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321 zyx

grad

grad is a vector

q = - K grad h

321)( z

h

y

h

x

hhgrad

Darcy’s law

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321 zyx qqqq

q is a vector

t

hSW

z

q

y

q

x

qs

zyx

Transient mass balance equation:

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0

z

q

y

q

x

q zyx

The dot product of grad and q is

div q = 0

grad q = 0

Steady State Mass Balance Equation with W = 0

q = 0 (del notation)

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Wt

hS

z

hK

zy

hK

yx

hK

xszyx

)()()(

Anisotropic medium: K at a point varies with directionKx Ky Kz

orKx = Ky Kz

Heterogeneous medium: K varies in space

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Homogeneous, isotropic Homogeneous, anisotropic

A A B B

Heterogeneous, isotropic Heterogeneous, anisotropic

AA

B

B

Characteristics of K in two dimensions

Kz

Kx

Kx Kz

Kx = Kz = K

school.edhole.com

Homogeneous, isotropic Homogeneous, anisotropic

A A B B

Heterogeneous, isotropic Heterogeneous, anisotropic

AA

B

B

Characteristics of K in two dimensions

Kz

Kx

Kx Kz

Kx = Kz = K

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Wt

hS

z

hK

zy

hK

yx

hK

xszyx

)()()(

2

2

)(x

h

x

h

x

0

2

2

2

2

2

2

z

h

y

h

x

h

2 h = 0

Laplace equation

homogeneous, isotropic porous medium: Kx = Ky = Kz = K

steady state conditions: 0t

h

W = 0

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Toth Problem (2D)

Impermeable Rock

Groundwater divide

Groundwater divide

2D, steady state

02

2

2

2

z

h

x

h

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i, jj

i

The 5 point starin a regular grid

i+1, ji-1, j

i, j-1

i, j+1

x

y

x = y

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04 ,1,1,,1,1 jijijijiji hhhhh

4

11,1,

1,1,11

,

m

jim

jimji

mjim

ji

hhhhh

Solution by iteration

Gauss-Seidel Iteration

4/)( 1,1,,1,1, jijijijiji hhhhh

Iteration indices

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Iterationplanes

initial guessesm

solution

m+2

m+1

m+3

41,1,,1,11

,

mji

mji

mji

mjim

ji

hhhhh

4

11,1,

1,1,11

,

m

jim

jimji

mjim

ji

hhhhh

Gauss-Seidel Iterationschool.edhole.com

j

i

10

10

10

10

10

10

9 8

9 8

7

7

6

6

5

5

1 2 3 4

1

2

3

4

5

6

2D examplewith specified headboundary conditions

(Assume an initialguess of h =10 forall interior nodes)

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j

i

10

10

10

10

10

10

9 8

9 8

7

7

6

6

5

5

1 2 3 4

1

2

3

4

5

6

Examplewith specified headboundary conditions

9.75

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j

i

10

10

10

10

10

10

9 8

9 8

7

7

6

6

5

5

1 2 3 4

1

2

3

4

5

6

Examplewith specified headboundary conditions

8.44

9.75

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j

i

10

10

10

10

10

10

9 8

9 8

7

7

6

6

5

5

1 2 3 4

1

2

3

4

5

6

Examplewith specified headboundary conditions

9.75

9.94

8.44

school.edhole.com

j

i

10

10

10

10

10

10

9 8

9 8

7

7

6

6

5

5

1 2 3 4

1

2

3

4

5

6

Examplewith specified headboundary conditions

9.94

school.edhole.com

i, jj

i

Gauss-Seidel Iteration

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C4=(C3+C5+B4+D4)/4school.edhole.com

4

33243

ECDDD

Example of spreadsheet formula

Toth Problem

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