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]

mailto:[email protected]

05/03/2023 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

http://www.urbandictionary.com/author.php?author=CharYu


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)


Subsurface Heterogeneity (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.

Subsurface Heterogeneity (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 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


)( 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)


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


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

pdf

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



0 2 4 6 8 10Hydraulic Conductivity (m /day)

0

0.2

0.4

0.6

0.8

pdf

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


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

pdf

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



0 4 8 12 16 20H ydraulic C onductiv ity (m /day)

0

0.2

0.4

0.6

0.8

pdf

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


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

pdf

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


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

L x

L yB

H d


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.


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 ) variab ility


Uncertainty Profiles in Hydraulic Head

L x

L yB

H d


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

pdf

0 4 8 12 16 20H ydrau lic Conductiv ity (m /day)

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 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


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

Engineering

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