A BRIEF REVIEW OF SOIL WATER, SOLUTE TRANSPORTAND REGIONALIZED VARIABLE ANALYSIS

D.R. NIELSEN1; J.W. HOPMANS1; M. KUTÍLEK2; O. WENDROTH3

1 Professor, Department of Land, Air and Water Resources, University of California, Davis, California 95616.2Professor, Faculty of Civil Engineering, Technical University Prague, Thakurova 7,166 29 Prague 6 - Dejvice, CzechRepublic.3Research Soil Physicist, Center of Agrolandscape and Land Use Research, institute for Soil Research, EberswalderStrasse 84,15374 Muncheberg, Germany.

ABSTRACT: We initially review basic concepts of the forces acting on soil water, soil water potential and soil waterretention, and equations to describe soil water movement under water-saturated and unsaturated conditions.Processes of infiltration, evaporation and redistribution of water will be presented for simple initial and boundaryconditions occurring within homogeneous soil columns. Next we consider the physical, chemical and biologicalprocesses within a soil profile that distribute, dilute or concentrate solute species within the liquid phase of a soil Therelative concentration of solutes in the liquid phase governs not only the retention and transport of water within soilsbut also contributes to our understanding of managing the quality of water within soils and that moving below therecall of plant roots deeper into the vadose zone. A complete set of references about this subject is available inKutflek & Nielsen (1994). In the last section we recall the differences between classical statistical concepts and thosethat utilize the coordinates of space and time at which state variables across the landscape are observed. Ourpresentation win cover the basic Ideas about autocorrelation, crosscorrelation, applied time series analyses, statespace analyses and similar techniques currently being used to enhance field research and investigations of land andwater managementKey Words: Solute transport, regionalized variable

UMA BREVE REVISÃO DE ÁGUA NO SOLO, TRANSPORTE DE SOLUTOE ANÁLISE DE VARIÁVEL REGIONALIZADA

RESUMO: Inicialmente foram revisados os conceitos básicos das forcas que atuam sobre a água no solo, potencial eretenção de água no solo, e as equações que descrevem o movimento de água sob condições de solos saturados e nãosaturados. Os processos de infiltração, evaporação e redistribuição de água foram apresentados para condições iniciale de contorno que ocorrem dentro de colunas de solo homogêneo. Posteriormente, foram considerados os processosfísicos, químicos e biológicos dentro de um perfil do solo que distribuem, diluem ou concentram espécies de solutos nafase liquida do solo. A concentração relativa de solutos na fase líquida não governa somente a retenção e o transporteda água no solo, mas também contribui para a nossa compreensão para o manejo da qualidade da água no solo edaquela que se move abaixo das zonas radiculares. Um conjunto completo de referencia concernente a esse assuntoestá disponível em Kutilek & Nielsen (1994). No final do presente trabalho, foram recodadas as diferenças entre osconceitos da estatística clássica e aqueles que utilizam as coordenadas de espaço e de tempo nas quais asvariabilidades de espaço foram observadas ao longo da paisagem. A apresentação cobre as idéias básicas sobreautocorrelação, crosscorrelação, análise de série temporal aplicada, análise de state-space e técnicas atualmenteutilizadas para subsidiar a pesquisa de campo, investigações sobre o solo e manejo de água.Descritores: Transporte de solutos, variável regionalizada

BASIC SOIL WATER CONCEPTS

Forces acting on soil water: Here we study thesystem of soil and water in an equilibrium state,characterized by zero fluxes in the soil. The soilwater content does not change in time and thewater and its solutes are in equilibrium withinternal and external forces acting on the system.Although we do not pay attention to the rate atwhich the equilibrium state was reached, we

should know how the equilibrium was obtained. Itis usually important to know the recent "history"of the system, e.g. was the equilibrium reached bywetting or drying of the soil? The forces may begrouped according to their nature into separatecategories - adsorption, capillarity and swelling.

Adsorption: All three phases - solid, liquid andgas - usually occur simultaneously in most allsoils, is the most frequent. Force fields existing at

the interfaces of solid-liquid, liquid-gas and solid-gas influence the behavior of soil water. Solutesin the soil water contribute to and modify theaction of these force fields. The action betweenthe surface of a solid soil particle including itssorbed ions and water takes place in closeproximity of the surface. This portion of soilwater is commonly called adsorption water. It canbe most easily studied in the absence of otherforces and is usually observed when limitedamounts of water vapor are adsorbed on the solidsoil particles. The adsorptive force field decreaseswith the distance from the solid surface. In Figure1, the adsorbed water can be visualized as thinfilms covering the soil particle surfaces whilecapillary water to be discussed next occupies thewedges between the particles.

Capillarity: Water filling only parts of thesmaller spaces in an unsaturated soil formscurved interfaces between air and water.Capillary forces arise from these interfaces. Whenwater meets a solid surface, a contact angle g isformed, see Figure 2. This contact angle isobserved when a drop of water is placed on aplane solid surface, or when the solid plate issubmerged into water. The magnitude of g is usedto distinguish three classes of wetting for thesolid surface. For g= 0°, the surface is completelywet and the solid is fully hydrophilic. A non-complete wetting of the surface occurs for O < y<90° and the solid is partly hydrophilic. A non-wetting surface exhibits y 2 90° and the solid ishydrophobia.

Let us assume that the walls of acapillary of radius r are hydrophilic. Water risesinto the capillary when its lower end is placedinto a pool of water. The height h of this capillaryrise is

where s and r are the surface tension and densityof water, respectively, and g the gravitationalacceleration at the earth's surface. For water at

20°C and with g = 0°, |h| is approximately equal

to 0.15/r for length units in cm. Note in Figure 3the magnitude of h for the concave meniscus isless than zero. If we installed instruments formeasuring the hydrostatic pressure, a straight linepassing through zero at the plane water levelwould be observed. Below the water level, thehydrostatic pressure is positive, and above it inthe water column of the capillary, the pressure isnegative. The atmospheric pressure is taken asthe reference.

Similarly, when the capillarity of awater lens between spherical particles is studied(Figure 4), the convex-concave air-water interfaceyields a value of pressure p defined by

where R1 is positive and R2 is negative. From thisdiscussion it follows that the soil water pressureis negative above the free (ground) water level.

Swelling.: When water penetrates between twoparallel clay particle surfaces, the surfaces tend toshift apart - a phenomenon called swelling.Swelling depends upon the clay content,mineralogical composition of clay and upon theexchangeable cations. Monovalent exchangeablecations induce greater swelling than divalentcations. When the clay fraction is dominated bymontmorillonite, soils swell more than thosedominated by chlorite or illite. The swelling ofkaolinitic soils is negligible. An explanation ofswelling starts with simple model of two parallelplates, which can demonstrate the behavior ofplates of clay. Iwata et al. (1988) describe indetail the repulsive and attractive forces actingupon two parallel plates. In the foregoingpresentation, although we shall neglect swellingand shrinkage, it must be remembered that suchprocesses are important to the behavior of waterand solute movement in field soils.

Soil Water Potential: The sum of the action ofthe above forces are all included within theconcept of the soil water potential. Generally, thedifference of the potential at two separate pointsrelates to the driving force of the flow of soilwater similar to the difference of temperature atthe distant ends of a metallic rod that producesheat flow. Soil water potential is the pivotalconcept of soil hydrology which treats the theoryof soil water flow and retention. The potentialenergy of soil water is quantified relative to areference state. The reference state is defined asthe potential energy of pure water (with noexternal forces acting on it) at a referencepressure (which here we assume is theatmospheric pressure), a reference temperatureand a reference elevation. Soil water potential is

then the potential energy per unit quantity ofwater relative to the reference potential of zero.

The total potential is the summation ofthe component potentials

where fg is the gravitational potential, fw the soilwater potential, f0 the osmotic potential, 4>a thepneumatic potential and fe envelope potential.

Gravitational Potential: The gravitational forceof attraction between any two bodies of mass mand M is

where r is the distance between the centers of thebodies and G is the universal gravitational cons-tant equal to 6.67 10-8 dyne.cm2.g-1. If we assumethat M is the mass of the Earth and m is that ofwater, the gravitational potential fg is defined by

where R is the radius of the Earth (6.4108 cm)and z is the vertical coordinate in the soil (in theorder of 102 cm). Integrating (Equation 4), wehave

considering that R >> z. Note from Equation 4that F m-1 = GMR-2 which is g, the gravitationalacceleration of 980 cm sec-2 when evaluated at theEarth's surface (r = R). Hence, the gravitationalpotential increases linearly with elevationaccording to

Soil Water Potential: Soil water potential isdefined as the amount of work per unit quantity ofpure water that must be done by external forces totransfer reversibly and isothermally an infintesi-

mal amount of water from the standard state tothe soil point under consideration. The use of thedefinition is illustrated in Figure 5 where the soilwater retention curve is constructed.

The soil water retention curve or the"soil water characteristic" relates the volumetricsoil water content to the soil water potential.Notice that the individual forces associatedwith adsorption, capillarity, adhesion, cohesionetc. are not individually or collectively measured.Rather, the potential energy associated with their

combined effects is estimated over a small soilvolume. From the concept given in Figure 5, wesee in Figure 6 how soil water potential ismeasured in the laboratory and the field.

Osmotic Potential: Osmotic potential (f0 isowing to the difference in the chemicalcomposition of the soil solution related to free,pure, bulk water at the same elevation. Richards(1965) explained the meaning of the osmoticpotential with a set of pools, and a manometer ortensiometer attached to each of the pools asillustrated in Figure 7.

The pools are separated bysemipermeable membranes permeable only tospecified substances. The central pool contains anunsaturated soil with water actually being asolution. The soil is separated by a membranepermeable only to water from the pool on theright containing pure water (and by a membranepermeable only to soil solution from the pool onthe left containing soil solution. This latter poolcontaining the solution of the same concentrationas the soil water is separated by a membranepermeable only to water from the next pool to itsleft containing pure water. The differentialmanometer on the left shows a pressure heightidentical to the osmotic potential (f0, while themanometer attached to the pool with the soilsolution shows the sum of the gravitational and

the soil water potential (fg + fw). The same valueof (fg + fw) is read if a tensiometer is insertedinto the soil inasmuch as its porous cup ispermeable to both water and its dissolvedsubstances. The manometer attached to the poolwith pure water shows the sum of the potentials(fg+ fw+fo).

Pneumatic Potential: The pneumatic potential faaccounts for air pressure inside the soil poresbeing different from the atmospheric air pressureacting upon the reference water.

Envelope Potential: When an externalmechanical pressure such as the overburdenpressure of the topsoil layers acts upon the soil,the magnitude of change of the total potential isexpressed by the envelope potential fe. Thecontribution of fe is usually negligible for sandysoils and becomes more important for soils havinggreater clay contents. In sands, because theexternal pressure is transmitted almostcompletely by the direct contacts of the grains,especially in pre consolidated sands, the potentialof the pore water is not altered by the externalpressure. When the external pressure istransmitted from particle to particle via the waterfilm envelopes around the particles, the porewater is greatly influenced and the value of fe issignificant.

We note here that the potentialmeasured with a tensiometer is fw + fa + feprovided that the solution within the tensiometercup is at chemical equilibrium with that in thesoil. The provision is not fully met in saline soilsor even in non saline soils whenever theconcentration of the soil solution changes abruptlyin the vicinity of the tensiometer cup.

In the majority of situations the simplestdefinition of the total potential is

and with the potential expressed as energy perunit weight of water

the total head H is simply

where x and y are horizontal coordinates, z thevertical distance from the reference level, positiveupwards and t time. For soils not fully saturatedwith water, h <0.

When Does Water Not Move? According toNewton's laws, a body is at rest or moving at aconstant velocity when the sum of forces acting onit is zero. Because we do not measure the forcesacting on soil water, we must calculate the totalforce. What must be the condition to have no flowin the horizontal directions? There should be noforce acting on the water in the horizontaldirections. Hence, the partial dervatives ofEquation 9 with respect to both x and y must eachbe identically zero. In other words,

In as much as B is not zero, the valuesof h must be constant in both directions.

What must be the condition to have noflow in the vertical direction? There should be noforce acting on the water in the vertical direction.Hence, the partial dervative of Equation 9 withrespect to z must be identically zero.

Because the parenthetic value must be nil,

where c is a constant of integration. When nowater flows vertically, the values of h and zthroughout the soil must equal but opposite insign.

Two examples are illustrated in Figure8 when water tables exist at a soil depth of 100cm and at the soil surface. Note that under thewetter condition with the water table at the soilsurface the magnitude of the total potential - isgreater, but its derivative remains nil.

Rate of Soil Water Flow: The rate at whichwater moves through a soil is proportional to theforce acting on the water,

where q is the flux density vector equal to the

volume of water moving per unit time through aunit surface area normal to the direction of flow.If we consider the flux density in only the verticaldirection, Equation 13 becomes

or

If we assume the flux density q and theforce given by the last term in Equation 15 arerelated equal by a constant for a particular valueof h or 9, we have the Darcy-Buckinghamequation

where the hydraulic conductivity K(h) equalskrgm-1 with k being the intrinsic permeabilityreflecting the geometry of the soil pores, and pand the density and viscosity of the soil water,respectively.

A simple example of unsaturated flow isdemonstrated in Figure 9. The cylinder containingthe soil has small openings within its wallsleading to the atmosphere. Semipermeablemembranes, permeable to water but not to air,separate the soil from free water on both sides ofthe cylinder. The pools of water are connected tothe cylinder with flexible tubes. Full saturation ofthe soil is first achieved when both pools, lifted tothe highest point of the soil, displace the soil airthrough the openings on the top side of thecylinder. At this moment, there is no flow in thesystem and the soil is assumed water saturated.With the pool on the left side of the cylinderlowered to the position h~ and the pool on theright side to position h2, air enters into the soilthrough the openings as the soil starts to drain in

a manner similar to a soil placed on a tensionplate apparatus. Although water flows from theleft pool to the right pool, the rate of flow isreduced significantly compared with that whenthe soil is water saturated. If the water level ineach of the pools is kept at a constant elevationwith time, steady flow will eventually be reachedwith the water content at each point within thesoil remaining invariant. At this time, the fluxdensity q will depend upon the hydraulic gradientand be governed by an equation similar toEquation 16 with the force of gravity omitted

because water is flowing only horizontally withthe gravitational head being every where the same

Inasmuch as the soil is not water saturated andflow occurs primarily in those pores filled withwater, the value of K will be smaller than that ofthe saturated hydraulic conductivity Ks for thesame soil. The unsaturated hydraulic conductivityK [h(q)] is physically dependent upon the soilwater content q because water flow is realizedprimarily in pores completely filled with water.Coarse-textured soils generally have greatervalues of K than those of fine-textured soils nearwater saturation (-h ® 0). On the other hand,fine-textured soils usually have the greater valuesas the value of h diminishes. See Figure 10.

The experimental values of K satisfy theempirical relation

where a, b and n are constants. Values of n forclays are about 2 while those for sands are 4 orgreater. Note that when h 2 0 in Equation 18 theratio of a and b is the value of the saturatedhydraulic conductivity Ks.

Direction of Soil Water Flow: The direction ofvertical flow depends upon the magnitude and

sign of the derivative of h compared to 1 inEquation 16. For q > 0, water flows upward andfor q < 0, water flows downward. See Figure 11.

Rate and Direction of Soil Water Flow: Whenthe Darcy-Buckingham law contains thefunctional form of K(h) for a particular soil, and ifthe steady-state boundary conditions arespecified, Equation 16 may be integrated to yieldthe magnitude and direction of water flow as wellas the distribution in(z) within the soil profile. Asan example, we choose a loam soil having valuesof a, b and n in Equation 18 equal to 200 cm3h-1,100 cm2 and 2, respectively. We specify that awater table exists at the 3 m depth. If we specifythe flux density, we may calculate the distributionh(z). To avoid complications of signs, we shallselect t = -h where t is always positive in ourcalculations. Hence, we designate the totalpotential head H = (t+ z) with the reference levelfor gravitational potential being zero at the watertable z = O.

Substituting Equation 18 into Equation16 and integrating with respect to the aboveboundary conditions, we have

Note that if q is known, the equationcan be solved for its only unknown quantity ~ atthe soil surface. On the other hand if ~ isspecified, the value of q can be calculated.

Depending on the sign of q, and therelative magnitudes of a and b, the integral inEquation 19 is of the form

For infiltration with q < 0, the first formis appropriate and becomes

Upon integration, Equation 21 becomes

For steady infiltration rates q equal to -0.001, -0.01, -0.1 and -1 cmh-1, the values of h atthe soil surface are -355, -262, -137, -43.6 and -9.99 cm, respectively. The correspondingdistributions of z(h) given by Equation 22 areillustrated in Figure 12.

For evaporation with q >O, the secondform Equation 20 becomes

and upon integration is

The distribution by Equation 24 is givenin Figure 12 for an evaporation rate q = 0.001cmh-1. At that rate, the value of h at the soilsurface is -355 cm. We note that a maximumevaporation rate of 0.00547 cmh-1 is reached byallowing the value of h at the soil surface todecrease and approach - ¥. Integrating Equation19 for other soils (n = 3, 4, 5 etc.) for soils thatare progressively more sandy, it can be easilyshown that the maximum rate of evaporation isproportional to L-n where L is the depth of thewater table.

Steady Infiltration into a 2 - Layered Soil: Thesimplest case is the crust-topped profile. Rainfallfrequently destroys soil aggregates within a soilsurface. Tillage also induces compaction. Bothprocesses processes often denoted by sealing,crusting or compaction results in the formation ofa less permeable soil surface layer. Thecharacteristics of the topsoil will be denoted bythe index 2, while those of the subsoil will begiven the index 1, see Figure 13.

The origin of the z-axis is identical withthe position of the ground water level which iskept constant. The thickness of the subsoilbetween the ground water level and the topsoil isL1, the thickness of the topsoil L2 and the depth ofwater on the soil surface ho. For steady-stateflow, q1 = q2, and we have

If KsI>>Ks2, and KI(hI)>> K2(hI) where hI is thevalue of h at the interface, we have

and because H - (h + z)

This condition of a larger gradient of hoccurring in the topsoil (layer 2) demands asufficiently small value of h/ including h1 < 0.Because we assume that just below the interfacein the subsoil (layer 1), dH/dz » 1, we can write

For the topsoil assuming it remainswater-saturated,

The value of hI is obtained by equatingEquation 28 and Equation 29. The subsoil can be

unsaturated if |q| < KS1, Under the assumptions

here, the subsoil becomes unsaturated below thetopsoil when

For more general conditions than thoseassumed here, the bottom of the topsoil above theinterface can also become unsaturated. For suchcases, the above approach has to be modified, seee.g. Takagi (1960) and Srinilta et al. (1966).

Unsteady Infiltration into a Soil: Let us assumethat the soil is initially at some uniform watercontent qn. Water can enter the soil surface as aresult of rainfall or some sort of irrigationtechnique. The soil water potential or water fluxdensity existing at the soil surface dominates theinfiltration process. If the soil surface is suddenlyand continuously flooded with a negligibly smalldepth of water (h = 0), the soil near or at thesurface can be assumed water saturated. It is also

possible to maintain a constant soil waterpotential head h < 0, which q is maintainedconstant and less than qS. Such a boundarycondition is known as Dirichlet's boundarycondition. For such a condition, the water fluxdensity at the soil surface changes as infiltrationoccurs. Qn the other hand, if the flux density atthe soil surface is assumed known, Neuman'sboundary condition exists. It can describeconstant rainfall at rates less than as well asgreater than Ks, and nonconstant rainfall. It also isappropriate for controlled sprinkler and other fluxcontrolled techniques for applying water to thesoil surface. Here we give a couple of examplesusing only the Dirichlet condition.

The primary infiltration data usuallymeasured are values of cumulative infiltration I(cm) as a function of time. The values representthe total amount of water infiltrated into the soilfrom the beginning of the infiltration test at t = 0.A typical l(t) relationship (Figure 14) is a smooth,monotonically rising curve.

The infiltration rate q0 = dl/dt where thesubscript o refers to the soil surface at z = 0. Thevalue of qo initially decreases rapidly with timeand eventually approaches a constant value. Fort = 0, q0 ® ¥, and for t ® ¥, q0 = constant.Theoretically, qo ® Ks as t ® ¥. Practically, theinfiltration rate starts to be constant for coarsetextured soils after only decades of minutes while

When unsteady soil water flow exists,two equations are needed to describe the waterflux density and the rate of change of soil watercontent in time. The flux density described by the

Buckingham-Darcy equation and the rate offilling or emptying of the soil pores described bythe equation of continuity are combined into theRichards' equation

for vertical, one-dimensional flow where z ismeasured positively in the downward direction.

If the soil is only wetting, 6 will beuniquely dependent upon only h and

which leads to the diffusivity form of Richards'equation

where the Darcy-Buchingham equation has thediffusivity form

and the soil water diffusivity D is the termderived from

The main reason for the derivation ofEquation 33 is the reduction of the number ofvariables from 4 to 3.

The first term on the right hand side ofEquation 33 accounts for the forces associatedwith differences of the soil water potential asillustrated in Figure 5. The second term on theright hand side of Equation 33 accounts for thegravitational force because water is movingvertically. During early stages of infiltration thesecond term is negligible, and during late stages itdominates the process. Hence, here we shall firstneglect it and assume water flows horizontally notinfluenced by gravity, and later consider itsinfluence.

Our horizontal soil column, initially atan unsaturated water content qn, has its end at x =0 maintained at water saturation qs. Hence, for

we solve Equation 33 without the gravitationalterm

It is only here for a homogeneous soil(i.e. not layered) that the gradient of 0 representsthe driving force of the process. Here when D is afunction of q, we must transform Equation 38 intoan ordinary differential equation using theBoltzmann transformation.

The transformed equation has a newvariable h instead of the two original variables xand t. The new variable h defined by theBoltzmann transformation

transforms Equation 38 to

with transformed boundary conditions

The solution for which we search issimply q(h), see Equation 39 and Figure 15.Measured soil water profiles q[x(t1)], q[x(t2)],q[x(t3)] etc. are thus transformed into the uniqueq(h) relationship by merely dividing x by t1

1/2 forthe first profile, t1

1/2, for the second profile etc.Note that for t = l, x = h. Hence, the physicalreality of q(h) is the soil water profile q(x) whenthe infiltration time is unity.

The amount of water infiltrated into theprofile is

or with Equation 39,

With the sorptivity S being defined as

we have the cumulative infiltration

Because the infiltration rate

we also have

Here, we note that the sorptivity isphysically the cumulative amount of waterinfiltrated at t = 1, and at that time, theinfiltration rate has diminished to one-half the

value of S. Sorptivity depends not only upon theD(0) function but upon qn. The value of Sdecreases with increasing qi and as q1 ® qs, S® 0.

Sorptivity is an integral part of mostinvestigations describing vertical infiltration aswe shall learn below.

The solution of Equation 33 subject toEquation 38 and Equation 39 for verticalinfiltration is

The cumulative infiltration I is

where Kn is K(qn). Note that Knt expresses thecumulative water flow with dH/dz = -1 at q = qn.The series Equation 50 converges for short andintermediate times of infiltration and theinfiltration rate q0(t) obtained by differentiation is

For large times, Equation 50 does notconverge. Inasmuch as the shape of the wettingfront remains invariant at large times, the wettingfront moves downward at a rate

while the infiltration rate for t ® ¥ is

Soil water content profiles measured atthree different infiltration times in the laboratoryfor a homogeneous column of Columbia silt loamare given in Figure 16. The solid lines arecalculated using measured values of K(~) andD(~) in the above equations. The extra depth towhich the soil water profile advances owing to theforce of the gravitational field is illustrated inFigure 17 where the distance to the wetting frontfor horizontal and vertically downward flows areplotted against the square root of time.

During the first 25 min, the depths ofwetting for both horizontal and verticalinfiltration are nearly identical. After 400 min,the vertical profile is more than 12 cm deeperthan that of the horizontal profile, and theinfiltration rate is nearly constant and equal to Ks

according to Equation 53.

Redistribution of Water after Infiltration:After infiltration ceases, the soil water contentgradually decreases even when the soil surface isprotected by a cover allowing no evaporation. Thedecrease of 9 within the originally wetted topsoilis caused by a downward flow of soil water.When water draining from the wetted topsoil wetsthe originally drier subsoil, we speak of soil waterredistribution. At that time a relatively large soilwater potential frequently near h = 0 exists withinthe topsoil down to the depth of the wetting frontZf. Below the wetting front, the value of h is very

small, and hence, the soil water profile is in adynamic state rather than one of equilibrium.When we discuss this non-equilibrium processcaused by infiltration, we denote the time ofcessation of infiltration to = 0. The non-equilibrium produces a downward water fluxwithin the soil profile without a contribution tothe flux from the surface, i.e. at z = 0, q0 = 0.With the soil between z = 0 and z = Zf beingdrained, water flows below Zf forming a newredistribution wetting front at Zr, see Figure 18a.

The soil water content profile q(z) at tois the profile at the end of infiltration.Subsequently, at times t1, t2, t3, .., redistributionprofiles are observed. The cumulative drainage Idand the cumulative wetting Iw below Zf are equalfor the time interval (0, t1), see the hatched areasin Figure 18a. The rate of movement dependsupon the hydraulic conductivity and the depth ofthe original wetting front from the infiltration.The flux density across the redistribution frontdecreases in time owing to two factors. First, thepotential in the wetted topsoil decreases with asimultaneous decrease of the gradient of the totalpotential dH/dz. Second, the value of thehydraulic conductivity decreases strongly withonly slight decreases of q.

Soil water profiles during redistributionin Columbia silt loam given in Figure 18bresemble those in Figure 18a. At time to, 6.1 cmof water had just infiltrated the soil surface withthe wetting front reaching 17 cm. Soil watercontents near the soil surface decrease gradually

and uniformly by developing zones of nearlyconstant water content extending from the soilsurface downward. After 30 d of redistribution,the wetting front has reached nearly 40 cm andthe water content at the soil surface hasdiminished to 0.21 cm3 cm-3. At that time theinitially abrupt wetting front into the dry soil is nolonger evident, indicative of water movement atextremely small water contents. Othermeasurements of redistribution in Columbia siltloam showed that rate of redistribution isinversely related to the initial depth of wettingwhich is just the opposite for sands.

Water Vapor Transport in Dry Soils: Inrelatively dry soils when a great portion of thepores are filled with air and liquid water exists invery thin layers on the soil particle surfaces, theflux density of water is composed primarily ofvapor, not liquid water. For nonisothermalconditions, the transfer of water vapor is evengreater. Jackson (1964) performed a simpleexperiment to demonstrate how water vapormoves through soils. Theoretically for an inertporous medium, he assumed that the steady statewater vapor flux density qv4, was described byFick's first law of diffusion

where Dv is the soil water vapor diffusivity and pthe water vapor density in the soil air. CombiningEquation 54 with the equation of continuity

we have for transient conditions in an inertmedium

where e is the air-filled porosity. Because soil isnot inert, it can absorb the water vapor and haveit condense on its particles' surfaces as well as thereverse - contribute to the vapor density of its airfilled pores from the liquid water on its particlesurfaces. At equilibrium the adsorption isotherm

r(q) expresses the relation between the water inthe soil air to that in the liquid state. Recognizingthat p is a function of the soil water contento,Equation 54 becomes

where Dqvap is the product of Dv and the slope of

the vapor adsorption isotherm r(q). For a soil,Equation 56 becomes

where the second term on the right-hand side ofEquation 58 accounts for the water vapor beingadsorbed on the soil and changing the soil watercontent q. If

Equation 58 becomes

which is identical in form to Equation 38. By onlymeasuring soil water contents within a soilcolumn, a separation of that water moving asvapor and that as liquid cannot be achieved.Hence, we rewrite Equation 59 as

where Dqv accounts for both vapor and liquid

transfers and is defined as

Jackson used pressure dependence ofDqvap to ascertain the relative magnitudes of

Dqvap and Dqliq

The experimental set-up to study watervapor diffusion is shown at the top of Figure 19.A homogeneous soil initially at water content qn

was subjected to a constant vapor density at x = 0.The constant vapor density at x - 0 gives rise to aconstant soil water content q0 If q0 > qn, the soilabsorbs water. If q0 < qn, the soil loses water. Thetop graph in Figure 19 shows soil water contentprofiles of a soil having qn = 0.02 g g-1 and beingsubjected to q0 = 0.05 gg-1 for times of 7,135 and12,945 min (about 5 and 9 d). Water moves intothese soil columns. The bottom graph in Figure19 shows soil water content profiles of a soilhaving qn = 0.06 g g-1 and being subjected to q0 =0.01 gg-1 for times of 10,260 and 20,020 min(about 7 and 14 d). Water moves out of these soilcolumns.

The graphs in Figure 20 show thatthese respective soil water profiles at differenttimes are coalesced into one curve by theBoltzmann transform Equation 39. Figure 21shows the relative amounts of vapor and liquidtransported during under isothermal conditions.Figure 22 shows the temperature dependence of

Dqv

TRANSPORT OF SOLUTES IN SOILS

First we consider the physical andchemical interactions of solutes that play animportant role in soil salinity and fertilizer

SOLUTE INTERACTIONS

Molecular Diffusion: Thermal energy provides acontinual, never ending movement of liquidphases of the soil system. Owing to moleculardiffusion, solutes in the soil solution obey Fick'sfirst law similar to that of Equation 54 above forwater vapor diffusion. For solute diffusion, thesolid matrix of the soil complicates matters byaltering both the diffusion path length and thecross sectional area available for diffusion as wellas providing an electric field and reactive surfacesthat further alter molecular movement. The steadystate solute flux density is given by

where Dm is the molecular diffusion coefficientand C the concentration of the solute.

Electric Force Fields: Electric force fieldsalways exist within the pore structure of soilsowing to the electric charge possessed by thewalls of soil pores. The charge per unit pore wallarea is caused by isomorphous substitution ofatoms in the tetrahedral and octahedral layers ofthe clay minerals as well as the presence of theSi-O-H (silanol) group on quartz, kaolin mineralsand other surfaces like organic matter (-OH and-COOH). The magnitude of the former is fixedwhile that of the latter depends upon pH andconcentration of the soil solution. In general,small highly charged ions cause the viscosity ofthe soil solution to increase while largemonovalent ions cause the viscosity to decrease.The electrostatic fields of the ions causepolarization and a binding of surrounding watermolecules which alter the kinetic properties ofsoil water. The hydrophilic nature of most soilparticles is attributed to the attraction of hydratedcations by the electrostatic field of soil particlesand to the hydrogen bonding of water to the clays(Low, 1961). The mobility of both water and ionsin the region of the pore walls is reduced below

that in bulk solutions (Kemper, 1960; Dutt &Low, 1962). The impact of the electric field onions and water is more pronounced in clayey soilsand depends upon the ionic concentration anddistance from the pore wall.

The distribution of cations as a functionof distance from a negatively charged fiat surfaceis explained by the electrostatic force causingcations to move toward the surface iscounteracted by the thermal motion of the solutescausing them to diffuse away from the surface.We see in Figure 23 that the extent of the unequaldistribution of cations and anions away from thesurface depends inversely upon the totalconcentration CO of the solution. And, we notefrom Fig 24 for cylindrical pores with a wallhaving a net negative charge and filled with asolution of concentration CO that theconcentration distribution across the poresdepends upon the magnitude of the pore radius. Inthe center of large pores the concentrations ofcations and anions are identical while in thecenter of small pores owing to the electric field,the cationic concentration exceeds that of anions.

As water moves through pores, cationsand anions unequally distributed across the poresbecause of the negatively charged pore walls areswept along with the water. Consequently, adifferential charge builds up along the length offlow which tends to retard water flow. This

differential charge is called streaming potential.Similarly, if an electrical potential difference isestablished across a soil, ions moving within theelectric field will create a water flux. Eachprocess contributes to the behavior of solutes andwater at the pore scale and offers an opportunityfor understanding and managing solute movementand retention in soil profiles.

Other Interactions: Constituents in the gas,liquid and solid phases of soil continually reactingwith each other through a variety of chemical andbiological pathways contribute to the presenceand behavior of particular solutes in soil profiles.Applicable equilibrium and non-equilibriumchemical concepts are those of oxidation-reduction, solubility-precipitation, association-dissociation, acid-base and exchange-adsorption.Microbiological reactions as well as thoseinvolving root systems of higher plants contributeto the behavior of solutes in field soils. A fullunderstanding of solute transport begins with aknowledge of the above interactions.

Miscible Displacement: The term miscibledisplacement refers to the process of one fluidmiscible with another invading and displacing thelatter. With the fluids being miscible, they areable to completely mix, dissolve and dilute eachother. Here after considering miscibledisplacement in a capillary tube withoutmolecular diffusion, we progress to laboratoryexperiments displacing solutes having differentkinds of soil that undergo various chemical andphysical reactions. Simple theoretical conceptsare used to explain the results. Our understandingof processes occurring during miscible displace-ment will be illustrated during the workshop with

measurements of leaching and fertilizermovement in field soils.

In a Single Capillary without Diffusion: Fromthe definition of viscosity, the velocitydistribution of a liquid within a capillary tube ofradius a during steady, uniform flow is derived tobe the well-known parabolic velocity distribution

where v(0) = 2v0. The average velocity v of theliquid is vO. Integrating v(r) with the areal cross-section of the capillary, the volumetric flow rateQ (cm3 s-1) through the capillary is

If we rely solely on Equation 63 todescribe the fluid velocity, what will be thedistribution of a second fluid of concentration COas it displaces a fluid of zero concentrationinitially within the capillary of length L? Assumethat the solution CO enters the tube at x = 0 attime t = 0. The concentration C averaged over thecross section of the capillary at distance x is

or as a function of distance and time is

In Figure 25, the paraboloid of thedisplacing fluid CO within the capillary gives riseto a linear concentration distribution. When theinvading front of CO has reached a distance 2L,the average concentration across the plane normalto the capillary at a distance of L is Co/2.Interestingly, the average concentration of thefluid moving across the plane L at that instant isnot Col2 but 3Co/4. The averageconcentration of fluid moving past x = L (seeFigure 26a) is

Eq. 67 leads to

where p = v0t/L and is the ratio of the volume offluid passing x = L to the volume of the capillarybetween O £ x £ L. Pore volume of effluent orsimply pore volume is the name commonly usedfor p. The value of C = 3Col4 for p = 1 andapproaches unity as p ® ¥, see Figure 26b. Evenfor such a simple geometry as a capillary tube, theconcentration distribution within the tube (Figure25b) is not easily reconciled with the shape of theconcentration elusion curve (Figure 26b).

Convective Transport of Solutes: The spreadingor dispersion of the solute caused by convectivetransport with the water can be qualitativelyvisualized in Figure 27 for a simplified soil. Theinvading stream of solute partitions itselfaccording to the microscopic pore water velocitiesoccurring between the soil particles. At still asmaller scale, the water velocity is zero at theparticle surface, departs markedly from the meanflow direction and approaches a zero value in thevicinity of dead-nd pores. These pathways andpore water velocities, severely altered with slightchanges of water content, have yet to bequantitatively evaluated. In the near future,computer-aided micro tomography and nuclearmagnetic resonance techniques will provide anopportunity to ascertain the exact nature of thevelocities at the pore scale.

For water infiltrating into a deep,homogeneous, water-saturated soil, we see inFigure 28 that a solute of concentration C0

maintained at a point on the soil surface isdispersed vertically and horizontally. The velocityof the soil solution varies in both magnitude anddirection owing to the distribution of irregularlyshaped pores within the soil. Along transect A-A',the initial concentration C0 at z = 0 graduallydiminishes to zero. Similarly, the concentrationdistribution normal to the average flow alongtransect B-B' gradually broadens with soil depth.

Laboratory Observations: One dimensional soilcolumns studied in the laboratory provide asimple means of quantifying the mixing,spreading or attenuation of the soluteschematically presented in Figure 28. Anapparatus is required to maintain steady stateflow and invariant soil water content conditionswhen the initial soil solution is invaded andeventually displaced by a second misciblesolution. No mixing of the two solutions shouldoccur at their boundary before entering the soilcolumn, and samples of effluent to be analyzedfor solute concentration has to be collectedwithout disturbing the steady state flowconditions. A cross sectional sketch of a typicalapparatus is given in Figure 29.

Breakthrough Curves: Let the volume of thesoil column occupied by soil solution be V0 andthe rate of inflow and outflow of the soil solutionbe Q. If the initial soil solution identified by asolute concentration C, is suddenly displaced byan incoming solution C0, the fraction of thisincoming solute in the effluent at time t will be(C - C1)/(C0 - C1), or for an initial concentration ofzero, simply C/C0. Plots of C/Ca versus porevolume of effluent (Qt/V0), commonly calledbreakthrough curves, describe the relative timestaken for the incoming solution to flow throughthe soil column. Note that the definition of pore toall soil water contents. Any experimentallymeasured breakthrough curve may be consideredone or a combination of any of the five curvesshown in Figure 30.

For Figure 30a-c, the solute spreadsonly as a result of molecular diffusion andmicroscopic variations of the velocity of the soilsolution, i. e. there is no interaction between thesolute, water and soil particle surfaces. In thesecases

regardless of the shape of the curve. Thisequation expresses the fact that the original soilsolution occupied exactly one pore volume or thatthe quantity of solute within the soil column thatwill eventually reach a chemical equilibrium withthat in the influent and effluent is C0V0.

Note also that the area under thebreakthrough curve up to one pore volume (areaA in Figure 30b and c) equals that above the

curve for all values greater than one pore volume(area B), regardless of the shape of the curve.This latter statement is a direct result of Equation30, i.e.

Danckwerts (1953) defined holdback Hb

as the left hand term of Equation 71 having arange 0 < Hb < 1 for non reacting solutes. Theconcept of holdback is a useful qualitativedescription whenever interactions between solute,water and soil solids are mínimal. It indicates theamount of the soil water or solutes not easilydisplaced. Values of Hb for unsaturated soils havebeen evaluated to be 3 to 4 times greater thanthose for saturated soils.

Piston flow (Figure 30a) never occursowing to solute mixing that takes place bymolecular diffusion and variations in watervelocity at the microscopic level within soil pores.The breakthrough curve shown in Figure 30b ischaracteristic of the longitudinal spreading of asolute as it is displaced through a soil having anormally distributed sequence of pores. Evidencefor lack of solute-solid interaction is the fact thatthe areas A and B described by (71) are identical.A water-saturated soil composed of equal-sizedaggregates manifesting a bimodal pore watervelocity distribution typically yields thebreakthrough curve given in Figure 30c. With the

areas A and B of this curve being comparable, anyinteraction between the solute and the soilparticles is negligible.

An appearance of a breakthrough curveto the left of one pore volume as shown in Figure30d (area A > area B) results from the incomingsolution not displacing water in stagnant anddeadend pores or the incoming solute beingrepelled from the soil particle surfaces as in thecase of a anion passing in the vicinity of anegatively charged soil particles. A solute havinga large diffusion coefficient mixes morecompletely with the water in stagnant and slowlyconducting zones, thus delaying its appearance inthe effluent. It should not be expected that therelative behavior of solutes be the same for allvelocities and different soils. For example, if onesolute has a diffusion coefficient much greaterthan the other, it would be possible for it to notonly invade the nearly stagnant zones but alsodiffuse downstream ahead of the other solute. Inthis case the faster diffusing solute will appear inthe effluent earlier that the more slowly diffusingsolute. An appearance of a breakthrough curve tothe right of one pore volume as shown in Figure30e (area A < area B) results from adsorption andexchange of the solute on the soil particlesurfaces as well as a chemical reaction in the soilthat serves as a sink for the incoming solute.Unsaturating a soil alters the pore water velocitydistribution, allows some of the solute to arrivedownstream earlier and increases the magnitudeof holdback manifested by area A in Figure 30.Desaturation eliminates larger flow channels andincreases the volume of water within the soilwhich does not readily move. These almoststagnant water zones act as sinks to moleculardiffusion. Later we shall discuss the opportunityafforded by controlling the water content and porewater velocity to change the leaching efficiency offield soils.

A Theoretical Description: Several differentkinds of theories have been proposed andequations derived to describe solute transport inboth inert and reactive porous media. Acomprehensive treatment would include transfersof the solute in all three phases of the soil - gas,liquid and solid. Here we consider thecornerstone of most theoretical descriptions - theconvective-diffusion equation. Although it iscommonly used to describe miscibledisplacement, it is nevertheless fraught withdifficulties and approximations not easilyresolved in the laboratory or the field.

We begin with a prism element havingedges of length Dx, Dy and Dz. The differencebetween the mass of solute entering the prism andthat leaving the prism equals the difference of thesolute stored in the prism in time Dt providingthat we account for any appearance (source) ordisappearance (sink) of the solute within theprism by mechanisms other than transport. Thesolute flux density into the prism in the directionof the x axis is Jx. If we assume the change in Jx

is continuous, the solute flux density out of theprism in the same direction is [Jx + (¶Jx/¶x) Dx].The solute mass into the prism is JxDyDzDt andthat out of the prism is [Jx +(¶Jx/¶x)Dx]DyDzDt.The difference between the mass into and out ofthe prism is

or

Similar equations are derived for thedirections of the y and z axes. The sum of thedifferences in three directions equals the changeof the solute content of the prism. Provided thatthe total mass of solute associated with both theliquid and solid phases per unit volume of prismS(t) has a continuous derivative for t >0), weobtain the equation of continuity

If we further take into considerationirreversible sources or sinks (i occurring withinthe prism during the time period over which theequation applies, we have

In general, soil solutes exist in bothgaseous and aqueous phases as well as beingassociated with the solid organic and inorganic

phases of the soil. Here, we neglect the fact thatnon aqueous polar and non polar liquids can also

reside in soils and participate in the displacementprocess. We have also assumed that the solutes inthe soil solution are not volatile and have ignoredtheir content and transport in the gaseous phase.Hence, the total solute concentration S inEquation 75 is

where pr is the soil bulk density, Cs the soluteadsorbed or exchanged on the soil solids and Cthe solute in solution.

The solute flux density J in Equation 75relative to the prism AxAyAz is difficult to defineunambiguously owing to the fact that therepresentative elementary volumes of each of theterms in J and S are not necessarily equal norknown, particularly for structured field soils.Each of the directional components of J iscomprised of contributions of solute movementwithin the liquid phase as well as along particlesurfaces of the solid phase. We assume that solutemovement along soil particle surfaces is nil or canbe accounted for by functions relating theconcentration of solutes in solution to thatassociated with the solid phase in Equation 76.Hence, the solute flux density consists of twoterms, one describing the bulk transport of thesolute moving with the flowing soil solution andthe second describing the solute moving bymolecular diffusion and meandering convectivepaths within the soil solution. For the z-direction,we have

where qz is the Darcian soil water flux, DC andDm are the coefficients of convective dispersionand molecular diffusion in the soil solution,respectively. Equations for Jx and Jy are identicalto Equation 77 when z has been replaced by x andy, respectively. We continue the analysis here foronly the vertical soil profile direction.

Substituting Equations 76 and 77 intoEquation 75, we obtain for a solute of the soilsolution that does not volatilize into the soil air

The first term of Equation 78 describesthe rate at which a solute reacts or exchangeswith the soil solids. With Da = (Dc + Dm ),Equation 78 reduces to

The source-sink term fi in Equation 78or Equation 79 is often approximated by zero orfirst-order rate terms

where g and gs are rate constants for zero-orderdecay or production in the soil solution and solidphases, respectively, and m and ms are similarfirst-order rate constants for the two phases. Forradioactive decay, physicists may safely assumethat m and ms are identical as well as assumingthat both g and gs are nil. Microbiologists,considering organic and inorganic transformationsof soil solutes in relation to growth, maintenanceand waste metabolism of soil microbes as aMichaelis-Menten process, often simplify theirconsiderations to that of (i in Equation 80.McLaren (1970) provided incentives to study suchreactions as functions of both space and time insoil systems - a task not yet achieved by soilmicrobiologists, especially when the individualcharacteristics of each microbial species isquantified and not lumped together as aparameter of the entire microbial community.Agronomic or plant scientists consider (i as anirreversible sink and source of solutes takingplace in the vicinity of the rhizosphere ofcultivated or uncultivated plants as a function ofsoil depth and time as well as some empiricalfunction defining the root distribution.

For a solute that does not appreciablyreact with the soil particles, does not exist in thesoil air and does not appear or disappear in

sources or sinks, respectively, Equation 79reduces to

For steady state flow in a homogeneoussoil at constant water content, Equation 81reduces still further to

which has been extensively used to developempirical relations between the apparentdiffusion coefficient Da and the average porewater velocity v.

Numerical and analytical solutions forthe above equations subject to initial andboundary conditions are available and will bediscussed in the lectures of the course for selectedlaboratory and field experiments.

Theoretical Implications: The majority ofinorganic cations, anions and solutes in soilsolutions have diffusion coefficients in the orderof 10-5 cm2.s-1 while organic cations, anions andsolutes usually manifest much smaller values.These coefficients are moderately temperaturedependent and slightly concentration dependent.The importance of their different magnitudes isapparent only at relatively small pore watervelocities.

The existence of concentration gradientsof inorganic salts in the soil solution responsiblefor solute transfer by diffusion or as a result ofconvection guarantees that the displacing anddisplaced solutions do not generally haveidentical values of density or viscosity no matterhow close their values. In soils, it is notuncommon to experience solutions of unequaldensity and viscosity. During the extraction ofwater from soil profiles by plants or byevaporation at the soil surface, the density andviscosity of the soil solution increase continually.Conversely the infiltration of rain or manyirrigation waters causes the soil solution to bediluted. Fertilizers and other agrochemicals alsoalter these properties of the soil solution. Thedensity and viscosity of the soil solution alsodiffer from those of the bulk solution owing to the

interaction of water and the soil particle surfacesespecially in unsaturated soils or chose soilshaving large clay contents.

Although arid soils usually aredominated by constant charge colloids andtropical soils by chose of constant potential, allsoils are mixtures of both, and hence pH cannotbe ignored. For soils containing substantialamounts of clay having a variable surface charge,the pH will depend upon solute concentration. Forsuch soils differences in leaching characteristicsare a result of the concentration of the soilsolution rather than strictly being caused byhydrodynamic and geometric aspects of the flowregime.

The mixing and attenuation of a soluteby convection depend upon the pore sizedistribution and the number of bifurcationsexperienced by the soil solution as water flowsthrough its system of microscopic pores (recallFigure 27). The greater the total macroscopicdisplacement length, the greater will be theopportunity for both convective and diffusivemixing. As the displacement length increases,both the number of bifurcations in the poresystem and the time for molecular diffusionincrease. Hence the deeper the displacement, thegreater the spreading of the solute.

Soil and Water Management Implications:Although our understanding and theoreticaldescription of solute transport in soils remainincomplete, we have nevertheless sufficientknowledge to derive a few principles orguidelines for managing solute retention orleaching in the field. Whether solutes accumulateor leach depends primarily upon the processes bywhich they enter, react and leave the soil profilerelative to their association with water.Summarizing the more important points of thissection regarding the relative movements of soilwater and its dissolved constituents, we make thefollowing conclusions: a. As water moves moreslowly through a soil, there is a greateropportunity for more complete mixing andchemical reactions to take place within the entiremicroscopic pore structure owing to the relativeimportance of molecular diffusion compared withthat of convection, b. Microscopic pore watervelocity distributions manifest their greatestdivergence for water-saturated soil conditions.Hence, under water-saturated conditions, thegreatest proportion of water moving through the

soil matrix occurs within the largest poresequences, c. Under water-saturated soilconditions, when the average pore water velocityis large compared with transport by moleculardiffusion, the relative amount of solute beingdisplaced depends upon the solute concentrationof the invading water, d. The concept ofpreferential flow paths occurs at all degrees ofwater-unsaturation even though their existence isusually only demonstrated for macropores nearwater-saturation. At each progressively smallerwater content, the larger pore sequencesremaining full of water establish still another setof preferential flow paths, e. Any attempt tomeasure the solute concentration based onextraction methods carried out either in thelaboratory or the field will be dependent upon therate of extraction and the soil water contentduring the extraction process, f. Inasmuch asrainfall infiltration usually occurs at greater soilwater contents and greater average pore watervelocities than does evaporation at the soilsurface, the amount of solutes transported nearthe soil surface per unit water moving through thesoil surface is greater for evaporation than forinfiltration.

STATISTICAL CONCEPTS

Classical Statistical concepts: In classicalstatistics used commonly in the agriculturalsciences, only the magnitudes of the observationsare involved in the calculations while theircoordinate values of space (or time) are frequentlyneglected. For example, observations of soilnitrogen taken in differently treated plots areexpressed as

where Z is the ith observation of treatment -having an expected mean m, a treatment deviationa and an error e. Usually, the probabilitydistribution of the observations is assumed to beknown without adequate experimentalverification, and an analysis of varianceconducted to ascertain the magnitude of aperturbation caused by a particular treatment.

Spatially Related Observations: An observationof a soil or crop attribute can also be consideredin the form

where x is the spatial coordinate (in 1, 2 or 3space dimensions or that of time). The value of nacross the landscape need not be constant andobservations Z will reflect both local and regionalvariations as well as depending on sample size.This latter concept, being able to sample differentattributes of the landscape over different scales ofspace and time, provides a panacea of new fieldresearch opportunity for crop and soil scientists.There has been already a remarkable amount ofresearch in agriculture and agroecology using thisregionalized variable approach to analyzeanthropogenic activities or those occurringnaturally within various kinds of ecosystems.Excellent texts are available to explain theassumptions upon which the theories and methodsare based as well as the limitations of theirapplication to a particular situation (e.g., Clark,1979; Davis, 1973; Joumel and Huijbregts, 1978;Shumway, 1988; and Webster and Burgess,1983).

Autocorrelation: A concept fundamental togeostatistics and regionalized variable analysis isthat of spatial dependence of soil or cropattributes. Intuitively, we do not expect fieldobservations of soil properties to be necessarilyspatially independent. We would expectmeasurements made close together to yield nearlyequal values, and measurements made a smalldistance apart to yield values correlated with eachother. We would also expect a partiallyrepetitious behavior of soil observations as aresult of cyclic tillage traffic and croppingpatterns in cultivated fields, sequences of low andhigh topographical positions giving rise to cycliclocations of greater and lesser degrees ofleaching, and sequences of soil mapping units notrandomly located within a landscape owing to soilformation processes that are linked spatially tothe coordinates of the soil surface. We describeand illustrate below the concepts of spatialdependence and probability distributions ofobservations which are often confused andconfounded by soil and crop scientists.

A measure of the strength of the linearassociation between pairs of observations isuseful in defining the separation distance betweenobservations beyond which there is no correlationbetween pairs of values. This concept of spatial

dependence or autocorrelation is helpful toanswer the question, "How far apart shouldobservations be taken, how large an area isrepresented by a single observation, or what sizeof observation is most appropriate?" Similarquestions in time are also relevant. Consider a setof n observations O(x) taken along aunidirectional transect across a field at a spacingh (h = xi+1 - xi). Analogous to calculating thelinear regression coefficient r for a set ofmeasurements of crop yield as a function ofapplied nitrogen fertilizer, the spatial autocorrelation coefficient r is calculated from

where cov is covariance and var is variance.When observations O are taken 1 unit apart, r(1)is the value of the linear regression coefficient forh = 1 (lag 1) when values of O (x + 1) are plottedagainst values of O(x). In other words, nearestneighbors are plotted against each other.Similarly, when h = 2, r(2) is the value of thecoefficient when values of O are plotted againstother values observed a distance of 2 units away.

The functional relation r(h) has beenexpressed with several empirical formulae. Oneof the most commonly used expressions is

where l is selected in order that the sum of themeasured deviations of r from the aboveexpression is zero. Note that the value of l isequal to that distance h between measured valuesfor which their correlation coefficient is l/e. l iscalled the autocorrelation length or the scale ofobservation. A liberal interpretation of l is that itrepresents the radial distance within a fieldcharacterized by a single observation.

Figure 3la presents soil water content qmeasured in a field at l m intervals with aneutron moisture meter at the 50 cm soil depthwithin a 160 m long transect. Neglecting thelocations of the observations (n = 160), Figure31b is a histogram showing that the mean m andstandard deviation s of the 160 observations are

0.136 and 0.0162 cm3.cm-3, respectively. Theheight of each vertical bar represents the numberof observations found in a range AB of 0.01cm3.cm-3. The smooth curve shown in the figure isfor a normal probability density function equation

where N is the number of observations of 0within q ± 2-1Dq. Having concluded that theseobservations are normally distributed, it ispossible to ascertain the number of observationsrequired to estimate the mean soil water contentacross the transect within prescribed levels ofprobability. On the other hand, the above formulasays nothing about how far apart the observationsshould be taken. Figure 31c, the autocorrelogramof the 160 observations, shows that theautocorrelation length l is about 6 m. Sampling atintervals less than 6 m is somewhat unnecessarybecause the observations are related to each other.Sampling at intervals greater than 6 m does notallow meaningful interpolation betweenneighboring observations. It should be obviousthat the functional relation between r and hdepends upon the size of the sample, and that ingeneral' the greater the sample size, the greaterthe value of the autocorrelation length X. For anyparticular study, the investigator should considerthe minimum distance between samplinglocations (h = 1) in relation to the objectives ofthe experiment and the potential utility of thevalues of l for each kind of observation.

The utility of using both of the aboveconcepts in crop and soil science is great. Neitheris a substitute for the other. The approximatelynormal frequency distribution of 0 given in Figure31b does not quantify the variability of the soilwater content as regards the spatial arrangementof 0 but merely treats the values in terms of theirmagnitudes independent of their spatial position.The two concepts (frequency distribution andspatial variability) should not be confused. Usingthe first concept, it is possible to calculate theprobability of an observation occurring withinspecified confidence intervals. This calculation,however, provides identical information forexpected values for all locations. Differentexpected values for specific locations cannot beobtained. On the other hand, using the secondconcept, the autocorrelation function providesinformation about the separation distances withinwhich observed values are related to each other.The spatial structure of a set of observations doesnot provide information about the frequencydistribution of the observations. The fact that aset of observations manifests no spatial structuredoes not imply that the observations are normallydistributed, and vice versa. The soil water contentdata in Figure 3la constitute a set of observationsbeing normally distributed, yet those observationsseparated by less than 6 m are dependent on eachother as expressed by l in Figure 31c. Theconcept of randomness associated with the normaldistribution should not lead to the conclusion thatlocations for field observations of soil waterproperties should be selected randomly.

Crosscorrelation: The concept of spatialautocorrelation is easily extended to two kinds ofobservations (O1 and O2) to calculate thecrosscorrelation coefficient rc

The utility of spatial or temporal cross-correlation should be obvious - it allows theinvestigator to consider the spacing and size ofone set of observations with those of another. Forexample, at what distance should a tensiometerbe placed in relation to a neutron meter accesstube? Or, at what distance should a soil sample betaken in relation to the crown of a crop plant?

Spectral and Cospectral analyses: Above, theinterpretation of the correlation coefficients r andre were applicable to observations compared moreor less in near vicinity to each other (i.e., for hmuch, much less than the width of the field beingsampled). An opportunity to discern repetitiousirregularities or cyclic patterns in soil or plantcommunities across a field exists with spectralanalysis that utilizes the function r (h), or withcospectral analysis that utilized the function rc(h).A spectral analysis identifies periodicities and canbe calculated by

where f is the frequency equal to p-1 and p is thespatial period. In a similar manner, rc (h) is usedto partition the total covariance for two sets ofobservations across a field. A cospectral analysisis made by

Spectral and cospectral analyses arepotentially powerful tools for managing andincreasing our knowledge of land resources. Withthem, we can spatially link observations ofdifferent physical, chemical, and biologicalphenomena. We can identify the existence andpersistence of cyclic patterns across thelandscape. In some cases, the cyclic behavior ofsoil attributes may be of more or equalimportance than the average behavior. From aspectral analysis, some insights may be gainedrelative to the distances over which a meaningfulaverage should be calculated. And, with spectralanalyses, it is possible to filter out trends across afield to examine local variations more closely, orvice versa. The following example illustrates theadditional information gained from a spectralanalysis of a set of solute observations taken froma soil profile by Gelhar et al. (1980). The soil wassampled at 30 cm intervals to a depth of 22 m(See Figure 32a). The spectrum S(0 [equation

(89)] shown in Figure 32b has a peak at f - 0.37m-1, which indicates a period of 2.7 m. Hence, thefluctuations of solute concentrations in Figure 31aare indicative of annual cycles of leachingassociated with irrigation in the desertenvironment of New Mexico. Disregarding thespatial distribution of the concentration valueleads only to a standard deviation of about 0.5mg.l-1 - a classical parameter that adds little toour understanding of the leaching process.

State-Space Analysis: Thirty-seven years agoKalman (1960) introduced the use of a linearestimation theory with stochastic regressioncoefficients to filter noisy electrical data in orderto extract a clear signal or to predict ahead of thelast observation. Kalman filtering iscommonplace in electrical and mechanicalengineering. Shumway & Staffer (1982) extendedthe concept of Kalman filtering to smooth andforecast relatively short and non stationaryeconomic series observed in time. Using thisstate-space model, a series of observations can besmoothed, missing observations can be estimated,and values outside the domain of observation canbe predicted. For the purpose of prediction onecan calculate confidence intervals to demonstratethe reliability of the forecasting. Reliability of theprediction will depend on the adequacy of theavailable model and number of predictionsoutside the domain of observations. Morkoc et al.(1985) utilized the state-space model to analyzespatial variations of surface soil water content andsoil surface temperature as a dynamic bivariatesystem. Following the suggestion of Shumway &

where

Stoffer (1982), Morkoc et al. described thevariables of the observation model as

where Mi is a known q x p observation matrixwhich expresses the pattern and converts theunobserved stochastic vector Z, into q x lobserved series Y¡, and ni is the observation noiseor measurement error. Equation (13) written inmatrix form for the bivariate system of Morkoc etal. is

where Wi0 and Ti

0 are the ith soil water

content and the ith soil surface temperatureobservations, respectively. Equation (92)indicates that an observation of either soilvariable consists of two parts. The observationalnoise or measurement errors may be generated

either by errors in measuring Wi0 or Ti

0 or by

ignoring other variables which affect either soilwater content or soil surface temperature. Forexample, soil temperature is not only affected byincoming solar radiation but also by mineralogicalcomposition, microtopography, texture, structureand color of the soil surface. Hence, any omittedvariables such as texture or color enter into thestate-space system as observational noise ormeasurement error v,.

Values of soil water content and surfacetemperature were modeled as a first-ordermultivariate process of the form

where f is a p x p matrix of state-spacecoefficients. Many other functions of multivariateprocesses equally applicable to site-specific cropmanagement await future investigation. Thematrix form of the above equation is

Values of fij and the error terms of theabove equation can be calculated using recursiveprocedures (Shumway & Stoffer, 1982).

Details and the results of the state-spaceanalysis of water content and temperaturerelations of a soil surface (without vegetation andhaving been non-uniformly sprinkler irrigated)are available (Morkoc et al, 1985). More directlyrelevant to the topic of this workshop is theapplication of equations similar to (91) and (93)to improve our understanding and management ofbiological processes as they occur in the field andto prioritize those soil attributes that contributethe most to sustainable agriculture.

REFERENCES

CLARK, I. Practical Geostatistics. Applied SciencePublish, London. 1979.

DANCKWERTS P.V. Continuous flow systems.Chemical Engineering Science, v.2,p.l-13, 1953.

DAVIS, J. C. Statistics and Data Analysis in Geology.John Wiley & Sons, Inc., New York. 1973.

GELHAR, L. W.; WIERENGA, P.J.; DUFFY, C.J.;REHFELDT, K. R.; SENN, R. B.; SIMONETT, M;YEH, T.C.; GUTJAHR, A. L.; STRONG, W. R.;BUST AMANTE, A. Irrigation return flow studies atSan Acacia, New Mexico: Monitoring modeling andvariability. Technical Progress Report No. H-3, NewMexico Institute of Mining and Technology, Socorro.1980.

IWATA, S.; TABUCHI, T.; WARKENTIN, B.P. SoilWater Interactions. M. Dekker, New York andBasel. 1988.

JACKSON, RD. Water vapor diffusion in relatively drysoil I. Soil Science Society of AmericanProceedings, v.28, p.172-176,1964.

JOURNEL, AG.; HUIJBREGTS, C.H.J. MiningGeostatistics. Acad. Press. New York. 1978.

KALMAN, ILE. A new appro^h to linear filtering andpredicting problems. Trans. ASME of Basic Eng. 8:35-45. 1960.

KUTILEK M.; NIELSEN, D.R. Soil Hydrology. CatenaVerlag, Cremlingen-Destedt, Germany. 1994.

MCLAREN A.D. Temporal and vectorial reactions ofnitrogen in soil: A review. Canadian Journal of SoilScience, v.50, p.97-109, 1970.

MORKOC, F.; BIGGAR, J.W.; NIELSEN, D.R.;ROLSTON, D.E. Analysis of soil water content andtemperature using state-space approach. Soil ScienceSociety of America Journal, v.49, n.4, p.798-803,1985.

RICHARDS L.A. Physical condition of water in soil. In:C.A. Black (Ed): Methods of Soil Analysis, Part 1,Physical and Mineralogical Properties, AgronomyMonograph Series No. 9, ASA, Madison, p. 128-152.1965.

SHUMWAY, R.H. Applied Statistical Time SeriesAnalysis. Prentice-Hall, Englewood Cliffs, New Jersey,1988.

SHUMWAY, R.H.; STOPPER, D.S. An approach to timeseries smoothing and forecasting using the EMalgorithm. Journal of the Time Service Analisys,v.3:,p.253-264,1982.

SRINILTA S.A; NIELSEN, D.R.; KIRKHAM, D.Steady flow of water through a two-layer soil. WaterResources Research, v.5, p. 1053-1063,1969.

TAKAGI, S. Analysis of the vertical downward flow ofwater through a two-layered soil. Soil Science, v.90,p.98-103,1960.

WEBSTER, R.; BURGESS, T.M. Spatial variation in soiland the role of kriging. Agricultural WaterManagement, v.6, p. 111-122,1983.

WILLIS W.O. Evaporation from layered soils in thepresence of a water table. Soil ScLenceSocletyAmerican Proceedings, v.24, p.239-242,1960.

Recebido para publicação em 25.04.97Aceito para publicação em 15.05.97