The document presents a method for calculating the coefficient of permeability (Kw) as a function of volumetric water content using the soil-water characteristic curve. It describes Kunze's model which divides the curve into equal intervals of volumetric water content and calculates the permeability for each interval using equations 1 and 2. The example applies this method to calculate Kw for 20 intervals of a sample soil-water characteristic curve by first determining the saturated permeability (Ks), saturated coefficient of permeability (Ksc), and matric suction values, then using these in equation 1 to find Kw over the range of volumetric water contents.

1. COMPUTATIONOF COEFFICIENT OF PERMEABILITY FUNCTION ,Kw() USING
SOIL WATER CHARACTERISTIC CURVE:
A statistical models use afew discrete points ratherthan a continuous
mathematical model to represent the permeability functionof unsaturatedsoil,
the accuracy of statisticalmodels is highlydependent on the numbers and locations of
these discrete points.Onthe other hand, the locations of these discrete
points are dependent on the manner in which the entire range of
suction is discretized into divisions. Kunze et al. (1968) proposed
equally dividing the volumetric water content,
θw, into intervals, Δθw, and calculating the interval of matric suction,
Δψ, accordingly. Equal divisionofthe volumetric watercontent,
θw, makes the density, f(r), unique for all pore radii, meaning that the pore-
size distributionfunctionfollowsauniform distribution.With improvements
to the Childs and Collis-George (1950) and Marshall
(1958) equations, Kunze et al. (1968) presented a simple Eq. (1) for the
calculationof the permeabilityfunction:
𝐾( 𝜃𝑤) =
𝐾𝑠
𝐾𝑠𝑐
𝐴𝑑 ∑ {(2𝑗 + 1 − 2𝑖)( 𝑈𝑎 − 𝑈𝑤) 𝑗
−2
}𝑚
𝑖=𝑖 ……1
Where i=1,2,….m,
kw(θw)i predicted coefficient of permeability for volumetric water
content;
ua airpressure (kPa);
uw pore-waterpressure (kPa);
(θw)i corresponds to the ith interval (m/s);
i = i nt e rval numbe r t hat i ncre as e s as t he vol ume t ri c wat e r co
ntent decreases;
j a count from “i” to “m”;
m= t ot al numbe r of i nt e rval s be t we e n t he s at urat e d vol ume t ri
c watercontent,
θs=
,and the lowest volumetric water content,
θL
;ks measured saturatedcoefficient ofpermeability(m/s);
ksc calculatedsaturatedcoeffcient ofpermeability(m/s);
Ad adjustingconstant;
𝑇𝑆
2
𝜌 𝑤 𝑔𝜃 𝑠
𝑝
2𝜇 𝑤 𝑁2
Ts= surface tension of water
𝜌 𝑤 = 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

2. G =gravitational acceleration
𝜇 𝑤 = 𝑜𝑏𝑠𝑎𝑙𝑢𝑡𝑒 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
𝜃𝑠 = 𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑤𝑎𝑡𝑒𝑟 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑎𝑡 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑟 𝑧𝑒𝑟𝑜 𝑠𝑢𝑐𝑡𝑖𝑜𝑛
P=a constant whichaccountfor interactionof poresof vorous sizes,the magnitudeof pcan besetto2
N= total numberof intervalscomputedbetweenthe saturatedvolumetricwatercontent
,𝜃𝑙, 𝑎𝑛𝑑 𝑧𝑒𝑟𝑜 𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑤𝑎𝑡𝑒𝑟 𝑐𝑜𝑛𝑡𝑒𝑛𝑡
(Ua-Uw)j=matricsuctioncorrespondingtothe jthinterval
𝐾𝑠𝑐 = 𝐴𝑑 ∑{2𝑗 + 1 − 2𝑖}( 𝑈𝑎 − 𝑈𝑤) 𝑗
−2
}
𝑚
𝑗=1
i=0,1,2,…m
substitutingthe matricsuctionscorrespondingtothe mid-pointsalongsoil-watercharacteristiccurve
intoequation2 andassumeing Ad=1we can getthe value of Ksc. Thenwe findthe ratiobetweenKsto
Ksc. Thenwe can conepute Kwbysubstitutingall these valuesinequation1.
The followingexample istakentoilluzlrate the technique bywhichcoefficientof permeability 𝐾 𝑤(𝜃 𝑤)
can be computedasfunctionof watercontent.
Let usconsidera soil watercharacteristiccurve.The curve isdividedintom=20 intervals.If volumetric
watercontentas showninfigure.

3. It can be seenthatthe curve has a maximumandminimumvolumetricwatercontentsof 0.3877 and
0.1091 respectively.The firstvolumetricwatercontentcorrespondstosaturatedcondition,ie (Ua-Uw)
equalstozero.
Then,the value of Ksc iscomputedusingequation2and the value of Ksc wasfoundto be 0.977042m/s.
Ie;Ksc=0.9770662 m/s
The computationof Ksc can be shownina tabularform inTable 1.
The Ks value istakenas 5.83 ∗ 10−8(
𝑚
𝑠
).
Thisis independentlymeasuredinthe laboratory.
Ad termisassumedas unity,ie Ad=1
𝐾𝑠
𝐾𝑠𝑐
=
5.83∗10−8
0.9770662
= 5.96684 ∗ 10−8Keepingthese valuesof Ad,
𝐾𝑠
𝐾𝑠𝑐
and correspondingmatricsuctions inequation1, the permeabilityvaluesare computed

4. Matric suction (Ua-Uw) KPa Coefficientof permeability,(𝐾 𝑤(𝜃 𝑤)(m/s)
0 5.83*10^-8
5.4 4.63 ∗ 10−8
10.8 3.94 ∗ 10−8
12.55 3.35 ∗ 10−8
14.3 2.85 ∗ 10−8
15.5 2.42 ∗ 10−8
16.7 2.04 ∗ 10−8
17.65 1.7 ∗ 10−8
18.6 1.41 ∗ 10−8
19.5 1.16 ∗ 10−8
20.4 9.35 ∗ 10−9
21.4 7.42 ∗ 10−9
22.4 5.778 ∗ 10−9
23.35 4.38 ∗ 10−9
24.3 3.2 ∗ 10−9
25.25 2.24 ∗ 10−9
26.2 1.48 ∗ 10−9
27.65 8.88 ∗ 10−10
29.1 4.66 ∗ 10−10
32.65 1.93 ∗ 10−10
36.2 4.55 ∗ 10−11