WATER RESOURCES RESEARCH, VOL. 33, NO. 1, PAGES 259-260, JANUARY 1997
Designof interfaceshapefor protectivecapillarybarriers
JohnSelker
Department
ofBioresource
Engineering,
Oregon
State
University,
Corvallis
Abstract. Slopinginterfacesof fine overcoarseporousmaterialhavebeenconsidered
for
useasbarriersto infiltrationfor manyyears.Previousliteraturehasdevelopedanalytical
solutions
for flow over suchinterfaces,numericalsimulationof suchflow,and the effects
of anisotropy
on the diversioncapacityof sucha system.
In all of theseanalyses,
however,
theinterfacewas assumedto consistof a constantslope,facilitatingcertaincalculations.
In thisnote it is shownthat both from a performanceperspective(as measuredby the
ability
to divertfluid)anda design
perspective
(easeof computing
diversion
capacity)
the
shape
oftheinterface
should
becurved,
typically
in a parabolic
shape.
Useofthisshape
is
predicted
to double
barrierdiversion
capacity.
Interfaces
madeupof easy
to construct
segmented
approximations
to theparabola
arepredicted
to provide
significantly
enhanced
performance
aswell.
the interface.For a givenlocationthe lateralrateof flowQ is
In thedesignof a capillarybarrierprotectivecoverthereare
several
parameters
whichmustbe established:
(1) Whatisthe given by
maximum
allowableflux throughthe system(leakage)?(2)
What is the horizontal distance over which water is to be
Q=
diverted?
(3) What is the maximumsustained
flux that the
system
will experience
duringthe design
life?and(4) What
materials
are availablefor use?In thispaperwe will discuss
the
K(z') dz'
K(z')S dz' = S
(1)
o
where z' is the vertical distance from the interface, T is the
implications
of the firstthreeconsiderations.
thickness
of the fine material,K(z') is the hydraulicconducWhena capillarybarrierexperiences
uniformandconstant tivitya distance
z' abovetheinterface,andS isthelocalslope
application
of water,the diverted
fluxincreases
linearlywith of the interface.In our casethe vertical moistureprofile is
horizontal
positiondowndip.
In thecaseof a barrierwherethe identicalat all locations,sothe valueof the integralis fixedfor
slope
in the downdipdirectionis constant,
firstfailurewill the entire interface.Thus (1) maybe written
always
be nearthe downdipendof the interface,whilethe
upper
portionof theinterface
couldhandlemuchmoreflow
without
localpenetration.
Indeed,the upperportionof the
Q = CS
(2)
T
(3)
where
interfacecould divert the incident flux even if the interface had
a lowerlocalslope,whileif the slopeweregreatertowardthe
downdip
portion,the diversion
couldbe increased
prior to
breakthrough.
It wouldappearlogicalto increase
theslopeof
thebarrierin proportionto the expected
local flux,which
C =
K(z') dz'
increases
linearlywith down-slopedistance.
This integralmaybe computed
for isotropicor anisotropic
Froma designperspective
it seemsreasonable
to suppose mediausingtheresults
ofRoss[!990],Steenhuis
etal.[1991],or
thatthe optimaluseof the total fall in elevationhasbeen Stormont
[1995]for the conductivity
profileat the downdip
achieved
whenthe breakthrough
fluxthroughthe interfaceis limit with the incidentflux setequalto the maximumpenetraconstant
alongthe entirelengthof the interface.It will be tion flux.
shown
thatthisassumption
immediately
dictatesthegeometry In the caseof the linear-roofdesign(Figure 1), Q = qx,
of theinterfaceandgreatlysimplifies
the hydraulic
analysis. wherex is the horizontaldistance
fromthe up-slopecrestand
Consider
thelimitingcondition
wherethemaximum
allowable q istheincident
flux.Putting
thisinto(3),weimmediately
find
fluxispenetrating
thebarrier.Letusnowposittohaveselected that
abarrier
geometry
suchthatthisfluxpenetrates
thebarrierat
xq
allpoints.
Thisuniformleakagefluxcondition
impliesthatthe
S=•(4)
pressure
isalsouniformalongthetexturalinterface.
Sincethis
condition
existsat all pointsalongthe barrier,the vertical
isparameterized
bycoordinates
(x*, z*), then
moisture
profilein the uppermediais uniformacrossthe If theinterface
barrier,
andtherewillbenogradients
in pressure
in directions
dz *
tangent
to the interface.Thusall barrierparallelfluxwill be
driven
bygradients
in thegravitational
potential,
which,given
theuniform
vertical
pressure
profile,
equaltothelocalslope
of
Copyright
1997bytheAmerican
Geophysical
Union.
Papernumber96WR03138.
0043-1397/97/96WR-03138509.00
259
s: ax*
(5)
whichmaybe substituted
into (4)
dz*
x*q
dx*
C
(6)
260
SELKER: TECHNICAL
NOTE
Figure 1. Definitionsketchesfor (left) circularand (right) linear-roofcapillarybarrier designs.
which may be solvedfor the interfaceelevationasa functionof
horizontal
distance from the crest
couldbe extendedto arbitraryroof geometryby setting
t-he
localinterfaceslopeto be a constant
timesthe areaof capture
that contributesto the flow at eachpoint.
x*2q
2c
(7)
Equation(7) suggests
that the useof a parabolicinterfacefor
linear-roofcapillarybarrierswill provideoptimaldiversion.
To estimatethe improvementin barrier performanceobtained usingthe curvedinterface,considera barrierwhich has
a totalfall of H overa horizontaldistanceL (Figure2). For the
straightbarrier, (2) suggestsa diversioncapacityof CL/H,
while for a parabolic system,(7) suggestsa diversion of
2CL/H; the parabolicinterfaceprovidesa factorof 2 increase
in diversioncapacityfor a giventotal elevationloss.
Within the uniform leakageframeworkit is alsostraightforward to computethe optimumshapeof a circular-roofdiversion. In sucha setup,for any radial distancer from the peak,
In practical
terms,the construction
of a smoothly
changing
slopepresents
logistical
difficulties.
Noticethat the curved
system couldbe approximated
by a seriesof n straightsegments
of equalhorizontalextentandlinearlyincreasing
slope.Using
suchan approximationto a barrier with total horizontalextent
H and total fall L, the slopeof the most downdipsegment
would be
Hn 2
s=
(8)
L•i
i•.l
In sucha segmentedsystem,first leakagewould occurat the
lowestend of eachsegmentsimultaneously,
sincethe diverted
thetotalfluxintercepted
bythebarrierwillbe ,rr2qwhichwill flux/sloperatio will be minimal and equal for all segments
at
transecta circumferenceof media with total length2,rr. The thesepoints.Using three and five straightsegments,the diverlocal lateral flow will be qr/2, which, by comparisonto the sioncapacityis predictedto be at least75% and 83%, respeclinear-roof analysis,implies an optimal interfaceparameter- tively,of the diversionof a smoothlycurvedinterface,demonizedbyz = r2q/C.The slopeshouldbe halfthatof a linear- stratinga rapid convergence
to the optimalperformance.
roof for anygivendistancefrom the crest.Clearly,thisanalysis
References
Ross,B., The diversioncapacityof capillarybarriers,WaterResour.
Res., 26, 2625-2629, 1990.
z
Steenhuis,
T. S.,J.-Y. Parlange,andK.-S.J. Kung,Commenton"The
diversioncapacityof capillarybarriers"by B. Ross, WaterResour.
(0,
•
Figure 2.
,•z'
Fine
(x',ϥ
Media
Res., 27, 2155-2156, 1991.
Stormont,J. C., The effectof constantanisotropy
on capillarybarrier
performance,WaterResour.Res.,31, 783-785, 1995.
J. Selker,Departmentof BioresourceEngineering,OregonState
Coarse
University,Corvallis,OR 97331-3906.(e-mail: selkerj@ccmail.
Media
orst.edu)
Parameter identification.
(ReceivedMarch28, 1996;revisedSeptember21, 1996;
acceptedOctober9, 1996.)