Lecture Packet 9.Numerical Modeling of Groundwater Flow

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1.72, Groundwat er Hydrology Prof. Charles Harvey

Le ct u r e Pa ck e t # 9 : N u m e r ica l M ode lin g of Gr oun dw a t e r Flow

Sim ulat ion: The predict ion of quant it ies of int erest ( dependent variables) based upon an equat ion or series of equat ions t hat describe syst em behavior under a set of assum ed sim plificat ions.

Gr ou n dw a t e r Flow Sim u la t ion

• Predict hydraulic heads ( 1D, 2D, 3D)

• For part icular condit ions – confined, unconfined, isot ropic, anisot ropic, hom ogeneous, het erogeneous, infinit e, finit e, st eady, t ransient .

• Varying levels of com plexit y:

Analyst ic solut ions – Theim , Theis, et c.

Advant ages: exact , sim ple, cheap, can provide sufficient insight Analog sim ulat ion

Physical m odels – scale m odels of aquifers Num erical Sim ulat ion

N u m e r ica l Sim u la t ion

• Given a PDE and appropriat e I Cs and BCs

• Discret ize t he syst em

• Approxim at e t he PDE corresponding t o t he discret izat ion

• Solve t he approxim at ed PDE on a com put er

• Com m only finit e differences or finit e elem ent s for GW flow

• Advant ages: can handle com plex geom et ries, I Cs, and C condit ions; can be used for nonlinear syst em s.

Be w a r e of t h e t e r m “M ode l” a n d h ow it is u se d

• “ A m odel should be used as sim ple as possible, but not sim pler”

• Mat hem at ical “ m odel” – a PDE

• Num erical “ m odel” – a part icular t echnique is applied

• Com put er or sim ulat ion “ m odel” – a code

Fin it e D iffe r e n ce s

A num erical m et hod t hat approxim at es t he governing PDE by replacing t he derivat ives in t he equat ion w it h t heir respect ive difference represent at ions. Procedure involves: Grid ( or Mesh) and Equat ion

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Equ a t ion – D iffe r e n ce Appr ox im a t ion of D e r iva t ive s

2 2

∂

h

∂

h

S

∂

h

∂

x

2

+

∂

y

2

=

T

∂

t

2D flow equat ion

Approxim at ing t he Tim e Derivat ive:

Back w ar d Differ en ce:

⎛ ∂

h

_i_,_j

⎟⎟

⎞

≈

∆

h

≈

h

i , j ,n

−

h

i , j ,n − 1

⎜⎜

⎝ ∂

t

_⎠

_∆

∆

t

n

∆

t

∆

h

∆

t hn- 1

hn head

t im e _tn- 1

tn End of

w here, t im e st ep

n = t he current t im e st ep index

∆ t = t im e st ep

i,j = x and y coordinat e indices Approxim at ing t he Space Derivat ives:

Consider a 2D discret izat ion, if w e assum e t hat t he grid spacing in t he x- direct ion and y- direct ion are t he sam e, our discret ized grid for an int ernal node w ill be:

h h

h

∆ x

∆ y y

x i+ 1,j

i,j i- 1,j

i,j + 1

i,j - 1

i- 1/ 2 i+ 1/ 2

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h

_i₋_,₁_j

+

h

_i₊_,₁_j

+

h

_i_,_j₋₁

+

h

_i_,_j₊₁

−

4 h

_i_,_j

_{S h}

_i_,_j_,_n

−

h

_i_,_j_,_n₋₁

=

∆

x

2

T

∆

t

What does t he finit e- difference equat ion indicat e for st eady st at e condit ions?

S

∂

h

≈

S

_⎜⎜

⎛

h

i , j , n

−

h

i , j , n −1

⎞

_⎟⎟

=

0 ∂

t

_⎝

∆

t

h

⎠

i −1 ,j

+

h

i +1 ,j

+

h

i , j −1

+

h

i , j +1

−

4 h

i , j

=

0 ∆

x

2

h

_i₋₁_,_j

+

h

_i₊₁_,_j

+

h

_i_,_j₋₁

+

h

_i_,_j₊₁

−

4 h

_i_,_j

=

0

i −1 ,j

+

h

i +1 ,j

+

h

i , j −1

+

h

i , j +1

=

h

_i_,_j

4

h h

h

h 10 10 10

9 11

100 65

i+ 1,j i,j

i- 1,j

i,j + 1

i,j - 1

Flow

Value of head at node is t he average of t he surrounding nodes ( for SS isot ropic hom ogeneous case)

Consider t he 1D st eady- st at e flow equat ion w it h a sink is:

2

H

d

T

−

w

'

=

0

or

dx

2

H

d

w

'

=

dx

2

T

∆ x

h1 h2 h3

H= 10 H= 9

N ode

0 1 2 3 4

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The nodal finit e difference equat ion is:

h

i , j −1

−

2 h

i , j

+

h

i , j +1

w

'

=

∆

x

2

T

Or

( )

2

∆

x

w

'

=

h

_i_,_j₋₁

−

2 h

_i_,_j

+

h

_i_,_j₊₁

T

Node 1:

1H0 + - 2h1 + 1h2 = 0 1( 10) + - 2h1 + 1h2 = 0 - 2h1 + 1h2 = - 10 Node 2:

1h1 + - 2h2 + 1h3 = 0 Node 3:

Node 3 has a forcing t erm due t o t he pum ping of t he w ell. This t ranslat es int o an init ial r ight hand side of t he equat ion:

2

∆

x

w

'

( 1

0 )

.

2

1

Given ∆x = 0.1,

( )

=

1 T

.

01

1h2 + - 2h3 + 1H4 = 1 1h2 + - 2h3 + 1( 9) = 1

1h2 + - 2h3 = 1- 9 = - 8

The syst em of finit e- difference equat ions consist s of 3 equat ions and 3 unknow ns

−

2 h

₁

1 h

₂

=

−

10 h

₁

−

2 h

₂

1 h

₃

=

0

1 h

₂

−

2 h

₃

=

−

8

Un k n ow n s Kn ow n s

Or in coeffiecient m at rix form

⎛−

2

1 ⎞⎛

h

1

⎞

⎛−

10 ⎞

⎜

⎟⎜ ⎟ ⎜

⎟

⎜

1 −

2

1 ⎟⎜

h

2

⎟ = ⎜

0 ⎟

⎜

⎟⎜

⎜

⎝

1 −

2 _⎠⎝

h

3

⎠

⎟

⎝ −

8 ⎠

⎟

FD Coeff. Unknow n RHS cont aining Mat rix Heads boundary

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Or in m at rix not at ion it can be w rit t en as Ah = b’

A is t he m at rix of difference coefficient s H is t he vect or of unknow n heads b’ is t he RHS vect or of know n quant it ies

A com put at ional linear solve yields t he vect or h:

2

3

8.5

1 2 3 4

Tr a n sie n t Sim u la t ion Usin g Fin it e D iffe r e n ce s

Procedure: March Through Tim e

h

₁

h

10

5 .

9 ⎛

⎜

⎛

⎜

⎞

⎟

⎞

⎟

9

9.5

=

5 .

8 ⎝

⎠

⎝

⎠

9.0

• St art w it h init ial condit ions ( t hese are known)

• Solve for heads at end of first t im e st ep ∆t ; t his give t he spat ial dist ribut ion of head ( a m ap) aft er a sm all t im e increm ent .

•

Given known heads at end of first t im e st ep solve for heads at t he end of t he second t im e st ep.

Wit h known value at t he end of t im e st ep solve for next t im e st ep – t his is called m arching t hrough t im e

Ahn = b* w here b* = b’ + hn- 1

The right - hand side alw ays cont ains know ns. The m at rix A is t he m at rix of finit e difference coefficient s reflect ing t he syst em param et ers and discret izat ion.

So if you can solve spat ial equat ions for one t im e st ep. Then you can solve it for as m any t im e st eps as you like.

The t im e st ep m ust be sm all w hen changes in heads are rapid – such as, w hen you st art t o pum p a w ell, ∆t m ust be seconds or m inut es. I t can be increased as changes in head becom e sm aller.

FD Sim u la t ion M ode ls: codes t hat solve t he above syst em of linear algebraic equat ions – fairly robust .

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Tim e St e ppin g

⎧

h

_i₋₁

−

2 h

_i

+

h

_i₊₁

⎫

S h

i ,n

−

h

i ,n − 1

⎨

⎬

=

T

∆

t

⎩

∆

x

2

⎭

_{n? n- 1?}

Som et hing else?

head

Ex plicit Cr a n k

-N ich olson I m plicit

∆

h

∆

t hn- 1

hn

tn- 1 tn

∆

t

T

−

2 h

i

+

h

i + 1

)

=

h

, ,

(

h

i − 1 i n

−

h

i n − 1

s

∆

x

2

Ex plicit : hi,n = chi- 1,n- 1 + ( 1- 2C) hi,n- 1 + chi+ 1,n- 1

t

T

∆

Where:

c

=

s

∆

x

2

Easy t o calculat e – no linear algebra, but unst able if t im e- st eps are t oo large. I m plicit : chi- 1,n - ( 1+ 2C) hi,n + chi+ 1,n = - hi,n- 1

Result s in syst em of linear equat ions t hat m ust be solved sim ult aneously. St able, but issues of accuracy.

c

h

Cr a n k N ich olson :

2

i − ,1 n

−

(1

+

c

)

h

i ,n

+

c

h

_i₊_,₁_n

=

(

c

−

1)

h

_i₋_,₁_n

−

c

(

h

_i₋₁_,_n₋₁

+

h

_i₋_,₁_n₋₁

)

2

Not m uch m ore num erical expense t han fully im plicit , but m ore accurat e Bou n da r y Con dit ion s:

Const ant Head – Don’t w rit e equat ion for const ant head node. Use value of const ant head in equat ions for neighboring nodes.