A MATLAB Program for Estimation of Unstaurated Hydraulic Soil ...
A MATLAB Program for Estimation of Unstaurated Hydraulic Soil Parameters Using An Infiltrometer Technique Presenter: Yi-Hsien Li Adviser: Chuen-Fa Ni Date:2008/12/04
1
Outline • • • • • • •
Introduction Objective Theory Methods Example Results and discussion Conclusions 2
Introduction • Numerical simulations of water and contaminant movement in the unstaurated zone at fieldscale have become increasingly important.
3
Introduction • Richards’ equation ∂θ ∂q = ∂t ∂z ∂θ ∂ ∂h ∂K (K )− = ∂t ∂z ∂z ∂z
4
Introduction ∂ ∂h ∂K ∂θ = (K )− ∂z ∂z ∂z ∂t
• 1D • The spatial heterogeneities
5
Introduction • Campbell/Mualem model ¾It is simpler than the corresponding models by van Genuchten (1980) and Brooks and Corey(1964).
6
Introduction
• Nonlinear • θ＝θ（h）
⎧⎛ hb ⎞b ⎪⎜ ⎟ for h < hb ⎪⎪⎝ h ⎠ θ⎨ ⎪θ for h ≥ hb ⎪ s ⎪⎩
• K ＝K（h） k = K s S e
2.5 b ( 2+ ) b
7
Introduction
• Zou et al. (2001) • an inverse routine »initial water content »a final steady-state water content
8
Introduction
• Wetting front
9
Objective
• The objective was to investigate the accuracy of the hydraulic parameter estimates.
10
Theory
• Solution of Richards’ equation ⎧ hi ⎪ h=⎨ ⎪h ⎩ 0
for
t =0
z>0
for
t≥0
z=0
11
Theory
• Solution of Richards’ equation t grav
⎛ C ⎞ ⎟⎟ = ⎜⎜ ⎝ K0 − Ki ⎠
2
K 0 : K (ho ) K i : K (hi )
C : sorptivity
12
Theory
• Solution of Richards’ equation ¾Philip(1969) » t≦tgrav
13
Theory
• The power series • Cumulative infiltration 1 2
3 2
I = C1t + C2t + C3t + C4t 2 + ....... + Cmt
m 2
14
Method
• Inverse method – Levenberg-Marquardt algorithm O ( x) =
1 N
∑ [w ( I N1
i =1
1
]
2
1,meas (t i ) − I 1,mod el ( x, t i )) +
1 N
∑ [w ( I i =1
2
]
2
N2
2 ,meas
(ti ) − I 2 ,mod el ( x, ti ))
15
Method
• Inverse method – The unknown parameters X = [Ks
b]
T
hb or
X = [Ks
hb
b
θs ]
T
16
Example
• hi=-100cm • Experiment(1) h0=0cm • Experiment(2) h0=-20cm
17
Example
• Experiment(1) • 427s cm 3 θs = 0.475 3 cm
18
Example
• Experiment(2) • 2973s cm 3 θs = 0.396 3 cm
19
Example
• Experiment • wetting front
20
Results and discussion
• To investigate the necessity for two infiltration experiments in the estimation procedure.
21
Results and discussion
• The six parameter planes • (ks-hb, ks-b, ks-θs , hb-b, hb-θs, b-θs, )
O ( x) =
1 N
∑ [w ( I N1
i =1
1
]
2
1,meas (t i ) − I 1,mod el ( x, t i )) +
1 N
∑ [w ( I i =1
2
]
2
N2
2 ,meas
(ti ) − I 2 ,mod el ( x, ti ))
22
Conclusions
• This in general reduces the uncertainty of the estimates, if the initial water content is estimated with sufficient accuracy. 23
Conclusions
• This in general reduces the uncertainty of the estimates, if the initial water content is estimated with sufficient accuracy. 24
Conclusions
• Water content measurement error of up to ±10% had in general a large effect on the estimated three unknown parameters. 25
Conclusions
• This finding indicates that in practice the four-parameter option would be preferable.
26
20081114
Double ring infiltrometer
27
20081120 Double ring infiltrometer
28
20081120 Double ring infiltrometer
29
20081120 Double ring infiltrometer
30
Thanks for your attention
1
Outline • • • • • • •
Introduction Objective Theory Methods Example Results and discussion Conclusions 2
Introduction • Numerical simulations of water and contaminant movement in the unstaurated zone at fieldscale have become increasingly important.
3
Introduction • Richards’ equation ∂θ ∂q = ∂t ∂z ∂θ ∂ ∂h ∂K (K )− = ∂t ∂z ∂z ∂z
4
Introduction ∂ ∂h ∂K ∂θ = (K )− ∂z ∂z ∂z ∂t
• 1D • The spatial heterogeneities
5
Introduction • Campbell/Mualem model ¾It is simpler than the corresponding models by van Genuchten (1980) and Brooks and Corey(1964).
6
Introduction
• Nonlinear • θ＝θ（h）
⎧⎛ hb ⎞b ⎪⎜ ⎟ for h < hb ⎪⎪⎝ h ⎠ θ⎨ ⎪θ for h ≥ hb ⎪ s ⎪⎩
• K ＝K（h） k = K s S e
2.5 b ( 2+ ) b
7
Introduction
• Zou et al. (2001) • an inverse routine »initial water content »a final steady-state water content
8
Introduction
• Wetting front
9
Objective
• The objective was to investigate the accuracy of the hydraulic parameter estimates.
10
Theory
• Solution of Richards’ equation ⎧ hi ⎪ h=⎨ ⎪h ⎩ 0
for
t =0
z>0
for
t≥0
z=0
11
Theory
• Solution of Richards’ equation t grav
⎛ C ⎞ ⎟⎟ = ⎜⎜ ⎝ K0 − Ki ⎠
2
K 0 : K (ho ) K i : K (hi )
C : sorptivity
12
Theory
• Solution of Richards’ equation ¾Philip(1969) » t≦tgrav
13
Theory
• The power series • Cumulative infiltration 1 2
3 2
I = C1t + C2t + C3t + C4t 2 + ....... + Cmt
m 2
14
Method
• Inverse method – Levenberg-Marquardt algorithm O ( x) =
1 N
∑ [w ( I N1
i =1
1
]
2
1,meas (t i ) − I 1,mod el ( x, t i )) +
1 N
∑ [w ( I i =1
2
]
2
N2
2 ,meas
(ti ) − I 2 ,mod el ( x, ti ))
15
Method
• Inverse method – The unknown parameters X = [Ks
b]
T
hb or
X = [Ks
hb
b
θs ]
T
16
Example
• hi=-100cm • Experiment(1) h0=0cm • Experiment(2) h0=-20cm
17
Example
• Experiment(1) • 427s cm 3 θs = 0.475 3 cm
18
Example
• Experiment(2) • 2973s cm 3 θs = 0.396 3 cm
19
Example
• Experiment • wetting front
20
Results and discussion
• To investigate the necessity for two infiltration experiments in the estimation procedure.
21
Results and discussion
• The six parameter planes • (ks-hb, ks-b, ks-θs , hb-b, hb-θs, b-θs, )
O ( x) =
1 N
∑ [w ( I N1
i =1
1
]
2
1,meas (t i ) − I 1,mod el ( x, t i )) +
1 N
∑ [w ( I i =1
2
]
2
N2
2 ,meas
(ti ) − I 2 ,mod el ( x, ti ))
22
Conclusions
• This in general reduces the uncertainty of the estimates, if the initial water content is estimated with sufficient accuracy. 23
Conclusions
• This in general reduces the uncertainty of the estimates, if the initial water content is estimated with sufficient accuracy. 24
Conclusions
• Water content measurement error of up to ±10% had in general a large effect on the estimated three unknown parameters. 25
Conclusions
• This finding indicates that in practice the four-parameter option would be preferable.
26
20081114
Double ring infiltrometer
27
20081120 Double ring infiltrometer
28
20081120 Double ring infiltrometer
29
20081120 Double ring infiltrometer
30
Thanks for your attention
