conceptual model a descriptive representation of a groundwater system that incorporates an...
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Conceptual Model
A descriptive representationof a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget. May include information on water chemistry.
a set of equations that describesthe physical and/or chemicalprocesses occurring in a system.
Mathematical Model
R x y Q
y
x
z
1. Consider flux (q) through REV2. OUT – IN = - Storage3. Combine with: q = -KK grad h
q
Derivation of the Governing Equation
*)()()( Rt
hS
z
hK
zy
hK
yx
hK
xszyx
Rt
hS
y
hT
yx
hT
xyx
)()(
Rt
hS
y
hhK
yx
hhK
xyx
)()(
2D confined:
2D unconfined w/Dupuit assumptions:
Storage coefficient (S) is either storativity or specific yield.S = Ss b & T = K b
General 3D equation
Types of Boundary Conditions
1. Specified head
2. Specified flow (including no flow)
3. Head-dependent flow
Toth Problem
Laplace Equation
2D, steady state
02
2
2
2
z
h
x
h0
x
h0
x
h
0z
h
h = c x + zo
Cross section through an unconfined aquifer.
Water table formsWater table formsthe upper boundary conditionthe upper boundary condition
b
h
ocean ocean
groundwater divide
“Confined” Island Recharge Problem
R
x = 0 x = Lx = - L
We can treat this system as a “confined” aquifer if we assume that T= Kb.
datum
T
R
y
h
x
h
2
2
2
2
Arealview
Water table is the solution.Water table is the solution.
Poisson’s Eqn.
2D horizontal flow through an unconfined aquifer where T=Kb.
b hocean ocean
groundwater divide
R
x = 0 x = Lx = - L
datum
Unconfined version of the Island Recharge Problem
K
R
y
h
x
h 22
22
2
22
Water table is the solution.Water table is the solution.
(Pumping can be accommodated by appropriatedefinition of the source/sink term.)
2D horizontal flow through an unconfined aquifer underthe Dupuit assumptions.
02
2
2
2
y
h
x
h
02
2
2
2
z
h
x
h
Vertical cross section through an unconfinedaquifer with the water table as the upper boundary.
2D horizontal flow in a confined aquifer; solutionis h(x,y), i.e., the potentiometric surface.
02
2
2
2
y
v
x
v
2D horizontal flow in an unconfined aquiferwhere v= h2. Solution is h(x,y), i.e., the water table.
All three governing equations are the LaPlace Eqn.
t = 0
t > 0
BC:h (0, t) = 16 m; t > 0h (L, t) = 11 m; t > 0
datum
0 L = 100 mx
IC: h (x, 0) = 16 m; 0 < x < L(represents static steady state)
t
h
T
S
x
h
2
2
Reservoir Problem
1D transient flow through a confined aquifer.
Three options:
• Iteration
• Direct solution by matrix inversion
• A combination of iteration and matrix solution
4
11,1,
1,1,11
,
m
jim
jimji
mjim
ji
hhhhh
Gauss-Seidel Formula for 2D Laplace Equation
General SOR Formula
)( ,1
,,1
,mji
mji
mji
mji hhhh
Relaxation factor= 1 Gauss-Seidel< 1 under-relaxation>1 over-relaxation, typically between 1 and 2 (e.g., 1.8)
T
Rahhhhh
mji
mji
mji
mjim
ji 44
211,1,
1,1,11
,
Gauss-Seidel Formula for 2D Poisson Equation
SOR Formula
)( ,1
,,1
,mji
mji
mji
mji hhhh
Relaxation factor= 1 Gauss-Seidel< 1 under-relaxation>1 over-relaxation
(Eqn. 3.7W&A)
n
n+1
n+2
n+3
Steady state
t
t
t
Initial conditions(at steady state)
Transient Problemsrequire time steps.
Time levels
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
211
)(
2
Explicit Approximation
t
h
T
S
x
h
2
2
t
hh
T
S
x
hhh ni
ni
ni
ni
ni
1
2
11
111
)(
2
Implicit Approximation
Or weighted average
• Explicit solutions do not require iteration but are unstable with large time steps.
• We can derive the stability criterion by writingthe explicit approx. in a form that looks like the SORiteration formula and setting the terms in theposition occupied by omega equal to 1.
• For the 1D governing equation used in the reservoirproblem, the stability criterion is:
1)(
22
xS
tT <
T
xSt
2)(5.0
<or