Inflow = Outflow ? S

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Transient Water Balance Eqn.
Inflow Outflow ? S
Recharge
Discharge
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OUT IN
?x ?y ?z
- ?V/ ?t
change in storage
Ss ?V / (?x ?y ?z ?h)
?V Ss ?h (?x ?y ?z)
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OUT IN
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Law of Mass Balance Darcys Law
Governing Equation for Groundwater Flow
-------------------------------------------------
-------------- div q - Ss (?h ??t) W
(Law of Mass Balance) q - K grad h
(Darcys Law) div (K grad h)
Ss (?h ??t) W

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Storage coefficient is either storativity (S) or
specific yield (Sy). S Ss b ( Tx Kx b Ty
Ky b) Ss is specific storage.
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End
Transient problems require multiple arrays of
head values
n3
Time planes
?t
n2
?t
n1
?t
Initial conditions
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2D confined
1D, transient, homogeneous, isotropic, confined,
no sink/source term
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Reservoir Problem
t 0
Confined Aquifer
1D, transient
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t 0
datum
x
0
L 100 m
t 0
IC h (x, 0) 16 m 0 x L (represents
static steady state)
Modeling rule Initial conditions should
represent a steady state configuration of heads.
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h1
h2
datum
x
0
L 100 m
At t tss the system reaches a new steady
state h(x) ((h2 h1)/ L) x h1
analytical solution equation 4.12 (WA)
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Governing Eqn. for transient Reservoir Problem
1D, transient, homogeneous, isotropic, confined,
no sink/source term
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Explicit Approximation
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(No Transcript)
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• Explicit approximations are unstable unless
• small time steps are used.

• We can derive the stability criterion by
  writing
• the explicit approx. in a form that looks like
  the SOR
• iteration formula and setting the terms in the
• position occupied by ? equal to 1.

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For the reservoir problem T 0.02 m2/min S
0.002 ?x 10 m
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Explicit Solution
Spreadsheet
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?t 5 min
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Transient Water Balance Eqn.
Inflow Outflow ?Storage
? Storage V2 V1
Recharge
Discharge
V2 gt V1
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Transient Water Balance Eqn.
Inflow Outflow ?Storage
? Storage V2 V1
Recharge
t1
Discharge
V2 lt V1
Inflow -?Storage Outflow
Inflow abs(?Storage) Outflow
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Water Balance- version 1

• ?Storage V(t2)- V(t1)
• IN gt OUT then water going into storage
• and ?Storage is
• OUT gt IN then water is coming out of storage
• and ?Storage is
• IN OUT ?Storage

MODFLOW Convention Water coming out of storage
goes into the aquifer (IN column). Water going
into storage comes out of the aquifer (OUT
column).
ABS(Flow out)
Flow in
ABS(?Storage)
?Storage
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Water Balance- version 2

• ?Storage V(t1)- V(t2)
• IN gt OUT then water going into storage
• and ?Storage is -
• OUT gt IN then water is coming out of storage
• and ?Storage is
• IN - OUT ?Storage

Convention Water coming out of storage goes
into the aquifer (IN column). Water going into
storage comes out of the aquifer (OUT column).
Total IN ABS(Total Out)
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Reservoir Problem Water Balance
t gt 0
IN
OUT
?Storage
IN change in storage OUT
(where here the change in storage term is
intrinsically positive)
Flow out
Flow in
Convention Water coming out of storage goes
into the aquifer ( column).
?Storage
Total IN ABS(Total Out)
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Units
Flow out
Flow in
?Storage
L3/T (m3/min)
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OUT IN
?x ?y ?z
- ?V/ ?t
change in storage
Ss ?V / (?x ?y ?z ?h)
?V Ss ?h (?x ?y ?z)
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Storage Term for the Water Balance
?V Ss ?h (?x ?y ?z)
3D problem
?t
?t
In 1D Reservoir Problem, ? y is taken to be equal
to 1.
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Water Balance
Error ABS(Total IN- Total Out) / Min(Total
INTotal Out)
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Stablity criterion for explicit solution
For the reservoir problem T 0.02 m2/min S
0.002 ?x 10 m
Judging by the water balance error, ?t 5 min
may be too large.