WATER RESOURCES RESEARCH, VOL. 35, NO. 5, PAGES 1685-1688, MAY 1999
Green and Ampt infiltration into soils of variable pore size
with depth
John S. Selker
Departmentof BioresourceEngineering,Oregon State University,Corvallis
Jinfan
Duan
Departmentof ForestEngineering,OregonStateUniversity,Corvallis
Jean-YvesParlange
Departmentof Agriculturaland BiologicalEngineering,Cornell University,Ithaca, New York
Abstract. Most expressions
for infiltrationrely upon the assumptionof vertical
uniformityin soiltextureand hydraulicproperties.We presentan extensionof the results
of Beven[1982, 1984] for infiltrationand lateral flow in soilswith decreasingpermeability
with depth. Unlike Beven,we basethe derivationsupon joint changesin soil properties
basedon an overall changein characteristicpore sizevia Miller scaling.A set of very
simpleexpressions
for the time rate of infiltrationare obtainedusinga Green and Ampt
approachfor soilswith permeabilitythat decreases
with depth followinglinear, power law,
and exponentialrelationships.
1.
Introduction
It is widely recognizedthat soil hydraulicconductivitytypically decreasesfrom the surface,yet most infiltration models
ignorethis fact. Beven[1982, 1984] introducedthe possibility
that analyticalexpressions
could be derivedfor small catchmentsbasedon soildescriptions
whichvarywith depth.In this
paperwe build on theseresultsto obtainequationswhichmay
be usefulin a variety of hydrologicsettings.
The basicnotionthat we wouldlike to pursueis the derivation of equationsfor infiltrationthat explicitlyincludeissuesof
changingsoil propertieswith depth.Beven[1982]presents
hillslopehydrologyand generateda seriesof hydrographs.
In
the presentpaperwe refinethe applicationof the abovestated
relations
2.
in the context of infiltration.
Analysis
First we would like to revisitthe data of Childsand Bybordi
[1969] as presentedby Beven [1984] to provide a physical
frameworkfor our analysis(Table 1). We start by askingthe
question:Why do soilshavelowerpermeabilitywith depth?Is
it dueto decreasein porositydueto greaterpackingdensityof
particlesof essentiallyhomogeneous
particlesize?This is not
Ks(z) =K,(D -z) n
(1) supportedby the data in Table 1, wherewe observemonotonicallyincreasingporositywith depth,while conductivitydrops
by
a factor of 26. In fact, Figure 2 of Beven[1984] dispelsthe
Os(Z)= O, (D - z) m
(2)
notion of a strongcorrelationbetweenthe changein conducto describethe vertical profilesin saturatedconductivityKs tivity andthe changein porositywith depth.To obtaina more
and saturatedmoisturecontent Oswith increasingdepth z physicallyreasonableconnection,we recall that muchof spa(positivedownward)for a soil with total depthD abovean tial variabilityin soilscanbe explainedby appealingto Miller
impermeablelayer. Here K,, 0,, n, and rn were taken to be similarity[Miller and Miller, 1956]. Miller scalingprovidesa
parametersto be fittedto the sitedata,whereBevensupposed quantitativeformulationto relate the hydraulicpropertiesof
that n • 2m on the basisof scalingrelationsfor permeability soilsthat have particle and pore size distributionswhich are
geometricallysimilar but that have dissimilarmean size. Reversusporosity.
Later, when developing a Green and Ampt infiltration cent examplesthat demonstratethe utility of this approach
model, Beven [1984] employedan exponentialrelationship includethe worksof Rockholdet al. [1996]and Warrick[1990].
Let us then supposethat the characteristic
pore sizevaries
givenas
with depthand that this is the primaryfactor affectingpermeKs = Ko exp (fz)
(3) ability.We wouldthen expectthat asthe pore sizedecreased,
conductivity
woulddropwith the squareof pore sizeand that
Os= 0oexp (#z)
(4) the Green and Ampt wetting front pressurewould increase
wheref and # are fitting parameterswhichwere seento be linearly.Table 1 presentsthe data of ChildsandBybordi[1969]
unrelated when fit to real soils. Within this framework, Beven with an additional computed column listing the product
•/2½ws,
aparameter
thatwould
bepredicted
tobeconstant
if
went on to constructa very interestingconceptualmodel for Ks
Miller similaritywasvalid.Thisnotioniswell supportedby the
Copyright1999by the AmericanGeophysicalUnion.
observation
that thisproductchangesby <4% in the first 1.5 m
of the soil profile, where both K and the Green and Ampt
Paper number1999WR900008.
0043- ! 397/99/! 999WR900008509.00
wettingfrontpotential½ws
varyby 50 timesthisamount.
1685
1686
SELKER
ET AL.:
TECHNICAL
which may be integratedfrom the surface,where H = z = 0
to any depthz, to obtain
Table 1. Soil PropertiesWith Depth
Layer,
m
0-0.3
0.3-0.6
0.6-0.9
0.9-1.2
1.2-1.5
1.5-1.8
K•,
m h
xxs
13.2
7.5
4.2
2.9
1.7
0.5
NOTE
-0.06
-0.08
-0.10
-0.125
-0.159
-0.178
0.35
0.3355
0.36
0.36
0.365
0.37
•gwf
-0.218
-0.219
-0.205
-0.213
-0.207
-0.126
q
H = 213Ko
[1- exp
(2/3z)]
(lO)
Writing thisin termsof the pressurepotentialh, we recallthat
for positionmeasuredpositivein the downwarddirectionwe
haveH
= h -
z'
q
Data are from Childsand Bybordi[1969] as presentedby Beven
[1984].Each soil layer was characterizedby a saturatedconductivity
h= 213Ko
[1- exp
(2/3z)]
+z
(11)
Ks, a GreenandAmptwettingfrontpotential½wf,andthe available
pore spaced0. The final columnlistsa productwhichwouldbe constantif the depth-varyingsoilswere similarin the senseof Miller and
Miller [1956].
If we take the depthof the wettingfront to be z*, we mayuse
(7) to obtainthe potentialat the wettingfront h(z*)'
h (z * ) = ½wSo
exp(/3z* )
There is greatadvantagein employingMiller scaling,in that
it providesa physicallink between the vertical variation in
conductivityand more widelyreportedvaluesof variabilityin
particle size.Further, it allowsa reductionin the numberof
parameters,aswe maylink n andrn aswell asf and# usedby
Beven, as shown below.
(12)
Combining(11) and (12) and solvingfor flux,we find
q=213Ko(
½wIo
(13z*)
-)z*)
1 -exp
exp
(2/3z*
(13)
A simplecheckon (13) is to let/3 go to 0, whichyieldsthe
expected rate of infiltration for uniform media q =
We desiresimpleexpressions
in soilswith verticallydecreas- -Ko(½wfo- z*)/z*.
ing conductivityfor infiltrationrates as a functionof time. To
Recognizingthat for Green and Ampt infiltration,
achievetheseends,we solvefor Green and Ampt infiltration
dz *
for soilswhichobeyeither (1) or (3), with changesin permeq
=
A
O
d•(14)
ability explainedby changesin pore size which obey Miller
scaling[Millerand Miller, 1956].
we may calculatethe depthof wettingas a functionof time as
First we will considera verticallyfining soil suchthat the
characteristic
microscopiclengthscaleX followsthe relationship
(15)
it(z) = )toexp (-/3z)
where/3 is a scaleparameterwith unit of inverselength.From
Miller scalingwe then knowthat the saturatedconductivitywill
vary with depth followingthe relationship
K(z) = Ko exp (-2/3z)
t(z*)=213Ko ½w•o
exp
(/3z)
-z
(5)
(6)
whichmay be evaluatednumericallyas needed.
If the soilfinesfollowa powerlaw, suchas suggested
in (1),
the same calculationsmay be carried forward. The simplest
caseis a linear declinein particle size,in which casewe have
whereKo is the saturatedconductivityof the uppermostsoil.
Similarly,the Green and Ampt wettingfront potential,which
in generalwill be negative,will follow
½w•(
Z) = ½wio
l3Z
(16)
Ks(z) = go•3-2z
-2
(17)
½w•(Z)
= ½wSo
exp(/3z)
where/3 is a scaleparameterwith unit of inverselength.Computingasbefore,we find that for Green andAmpt infiltration,
(7)
WenotethatBeven
[1984]heldtheproduct
AO½wSo
constant
to
easecomputation,which,as notedby Bevenand illustratedin
Table 1, is not in keepingwith observations.
For vertical Green and Ampt infiltrationq, we know from
Darcy'slaw that
3Ko(13Ow/o1)
q= -
z,2/32
(18)
and that the wetting front movesin with time following
•32A
0
(9Ko(13½wfo-1)t)
•/3 (19)
dH
2:*= --
q =-K(z) dz
dH
= -Koexp(-2/3z)dz
(8)
which is, surprisingly,simpler than the implicit logarithmic
form obtainedfor verticalinfiltrationinto a homogeneous
profile.
This computationcan be carried out for other powersof
whereH is the total potential(pressureplus elevation)measuredin unitsof hydraulichead.The wettingfront is assumed depthaswell. Of particularinterestis the very generalpower
byDuan andMiller
to be sharp;therefore the flux is constantwith depth behind law relationshipintroducedsimultaneously
the wettingfront but will vary in time. Solvingfor total poten- [1997] and Iorgulescuand Musy [1997]. Both investigations
showthat the power law model may be employedin the hilltial, we find
slopemodelsof Ambroiseet al. [1996],and here we demonq
stratethat the powerlaw modelis easilyadaptedto the Millerdm= - exp(2/3z)
dz
(9) similarGreen and Ampt infiltrationapproach.
SELKER
ET AL.: TECHNICAL
In the notationof the presentdiscussion,
the generalmodel
may be obtainedfrom the soil scaleparameter
•. = •.o(1-- [3girt)n
NOTE
1687
2.5
(20)
A
2
wheren > 0. It is interestingto note that (20) reducesto the
exponentialmodelgivenin (3) for n = m. Appealingto Miller
scaling,we may deducethe expectedconductivityandwetting
front potentialsfor a soilwhichobeys(20) to be
Ks(z)= go(1 - [3girt)
2n
(21)
ddwf(Z
) = ddwfo(1
- •3Z/H)
-n
(22)
Equations(21) and (22) are physicallymeaningfulat depths
wherez < n/ill, whichmightbe interpretedasthe depthto an
impermeablelayer.Computingasbefore,the flux asa function
of wettingfront depthis givenby the simplealgebraicexpres-
to 0.5
sion
0
,
0.0001
q =Kol3
.
0.001
(23)
As before,we canwrite the relationshipbetweenelapsedtime
and depthof infiltrationusing(14). Proceedingformally,in
this casethisyields
,
0.01
,
0.1
Time (d)
Figure 2.
Predicted cumulative infiltration based on the
three models'fit to the data of ChildsandBybordi[1969].The
heavysolidline is the linear model, the light solidline is the
power law model, and the dashed line is the exponential
model.
AOnfo
z*1- (1-13z/n)
1-2n
t(z*)=gol3(2n
- 1) q'W/o(1
--[•Z/H)
-n--Zdz
3.
(24)
Example of Application
To illustrate the results,we can fit the three models to the
data givenin Table 1 andpredictthe cumulativeinfiltrationin
time. To fit the models,we took 0.36 to be the averagevalueof
A0, fit the conductivityfunction to the data of Childs and
Bybordi[1969] to obtain valuesof Ko, 13, and n, and then,
Optimalfit wastakenasthe parametersetthat providedminimum sum of squareerror dividedby the squareof the measuredvalues.This approachavoidedgivingundue weight to
the large permeabilityvalues.Clearly,alternatefitting procedurescouldbe selectedas appropriate.The cumulativeinfiltration was then predictedusing(15), (19), and (24).
The fit conductivities
showthat the more flexibleexponen-
tial and powerlaw modelsfit the databest,while the linear
finally,fit qtwf
øto theChilds
andBybordi
[1969]datasetusing modelwasunableto fit the permeabilitydatawell (Figure 1).
the other parametersfound in fitting the conductivitydata.
1 oo
•'
10
A
The predictedinfiltrationsof the threemodels(Figure2) have
severalnotablecharacteristics.
The earlytime infiltrationpredicted by the linear model significantlyexceedsthat of the
other models,predictinga factor of 2 more infiltrationin the
first0.5 hours.Thiswouldgiveriseto significantdiscrepancies
in short-timerunoffpredictions.The modelsarewithin 30% of
each other at intermediatetimes (0.1-0.50 days).The power
law model divergesfrom the other modelsat longertimes as
the depth of the presumedimpermeablelayer is approached
(t > 0.6 days).
>
4.
o
n,
We havefoundthat the useof the Miller similarityallowsfor
exact solutionsto the Green and Ampt vertical infiltration
problemfor a varietyof continuously
varyingsoiltextureprofiles.The expressions
obtainedare physicallyreasonable,simple to apply,andsufficiently
flexibleto be fit to a widerangeof
soilprofiles.They can be appliedin the full rangeof settings
1
0.1
0.'
,
,
1
10
Summary
100
Model Value (m/h)
Figure 1. Model versusmeasuredpermeabilitybasedon the
threemodels'fit to the dataof ChildsandBybordi[1969].The
where
theGreenandAmptapproach
hasproven
tobeauseful
quantitativemodel for infiltration.
References
solidline is the line of perfectfit, the trianglesare the linear Ambroise,B., K. Beven, and J. Freer, Toward a generalizationof the
model,the squaresare the powerlaw model,andthe diamonds
are the exponentialmodel.
TOPMoDEL concepts:Topographicindicesof hydrologicalsimilarity, WaterResour.Res.,32, 2135-2145, 1996.
1688
SELKER
ET AL.:
TECHNICAL
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Hydrol. Sci.J., 27, 505-521, 1982.
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NOTE
and tritium transportat the Las Crusestrench site, WaterResour.
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J. Duan, Department of Forest Engineering,Oregon State University, Corvallis,OR 97331-3906.
J.-Y. Parlange,Department of Agricultural and BiologicalEngineering, Cornell University, Riley Robb Hall, Ithaca, NY 14850.
Op58@cornell.edu)
J. Selker, Department of BioresourceEngineering,Oregon State
University,Corvallis,OR 97331-3906.(selkerj@engr.orst.edu)
1997.
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(ReceivedApril 13, 1998;revisedDecember31, 1998;
acceptedJanuary4, 1999.)