Solute Dispersion in Unsaturated Heterogeneous Soil at Field Scale: II. Applications1

ESHEL BRESLER AND GEDEON DAGAN2

ABSTRACTThe model for nonadsorbed solute spread by vertical steady-

state flow of water in a heterogeneous field soil, developed inPart 1, is applied to a Panoche soil with properties which havebeen investigated previously. Average solute concentration pro-files as functions of depth and time are computed for varioussurface rates of infiltration. The average concentration overa layer extending from the soil surface to a given depth isalso computed. It is shown that the concentration profile ex-tends over an apparent mixing zone which is much larger thanthat predicted by a conventional diffusion-convection (disper-sion) equation in a homogeneous fictitious field, and that thelatter lacks practical significance for prediction of solute spreadin the field. The implications of the results for the leachingof saline field soil and crop yield estimates as they are influ-enced by rate of surface recharge and its uniformity are alsodiscussed.

Additional Index Words: spatial variability, unsaturated hy-draulic conductivity, salinity, leaching, randomness of hydrau-lic properties.

Bresler, E., and G. Dagan. 1979. Solute dispersion in unsaturatedsoil at field scale: II. Application. Soil Sci. Soc. Am. T. 43:467-472.

IN PART 1 OF THE PRESENT STUDY (Dagan and Bresler,1979) a statistical model for solute concentration dis-

tribution in a heterogeneous field was presented. Toarrive at a simple model the following assumptionswere adopted in Part 1 (will be referred to herein asPI): (i) the direction flow is vertical; (ii) the soilproperties relevant to solute transport do not changealong any vertical profile but vary considerably inthe horizontal plane; (iii) this variation of the proper-ties can be described in terms of a probability distri-bution at each point; and (iv) the statistical structureis homogeneous (stationary), i.e., the frequency func-tions do not depend on coordinates. In addition, sev-eral simplifying assumptions were forwarded in PIfor a particular, but representative case: (i) the hy-draulic conductivity-water content relationship K(B)has a simple analytical structure; (ii) the randomnessof K stems from the stochastic nature of the saturatedconductivity Ks which depends quadratically on a

1 Contribution from Agricultural Research Organization, TheVolcani Center, Bet Dagan, Israel. 1978 Series, No. 194-E. Re-ceived 18 July 1978. Approved 4 Jan. 1979.

"Soil Physicist, The Volcani Center and Professor of FluidMechanics, Tel Aviv Univ., and Hydrologist, The Volcani Cen-ter, A.R.O., Bet Dagan, Israel, respectively.

scaling parameter 8 which in turn has a lognormal dis-tribution (i.e. Y = In 8 is normal with mean my andvariance "y3); (iii) the flow is generated by steadyrecharge applied on the surface at a random rate Rwith a rectangular distribution (average R, bandwidth2,d,R); (iv) the flow is steady so that the infiltrationvelocity V and the soil water content 0 do not changewith depth and time; (v) pore-scale dispersion maybe neglected; and (vi) the dimensionless concentrationC is initially zero throughout the profile and is equalto unity at the soil surface (z — 0) for any infiltrationtime t.

Under these simplifying assumptions a piston flowconcentration profile is obtained at any point in thefield, which means that C = 1 for z Vt and C — 0for z > Vt, with V being the downwards pore-watervelocity, constant along vertical lines but varying inthe horizontal plane. Hence at a given time and ahorizontal plane at depth z, part of the area of thefields which has been swept already by the frontZf = Vt, is at a concentration C = 1 while in the re-maining part solute concentration is zero (see PI, Fig.3). Hence, the properties of the frequency distribu-tion of C are shown in PI to be exhausted by the sin-gle parameter C(z,t), the average concentration. InPI we have derived a closed form analytical solutionfor C as a function of the dimensionless variable £ —z6s/tKs* and five parameters r •=. R/KS*, SR = dR/R,mY, "\, /8 where Os is the value of water content, 9, atsaturation, Ks* is a pondered average of the hydraulicconductivity for the entire field (defined by Ks* =[f(Ks)*f(Ks)dKs]V) and p is a constant coefficient inthe K(0) relationship as defined by Eq. [8] in PI. Forany given set of field properties and recharge distribu-tion R, the average concentration C becomes a functionof the dimensionless f. This C (|) function has threedifferent expressions for three ranges of £ in the inter-val 0<£<oo (see PI, Eq. [26], [27], and [28]). Thisdivision to three parts results from the fact that part ofthe entire field surface is under ponded conditions.Ponding occurs, under the preceding simplifying as-sumptions, where ever the value of the recharge Rexceeds the value of the saturated hydraulic conduc-tivity, Ks. While the saturated value of the velocityV is equal to KS/9S and does not depend on R beneaththe ponded area, the flow in the rest of the field is un-der unsaturated conditions with V = R/Q.

The purpose of this second part of the study is toapply the model formulated in PI to a particular

468 SOIL SCI. SOC. AM. J., VOL. 43, 1979

field, with Panoche soil series and uniform profiles,studied extensively by Nielsen et al. (1973) and War-rick et al. (1977a). This specific field was chosen inorder to assess quantitatively the ideas of our modelpresented in Part 1. As a first step the C profileis computed for various values of r and SR. Subse-quently, the volume averaged C**, defined as thesolute concentration averaged over the field volumeconfined between the soil surface and a given depthz, is computed from the appropriate C (f) functions,given by either one of Eq. [26] through [30] of Part1. As anticipated, we will demonstrate that pore scalehydrodynamic dispersion has a negligible effect ascompared with field scale dispersion caused by fieldheterogeneity, and that convection by average flowvelocity coupled with heterogeneity is the main mech-anism which governs the solute spreading in the field.

AVERAGE CONCENTRATIONDISTRIBUTION C

For the purpose of applying our methods to a heterogeneousPanoche field soil we have adopted the values of the distribu-tion parameters mT and ay of Warrick et al. (1977a) for thenormal distribution of Y = In S as

m? — —0.616; aT — 1.16. [1]

In addition, we have adopted for the K(S) function (Eq. [8] inPI) a value of 1//3 = 7.2 (Bresler et al., 1978). With thesevalues of the parameters mT, a?, and /3 we are able to computethe average concentration distribution function C (f) for variouscombinations of the parameters r = R/K,* and ss = dB/R,characterizing the rate of recharge at the soil surface. A fewsets of curves are represented in Fig. la for sa = 0 and ss = 1for three values of r (r=0.2, r=0.6, and r—l). Note that thecases where SR — 0 represent deterministic recharge R = R,while SB — 1 is the largest coefficient of variation of the re-charge R. For a given r and an intermediate value of ss (i.e.,0<$R<1) the corresponding profile lies between the two ex-treme profiles demonstrated in Fig. 1.

Consider the simplest case of SB = 0, i.e., deterministic uni-form recharge on the soil surface. In this case the concentrationprofile for the zone extending from the soil surface to f=r(i.e., f ^ r) is described by Eq. [29] of PI regardless of r. Thisis demonstrated by the curves lying above the breaking pointsof Fig. la and better by the curves above the correspondingchange of profile-shape point of f/r = 1 in Fig. Ib. In thiszone which lies between f — 0 and f — r, solute transport is

.8 LO

2.5 L (b)

controlled by the saturated flow beneath the ponded area. Forthe zone in which f > r the C (£) profile is described by Eq.[30] of PI, which pertains here to the unsaturated flow. Hence,the three SK = 0 curves of Fig. la differ mainly in the locationof the breaking point of change of shape of the profile at f =r and C(f) profiles for other values of r can easily be obtainedby translation. For cases where SK > 0, which are demonstratedby the curves of Fig. 1 corresponding to SB = 1, the C(f) pro-files are calculated by Eq. [26] through [28] of PI (Dagan andBresler, 1978). Here the curves are smooth as C, at a givendepth, is influenced by front translation of solute concentrationprofile either in the saturated or in the unsaturated zone be-cause of the simultaneous but independent, variation of K,and R.

Figure 1 shows that for a given SB the solute spreading overthe entire field is larger for large values of r than for small rvalues. This is due to the fact that a larger portion of thefield is ponded for large r and then the saturated flow zoneplays a dominant role. In this zone the profile front velocityvaries over a wide range because of relatively high K , variation.In contrast, in the unsaturated part of the field the variationof V depends on 6 (Eq. [18] of PI) which vary at a lesser de-gree than K, because of the algebraic relationship in Eq. [8]of PI. It is emphasized that for small values of 1//3, or if ftwould have also been considered as a random variable, solutespread should become even larger than that illustrated in Fig. 1.

The same six combinations of r and ss as in Fig. la have beenadopted in Fig. Ib where C is represented as a function of f / r= z6,/tR rather than as function of f . As a result of thischange of variable the C profiles become closer for SK — 0 andthe point of slope breaking is plotted on the fixed value of£/r = 1.

Applications of the results presented in Fig. 1 to the fieldbecome straightforward. For given values of 6, and K,* theaverage concentration profile at various t can be depicted as afunction of z by substituting the appropriate values of z and tin f. For example, for Panoche soil, Warick et al. (1977b)give expectation values of "saturated" water content E(6,) andhydraulic conductivity E(K,) of 0.43 and 0.93 cm/hr, respec-tively. Since K,* is defined by

we have K,* = 0.93 exp(— 1.46) — 0.22 cm/hr. Thus, for irri-gation application with r = 1 (R = 0.25 E(K,) and SR — 1,50% of the area is leached to a depth of z = 45 cm after atime of t — 400 hours has elapsed from the beginning of thewater application (C = 0.5 leads to f — 0.225 on curve r z= 1,stt - 1 of Fig. la).

FIELD SCALE DISPERSION

When comparing any of the C (£) profiles (Fig. la) with theC (z,t) profiles which could have been obtained by assumingthat C (z,J) satisfied the simplest solution of the dispersionequation for a constant average field pore-water velocity andwater content (Eq. [3] in PI), one can immediately see thatthe shape of the C (£) profile (Fig. 1) is far from the sigmoidcurve given (for «»Z>/F2) by

C(z,t) - i/2[l _ erf [2]

Fig. 1—The average cincentration profile: (a) C (f) and (b) C(g/r) ,for six combinations of r — R/K,* and ss — ds/R inPanoche soil.

Thus, it would be very difficult to fit the C curves of Fig. 1,using Eq. [2] with the average field velocity, in order to esti-mate an effective dispersion coefficient D,t, which would incor-porate the heterogeneity effects. If this is somehow done thevalue of Der would have varied with time. Indeed, in the solu-tion of the dispersion equation for homogeneous column (e.g.Eq. [2]), the transition zone expands as square root of time(t^2) whereas in our solution for heterogeneous field the transi-tion profile expands as t.

To clarify this idea let us define the length of the transition

BRESLER & DAGAN: SOLUTE DISPERSION IN UNSATURATED HETEROGENEOUS SOIL AT FIELD SCALE: 11. 469

Table 1—Computed values of ponded area and effectivedispersity X^/for various combinations of r = R/KS*

and SR =

Case

123456

SR000111

r

0.20.61.00.20.61.0

Fraction ofponded area

0.4350.6220.7020.3980.5730.650

v*..0.1610.7511.3870.7351.9603.342

Xe^(cm) for*<>.. = 1 m

16.175.1

138.773.5

196.0334.2

zone L* as the difference between the depth for which C =0.1 and C = 0.9, which similarly to Eq. [4] in PI yields

L* - (K.*/«.)(&.i - &.„)« [3]

with £01 and £<,.» functions of r, dx, mr, and <rr. Note that forthe values of mr and <rr given by Eq. [1] the dependence offo.i and £0.9 of Fig. 1 is on r and dK solely. In the equivalentprofile derived for column dispersion (Eq. [2]) the length of thetransition zone is given by Eq. [4] of PI as L = (13.54 Dt)^.Taking formally L — L* and Def = D, we arrive from thedefinition of L and Eq. [3] at the expression for Del as

13.54 [4]

which would have rendered the two transition zones of equallengths. To arrive at some estimates for the effective dispersiv-ity \ef let the average velocity V be taken as the velocity atwhich the C = 0.5 plane (i.e., £0.B) moves in a fictitious homo-geneous field,

V = Wt [51and X,, be defined by

X,, = D.r/V. [6]Using the definition f0 „ = z0.50,//f,*J, and substituting Eq.[5] and [6] into Eq. [4] yields after rearrangement

nef (go.l — £o.e)= 13.54 £"„.„ [7]

Values of X,f calculated from Eq. [7] for the same six com-binations of ss and r as represented in Fig. 1, are summarizedin Table 1, which incorporates also the ponded area values.

The last column in Table 1 which gives values of Xe/ (in cm)for z0.B=l m is particularly instructive. The striking results arethat the equivalent field effective dispersivity \ef is much largerthan the pore-scale dispersivity measured in laboratory columns(X ~ 10~l cm) or even as determined by Biggar and Nielsen(1976) under field conditions (X = 3 cm) by performing statis-tical analysis. For the values of mT, ar, da and r on which Ta-ble 1 is based, the pore scale dispersion plays a negligibly smallrole. Hence, the solute spreading at the field scale is dominatedby field heterogeneity. Obviously, if the values of ar weresmaller the spreading would have been smaller and the twomechanisms could have comparable effect. On the other hand,effective field dispersion even larger than that of Table 1 oc-curs when the value of I//? becomes smaller and/or when oyis larger.

An additional parameter of interest is the variance of theconcentration crc* as being defined in PI. For the simplifiedcase of our analysis the standard deviation is given by at, =[C (l_-C)]vil. This equation makes it possible to compute con-veniently the ff0(f) function for each 'C(f) profile present in Fig.1. Note that <rc = 0 at the surface where C — 1 by definitionand it also vanishes at large value of z where C—> 0. Maximumvalue of a, is at C = 0.5 where a, — [0.5(1—0.5)P = 0.5. Thismeans that the standard deviation is at a maximum at thedepth ZD.S where half of the field is leached by the water appliedto the surface.

Application of the present results to predict yield response tosalinity become obvious if one assumes the threshold concen-tration concept, or knows the crop response to salinity functionin the root zone. The results then permit one to compute theprobability of the soil salinity (as represented by C) to be within

(a)Fig. 2—Schematic representation of: (a) probability distribu-

tion, and (b) frequency distribution of C*.

a given range for a certain period of time and at a given depthin the root zone. A combination of the computation and theknowledge on crop response makes it possible to predict thefrequency distribution of the yield in the field. As the rootszone extends over a finite depth and since plants respond tothe average salinity over the main root zone, one is interestedin practice in the average concentration over a depth spanrather than at a fixed depth z. Such an average is evaluated inthe following section.

AVERAGED PROFILE CONCENTRATION

Let us define first C* (zi, z2, t; a\, . . . , UN) as theaverage solute concentration over the vertical intervalZi<z<z2 at a given point in the horizontal plane

C* - 'C(z,t; aN )dz [8]

and the volume averaged concentration C** as the av-erage of C* over the entire field (see Eq. [12] in PI)

C# # /_ —— i\ _____ I /^# f /*, - J. /"1#\yJ/~'# _____ fOI

\~\) *"2> 1) —- I ^ /v^lj ^2j f j *-* /"O z ["J

r r . . . r c*(Zu ^ « « , . . . , «,>/(«,Note that ai,..., an are the random parameters whichcharacterize the flow. For the simple case of pistonflow with C = H(Vt—z) it is immediately seen thatby Eq. [8] the profile averaged concentration C* isgiven by

C* = 0Vt ~Z

Z2 —C* = 1

(for

,'for [10]

(for Fi>z2).Hence, the probability distribution function of C*has the shape of Fig. 2. For the point a of Fig. 2a wehave from [10]

P(C*=e) = P(Vt=zi) = P(V= y) [11]

where £->0 and P is the probability.Recalling the definition of C (see Eq. [6] of PI) givesprecisely

P(C*=e) = 1—C(fi) [12]where £t — ZiOs/tKs*. The simple physical interpreta-tion, which is based on Fig. 3 of Part 1 is that C*is equal to zero for the area on the upper plane z =Zi which has not yet been reached by the piston front,

470 SOIL SCI. SOC. AM. J.j VOL. 43, 1979

.4 .6 .8 1.00 .2 .4 .6 .8 l£»AVERAGE PROFILE CONCENTRATION C**

Fig. 3—Probability distribution of average vertical profile concentration P(C**) as computed from Eq. [17] taking 6, = 0.43, K,*= 0.22 cm hr-1 and z, — 0.4m (i.e., f , — 80/t) for (a) ss — 0, r — 1 and (b) SK — 1, r — 1. The numbers labelling the lines arevarious tunes in hours.

leading from the definition of C* to the above Eq.[12]. Similarly for the point b of Fig. 2a we have

P(C* = l-£) = l-C(ft) [13]where this refers to the area on the lower plane z =z2 which has not yet been swept by the front (Fig. 3,PI). Finally, the curve a & on Fig. 2a is given withthe aid of Eq. [10] byP(C*) =

Hence,

(z2 - 1)] = P[V =Zi+C*(z2—z [14]

-ft)]- [15]The frequency function of C* which has been ob-tained by differentiation of P(C*), is representedschematically in Fig. 2b. This function has two spikesof areas 1 — C (ft) and C (ft) at C* = 0 and C* =1, respectively, and a continuous portion of equation

/(C*) = - (ft - ft) [16]

in between the two spikes.Generally, the main soil root zone of interest, from

the agricultural point of view, extends from the soilsurface to a given depth in the soil profile. Thus,Eq. [10] through [16] should be considered for theparticular case of z\ = 0 (ft = 0) for which the spikeat C* — 0 in Fig. 2b vanishes and its continuous por-tion becomes

P(C*) = l_C(C*ft)(forO< C*<1 —E)

/(C*) = -&(dC/d|).

[17]

[18]To compute the volume averaged concentration C**in this case we use integration by parts to obtain

[19]= C*P(C*) |1 _ f1 P(C*)dC*.

Substituting P(C*) of Eq. [17] into [19] yields

C**(0,ft) = J - q f t d f t [20]

Thus, the probability distribution P(C*) is a signi-ficant function one can use to estimate the crop yieldsand other entities which depend on the concentrationin the upper soil layer. Hence, if one is interestedin the relative area of the field for which the concen-tration, averaged over the profile extending from thesurface to a depth z2, is smaller than a critical value ofC* at a time t from the beginning of the leaching pro-cess he would have to compute ft = z28s/tKs*, to pickup the value of C(^) given by Eq. [26] through [28]of PI for the argument 77 = C*ft and to substitute itinto [17].

Such computations of Eq. [17] for the same Panochesoil discussed in the preceding section have been car-ried out by making use of the curves given by Fig. la.For instance, taking 0S = 0.43, Ks* — 0.22 cm-hour"1

(Warrick et al., 1977b) and z2 = 0.4m resulted in avalue of ft = 80/t. By taking a few values of t —20, 30, 40, 50, 60, 70, 80, 90, 100 hour, a set of curvesrepresenting P(C*) for each value of t is obtained.The graphs for r = l(R = 0.25E(Ks)sR = 0 arerepresented in Fig. 3a whereas those pertinent to r= 1, SR = 1 are depicted in Fig. 3b. For the purpose ofcomparison we have also represented P(C*) for valuesof z2 which are twice as much as the previous value,i.e., z2 = 0.8 m for the same combination of the para-meters r and SR. The results are represented in Fig.4 (a and b) for values t = 20, 40, 60, 80, 100, 120, 140and 160 hours.

For the given values of r and SR, of the soil para-meters 0S and Ks*, and depths z2, Fig. 3 and 4 containthe entire relevant information. For example, a valueof C* = 0.9 means leaching of 90% of the salt con-tained initially in the profile, or conversely, soil salin-ity increase to 90% of the inlet water salinity. InFig. 3 a (which corresponds to r = 1, SR = 0 and z2 =0.4m) it is seen that only when 40 hours have elapsedfrom the beginning of the water application there issome area for which C** > 0.9 and this area increasesto a value of 1 — P (0.9) = 0.34 after 100 hours ofcontinuous application of water to the soil surface.Thus, after 100 hours 34% of the field upper layer

BRESLER & DAGAN: SOLUTE DISPERSION IN UNSATURATED HETEROGENEOUS SOIL AT FIELD SCALE: n. 471

.4 .6 .8 LOO .2 .4AVERAGE PROFILE CONCENTRATION

.6 .8 1.0

Fig. 4—Probability distribution of the profile averaged concentration P(C**) for % = 0.8 m (i.e., |2=160/t). Values of 6,, Kt*, thenumber labelling the lines, (a) and (b) are as in Fig. 3.

has been leached, for all practical purposes, by theirrigation water. If for the same field we examine thecase with SR = I rather than SR = 0, other factorsequal, we see in Fig. 3b that leaching starts much ear-lier, but after the same hours of water application only24% of the area is practically leached.

In Fig. 4 in which the parameters are the same asthose in Fig. 3 but with a thicker layer (z2 = 0.8mrather than 0.4m). Only minute portions (5% for SR= 0 and 10% for SR = 1) of the field have beenleached and significant values could be achieved onlyfor larger periods. In Fig. 4a (SR = 0) the time atwhich in some area C** > 0.9 is 80 hours and thisarea increases to a value of 1 — P(0.9) = 0.31 after160 hours of continuous irrigation. For the same casebut with SR = 1 rather than SR = 0, again leachingstarts earlier but after the same 160 hours of continuouswater application only about 20% of the area of theentire field is leached (Fig. 4b). Note that very lowwater application rate (about 2 mm hour"1) was as-sumed in these examples.

As one may expect, a uniform distributed rechargeis ultimately, at large values of time of steady waterapplication, more effective in leaching saline soil bynonsaline water than a nonuniformly distributed irri-gation, provided that all other conditions are equal.It should be pointed out here that for a fictious homo-geneous soil with saturated hydraulic conductivityKs* and for value of r = 1, which means R = 0.25E (KS),/KS* a complete piston flow leaching would beattained after 80 hours of irrigation in the case rep-resented by Fig. 3 or after 160 hours in the case ofFig. 4. When accounting additionally for hydrody-namic dispersion very little change in this piston flowwill occur. The data of Fig. 3 and 4 suggest that sucha hydrodynamic dispersion model for homogeneoussoil grossly overestimates the leaching effectiveness offield soils.

One can derive additional related quantities of in-terest by integrating the probability distribution P(C*) in time or over C*. However, these computationsare not carried out here. The main result of thissection is that if the relationships between yields andsoil salinity are known and a solution of the flowequations in a homogeneous column can predict thesoil salinity distribution and therefore the crop yield,then a statistical analysis of the entire field would

predict the frequency distribution of the concentra-tion in the root zone and therefore the probabilitydistribution of the yield.

ANALOGY WITH TAYLOR'S THEORY OFDISPERSION IN FLOW THROUGH A TUBE

The relationship between the solute spreading proc-ess in the upper soil layer due to heterogeneity andthe one described by the conventional diffusion con-vection (dispersion) equation can be better under-stood by referring to dispersion of solute in solventflowing slowly through a tube. Taylor (1953) has in-vestigated the spread of a tracer injected continuouslyin the cross-section at the beginning of a tube in whichthe fluid flows steadily. The solute transport is gov-erned by the convection-diffusion equation, the ve-locity having a parabolic profile in the cross-sectionfor laminar flow. With C defined as average concen-tration over the cross-section of the tube, Taylor(1953) showed that convection tends to cause a largespread of C along the tube length (x) as the velocityis zero near the wall and it has a maximum at thecenter. In contrast, molecular diffusion manifests alateral mixing which tends to wipe out the largelateral gradients in the cross-section caused by convec-tion. The diffusion time, T<ut, defined as the time re-quired to attain mixing over the cross-section is ofthe order ps/D0, where p is the radius of the tube andDO is the coefficient of molecular diffusion. Twodifferent regimes, prevailing in two different rangesof time elapsed from, beginning of injection, are re-vealed by Taylor's analysis. At values of time smallerthan Tdif (i.e. t < Tlm) the solute spread process isdominated by convection. It can be readily shownthat in this incipient stage C(x,t) does not satisfy adiffusion-type equation and the transition zone in-creases as t. For large values of t, i.e. for i»Tdif, dif-fusion ensures effective cross-section mixing, and C isgoverned by the one-dimensional convection — disper-sion — equation in which the velocity v averaged overthe cross-section and a longitudinal dispersion coeffi-cient, derived by Taylor, are substituted. The disper-sivity is then related to the radius of the tube andC(x) profile has the usual sigmoid shape. The tworegimes can also be analyzed in terms of distancesXo.s =r vt covered by an observer moving with C = 0.5.

472 SOIL SCI. SOC. AM. J., VOL. 43, 1979

Indeed, the convection (without dispersion) dominatedprocess is valid for x0.5/v « Tdif, i.e. for XO.S/P« vp/D0, whereas the dispersion type process pre-vails for XO.S/P » VP/DQ (note that vp/D0 is thePeclet number).

Returning again to the case of solute transport inthe upper soil layer, we can distinguish between twosimilar regimes. When a tracer is applied at the sur-face, at the beginning the solute spread is governedby vertical convection. For a heterogeneous fieldthis means that a transition zone develops due to varia-tions of the vertical velocity throughout the field. Thisis precisely the case analyzed in the present paper,and, similarly to case of flow through tubes, G doesnot satisfy a dispersion type equation and the transi-tion length L* grows as t.

The second regime, of convection-dispersion type,could be reached in the soil if the depth is sufficientlylarge to ensure effective lateral mixing by lateral porescale dispersion. To obtain an estimate of the re-quired depth we can use a criterion similar to thatof tube flow but with D0 replaced by lateral pore-scaledispersion coefficient Diat, and with p replaced by themicroscale of the field heterogeneity. The latter isroughly defined as a horizontal length scale overwhich the hydraulic properties of the field, say Ks,do not change appreciably. A recent analysis by Russo(1978, Ph.D. thesis, under preparation) has shown thatfield scale p is of the order of a few meters. It may beconcluded therefore that with average field verticalvelocity of order 1 cm/hour and lateral dispersityAiat of order 1 cm (as suggested by Biggar and Nielsen,1976), the depth ZO.S/P is of order p/\iat» i-e., a fewhundreds. Consequently, only for depth in the soilof order of tens of hundreds of meters the solute spreadcould be described by the usual convection-dispersionequation leading to Eq. [2] in which the dispersivitycoefficient A. is very large (of order of p). Obviously,this is not the case for the upper soil layer and theanalogy with Taylor's dispersion in tube-flow castsadditional light on the inadequacy of the usual columndispersion equation to describe the case of heterogene-ous fields.

SUMMARY AND CONCLUSIONSIn the present study (Parts I and II) we have sug-

gested a conceptual model of solute transport in un-saturated soil for heterogeneous fields, and appliedit to compute concentration distribution under sim-ple flow conditions. The basic idea is that, unlikelaboratory columns experiments or hypothetical homo-

geneous fields, the properties of a heterogeneous fieldvary at random. Consequently, solute concentrationshould be regarded as a random variable which canbe defined in statistical terms. The frequency distri-bution of solute concentration has been computed forvertical steady flow caused by recharge on the soilsurface for a field in which the hydraulic conductivityand the recharge magnitude vary at random in thehorizontal plane. We have shown that for the Panochesoil the solute spread due to field heterogeneity ismuch larger than the spread caused by the conven-tional pore-scale hydrodynamic dispersion so that thelatter can safely be neglected. Therefore, the porescale dispersion lacks a practical significance for hetero-geneous field, where convection coupled with hetero-geneity is the main mechanism governing solute spreadin the field. Furthermore, the average concentrationprofile has a shape and evolves with time in a man-ner, which differs completely from the erfc profile pre-dicted by the usual diffusion (dispersion)-convectionequation.

This study serves as a first step only towards a com-prehensive investigation of solute transport processesin agricultural fields. Such an investigation shouldconsider the variability of parameters other than hy-draulic conductivity, flow regimes other than steadyuniform flow (e.g., redistribution) and more complexinitial and boundary conditions.

ACKNOWLEDGEMENTThis research was supported by a grant from the United

States — Israel Binational Science Foundation (BSF), Jerusalem,Israel.