A descriptive representation of a groundwater system that incorporates an interpretation of the geological

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Conceptual Model
A descriptive representation of a groundwater
system that incorporates an interpretation of the
geological hydrological conditions. Generally
includes information about the water budget.
2
Mathematical Model
a set of equations that describes the physical
and/or chemical processes occurring in a system.
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Components of a Mathematical Model

• Governing Equation
• Boundary Conditions
• Specified head (1st type or Neumann) constant
  head
• Specified flux (2nd type or Dirichlet) no flux

• Initial Conditions (for transient conditions)

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Mathematical Model of the Toth Problem
h c x zo
Laplace Equation
2D, steady state
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• 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)

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Toth Problem
h c x zo
z
Mathematical model
x
Analytical Solution
Numerical Solution
continuous solution
discrete solution
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Toth Problem
h c x zo
z
Mathematical model
x
Analytical Solution
Numerical Solution
h(x,z) zo cs/2 4cs/?2 ?
(eqn. 2.1 in WA)
z
x
continuous solution
discrete solution
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Toth Problem
h c x zo
z
Mathematical model
x
Analytical Solution
Numerical Solution
h(x,z) zo cs/2 4cs/?2 ?
(eqn. 2.1 in WA)
z
hi,j (hi1,j hi-1,j hi,j1 hi,j-1)/4
x
continuous solution
discrete solution
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OUT
IN
Q KIA
Hinge line
OUT IN 0
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?x?z ? Q K ?h
?x
z
?x
?z
1 m
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Mesh centered grid area needed in water balance
?x
(?x/2)
water table nodes
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?x?z ? Q K ?h
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Block centered grid area needed in water balance
No flow boundary
?x
?x
water table nodes
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K as a Tensor
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div q 0
Steady state mass balance eqn.
q - K grad h
Darcys law
z
q
equipotential line
grad h
grad h
q
x
Isotropic
Anisotropic
Kx Kz
Kx ? Kz
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div q 0
steady state mass balance eqn.
q - K grad h
Darcys law
Scalar 1 component Magnitude Head (h)
Vector 3 components Magnitude and direction q grad
Tensor 9 components Magnitude, direction and magnitude changing with direction Hydraulic conductivity (K)
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div q 0
steady state mass balance eqn.
q - K grad h
Darcys law
Assume K a constant (homogeneous and isotropic
conditions)
Laplace Equation
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Governing Eqn. for TopoDrive
2D, steady-state, heterogeneous, anisotropic
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global
local
z
z
bedding planes
x
?
x
Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
Kx 0 0 0 Ky 0 0 0 Kz
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q - K grad h
Kxx 0 0 0 Kyy 0 0 0
Kzz
qx qy qz
-
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q - K grad h
Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
K
K is a tensor with 9 components
Kxx ,Kyy, Kzz are the principal components of K
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q - K grad h
Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
qx qy qz
-
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This is the form of the governing equation used
in MODFLOW.
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global
local
z
z
bedding planes
x
?
x
Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
Kx 0 0 0 Ky 0 0 0 Kz
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Assume that there is no flow across impermeable
bedding planes
z
local
global
z
grad h
q
q
x
Kz0
?
x
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global
local
z
z
bedding planes
x
q
q
?
x
Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
Kx 0 0 0 Ky 0 0 0 Kz
K R-1 K R