Steady Groundwater Flow Simulation towards Ains in a Heterogeneous Subsurface

Steady Groundwater Flow
Simulation towards Ains in
a Heterogeneous Subsurface
Dr. Amro M. M. Elfeki
Water Resources Dept.,
Faculty of Meteorology, Environment and Arid Land
Agriculture,
King Abdulaziz University, Jeddah, KSA
E-mail: amro_elfeki@yahoo.com

9/16/2016 Dr. Amro Elfeki 2
Presentation Layout
• Typical Ains.
• Subsurface Heterogeneity.
• Modeling Heterogeneity.
• GW Model Equation in “FLOW2AIN”.
• Monte-Carlo Approach.
• Results.
• Conclusions.

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

Ain Longitudinal Section

Ain Cross-Section

How much water will flow to
Ain?

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)

(cont.)
Saudi Arabia Geological Survey Web Site

Space series from Mount Simon
sandstone aquifer: Gelhar, (1996).
Laboratory
Measurements:
Conductivity and
porosity.
Observation:
Variability of
hydrological parameters.
(cont.)

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

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

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)

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

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)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-400
-200
0
-10 -8 -6 -4 -2 0 2 4 -5.0 -3.0 -1.0 1.0 3.0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-400
-200
0
0.0 0.8 1.5 2.3 3.0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-400
-200
0
-8 -6 -4 -2 0 2 4
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Horizontal Distance (m)
-400
-200
0
Depth(m)
1 2 3 4
Log (Hydraulic Conductivity m/day) Log (Hydraulic Conductivity m/day)
Log (Hydraulic Conductivity m/day)
Log (Hydraulic Conductivity m/day)
(a) Non-Stationarity
in The Mean.
(b) Non-Stationarity
in The Variance.
(c) Non-Stationarity
in Correlation Lengths.
(d) Global Non-
Stationarity.
Geological Structure.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-400
-200
0

)( Z,Z= Covc jiij
...),(
.....
....
...),(
),(..),(
2
1
2
2
12
121
2
2
1

















p
i
Zp
Z
Z
pZ
ZZCov
ZZCov
ZZCovZZCov




C


i
j
X
Y
0
Z
Z
sij
1
p



2 3

Modeling Heterogeneity
(LU-decomposition method)

UL=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
T
21 },...,,{ p
εU=X
X+μ=Z

Realization of Variance
Ln(K)=0.1
0 1 2 3 4 5
Hydraulic Conductivity (m/day)
0
0.4
0.8
1.2
1.6
pdf
0 5 10 15 20 25 30 35 40 45 50
-20
-15
-10
-5
0
-0.3
-0.15
0
0.15
0.3

Ln(K)=0.5
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
pdf
0 5 10 15 20 25 30 35 40 45 50
-20
-15
-10
-5
0
-1.4
-1.15
-0.9
-0.65
-0.4
-0.15
0.1
0.35
0.6
0.85
1.1
1.35

Ln(K)=1.
0 4 8 12 16
0
0.2
0.4
0.6
0.8
pdf
0 5 10 15 20 25 30 35 40 45 50
-20
-15
-10
-5
0
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3

Ln(K)=1.5
0 4 8 12 16 20
0
0.2
0.4
0.6
0.8
pdf
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

Ln(K)=2.
0 4 8 12 16 20
0
0.2
0.4
0.6
0.8
pdf
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

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

Model Domain and
Boundaries
Lx
Ly
B
H d

Expected Values and
Uncertainty
x
1
1
( ) ( ),
MC
k
k
=
MC 
  x x
 
22
1
1
( ) ( ) ( )
MC
k
k
=
MC


   x x x
( )k x xis the hydraulic head at location x
in the kth realization, and
2
( ) x represents the uncertainty in the predictions.

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

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) variability

Uncertainty Profiles in
Hydraulic Head
Lx
Ly
B
H d

Expected Flux to Ain,
Variance and CV
0 0.4 0.8 1.2 1.6 2
Variance of Ln(K)
-5
0
5
10
15
20
25
ExpectedFluxtoAin(m^3/day/m'),VarianceinFlux
Expected Flux to Ain
Variance of Flux to Ain
CV of Expected Flux
0 1 2 3 4 5
0
0.4
0.8
1.2
1.6
pdf
0 4 8 12 16 20
0
0.2
0.4
0.6
0.8
pdf

86% Confidence Interval of
Expected Flux to Ain
0 0.4 0.8 1.2 1.6 2
Variance of Ln(K)
0
4
8
12
FluxtoAin(m^3/day/m') Expected Flux to Ain
Upper Limit: E(Q)+Q
Lower Limit: E(Q)-Q

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.

Steady Groundwater Flow Simulation towards Ains in a Heterogeneous Subsurface

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Steady Groundwater Flow Simulation towards Ains in a Heterogeneous Subsurface