Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

A Modeling Approach Supporting Air Sparging System Design 91
Linear function
In order to get a preliminary assess of the behavior of the air injected in soil when no
experimental data about the capillary pressure function are available, a linear relationship
between P c and the degree of saturation to water S l can be selected:
⎧
⎪Pcr
for Sl
≤ Slr
⎪
⎪ Sls
− Sl
PC = ⎨Pcr
for Slr
< Sl
< Sls
(16)
⎪ Sls
− Slr
⎪
⎪0
for Sl
≥ Sl
s
⎩
where P cr (Pa) is the minimum value of the capillary pressure may have.
Function of Pickens et al.
This is a nonlinear model based on the empirical equation proposed by King (Pickens et
al., 1979; van Genuchten et al., 1985) relating θ l to h:
⎡
χ
⎤
⎢ ⎪
⎧ ⎛ h ⎞ ⎪
⎫ θl
0
−θlr
cosh⎨
⎥
⎢
⎜
⎟ + ε⎬
− cosh ε
+ ⎥
⎢
⎪⎩ ⎝ h0
⎠ θl
0
θlr
θ
⎪⎭
⎥
l
= θl0
(17)
χ
⎢
⎪
⎧ ⎛ h ⎞ ⎪
⎫ θ −θ
⎥
l 0 lr
⎢cosh⎨
+ ⎬ +
⎥
⎢
⎜
⎟ ε cosh ε
+ ⎥
⎣ ⎪⎩ ⎝ h0
⎠ θl
0
θlr
⎪⎭
⎦
where θ l0 , h 0 , χ and ε are curve-fitting parameters of a set of experimental values. This
expression showed good results in fitting experimental data for several coarse- and finetextured
soils. Note that θ l0 and h 0 may refer to the saturated condition.
In case of ε = 0, the inversion of Eq. (17) results in:
⎧ ⎡
⎛
2 ⎞⎤⎫
⎪ ⎢ Sl
⎜ ⎛ Sl
⎞ ⎟
1+
⎥⎪
⎪ ⎢
⎜ ⎜1+
⎟
S
⎟⎥⎪
l 0
Sl
0
−Slr
⎨ ⎢
⎜ ⎜ Sl
0
Sl
0
−Slr
P
⎟
c
= P0
1−
1+
1−
⎟⎥⎬
(18)
⎪ ⎢
S + ⎜ ⎜ + ⎟
l
1−
Sl
0
S
S
lr
l
⎟⎥⎪
⎪ ⎢
⎜ ⎜ 1−
Sl
0
Slr
⎟
S
⎟
l 0
⎥⎪
⎝ Sl
0 ⎠
⎩ ⎣
⎝
⎠⎦
⎭
1/
χ
where P 0 is related to h 0 by Eq. (8), and S l0 and S lr are related to θ l0 and θ lr respectively,
through porosity.
Milly’s function
This function is based on the generalization of the soil water retention curve obtained by
Haverkamp for the Yolo clay soil, as reported in Milly (1982):