SW—Soil and Water: Transport of Particulate and Colloid-sorbed Contaminants through Soil, Part 2: Trapping Processes and Soil Pore Geometry
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Biosystems Engineering (2002) 83 (4), 387395w
sufciently large diameter to have their movement restricted by physical straining in the soil pores when waterdrains from the soil under gravity, but if the soil is very wet (near to saturation) even these will move withoutrestriction. # 2002 Silsoe Research Institute. Published by Elsevier Science Ltd. All rights reserved1. Introduction
Part 1 of this paper (McGechan & Lewis, 2002)discusses some general principles governing transportthrough soil of colloid-associated contaminants such asphosphorus, pesticides and other agrochemicals,plus a range of biological microorganisms. In particular,it was noted that there appear to be two importanttypes of restrictions physically limiting the colloidmovement in soil and other porous media: straining,where the physical size of the pore is smaller thanthe particle so the particle is unable to pass, andltration, which covers a range of mechanisms bywhich particles are captured in pores with dimensionslarger than the particles. Both types of capture, butstraining in particular, are very dependent on thesizes (or the size distributions) both of colloidalparticles and of soil pores. Consideration of sizes ofcolloidal particles can refer to the sizes of free-standing
indication of the extent of trapping of colloids bystraining.
2. Mathematical description of soil pore space
2.1. Requirement for description of soil pores
Restrictions on movement of colloids through soilscaused by a range of mechanisms (but particularly dueto straining) are likely to be related to the size ofparticles in relation to the size of soil pores. Since thesizes of both colloidal particles and of the soil pores canvary over a wide range, it is necessary to consider sizedistributions for both. The geometrical shape, whichtypically is not circular or spherical, also needs to beconsidered, particularly for soil pores.The size distribution of water-lled pores in soil can
be estimated from hydraulic properties using twodoi:10.1006/bioe.2002.0136, available online at http://wwSW}Soil and Water
Transport of Particulate and Colloid-sorTrapping Processes an
Environment Division, SAC, Bush Estate, Penicuik
(Received 14 September 2001; receive
It was established in Part 1 that potential impediments tinto straining and ltration, depending on whether aphysical trapping) or much smaller. Information abouPart 1. Owing to the dependence of colloid or particle caparticles and pores, information about pore sizeshydrological equations. Various approaches to extenparticles would be trapped by necks or irregularitiesprotozoan microorganisms (such as Cryptosporidium amicroorganisms, or of soil particles onto which con-taminants are sorbed, as discussed in Part 1. This paper(Part 2) explores a theoretical mathematical approachbased on equations of hydrology which might give an
1537-5110/02/$35.00 387.idealibrary.com on
bed Contaminants through Soil, Part 2:Soil Pore Geometry
H26 0PH, UK; e-mail: email@example.com
in revised form 2 September 2002)
o movement of colloids through soil can be subdividedparticle has a dimension similar to pores (leading tot size distributions of particles was also presented inpture processes on the relative size distributions of theistributions has been derived here (Part 2) fromding this mathematical treatment to indicate whenin the pore are also presented. Results suggest thatnd Giardia) are the only colloidal contaminants withfundamental equations in uid mechanics, the capillaryrise equation and Poiseuilles Law. Later in this paper,the approach of using these equations, or adaptations ofthem, to estimate the sizes of suspended particulate or
# 2002 Silsoe Research Institute. Published byElsevier Science Ltd. All rights reserved
M.B. MCGECHAN388so h=15 10 /rm.Poiseuilles law is stated as
where: Q is the water ux in m3 s1 due to a pressuredrop DP in Pa along length of tube L in m and m is thedynamic viscosity of water in Pa s. For water at 108C,m=1312 103 Pa s. From Eqn (2), the mean owvelocity %uu in m s1 through the tube can be derived:
2.3. Capillary bundle model
There have been a number of theoretical studies ofsoil and other porous media using these equations,
integral of the pore size distribution curve, or converselythat the pore size distribution is the slope of the waterrelease curve (Payne, 1988). Ball (1981) pointed out thatthis model is inaccurate in soils where pores ofcontrasting radius are joined together, since larger porescan drain only at the same time as the smaller pores thatthey link, and consequently the volume of small pores isoverestimated. However, in the absence of a satisfactoryalternative, the simple model is widely assumed. Itappears to be preferable to the approach adopted byNicholson et al. (1988), Kosugi and Hopmans (1998)and Kosugi (1999), who based similar analyses onvarious alternative standard frequency distributionequations for soil pore radius, rather than basing it onthe measured water release curve. Brooks and Corey(1966) derived an equation for a curve to be tted toexperimental water release data, based on an assumeddistribution of pore sizes. This equation includes a porecolloidal material which will either pass freely throughthe soil prole or be prevented from passing, is explored.
2.2. Fundamental equations
The equation for capillary rise (which is mostsignicant for a tube with small radius) can be statedfor a liquid in a tube with circular cross-section as
where: h is the height in m that water will rise in acapillary tube of radius r in m with which it forms a zerocontact angle; s is the surface tension of the liquid waterin kg s2; rw is the density in kgm
3; and g is theacceleration due to gravity in m s2. For water at 108C,s=00727 kg s2, rw=1000 kg m
3 and g=981m s2,6
A matching parameterd diameter of ltrate particle, md85 85% percentile of ltrate particle diametersD diameter of lter particle, mD15 15% percentile of lter particle diametersg acceleration due to gravity, m s1
h capillary rise, mi, j, n summation limitsk Kozeny constantk0 shape factorK(y) hydraulic conductivity of soil at volumetric
water content yL length of straight tube or porous medium, mLe actual path length through length L of
medium taking account of tortuosity, mm ratio of the pore space volume to the area of
Notabased on the analogy with a bundle of capillary tubes ofvarious diameters, e.g. that of Jury et al. (1991), who usethe term capillary bundle model. Analysis would bevery straightforward for a bundle of tubes all of thesame diameter, but this is not the case for soil, andvarious approaches have been taken to take account ofthe distribution of tube (or pore) diameters. Thisdistribution of pore diameters, along with other featuresof pore geometry such as shape and continuity, accountsfor the important hydraulic characteristics of a real soil,the water release curve (tension against water content)and the hydraulic conductivity (which declines withdecrease in water content or increase in tension).In the capillary bundle model, the distribution of tube
diameters is derived from a measured water releasecurve assuming that tension is related to the radius ofthe largest water-lled pore according to Eqn (1). Thisrelationship implies that the water release curve is the
the wetted surface (hydraulic radius), mDP pressure drop along length L of tube, PaQ water ux due to a pressure drop Dp, m3 s1
r radius of tube with circular cross-section, mS surface area per unit volume, m1
%uu mean ow velocity through tube, m s1
a scaling parametery volumetric soil water content, fractionys porosity, fractionl pore size distribution index (Brooks & Corey,
1966)m dynamic viscosity of water, Pa srw density of water , kg m
s surface tension of liquid water, kg s2
j soil water tension, Pa
TRANSPORT OF CONTAMINANTS THROUGH SOIL 389size distribution index parameter l, such that a largevalue of l indicates a narrow range of pore sizes and asmall value of l indicates a wide range of pore sizes.Arya and Paris (1981) present a model for predicting thesoil water characteristic from the particle size distribu-tion and bulk density, based on the similarity betweenthe particle size distribution and the water retentioncurve. Pore lengths based on spherical particles werescaled to natural pore lengths using a scaling parametera, taking an average value of 138. Later, Arya et al.(1999a) determined different values of a for different soiltypes by summing properties in particle size classes.
2.4. Theoretical basis for variation in hydraulicconductivity
2.4.1. Conductivity based on capillary bundle approachHydraulic conductivity is related to the whole pore
size distribution rather than the largest water-lled pore,since at a given soil water content all continuous water-lled pores contribute to the total ow of water throughthe soil. The electrical analogy for this is the currentscalculated by Ohms law for resisters in parallel. Thesimplest treatment of this is described by Jury et al.(1991), by summing the contributions from each water-lled tube in the bundle as given by the tube radiusdistribution derived from the water release curve. Thecontribution from tubes of a particular range of radii isderived from Eqn (2), with an additional tortuosityterm based on the assumption that each tube isconvoluted, so its length Le is longer than the wholebundle L, as discussed by Fatt and Dykstra (1951),Burdine (1953) and by Wyllie and Gardner (1958).
2.4.2. Adaptation of capillary bundle model tonon-circular cross-sections
A number of researchers (Kozeny, 1927; Fair &Hatch, 1933; Carman, 1937; Wyllie & Spangler, 1952)have modied the ow velocity equation further to takeaccount of non-circular cross-sections as well as thetortuosity effect, replacing r2/8 in Eqn (3) by m2/k0,where m is the ratio of the pore space volume to the areaof the wetted surface (called the hydraulic radius, inm), and k0 is a shape factor; m is describedmathematically as ys/S where ys is the porosity (frac-tion), S is the surface area per unit volume (in m1), andk0 is a component of the Kozeny constant k where
k k0 Le=L 2 4
Values of m can be calculated for regular-shaped pores,e.g. r/2 for a cylindrical tube, r/3 for a sphere and r/333for the space surrounding uniform diameter spherespacked with their centres in line in three dimensions.Values of k0 have been measured experimentally asbeing in the region of 25 for most irregular shapes,compared to theoretical values for regular cross-sectionsreported by Carman (1937) such as 20 for a circle and30 for a rectangle of innite width. Similarly, values ofaround 50 have been measured experimentally for k,implying a tortuosity (Le/L)
2 of around 20. Thehydraulic radius and shape factor have been incorpo-rated by Brooks and Corey (1966) into a procedure forderiving the decrease in hydraulic conductivity withdecrease in water content, from the pore size distribu-tion indicated by l in their equation for the water releasecurve.
2.4.3. Random joining of pores approachAnother complexity in deriving the conductivity curve
was rst introduced by Childs and Collis-George (1950),but later modied by Marshall (1958), and Millingtonand Quirk (1959, 1961), and further discussed by Greenand Corey (1971), Jackson (1972), Campbell (1974) andGavalas (1980). This treatment takes account of varioussequences of pores of different sizes when pores arerandomly arranged in a porous system. Mathematically,they assume that the bundle of tubes of various sizes iscut and randomly rejoined in a different position sosmall pores meet up with large ones and the ow isdetermined by the smaller pore at the join, whichrequires a summation for tension (pore size) class groupsof an equation of the form
K y i AXn
2j 1 2ij2j ; i 1; 2; :::; n 5
where: Ky is the hydraulic conductivity of soil atvolumetric water content y; i; j and n are summationlimits; A is a matching parameter; and j is the soil watertension in Pa.Equation (5) is similar to that presented by Marshall
(1958) who simplied the procedure by dividing up thepore size distribution into equal porosity intervals,unlike Childs and Collis-George (1950) who subdividedtension into class intervals which made the procedureeven more complex. Some authors have attempted topredict absolute values of conductivities taking accountof estimates of m, k0 andy (the subject of much of theongoing discussion by Marshall, 1958; Millington &Quirk, 1959, 1961; Green & Corey, 1971), the so-calledKozeny-Carmen equation. However, this equation hasbeen applied mainly to fairly uniform porous mediasuch as microscopic glass beads or sand beds rather thanagricultural soils. Other authors including Childs andCollis-George (1950), Brooks and Corey (1966) andJackson (1972) only predict the variation of conductivitywith water content, choosing a value of the matchingparameter A in Eqn (5) to t the relationship to the
Fig. 1. Water release characteristics, measured data points andBrooks and Corey (1964) fitted relationships for three soils:(a) clay loam tilled soil; (b) sandy loam tilled soil; (c) siltyclay loam grassland soil; (from McGechan et al., 1997). Radiusof largest water-filled pore according to Eqn (1) also indicated:&, 0.05 m depth (upper topsoil);}m}, 0.15 m depth (lowertopsoil); }*}, 0.5 m depth (subsoil); }^}, 0.8 m depth(lower subsoil, tilled sandy loam soil only); . . ... . ., below
0.95 m (C horizon, tilled sandy loam soil only)
M.B. MCGECHAN390measured conductivity at one particular water content,and this is the procedure which has been adopted herefor agricultural soils.
2.4.4. Other approaches to relating conductivity topore sizes from hydraulic equations
An alternative set of equations for determining thehydraulic conductivity curve from water retention datais presented by Campbell (1994). Vucovi!cc and Soro(1992) compared and assessed ten alternative theoreticalor empirical equations relating conductivities of porousmedia to grain size composition. Arya et al. (1999b)apply a similar approach to that which they used for thewater retention curve (determining a by summing inparticle size classes, Arya et al., 1999a, see Section 23)to determine the hydraulic conductivity function fromthe particle size distribution. Beven and Germann (1981)describe a two-domain adaptation of the capillarybundle model approach for separate representation ofmacropores and micropores. Further complicationsspecic to macropores, which require an extension tothe approach based on Eqn (2), are described by Chenand Wagenet (1992).
2.5. Measured and derived hydraulic conductivitiesfor real soils
Measured water release data for three layers each ofthree agricultural soils, as reported by McGechan et al.(1997), are shown in Fig. 1. Fitted water release curvesaccording to the Brooks and Corey (1964) equation witha transition to a linear relationship in the macroporousregion near to eld capacity (Janss...