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GROUND WATER MANAGEMENT. PROBLEM DESCRIPTION. Aquifer of irregular geometry Pond - source of recharge Choose disposal sites to minimize the pollutant impact at various locations. MATHEMATICAL FORMULATION OF THE SIMULATION MODELS. Pollutant advection dispersion equation - PowerPoint PPT Presentation
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GROUND WATER MANAGEMENT
PROBLEM DESCRIPTION
• Aquifer of irregular geometry• Pond - source of recharge• Choose disposal sites to minimize
the pollutant impact at
various locations
MATHEMATICAL FORMULATION OF THE SIMULATION MODELS
• Pollutant advection dispersion equation
• Steady state ground water flow equation
t
C
z
CD
zx
CD
yx
CD
x
dt
dCCv
zCv
yCv
x
zyx
inzyx
0z
hK
zy
hK
yx
hK
xR zyx
ADVECTION PROCESSES
• Seepage velocity
– K hydraulic conductivity
– n effective porosity
– h hydraulic head
• Advective flux
dx
dh
n
Kvx
nCvF xx
x
h
X2x1
h2
h1
vx
MATHEMATICAL FORMULATION OF THE MANAGEMENT MODEL
• Maximize the objective function
• subject to
R(f){c*}
{f}0• where {u}=(1,1, …, 1)• f vector pollutant disposal rates
• Steady state ground water flow equation
}f{}u{ZMax T
SOLUTION OF THE MANAGEMENT MODEL
• Transform the inequalities in equalities and modify the objective function– R(f)={c*}+dc
• where p(dc) is a negative function if dc<0 and p(dc) = 0 if dc>0 • Use a constrained optimization procedure to solve the modified problem
• Steady state ground water flow equation
)dc(p}f{}u{ZMax T
OPTIMIZATION TECHNIQUES
• Steepest descent method
• Example: F(x,y)=x2+2y2-4x-4y+6
0x00 x
F)xx()x(F)x(F
0x0new x
Fxx
0x
2
0new x
F)x(F)x(F
x
y
OPTIMIZATION TECHNIQUES (ctd.)
• Newton methods
• Example: F(x,y)=x2+2y2-4x-4y+6
00 x
2
22
0x
00 x
F)xx(
2
1
x
F)xx()x(F)x(F
00x
1
x
2
2
0new x
F
x
Fxx
40
02
y
F
yx
Fyx
F
x
F
x
F
2
22
2
2
2
x
2
2
0
EFFICIENT FORMULATION OF THE RESPONSE OPERATOR
• Use the physical models to generate a large number of simulations corresponding to various combinations of {f}
• Derive statistical relationships (universal functions) between {f} and {c} at the critical locations
EFFICIENT FORMULATION OF THE RESPONSE OPERATOR (ctd)
• Determine the set of parameters w that satisfies – F(w,fi)=R(fi) fi in {f}
• This is equivalent to
• Once F(w,*) is determined, solve the initial management problem
2N
1iii
w)f(R)f,w(Fmin
EXAMPLE
x
x*w_11+w_12
x*w_21+w_22
x*w_21+w_12
y1
y2
y3
y1*h_1+y2*h_2y3*h_3+h_4
WRAP-UP
• A mathematically rigorous formulation for groundwater management may be derived
• Mathematical tools for solving the groundwater management problems are available
• Universal approximations can be effectively used to reduce the computational load
