DIVISION S-l—SOIL PHYSICS

Approximate Soil Water Movement by Kinematic Characteristics1

ROGER E. SMITH2

ABSTRACTUsing the Richard's equation in the Fokker-Planck nonlinear diffu-

sion form, unsaturated soil water flow may be treated as a diffusion-convection wave process. If dO/az is assumed a function of 9 alone, theunsaturated flow equation may be solved by the method of character-istics, and when dd/dz becomes sufficiently small, the Peclet number isassumed large enough to treat unsaturated flow kinematically. Changesin 0 with depth in the soil profile are treated as waves, moving downward.Advancing and receding "waves" are treated differently in the approx-imate analytical technique described here, with advancing wetting frontsdescribed by kinematic "shocks." The method is compared to the com-plete solution to Richard's equation for a complex rain pattern and foundto predict well the location of deeper moving fronts and also general 8patterns. The kinematic method is also shown to apply to root waterextraction zones and to layered soil situations.

Additional Index Words: unsaturated flow, hydrology, infiltration,drainage.

Smith, R.E. 1983. Approximate soil water movement by kinematiccharacteristics. Soil Sci. Soc. Am. J. 47:3-8.

WATER moves through soil in response to two forces,capillary potential gradients and gravity. Most

mathematical treatment of unsaturated soil water flowto date has focused on the description of flow induced bycapillary potential gradients. This is reasonable becausewhen the upper boundary is suddenly saturated or a highflux is imposed, flow into a dry soil is dominated by cap-illary flow dynamics. For example, much of the famouswork of Philip (1957a; 1957b) concentrates on the prop-erties of the soil diffusivity, D, which varies with watercontent, 6, and he makes use of analogy to the body ofmathematics of heat flow equations, of which unsatu-rated flow is a nonlinear variant. Philip treated the ad-dition of gravity-induced flow by successive perturbationto the capillary potential induced flow (horizontal) caseto produce a series solution. Much of the work of Par-lange (1970, for example) explores the integral propertiesof the capillary relations of the media, such as D(6).

There are, however, several cases of interest in verticalporous media flow where gravity dominates the unsatu-rated movement of soil water. One of these cases is thevertical movement of water in relatively porous soils whenrainfall or surface fluxes are typically on the order of orless than the soil saturated hydraulic conductivity, Ks(6).This situation occurs in large areas of the world withrelatively porous soils and active recharge of a deepersaturated zone. Moreover, even in cases where flow nearthe surface may be dominated by capillary forces duringinfiltration, water percolating deeper into the soil profilemay be well described by gravity-induced flow.

The unsaturated water flow equation is treated as a

wave equation composed of diffusive and gravitationalcomponents. The 6 profile is considered to be a series ofwaves advancing down through the soil. A sample com-parison is made of the relative importance of the diffusiveand gravitational components. In cases such as those listedabove, the method outlined here provides a simple ana-lytical solution for following unsaturated water profiletranslation. When the assumptions are appropriate ar-bitrary temporal patterns of flux at the surface will pro-duce spatial patterns, or waves, of 6 moving through theprofile, whose movement through the soil may be de-scribed kinematically by the method of characteristics.The waves of 6 are kinematic in the sense that they aredescribed by an expression for velocity as a function of6, plus a differential equation of mass continuity. Theyare thus direct analogies of kinematic surface water waves(Lighthill and Whitham, 1955).

Gravity wave mechanisms alone can account for a cer-tain amount of profile attenuation, can fairly accuratelypredict wetting front movement, and can treat cases in-volving root extraction fluxes and layered soil situations.The receding 6 profile during uniform drainage was ap-proximated by Sisson et al. (1980), using the same char-acteristic assumptions used here. In this study the methodis applied to the general 6 profile translation.

MATHEMATICAL DEVELOPMENTDescription of unsaturated vertical flow commonly neglects

the flow of air and combines the differential conservation ofmass equation,

dt dz 'with flux q, described by Darcy's law,

< ? = -K(8) \-£~

to obtain Richards' equation (Richards, 1931):

dt dz dz dz

[1]

[2]

[3]

1 Contribution from the ARS-USDA. Received 27 Apr. 1982. Ap-proved 30 Sept. 1982.2 Research Hydraulic Engineer, ARS, Fort Collins, Colo.

where 8 = water content, z = depth, downward from the sur-face, L, K(8) = hydraulic conductivity, LT1"1, t = time, T,and \l/ = soil water capillary potential, L,

For analogy to a diffusion-convection equation, Eq. [3] isoften transformed by denning "diffusivity," D, as

[4]

[5]

[6]

D(d) = K(6) -±,

to obtain, with Eq. [2],

q = -D(6)—+ K(8) ,

anddz

SOIL SCI. SOC. AM. J., VOL. 47, 1983

For a kinematic approach, one may return to Eq. [1], and notethat if q is assumed a function of 0 alone, Eq. [1] or Eq. [6]can be written:

- + _ _ =

dt de dz [7]

Equation [7] may be considered as a wave equation and canbe solved by the method of characteristics (Abbott, 1966). InEq. [7] it is not necessarily assumed that diffusivity is zero, northat diffusive flux is negligible. With q assumed to be a functiononly of 0, Eq. [5] implies that dO/dz is closely a function of 8,and thus that 0-profile shape is well preserved through trans-lation. From Eq. [5], net flux is K(6) - D dd/dz.

In general, the movement of a location on the 6 profile of agiven 6 has a characteristic velocity, vc, of

da[8]

The characteristic solution of Eq. [7] amounts to translation ofeach value of 6 according to its velocity vc from Eq. [8]. Sincesoils generally exhibit a rapid increase of q with 8, 8 profilesevolve until at some time a doubled-value 6(z) solution occursfor regions of 6 decreasing with z. This produces what are calledshock fronts, described by Colbeck (1972) in relation to watermovement through snow, and by Kibler and Woolhiser (1972)for shallow surface water flow.

The kinematic characteristic method described herein makesapproximations which allow physically realistic treatment ofshock (advance) waves in soil water flow, and approximatesmotion of waves of opposite sense (8 increasing with z) usingEq. [7], and neglecting D 36/dz. Only the following flux-typeboundary condition will be considered:

q(z = 0) = q0(t).At the longer times more appropriate to this method, gravity-

dominated flow would make little distinction between thisboundary condition and a specified 80, since q0 = K(60). Theanalytic method described below does not require simple func-tional expressions for the soil relations between \j/, K, and 6. Ifused in tabular form, such relations must be continuous andhave a continuous dK/dd. A functional differentiable relationof K and 8 is enormously useful in applying the expressionspresented below. Figure la and Ib and the Appendix presentthe relations chosen for use in the examples herein.

Kinematic Soil Wetting WavesIn soils, where q increases rapidly with 6, steep changes of 8

with z are characteristic at wetting fronts, and such fronts areclosely approximated as kinematic shocks. Consider a region of

soil in which 6 decreases with depth z (positive downward) from0! to 82, 6\ > 62. Assume for simplicity that 8 = 0] for z < Z]and 8 = 02 for z > z2. The change in flux, q,, between 0, and82 is

9,1,2 = «(«,) - q(02) , [9]and the 8 profile between z, and z2 comprises a wave whichmoves with velocity vs(0,, 02) equal to

v, (0,, 82) = - q(82)\/(6l - 62) . [10]Referring to Eq. [5], diffusion causes a soil wetting front profileto lose steepness and thereby decrease I d0/dz I, while the mon-otonic K(6) causes I d0/dz I to increase with time. Both D(6) andgravitational flux K(6) increase with 0, so that a stable wavefront will evolve in which a balance in the two terms of Eq. [5]exists in their effect on dd/dz. Such a balance, which preservesa nearly constant total vc, is inherent in the profile at infinitydiscussed by Philip (1957b). For a leading wave generated bya flux qu with 0 increasing from 0,- to 8U across the wave, thenet wave flux is qu — qs, and from Eq. [10] the shock wavevelocity vc is constant at

0. - o,'From this and Eq. [5] and Eq. [8] we have

which may be integrated in two steps to obtain

z(0) _ P"Jet

D(d) d6K(8) - qu + q, + Zc, [11]

which is an integrated form of Philip's (1957b) Eq. [36], the"profile at infinity." Zc is a constant of integration. Kinematictreatment of this advancing wave assumes this balance discussedabove to have been approximately reached such that d0/dz isnot varying at any 6 with time. Equation [11] is the asymptoticcase in which Eq. [7] is exact, and describes a purely translatingwave, fully including diffusive flux.

Referring again to a simple advance case, if d0/dz is negligibleat locations z\ and z2 (one is a local maxima and the other isa local minima), q(8) is simply K(8) at z, and z2 and the motionof the advancing wave is treated as that of a shock with Eq.[10] becoming

[12](0, -

The opposite case, 0 increasing with z, is here referred to as

0.0210'

,(/, cm Water Content (

Fig. 1—Soil capillary characteristics. 9, is normalized water content, as a function of capillary tension, ,̂ in the left graph. The parameters shownrefer to the algebraic curve described in the appendix. The right graph illustrates the relation of hydraulic conductivity to water content, and itsrelation to kinematic soil water flow properties.

SMITH: APPROXIMATE SOIL WATER MOVEMENT BY KINEMATIC CHARACTERISTICS

a "trailing wave" and has quite different properties from theadvancing wave. Here both diffusive and gravitational forcestend to cause elongation of the trailing profile for monotonic Dand K functions, no shock forms, and vc(8) is described by Eq.[7] for all 0, < 8 < 82. vc(8) is also assumed monotonic. Thekinematic approach taken here assumes that since 68/dz becomessmall in these trailing wave regions 6{ < 8 < 82, one can saythat approximately

ve(«) = 4jL. [13]

Consider a simple pulse wave in which, over a region za to zb(zb > za), 8 increases from 8l to a maximum of 82 at someintermediate point z2 (?b > ^2 > zfl), and returns to 6{ at zb.It is simple to show that continuity is inherently preserved ifq, is described by Eq. [9], and the trailing region described byEq. [13], by calculating the net flux in the trailing wave fromza (6 = 0]) to z(62) = z2 (which must equal qs)\ using 5 as thevariable of integration:

za —

<7.,2 ="2 dK-ds = K(82) - [14]

EVALUATION OF THE KINEMATICAPPROXIMATION

Evolution of a Simple Pulse Wave FrontConsider the case of a flux q into a uniform soil with

initial condition 6 — 0,- and define dimensionless advancewave or shock velocity as

where Ks = Kty — 0). Assume the soil properties aredescribed by Fig. la and Ib, with K, = 0.2 cm/h. Theevolving profile may be obtained by a careful, fine-meshfinite difference solution to Eq. [3] (Smith and Parlange,1978). The resulting 0 profiles after 4 and 20 h are shownin Fig. 2 in terms of 8e and z', where z' = z — zn, with

-50

-40

-30

E -20

- 1 0

10

20*

30

r. = 0.5

,1*5 hrs

1.00.2 0.4 0.6 0.88g , Scaled Water Content

Fig. 2—Vertical flow profiles for dimensionless surface flux of 0.5 at 5,20, and oo h. Shorter time profiles end at the surface water contentdeveloped at that time, which slowly approaches the value for theprofile at t = infinity.

[15]

and

where 0r = residual soil water content, 0S = saturatedwater content, and 0U is obtained by using q for K in theinverse K(6) soil relation, Fig. Ib. By simple continuity,the kinematic shock assumption would make z' = 0 theshock or advancing wave location for each of the profilesin this figure. Also shown is the profile of infinity for thesame soil and surface flux, calculated from Eq. [11].

Figure 3 examines the flux components at t = 20 h inmore detail. The scaled water content profile is shown onthe left, and diffusivity, diffusive, and total velocity pro-files are plotted at the right. The total velocity increasesslightly with depth down to approximately z = 8 cm,after which it decreases at an increasing rate. The resultsshown here describe a profile whose shape is evolvingslowly toward the profile of infinity, as indicated in Fig.2. Diffusivity is seen to decrease rather steadily with depththrough the wetter (upper) part of the profile. Diffusivevelocity increases rather steadily from some 42% of thetotal velocity at z = 0 to nearly 100% at the bottom ofthe profile.

Since the upper boundary condition is employed byusing 0(q) in Eq. [12] or Eq. [13], and does not simulatean evolution of 0U with time, mass balance will insurethat wetting front position is underpredicted by themethod here at shorter times. This error in 6U may bemore significant than error in assumption of constant Vjacross the wetting front. At very large times, dO/dz l z_0approaches zero, the surface diffusive flux goes to zero,and wave travel is kinematic. At short times, dd/dz l z_0is large and diffusive flux dominates all across the profile.A measure of the relative accuracy of kinematic predic-tion of mean front location by Eq. [7] and Eq. [13] maybe obtained by using a porous media Peclet number, Pn,using concepts from hydrothermodynamics, defined forthis purpose as

p s" [16]Dd6/dz '

which is the ratio of convective to diffusive flux. SmallPeclet numbers indicate predominately diffusion flux and

8t , Scaled Water Content0.2 0.8 0

V e l o c i t y , c m / hr0.2 0.3 0.4 0.6

0 1 2 3 4 5 6 7 8 9 10 II 12 13Di f fus iv i ty , cm2/ hr

Fig. 3—Variations along the wetting front of flow properties for q, =0.5, at 20 h. The total velocity is relatively constant along the front,and diffusivity varies almost linearly.

SOIL SCI. SOC. AM. J., VOL. 47, 1983

conversely, large Pn implies predominately gravitationalflux. In any case, the mean front location, as illustratedin Fig. 2, is predicted within approximately 1 cm forerrors in predicting 0U on the order of 20%.

Evolution of a Simple Step Trailing WaveConsider a region of soil in which 0 increases with

depth from 03 to 82, d6/dz > 0, 02 > 03. From the 0(z)profile at t = 0, for monotonically increasing K(0), thekinematic motion of the profile causes elongation (Fig.Ib) as discussed above. For simplicity, assume that at t= 0 surface flux q0 dropped from q2 to q3 so that ac-cording to the K(9) relation for the kinematic approxi-mation, 0 dropped suddenly from 02 to 03. On this trailingwave thus generated, each 0 moves downward with ve-locity vc = dK/d0, which is indicated as the tangent tothe 9 — K curve in Fig. Ib, point 3. Since at t = 0, z= 0 for all 03 < 0 < 02, for this example, Eq. [8] maybe integrated first to yield

z = vc(0)t ,and differentiated with respect to 0 to obtain

d0 vc(0)[17]

From this relation and Eq. [13] one may describe theevolution of the trailing wave profile with time and dis-tance. As indicated above, Sisson et al. (1980) employedthis method for simple drainage of a saturated profile.

The analytic procedure is illustrated schematically inFig. 4. The kinematic shock wave front between 0{ and02, from Eq. [12], advances with a velocity given by thechord slope between points 1 and 2 on the K(0) relation(Fig. Ib). Since the characteristic velocity of each 0 onthe trailing wave moves as the tangent to the curve K(0),or dk/d0, a simple square wave generated at the surface,as shown in Fig. 4, will move and evolve until the point02 from the trailing wave peak intercepts point 02 on theshock wave front. In Fig. 4, after some time the originallysquare pulse has evolved to shape c-d-b, the originalwave front having progressed from a to b. At some furthertime, tt, points d and b will coincide. For t > tt, the shock

Water Content 8

i(b) Leading Wave , t = t.

Fig. 4—Schematic illustration of kinematic advance of a wetting profile,showing the advance of the wave position from time t, to time r2.

wave height will decay as points 0 < 02 on the trailingwave continue to intercept the front, and the decayingheight of the front will, while preserving the enclosedvolume of water, cause a slow deceleration. Routing ofthe decaying wave is a problem in preserving continuitywhile preserving the kinematic motion of all points onthe profile.

The decay of the wave peak for t > t, is a calculablefunction of the trailing wave profile. Consider a familyof simple pulse waves generated at the surface, all havingthe same volume / = qtq, and starting and ending at 0].The kinematic waves generated will have a range of"heights" 0, given by the K(0) relation as in Fig. Ib. Thepreservation of continuity during the decay of each waveproduces a relation between depth and wave height, 0p,dependent on soil characteristic relations and the valueof /. If time is measured from the beginning of the squaresurface flux pulse, the depth of the wave front at the timeof intersection, tt, is

z- = v t- f!81'I VSll • 1 1 0 J

t this

[19]

The trailing wave has moved the same distance at thisinstant, so that

z,. = vc(t - g .Equation [18] and Eq. [19] produce

~ v,) q(0)[20]

Let 6P represent the time variable peak 0 at the shockfront for t > ?,-, and 0a and 0b are two arbitrary pulseheights for the case described above, with 0a > 6b > 0p.Since the 0 wave produced by pulses of equal volume /,but higher 0a will decay to the same trailing wave profilewhen 0P = 0b, Eq. [20] is a relation describing the decayof Op with z for a given / and 0\. vs, vc are functions of6 and 0 } . Such a relation of 0p and z is plotted in Fig. 5,using the functional soil characteristic relations given inthe Appendix, with parameters as indicated, for pulsevolume / = 20 mm of water. It should be noted thatwhen 0 below the front is not constant, as when a frontencounters a trailing wave, a more complicated procedureis necessary to calculate front advance so that continuityis preserved.

0.3

0.2

O.I

Single Pulse Attenuation, Kinematic Unsaturoted Flow20 mm Water PulseSandy SoilKs = O. I35m/hr

10 20 30 40 50 602, Depth Below Sur face , cm

70 80

Fig. 5—Kinematic attenuation of a simple pulse wave is illustrated inthis figure, showing the wave height attenuation with depth for aninitially square wave of a given volume at initiation at the soil surface.

SMITH: APPROXIMATE SOIL WATER MOVEMENT BY KINEMATIC CHARACTERISTICS

6 Waves from Complex Surface Flux PatternsThe methodology presented above can treat the move-

ment of an arbitrary rainfall pattern through any depthof soil, with rainfall flux given, as is commonly the case,by a sequence of pulse rates. For cases, not emphasizedhere, where q* > 1, an infiltration model such as discussedby Smith and Parlange (1978) can predict the rate ofsoil water influx during a rainfall. When a leading waveof size 03 — 82 follows another of size 02 ~~ # i > w'th 63> 62 > ®\ the larger wave moves faster, and at sometime overtakes the slower moving wave to form a waveof height 03 — 0]. Figure 6 shows wave b-c moving toclose the distance between it and wave d-e. Here, as inother kinematic approximations, the shock waves aremerely representations of mean wetting front positions.Figure 6 also indicates how the characteristic velocityvc(0) along the trailing wave may be treated computa-tionally as a succession of steps, since in a finite differencesense, the velocity vc = dk/dd can be taken as

K(6) - K(6 - Afl)A0 [21]

Accuracy of the kinematic characteristic method isdemonstrated in Fig. 7 for a five-step rainfall pattern,wherein results of the kinematic procedure (Eq. [8], Eq.[12], and Eq. [17]) are compared with a rather fine-meshfinite difference solution to Eq. [3]. The soil used here isa fine sand. Maximum q* = 0.5 (Ks = 2 cm/h). At 6h, note that both the complete solution and the kinematicapproximation indicate a loss of much of the detail ofthe original pulse pattern, due to shock merging and de-cay, as well as diffusion. The surface saturation 00 doesnot return to Q(q) immediately, shown by the t = 6 so-lution of Eq. [3], but it has done so at t = 19 h. Thekinematic wave approximation predicts rather well thelocation of the wave front, as well as the general patternof 6 behind the front, for results at both 6 and 19 h.

0 0,

Water Content 9

C/4 o 2 ^3

——Jd.

Fig. 6—Schematic illustration of kinematic approximation of unsaturatedvertical flow, for both advancing and trailing waves. The trailing wavemay be approximated by a series of smaller square waves.

£ 10

4 6Time ( h r )

Woter Content0.12 0.16 0.24

25 -

Fig. 7—Illustrative example of the accuracy of the analytical solution(dashed line) as compared with precise numerical solution of Eq. [3](solid line) at two times. The rain pattern used is shown above. Ks is20 mm/h.

Unsaturated Flow Through a Root Extraction ZoneKinematic treatment of water movement through a root

zone is quite analogous to the kinematic movement ofsurface water over an infiltrating surface. Assume weknow the root soil water extraction rate per unit depth /[T~l] at some time t. This becomes a rate of 6 loss, r,of

= r(z.r),

Root _jZone I

Depthq, F luxa tz=0° ° z'

Fig. 8—The characteristic paths of the kinematic solution will curvedue to losses in a root zone, and the advancing wave will also slowwhen characteristics from trailing waves d-e and f-g intersect it.Segments 1-2 and 3-4 are straight lines.

8 SOIL SCI. SOC. AM. J., VOL. 47, 1983

where 0 is porosity. Loss rate r may vary with z, asindicated.

The characteristic solution requires that, along the pathdescribed by Eq. [7],

dO/dt = r(z) . [22]If Eq. [22] provides an analytic expression of r(z,t) itmay under certain forms be used in Eq. [7] for an an-alytical expression of the characteristic path in the z,tplane. Otherwise, use of Eq. [7] with Eq. [22] is a simplenumerical exercise. Figure 8 shows the z,t characteristicpath solution features for a simple flux pattern at z = 0and a small root extraction zone down to z = zr. Char-acteristic paths curve only in this region, with vc slowingas 0 decreases with time. Shock path curves when thefront height, 0U — 0,-, is decaying. The 6 is constant be-hind the front and shock velocity is constant only in seg-ments 1-2 and 3-4.

Kinematic Routing Through Layered SoilsKinematic treatment of unsaturated flow in layered

soils is also analogous to the kinematic surface watercase, where soil layers correspond to cascaded planes(Kibler and Woolhiser, 1972). The analytic method issimply applied within each layer, matching fluxes at eachinterface as boundary conditions. The limiting conditionis that for all TV layers in the profile, the flux must staybelow all Ks, i.e.,

q.j<l; [Kj<N].Not shown in Fig. 8 for clarity is the feature that layeredsoils would produce some manner of refraction of char-acteristics paths at layer interfaces.

SUMMARYThe methodology discussed above for kinematically

approximating the vertical movement of unsaturated soilwater is suggested for fast analytic calculation of unsat-urated water movement under one or more of the followingconditions:

1. Movement of water resulting from rain character-istically less than saturated conductivity.

2. Deep movement of water from arbitrary rainfallpatterns, in conjunction with a surface infiltrationfunction.

3. Calculation of influx patterns to shallow or perchedaquifers or groundwater recharge rate calculationsof any sort which occur in unsaturated flow.

It has also been indicated above that root water ex-traction and layered soil profiles lend themselves tostraightforward treatment within the kinematic charac-teristic method of soil water movement.

APPENDIXFunctional Soil Characteristic Relations

Quantification of the above relations require functional re-lations between K, 9, and \[/. In particular, Eq. [8], Eq. [11],and Eq. [12] require a relation K(6). Several are available inthe literature and the kinematic treatment of vertical unsatu-rated flow is independent of the function chosen, so long as it

is continuously differentiable in 8. Herein is used a modificationof the relation of Brooks and Corey (1964), as follows:

Let x = In (^/^k) where \pb is air entry pressure (Fig. la).Then define 8,, and set

0j ~ ®rf-= t*p[-\ F(x.c)] , [Al]

where F(x,c) = (x/2) + ^(x2/4) + c, and 8S = 8(t = 0),er = 0(t -> -oo).

The relation of K to Oe is as in Brooks and Corey (1964),K = Ks Of [A2]

where t = (2 + 3\)/X.The relations of K, 6e, and ^ with parameters c, X, and \l/b

are illustrated in Fig. la. Parameter c modifies the saturatedend of the i/< — 8 relation in accordance with most experimentalobservation, and Eq. [Al] reduces to the original Brooks andCorey relation as c is taken to 0 i.e.,

Hi)-\W > W) • [A3]

From Eq. [A2], Eq. [A3], and Eq. [4] we can show that (\l/bnegative):

\l/k 6 0+2*) A«•>--*•$!£=»••

[A4]

or using Eq. [Al] and Eq. [A2],

in which™--*j^Hw+i] [ASI

m)-xhsr-""-]which also reduces to Eq. [A4] as c —> 0.

Using Eq. [A2] above, Eq. [13] becomes

[A6]

and Eq. [17] describing the trailing wave profile may be solvedfor a segment 0 l t 62 to yield

^-fe)*-