u r ¼ 2 u 1 r 2 1 r 3 : ð3þ

3 GHEZZEHEI AND OR: TECHNICAL NOTE TNN 1-3 Figure 2. (a) Schematic representation of the transition from open pores to close pores. Note that for the open pores an close pores the flow paths are symmetrical, whereas the transition has an asymmetrical complex flow path. (b) Definition of close pore. [7] We consier the soil shell as incompressible flui, with any increase in relative ensity r attribute to ecrease in r 1. It follows, then, that the aheral shear strain rate is [Mackenzie an Shuttleworth, 1949] p _g ¼ ffiffi 2 _er 2.2. Constitutive Energy Relationships [8] Accoring to the theory of viscous flow, the rate of energy issipation in the soil volume between r 1 an r 2 accompanying the pore closure is given as _E V ¼ 4 p Z r2 r 1 ð4þ 3 _e 2 r h r 2 r; ð5þ where h is viscosity of the soil material in a generic form. The stress, P, acting upon the pore surface, 4 p r 1 2, oes mechanical work at a rate of _e s ¼ 4 p r 2 1 r 1 t P : [9] Then, the rate of pore closure can be given in general terms by equating the rate of energy issipation by viscous flow, _e V, to the rate of mechanical work input, _e S, r 1 t ð6þ ¼ 3 Z r2 r1 2 P _e 2 h r 2 r: ð7þ r 1 [10] Next, we consier two types of stress conitions applicable for agricultural soil ynamics: steay an transient.

4 TNN 1-4 GHEZZEHEI AND OR: TECHNICAL NOTE 2.3. Soil Densification Uner Steay Stress [11] Steay an nonirectional stresses coul arise in agricultural soils from, for example, the action of capillary forces uring slow rying, an overburen in the subsurface. Uner such stress conitions, soil rheology can be escribe by Bingham viscoplastic moel [Ghezzehei an Or, 2001], h ¼ h p þ = _g; ð8þ where h p (Pa s) is the coefficient of plastic viscosity, an (Pa) is the yiel stress in shear. Substitution of (2), (4), an (8), into (7) an integrating the right-han sie of (7) yiels r 1 t ¼ P r 1 4h p r p 1 þ ffiffiffi 2 ln½1 rš : ð9þ P Consiering that there are n pores per unit volume, each with the volume of 4 p r 1 3 /3, enables one to erive a irect relationship between the raius of the iniviual pore an the relative ensity, r an n (m 3 ), r 1 ¼ 3 1=3 1 r 1=3 1 4p r n : 1=3 ð10þ By application of the chain-rule we obtain u 1 = r 1 /t = r 1 /r r/t, which, combine with (10), simplifies (9) to r t ¼ 3 4 p ð Þ 1 þ ffiffi 2 ln½1 rš : ð11þ P P h p r 1 The relative ensity is obtaine by solving (11) with the initial conition r(t o )=r o as rðþ¼1 t exp p 1 ffiffi P þ p 1 ffiffi P þ lnð1 r 2 2 t o Þ y " exp 3 pffiffi #) ðt o tþ 2 ty : ð12þ 4 h p [12] The relative ensity is plotte as a function of scale time (t o t)p /h p for various values of the imensionless stress P /, as shown in Figure 3. The scaling of the time axis implies that the time require to reach a given soil ensity is irectly proportional to the stress P an inversely proportional to the plastic viscosity h p. The imensionless stress P / is irectly proportional to the maximum ensity that can be attaine at infinite time. For most practical applications, it is sufficient to know the maximum ensity. This can be explicitly obtaine by solving (11) when ensification ceases at r/t = 0, pffiffi 2 r max ¼ 1 exp 2 P : ð13þ [13] The relationship between the stress ratio an the maximum ensity is epicte in the inset in Figure 3. Figure 3. Viscoplastic ensification of isolate pore as a function of nonimensional time for various imensionless loa ratios. Inset shows the maximum strain as a function of the imensionless loa ratio (circles enote the loa-ratio values use in the main plot). Equations (12) an (13) imply that either a higher stress or a lower yiel stress results in higher maximal ensity Soil Densification Uner Transient Stress [14] Rheological properties uner transient flow conitions are often etermine by subjecting a sample to a sinusoial stress [e.g., Ghezzehei an Or, 2001] of the form: t ¼ t o sinðwtþ; ð14þ where t o is shear-stress amplitue an w is angular velocity. The resulting shear strain rate is also sinusoial: _g ¼ g o w sinðwt þ Þ; ð15þ with g o w being the amplitue of the strain rate an is a phase shift angle ue to viscous lagging of strain rate path behin the stress path. The ratio of the stress amplitue to the strain rate amplitue efines a complex viscosity term: h* ¼ t o g o w ð16þ which can be ecompose into its viscous (energy issipating) an elastic (energy conserving) components, respectively, as: h 00 ¼ h* sinðþ h 0 ¼ h* cosðþ ð17þ ð18þ [15] Transient stresses in agricultural soil are often inuce by passage of farm implements [Horn et al.,

5 GHEZZEHEI AND OR: TECHNICAL NOTE TNN 1-5 In Figure 4 the solution for time epenent ensity (23) is evaluate as a function of scale time tw/p for various values of imensionless stress P o sin ()/(h*w). The scaling of the time axis by w implies frequency epenence of the strain. The maximum ensity that can be attaine for a given frequency epens on the imensionless stress as shown by the inset in Figure 4. As the relative proportion of the viscous component of the rheological properties increases (increase in sin ()) the viscous eformation also increases. Similarly, higher stress an/or low viscosity can be translate to less resistance to flow an result in higher ensification. Figure 4. Viscous component of ensification resulting from transient stress as a function of nonimensional time for various imensionless loa ratios. Inset shows the maximum strain as a function of the imensionless loa ratio (circles enote the loa-ratio values use in the main plot). 1998]. The temporal pattern of such stresses can be approximate by a sinusoial function, P ¼ P o sinðwtþ; ð19þ where P o is amplitue stress. We split the total stress P into its viscous an elastic components by assuming linear relationships with sin 2 () an cos 2 (), respectively: P 00 ¼ P sin 2 ðþ; P 0 ¼ P cos 2 ðþ; ð20þ ð21þ 3. Illustrative Example [17] The methos presente in this note are illustrate using two examples that compare moel preictions with experimental measurements of ensification uner steay stress Soil Densification Uner Steay Stress [18] In this example, we focus on the ultimate ensity (13) that results at a given external stress. The moel performance is teste by comparison with isotropic compression of moeling clay reporte by [Davis et al., 1973]. Equal-size spheres were forme from moeling clay of plastic consistency. The spheres were arrange in cubic an rhombic packings insie a cylinrical rubber membrane of 100 mm iameter an 100 mm height. The samples were isotropically compresse insie a stanar triaxial cell. The volume ecrease of the sample in response to applie stress was etermine by monitoring the volume of air uner constant stress leaving the sample. The relationship between the applie isotropic stress an sample relative ensity are plotte in Figure 5. Davis et al. [1973] scale the stress by yiel stress of the moeling clay etermine using frictionless inentation with a sphere. For the subsequent compar- Thus P 00 = P for =90 (perfectly viscous material) an 0 P = P for = 0 (perfectly elastic material). The trigonometric ientity sin 2 () + cos 2 () = 1 satisfies the stress closure P 00 + P 0 = 1 for all viscoelastic materials 0 < <90. [16] Because volume reuction of a pore embee in soil matrix requires flow of material towar the pore, only the viscous component of the stress shoul play a role in ensification. The temporary strain that oesn t involve reuction of the pore (flow of material towar the pore) can be accommoate by temporary eformation of the matrix only. Thus subsequent analyses of ensification consier only viscous flow. Upon substituting (20) an (17) for P an h, respectively, in (7) an simplifying we get, r t ¼ 3 4 P o sinðþ ðr 1ÞsinðwtÞ: ð22þ h* The viscous component of the strain is obtaine by integrating (22) after separation of variables, an with the initial conition r(t o )=r o, rðþ¼1 t ð1 r o Þexp 3 P o sinðþ ð1 cosðwtþþ : ð23þ 4 h*w Figure 5. Densification of oil-base moeling clay uner isotropic stress for two packing angles, expresse in terms of relative ensity as a function of imensionless stress.

6 TNN 1-6 GHEZZEHEI AND OR: TECHNICAL NOTE isons, we consier the yiel stress etermine by the inentation experiments to be equivalent to the yiel stress in compression, s y = /2. Hence the nonimensional stress reporte by [Davis et al., 1973] is equivalent to 2 P /.A moel given in a companion paper [Ghezzehei an Or, 2003] escribes the initial ensification processes, by coalescence of the spheres at their contacts. The coalescence process is epenent on the packing angles. However, when the pores can be consiere as close (see Figure 1), the initial packing is of no consequence, as the ensification process involves only raial flow of material towar the pore center. Ieally, the maximum ensity of the system is equal to the ensity of the shell as the pores vanish (r =1). However, ue to physical constraints, such as air entrapment in isolate pores, this ensity may not be attaine. Hence we ajust the solution (13) to inclue a maximum relative ensity that coul be less than unity (r f <1) pffiffi 2 r max ¼ r f exp 2 P : ð24þ [19] In Figure 5, (24) is plotte as a function of the imensionless stress, with r f = It escribes the post pore-closure processes for both packing angles. At low ensity, the pore shape is an irregular one enclose by portions of spheres. At such low ensities, the specific pore surface area is less than that of a perfect sphere; hence the rate of work one by the external stress P is unerestimate. This is, in part, responsible for the unerpreiction of the ensity at a loa ratio of 2 P / Soil Densification Uner Constant Strain Rate [20] In an experimental setup, it is common to compress soil samples uner constant strain rate in orer to etermine stress-strain relationships. The theoretical framework above can be aapte to simulate such experiments. In the simplest case, we consier only one-imensional ensification in the z axis, then the strain e z an the relative ensity of the unit cell r are relate by, e z ¼ 1 r o =r ð25þ If the axial strain rate is constant, e z =t ¼ _e c, then, by chain rule we have r t ¼ r2 =r o _ec ð26þ Substituting (26) in (11) gives an implicit relationship between the relative ensity an the aheral stress, ( r 2 _e c ¼ 3 r o 4 P pffiffi ) 2 ty ðr 1Þ 1 þ ln½1 rš ð27þ h p P [21] Unlike the previous example, (27) epens on the strain rate, hence time an viscosity are important factors. In this example, we use experimental ata of uniaxial compression of a natural soil aggregate be reporte by Braunack an Dexter [1978], shown in Figure 6. Different sizes of aggregates of Urrbrae loam (Australia) soil were collecte by sieving. The ratio of the major: intermeiate: minor axes was 1.0:0.8:0.6 for all the aggregate classes (i.e., resemble spheres). The aggregates were first saturate an then raine to the require matric potential on a Figure 6. Densification of natural soil aggregates uner uniaxial stress at constant strain rate, expresse in terms of relative ensity as a function of imensionless stress. pressure plate. The wette aggregates were packe in a cylinrical compression cell of 80 mm iameter an 100 mm height, an compresse uniaxially at a strain rate of _e c ¼ 2: s 1. The height of the samples H was recore as a function of the axial stress P. The stress P was scale by the tensile yiel strength of the iniviual aggregates Y, measure by the force F require to crush them between parallel plates, an the yiel strength was etermine by an empirical relationship Y ¼ 0:576F= 2 ; ð28þ where is the mean aggregate iameter. For the illustrative comparisons, we assume that the magnitue of yiel strength Y is equivalent to s y. The ata use for this illustrative example was that of mm iameter aggregates at a water content of 0.3 kg kg 1 an matric potential of 10 kpa. The tensile yiel strength etermine by the proceure escribe above was 21 kpa. In Figure 6, the measure relative sample height (scale by the initial height H i = 100 mm) is plotte as a function of imensionless stress P/Y. In view of the large eformations observe in the experimental ata, the initial ensity of the aggregate be was approximate using cubic packing, that is r o = The initial eformation, while the interaggregate pores are consiere as interconnecte, was moele using the coalescence moel. [22] Because the actual viscosity of soil aggregates was not known, coefficient of plastic viscosity ( ETA p ) was use as a fitting parameter in (27). The value of ETA p that resulte in the best agreement was ETA p = 200 kpa. These comparisons are intene to emonstrate the plausibility of the propose moel. Further experimental work is require for quantitative comparison an moel testing. 4. Summary an Conclusions [23] This stuy aresses ensification of soil with low interaggregate porosity, where moels assuming the presence of istinct soil aggregates cannot be applie because of

7 GHEZZEHEI AND OR: TECHNICAL NOTE TNN 1-7 substantial overlapping of aggregate contact. The objective of this note was to exten the application of the iscrete aggregate coalescence moels presente in the previous stuies [Ghezzehei an Or, 2000, 2001, 2003; Or, 1996; Or et al., 2000] by consiering eformation of relatively ense soils. A self-consistent moel that consiers volume reuction of spherical pores surroune by a concentric shell of homogeneous soil matrix was employe. For simplicity, the aheral mean stress was consiere as the riving force for the ensification. This implies isotropic volume reuction of the pores, regarless of the anisotropy of the applie external stress. Solutions were provie for ensification uner steay an transient stress. The scenarios consiere for steay stress were: time-epenent ensity an ultimate ensity at a constant external stress, an time-epenent ensification at a constant strain rate. Uner transient stresses, only the viscous part was consiere in volume reuction of the spherical pores, whereas, the elastic part was attribute to temporary eformation of the soil matrix surrouning the pores. The geometry of the moel is more accurate at higher relative ensity. The transition zone from coalescence of iscrete aggregates to volume reuction of spherical pores is poorly explaine by either the aggregate coalescence moel or the raial volume reuction moel. The application of the moels was emonstrate using illustrative examples that compare favorably with measure ata from literature. The experimental ata, as well as the moel calculations, show that the rate of ensification ecreases as the relative ensity approaches unity (complete closure of pores), an the relative stress require to rive ensification increases exponentially with relative ensity. [24] When soils are subjecte to uniaxial stresses, such as overburen an traffic, it is likely that soil pores eform anistropically. Consequently the moel presente in this note requires further moifications to account for such anisotropic eformation by consiering, for example, ellipsoial pores. [25] Presently, moel calculations are base on a single representative pore. However, the current moel is now being extene to interacting ellipsoial pores [Eshelby, 1959; Mori an Tanaka, 1973; Mura an Cheng, 1977]. Such an extension provies a means for eriving bulk soileformation properties such as mouli an Poisson s ratio that are important for relating moel calculations with experimental measurements at large sample scales, an provie a link with traitional soil mechanics moels. Apart from applications for compaction of agricultural soils, the moel coul be useful to escribe eformation of untille ense soils, such as for engineering applications. [26] Acknowlegments. The partial funing by USDA/NRI through contract an the Research Fun (BARD) through contract is gratefully acknowlege. The work reporte herein was performe at Plants, Soils an Biometeorology Department, Utah State University, Logan, Utah. References Braunack, M. C., A. R. Dexter, Compaction of aggregate bes, in Moification of Soil Structure, eite by W. W. Emerson, R. D. Bon, an A. R. Dexter, pp , John Wiley, New York, Braunack, M. V., J. S. Hewitt, an A. R. Dexter, Brittle fracture of soil aggregates an the compaction of aggregate bes, J. Soil Sci., 30, , Davis, P. F., A. R. Dexter, an D. W. Tanner, Isotropic compression of hypothetical an syn-thetic tilths, J. Terramech., 10, 21 34, Eshelby, J. D., The elastic fiel outsie an ellipsoial inclusion, Proc. R. Soc. Lonon, Ser. A, 252, , Frenkel, J., Viscous flow of crystalline boies uner the action of surface tension, J. Phys. Moscow, 9, , Fröhlich, H., an R. Sack, Theory of the rheological properties of ispersions, Proc. R. Soc. Lonon, Ser. A, 185, , Ghezzehei, T. A., an D. 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Res., 32, , Or, D., F. J. Leij, V. Snyer, an T. A. Ghezzehei, Stochastic moel for post-tillage soil pore space evolution, Water Resour. Res., 36(7), , T. A. Ghezzehei, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. (taghezzehei@lbl.gov) D. Or, University of Connecticut, Storrs, CT 94720, USA. (ani@engr. uconn.eu)