Steady Groundwater Flow Simulation towards Ains in a Heterogeneous Subsurface
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Transcript of Steady Groundwater Flow Simulation towards Ains in a Heterogeneous Subsurface
Steady Groundwater Flow Simulation towards Ains
in a Heterogeneous Subsurface
Dr. Amro M. M. ElfekiWater Resources Dept.,
Faculty of Meteorology, Environment and Arid Land Agriculture,
King Abdulaziz University, Jeddah, KSAE-mail: [email protected]
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Presentation Layout• Typical Ains. • Subsurface Heterogeneity.• Modeling Heterogeneity.• GW Model Equation in
“FLOW2AIN”.• Monte-Carlo Approach.• Results. • Conclusions.
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Definition of Ains (Qanats)
• Qanats are underground tunnels, with a canal in the floor of the tunnel, which carries water.
• The difference between the qanat and a surface canal is that the qanat can get water from an underground aquifer.
Source: CharYu, Oz, Jun 21, 2005
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Ain Longitudinal Section
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Ain Cross-Section
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How much water will flow to Ain?
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Subsurface Heterogeneity
• One can easily experience the heterogeneity from most fields by observing huge variation of its properties from point to point (Gelhar, 1993)
• The heterogeneity of subsurface has been a long-existing troublesome topic from the very beginning of the subsurface hydrology (Anderson, 1983)
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Subsurface Heterogeneity (cont.)
Saudi Arabia Geological Survey Web Site
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Space series from Mount Simon sandstone aquifer: Gelhar, (1996).
Laboratory Measurements: Conductivity and porosity.
Observation: Variability of hydrological parameters.
Subsurface Heterogeneity (cont.)
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Modeling Subsurface: Deterministic, Random, or
Stochastic?Purely random?
No Regularity
Pure Random Process
Purely deterministic?
Deterministic Regularity
Pure Deterministic Process
Something in between?
Stochastic Regularity
Stochastic Process
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Why do we need the Stochastic Approach?
• The erratic nature of the subsurface parameters observed at field data
• The uncertainty due to the lack of information about the subsurface structure which is known only at sparse sampled locations
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Geostatistics
Kriging (stochastic interpolation)
Gaussian Random Field
Non-Gaussian Random Field
Simulation of Sedimentary Depositional
Process
a priori knowledge
sedimentary history
geometry of sedimentary structure
Site Specific Information
a priori knowledge
well logs
geophysical data
Koltermann and Gorelick (1996)
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Facts about Each Method• Descriptive
– Quantification is difficult• Process-Imitating
– Conditioning is difficult, too sensitive to initial condition, and computationally demanding
• Structure-Imitating– Lateral variability data is hard to get– Produce multiple, equally probable
images
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Structure Imitating Models
Gaussian Random Fields.
Indicator Random Fields.
Combined Fields.
Fractals Fields.
0 20 40 60 80 100 120 140 160 180 200
-40
-20
0
-3.3 -2.3 -1.3 -0.3 0.7 1.7 2.7
Y=Log (K)
0 200 40 0 600 8 00 1 000 1200 1 400 16 00 1800 2000-40 0
-20 0
0
-10 -8 -6 -4 -2 0 2 4 -5 .0 -3 .0 -1 .0 1 .0 3 .0
0 200 40 0 600 8 00 1 000 1200 1 400 16 00 1800 2000-40 0
-20 0
0
0.0 0 .8 1 .5 2 .3 3 .0
0 200 400 60 0 800 1000 1 200 14 00 1600 1800 2000-400
-200
0
- 8 - 6 - 4 - 2 0 2 4
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0
H o r i z o n t a l D i s t a n c e ( m )
-40 0
-20 0
0
Dep
th (m
)
1 2 3 4
Lo g (H ydra ulic C on ductiv ity m /d ay) L og (H yd raulic C ondu ctiv ity m /day)
Lo g (H ydra ulic C on ductiv ity m /d ay)L og (H ydr aulic C o nductiv ity m /day)
(a) N on-S ta tio na rity in T he M ea n .
(b ) N o n -S ta tiona rity in T he V a ria n ce .
(c) N o n -S ta tion a rity in C orre la tio n L eng th s.
(d) G lo bal N o n - S ta tion a rity.
G eo log ica l S truc ture .
0 200 4 00 600 8 00 1000 12 00 1400 1 600 180 0 2000-40 0
-20 0
0
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)( Z, Z = Cov c jiij
...),(............),(
),(..),(
21
2
212
1212
2
1
p
i
Zp
Z
Z
pZ
ZZCov
ZZCovZZCovZZCov
C
ij
X
Y
0
ZZ
1
p
2 3
Modeling Heterogeneity (LU-decomposition method)
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U L= C where, L is a unique lower triangular matrix, U is a unique upper triangular matrix, and U is LT , i.e., U is the transpose of L.
LU-Decomposition
T21 },...,,{ p
ε U= X
X + μ= Z
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Realization of Variance Ln(K)=0.1
0 1 2 3 4 5H ydraulic Conductiv ity (m /day)
0
0.4
0.8
1.2
1.6
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0
- 1 5
- 1 0
- 5
0-0
.3
-0.1
5 0
0.15 0.
3
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Realization of Variance Ln(K)=0.5
0 2 4 6 8 10Hydraulic Conductivity (m /day)
0
0.2
0.4
0.6
0.8
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0
- 1 5
- 1 0
- 5
0-1
.4-1
.15
-0.9
-0.6
5-0
.4-0
.15
0.1
0.35 0.
60.
85 1.1
1.35
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Realization of Variance Ln(K)=1.
0 4 8 12 16H ydraulic Conductiv ity (m /day)
0
0.2
0.4
0.6
0.8
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0
- 1 5
- 1 0
- 5
0
-3-2
.5 -2-1
.5 -1-0
.5 00.
5 11.
5 22.
5 3
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Realization of Variance Ln(K)=1.5
0 4 8 12 16 20H ydraulic C onductiv ity (m /day)
0
0.2
0.4
0.6
0.8
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0
- 1 5
- 1 0
- 5
0
-4 -3 -2 -1 0 1 2 3 4
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Realization of Variance Ln(K)=2.
0 4 8 12 16 20H ydraulic Conductiv ity (m /day)
0
0.2
0.4
0.6
0.8
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0
- 1 5
- 1 0
- 5
0
-4 -3 -2 -1 0 1 2 3 4
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Steady Groundwater Flow Model in “FLOW2AIN”
where is the hydraulic conductivity,
and is the hydraulic head at location
. ( ) ( ) 0K x x
( )K x
( ) x x
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Model Domain and Boundaries
L x
L yB
H d
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Expected Values and Uncertainty
x
1
1( ) ( ),MC
kk
= MC
x x
22
1
1( ) ( ) ( )MC
kk
= MC
x x x
( )k x x is the hydraulic head at location x
in the kth realization, and
2 ( ) x represents the uncertainty in the predictions.
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Simulation Parameters used in the Numerical Experiment
(MC)Parameter Numerical Value
Geometric mean of hydraulic conductivity
1 m/day
Variance of Ln (K) 0.1, 0.5, 1.0, 1.5, 2
Correlation length in both directions 2 m
No of Monte-Carlo 1000
Domain dimensions Lx=50. m, Ly=20. m
Domain discretezation Dx=dy = 1 m
Water table elev. in the ambient groundwater
1.0 m
Accuracy of computations 0.00001
Ain dimensions 5 m x 5 m
Water surface elevation in Ain 0. m
Ain dimensions H = 5 m. B = 5 m
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Expected Hydraulic Head and Variance
-1.4 -1 -0 .6 -0.2 0.2 0.6 1 1.4
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
-4 -3 -2 -1 0 1 2 3 4
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
Ln (K ) variab ility
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Uncertainty Profiles in Hydraulic Head
L x
L yB
H d
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Expected Flux to Ain, Variance and CV
0 0.4 0.8 1.2 1.6 2Variance of Ln(K )
-5
0
5
10
15
20
25
Exp
ecte
d Fl
ux to
Ain
(m^3
/day
/m'),
Var
ianc
e in
Flu
x
Expected F lux to A inVariance o f F lux to A inC V of Expected F lux
0 1 2 3 4 5H ydrau lic C onductiv ity (m /day)
0
0.4
0.8
1.2
1.6
0 4 8 12 16 20H ydrau lic Conductiv ity (m /day)
0
0.2
0.4
0.6
0.8
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86% Confidence Interval of Expected Flux to Ain
0 0.4 0.8 1.2 1.6 2Variance of Ln(K )
0
4
8
12Fl
ux to
Ain
(m^3
/day
/m')
Expected F lux to A in Upper L im it: E (Q )+Q
Low er L im it: E (Q )-Q
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Conclusions• FLOW2AIN has been developed to study the
influence of subsurface heterogeneity on hydraulic head and water flux to Ains.
• Increasing heterogeneity of the hydraulic conductivity leads to an increase in the hydraulic head uncertainty, and
• Increasing heterogeneity leads to an increase in the expected water discharge to Ain. This reflects the Log-normal distribution of K.
• For Ln(K) Less than 1.5 the uncertainty is relatively low, however, it increases drastically over this value.