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Infiltration Equations h( ), K(
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Infiltration EquationsFundamental Mass Balance Equation:
0
zq
yq
xq
tzyx
Darcy’s Law (z direction):
zhKq zz
Wherezh
Infiltration Equations
h(), K(
Infiltration Equations
h(), K(
Infiltration Equations
h(), K(
Infiltration Equations
)()()( zzz Kz
Kq
Vertical Darcy’s Law is then
Soil water diffusivity is defined
ddKD )()()(
combining
)()( zzz Kz
Dq
Infiltration EquationsIn three dimensions, using K)
zhK
zyhK
yxhK
xt)()()(
defining the specific moisture capacity as C() = d/d
1)()()()(
zK
zyK
yxK
xtC
Infiltration EquationsFor 1-D vertical, saturated flow:
01
zK
z s
Infiltration EquationsBoundary conditions for infiltration at ground surface:
ps ttttz
KzhKr
0),0(),0(1)()(
ps tttht ),0(),0( 0
Infiltration EquationsLower boundary conditions for infiltration
00),( ttL
0)( tKqz
Infiltration EquationsMixed form of Richards Equation
1)(
zK
zt
Infiltration EquationsParameters (van Genuchten, 1980):
0
01)(
s
mnrs
r
2/111)(mm
eles SSKK
where m = 1-1/n for n > 1Se = ( - r)/(s - r)r is the residual volumetric moisture content l, and n are van Genuchten model parameters
Infiltration EquationsParameters: Rosetta by Marcel G. Schaap (1999)
Soil textural classes;Sand, silt and clay percentages;Sand, silt and clay percentages and bulk density;Sand, silt and clay percentages, bulk density and the value of at = 330 cm (33 kPa); andSand, silt and clay percentages, bulk density and the value of at = 330 and 15,000 cm (33 and 1500 kPa).
Infiltration Equations
Infiltration Equations
LLF i )(
Green-Ampt
L
LKL
LhKf )(0
F
FKf
from conceptualization:
Darcy’s Law:
combining:
Infiltration Equations
KttFtF
)(1ln)(
since f = dF/dt
how is this solved?
Infiltration EquationsGreen-Ampt ponding time
)( KiiKt p
for t < tp, how are f and F computed?
Infiltration EquationsGreen-Ampt with a hyetograph
• discrete, steady pulses of rainfall of t duration is used to describe i(t)• F(t + t) = F(t) + i(t)t while f < i• when the surface is ponded throughout the time increment:
tKtF
ttFtFttF
)()(ln)()(
Infiltration EquationsGreen-Ampt with hyetograph continued...
• If f(t + t) < i(t), then ponding occurs during the time interval and using f(t) = i(t) and F(t) = Fp :
KtiKFp
)(
• then tp = t + t' where
)()(
'ti
tFFt p
after Chow et al., 1988
Infiltration EquationsGreen-Ampt parameters, after Chow et al., 1988
Infiltration Equations
Kostiakov
tKf k
Infiltration Equations
Horton
tcoc effff )(
Infiltration Equations
Holtan
ca fSAGIf 4.1
Infiltration Equations
Phillip
sKStf 5.05.0
Infiltration Equations
Smith-Parlange
1)/exp(
1BF
Kf s
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