Lecture Packet 9.Numerical Modeling of Groundwater Flow

  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 particular conditions – confined, unconfined, isotropic, anisotropic, hom ogeneous, het erogeneous, infinit e, finit e, st eady, t ransient .
• Varying levels of com plexity: 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 appropriate I Cs and BCs
• Discretize the system
• Approxim ate the PDE corresponding to the discretization
• Solve the approxim ated PDE on a com puter
• Com m only finite differences or finite elem ents for GW flow
• Advantages: can handle com plex geom etries, I Cs, and C conditions; 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”
• Mathem atical “ m odel” – a PDE
• Num erical “ m odel” – a particular technique is applied
• Com puter or sim ulation “ 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 Grid – a represent at ion of t he physical dom ain t hat enables one t o account for t he boundaries and int ernal feat ures

  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

• =

  2D flow equat ion

  2

2 T

  ∂ x ∂ y ∂ t

  Approxim at ing t he Tim e Derivat ive:

  Back w ar d Differ en ce: h − h

  ⎛ ∂ h ⎞ ∆ h i , j i , j ,n i , j ,n − 1

  ≈ ≈ ⎜⎜ ⎟⎟ ∆ t ∆ t ⎝ ∂ t ⎠ n t

  ∆ _n

  h head h _{n- 1} ∆ h t _{n- 1 n} ∆ t im e End of t t w here, t im e st ep n = t he current t im e st ep index

  ∆ t = tim e step 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 i,j + 1 h i,j h h i+ 1,j _{i- 1,j}

i- 1/ 2 i+ 1/ 2

  y

  h i,j - 1 ∆ y

  x

  ∆ x

  I n t h e x - dir ect ion

  : Approxim at e derivat ive at locat ion h : _{i,j 2}

  ⎛ ∂ h ⎞ ₂ ⎜⎜ ⎟⎟ x

  ∂ ⎝ ⎠

  Approxim at e t he se con d spat ial derivat ive at i,j as follow s:

  h h ∂ ∂

⎛ i 1 / ,

  2 j i 1 / , 2 j ⎞ − + −

  2 ⎜⎜ ⎟⎟ ∂ x ∂ x

  ⎛ h ⎞ h ∂ ∂ ∂ ⎛ ⎞ ⎝ ⎠

  = ≈ ⎜ ⎟

  2 ⎜⎜ ⎟⎟ x ∂ x ∂ x ∆ x

  ∂ ⎝ ⎠ ⎝ ⎠

  We can approxim at e t he fir st spat ial derivat ive at i- 1/ 2,j and i+ 1/ 2,j as follows:

  ∂ h h − h ∂ h h − h ⎛ i / , j ⎞ ⎛ ^{1 2 i, j i , 1 j ⎞ ⎛ i 1 / , 2 j ⎞ ⎛ i , 1 j i, j ⎞} − − + + and

  ≈ ≈ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ∂ x ∆ x ∂ x ∆ x ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

  Subst it ut e t he above equat ions t o obt ain:

  h − h h − h ⎛ ⎞ ^{i , 1 j i , j i , j i ,} − ^{1 j}

• ₂

  − ⎜⎜ ⎟⎟ ⎛ ⎞ ∂ h ∆ x ∆ x _{i , j} ⎝ ⎠

  ⎜ ⎟

  or ₂ ≈ ⎜ ⎟

  ∂ x ∆ x ⎝ ⎠ ₂ ⎛ ⎞

  2 ∂ + h h − h h _{i , j i , 1 j i , j i 1 j}

  − + , ⎜ ⎟ ₂ ≈ ₂ ⎜ ⎟

  ∂ x ∆ x ⎝ ⎠

  I n t he y- direct ion: ₂

  ⎛ ⎞

  2 ∂ + h h − h h ^{i, j i , j} − ^{1 i , j i , j 1}

  ⎜ ⎟

• ₂

  ≈ ₂ ⎜ ⎟ ∂ y ∆ y ⎝ ⎠

  Com bining Flow Equat ion Term s:

  2

  2 ∂ h ∂ h ∂ h ⎛ ⎞

• i j , i j , S i j ,

  =

  2

  2 ⎜⎜ ⎟⎟ x y T t

  ∂ ∂ ∂ ⎝ ⎠ η t ∆

  For t he backwards difference t im e derivat ive ( not e: all left hand side values of h are for t im e = n- 1) Linear diff. eq’n. This eq’n is

  − 2 h + h − 2 h + h −

  h h h _{i − , 1 j i j , i + , 1 j i j , − 1 i j , i j , +1 S h i j , , n i j , , n −}

₁

  = + _{2 2} not explicit for h at any ∆ x ∆ y T ∆ t

  part icular t im e I f ∆ x and ∆ y are equal – this is the finite difference groundwater flow equation

  _{i 1 j i 1 j i , j i , j i , j i , j , n i , j , n} + + + h h h h − 4 h − h − , + , − 1 + 1 S h − 1 ₂ = ∆ x T ∆ t

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

  h − h ⎛ ⎞ ∂ h i , j , n i , j , n − 1

  S ≈ S ⎜⎜ ⎟⎟ = 0 ∂ t ∆ t ⎝ ⎠ _{i 1 j i 1 j i , j i , j i , j} + + + h h h h − 4 h

  − , + , − 1 + 1 ₂ = 0 ∆ x h + h + h + h − 4 h = 0 _{i − , 1 j i + , 1 j i , j − 1 i , j + 1 i , j i 1 j i 1 j i , j i , j} + + + h h h h

  − , + , − 1 + 1 = h _{i , j}

  4

  h _{i,j + 1}

  10

  65 100 h _{i,j h i+ 1,j} h i- 1,j

  11

  10

  9 h i,j - 1

  10 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

  d H '

  T − w = 0 or _{2 2} dx ' d H w ₂ = dx T ∆ x

  h ^{1 h 2 h 3} H= 9 H= 10

  N ode

  0 1 2 3 4 The nodal finit e difference equat ion is:

  − 2 ' −1 +1 ₂ =

• h h h _{i , j i , j i , j w}

  T ∆ x

  Or ₂

  ' x w

  ( ) ∆ h − 2h + h = _{i , j i , j i , j}

  −1 +1 T

  Node 1:

  1H + - 2h ^{1 + 1h 2 = 0} 1( 10) + - 2h ^{1 + 1h 2 = 0}

• 2h + 1h = - 10
_{1 2} Node 2: 1h ^{1 + - 2h 2 + 1h 3 = 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 2} ' ( 1 0 ) .

  1 ( ) ∆ x w

  ∆x = 0.1, Given = = 1 .

01 T

  1h ^{2 + - 2h 3 + 1H 4 = 1} 1h + - 2h + 1( 9) = 1 _{2 3}

  1h ^{2 + - 2h 3 = 1- 9 = - 8} The syst em of finit e- difference equat ions consist s of 3 equat ions and 3 unknow ns

  1 − 2 h h = −10 _{1 2}

  1 h − 2 h h = _{1 2 3}

  1 h − 2 h = − 8 _{2 3} Un k n ow n s

  Kn ow n s

  Or in coeffiecient m at rix form

  1 ⎛− 2 ⎞⎛ h ⎞ ⎛−10⎞ ₁ ⎜ ⎟⎜ ⎟ ⎜ ⎟ 1 ⎜ 1 − 2 ⎟⎜h ^{2 ⎟ = ⎜ 0 ⎟} ⎜ ⎟⎜ ⎟ ⎜ ⎟

  1 ₃ − 2⎠⎝h ⎝ ⎠ ⎝ − 8 ⎠

  FD Coeff. Unknow n RHS cont aining boundary Mat rix Heads condit ions and know n pum ping 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 :

  10 9 .

  5 h

  ⎛ ⎞ ⎛ ⎞ ₁ ⎜ ⎟ ⎜ ⎟

  9.5

  9 h =

  ⎜ ^{2 ⎟ ⎜ ⎟} ⎜ ⎟ ⎜ ⎟ 8 .

  5 h ₃

  9.0 ⎝ ⎠ ⎝ ⎠

  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

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

  ∆t; this give the spatial distribution of • Solve for heads at end of first t im e st ep 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

  Ah n = b* w here b* = b’ + h n- 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 inutes. 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 .

  Tim e St e ppin g

S h − h

i , n i , n

  − 1 − 2 h + h ⎧ h i ⎫ ^{1 i i 1}

  − =

• ⎨ ^{2 ⎬}

  T ∆ x ∆ t ⎩ ⎭

  n? n- 1? Som et hing

  Cr a n k -

  else?

  I m plicit N ich olson _n Ex plicit

  h head ∆ h _{n- 1} h _{n- 1 n} ∆ t t t

  T ∆t

  ( h − 2 h + h ) = h − h _{i i i i , n i , n 2} − 1 + 1 − 1 s ∆ x

  Ex plicit : h i,n = ch i- 1,n- 1 + ( 1- 2C) h i,n- 1 + ch i+ 1,n- 1 T ∆ t

  Where: c

  = ₂ s ∆ x Easy t o calculat e – no linear algebra, but unst able if t im e- st eps are t oo large.

  I m plicit : ch i- 1,n - ( 1+ 2C) h i,n + ch i+ 1,n = - h i,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 c c Cr a n k N ich olson : ) h − (1+ h c h = ( c − 1)h + − h + h _{i 1 n i , n i 1 n i 1 n ( i 1 n i 1 n )}

  − , + , − , − , − 1 − , − 1

  2

  2

  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. Flux Boundary – replace first derivat ive t erm w it h const ant

  h − h h − h ⎛ ⎞ _{i − , 1 j i , j i , j i + , 1 j 2} − ⎜⎜ ⎟⎟

  ⎛ ∂ ⎞ h ∆ x ∆ x _{i , j}

  ⎝ ⎠ ⎜ ⎟ ₂ ⎟ ≈ ∆ x ⎝ ⎜ ∂ x ⎠ h − h _{i , j i 1 j Q}

• ,

  = ∆ x T