Transcript Document

Mathematical Model
a set of equations that describes
the physical and/or chemical processes
occurring in a system.
Components of a Mathematical Model
• Governing Equation
• Boundary Conditions
• Initial conditions (for transient conditions)
problems)
Mathematical Model of the Toth Problem
h = c x + zo
h
0
x
Laplace Equation
 2h
 2h

0
2
2
x
z
h
0
z
2D, steady state
h
0
x
Types of Solutions of Mathematical
Models
• Analytical Solutions: h= f(x,y,z,t)
(example: Theis eqn., Toth 1962)
• Numerical Solutions
Finite difference methods
Finite element methods
• Analytic Element Methods (AEM)
Toth Problem
z
Math Model
h
0
x
h = c x + zo
 2h
 2h

0
x 2
z 2
h
0
z
Analytical Solution
continuous solution
h
0
x
x
Numerical Solution
discrete solution
Toth Problem
h = c x + zo
z
Math Model
h
0
x
 2h
 2h

0
x 2
z 2
h
0
z
Analytical Solution
h
0
x
x
Numerical Solution
h(x,z) = zo + cs/2 – 4cs/2 …
(eqn. 2.1 in W&A)
z
x
continuous solution
discrete solution
Toth Problem
h = c x + zo
z
Math Model
h
0
x
 2h
 2h

0
x 2
z 2
h
0
z
Analytical Solution
h
0
x
x
Numerical Solution
h(x,z) = zo + cs/2 – 4cs/2 …
(eqn. 2.1 in W&A)
z
hi,j = (hi+1,j + hi-1,j + hi,j+1 + hi,j-1)/4
x
continuous solution
discrete solution
div q = 0
Steady state mass balance
q = - K grad h
Scalar
1 component
Vector
3 components
Magnitude
Tensor
9 components
Magnitude,
direction and
magnitude
changing with
direction
Darcy’s law
Head (h)
Magnitude and q & grad
direction
Hydraulic conductivity
(K)
div q = 0
Steady state mass balance
q = - K grad h
q
Darcy’s law
equipotential line
grad h
Isotropic
q
grad h
Anisotropic
div q = 0
Steady state mass balance
q = - K grad h
Darcy’s law
Assume that K = a constant
Laplace Equation
 h
2
x
2

 h
2
y
2

 h
2
z
2
0
Governing Eqn. for TopoDrive

h

h

h
( Kx ) 
( Ky )  ( Kz )  0
x
x
y
y
z
z
Steady-state, heterogeneous, anisotropic
q = - K grad h
K is a tensor with 9 components
Kxx Kxy Kxz
K=
Kyx Kyy Kyz
Kzx Kzy Kzz
Principal components of K
h
h
h
qx   Kxx
 Kxy
 Kxz
x
y
z
h
h
h
qy   Kyx
 Kyy
 Kyz
x
y
z
h
h
h
qz   Kzx
 Kzy
 Kzz
x
y
z